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{
"caption": "Alright so with that, I'll see you guys next video, and I think you're really going to like where this is going.",
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"text": "Alright so with that, I'll see you guys next video,"
},
{
"end_time": "00:13:58.100000",
"segment_index": 215,
"start_time": "00:13:49.078000",
"text": "and I think you're really going to like where this is going."
}
],
"title": "Binary, Hanoi and Sierpinski, part 1",
"url": "https://youtube.com/watch?v=2SUvWfNJSsM",
"video_id": "2SUvWfNJSsM"
}
|
2SUvWfNJSsM_214-215
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
|
|
{
"caption": "After that, you can move disk 1, but it has to go on whatever peg doesn't currently have disk 0, since otherwise you'd be putting a bigger disk on a smaller one, which isn't allowed. If you've never seen this before, I highly encourage you to pause and pull out some books of varying sizes and try it out for yourself. Just kind of get a feel for what the puzzle is, if it's hard, why it's hard, if it's not, why it's not, that kind of stuff.",
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"text": "After that, you can move disk 1, but it has to go on whatever"
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"text": "peg doesn't currently have disk 0, since otherwise you'd be"
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"text": "putting a bigger disk on a smaller one, which isn't allowed."
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"end_time": "00:01:48.168000",
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"text": "If you've never seen this before, I highly encourage you to pause"
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"start_time": "00:01:48.168000",
"text": "and pull out some books of varying sizes and try it out for yourself."
},
{
"end_time": "00:01:55.142000",
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"start_time": "00:01:52.300000",
"text": "Just kind of get a feel for what the puzzle is, if it's hard,"
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{
"end_time": "00:01:57.940000",
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"start_time": "00:01:55.142000",
"text": "why it's hard, if it's not, why it's not, that kind of stuff."
}
],
"title": "Binary, Hanoi and Sierpinski, part 1",
"url": "https://youtube.com/watch?v=2SUvWfNJSsM",
"video_id": "2SUvWfNJSsM"
}
|
2SUvWfNJSsM_22-28
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
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|
{
"caption": "Now Keith showed me something truly surprising about this puzzle, which is that you can solve it just by counting up in binary and associating the rhythm of that counting with a certain rhythm of disk movements. For anyone unfamiliar with binary, I'm going to take a moment to do a quick overview here first. Actually, even if you are familiar with binary, I want to explain it with a focus on the rhythm of counting, which you may or may not have thought about before.",
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"text": "Now Keith showed me something truly surprising about this puzzle,"
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"text": "which is that you can solve it just by counting up in binary and"
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"text": "associating the rhythm of that counting with a certain rhythm of disk movements."
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"text": "For anyone unfamiliar with binary, I'm going to"
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{
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"text": "take a moment to do a quick overview here first."
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"text": "Actually, even if you are familiar with binary,"
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"text": "I want to explain it with a focus on the rhythm of counting,"
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"text": "which you may or may not have thought about before."
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],
"title": "Binary, Hanoi and Sierpinski, part 1",
"url": "https://youtube.com/watch?v=2SUvWfNJSsM",
"video_id": "2SUvWfNJSsM"
}
|
2SUvWfNJSsM_29-36
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
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{
"caption": "Any description of binary typically starts off with an introspection about our usual way to represent numbers, what we call base 10, since we use 10 separate digits, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. The rhythm of counting begins by walking through all 10 of these digits. Then, having run out of new digits, you express the next number, 10, with two digits, 1, 0.",
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"text": "Any description of binary typically starts off with an introspection"
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"text": "about our usual way to represent numbers, what we call base 10,"
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"text": "since we use 10 separate digits, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9."
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"text": "The rhythm of counting begins by walking through all 10 of these digits."
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"text": "Then, having run out of new digits, you express the next number,"
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"text": "10, with two digits, 1, 0."
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],
"title": "Binary, Hanoi and Sierpinski, part 1",
"url": "https://youtube.com/watch?v=2SUvWfNJSsM",
"video_id": "2SUvWfNJSsM"
}
|
2SUvWfNJSsM_37-42
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
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{
"caption": "You say that 1 is in the tens place, since it's meant to encapsulate the group of 10 that you've already counted up to so far, while freeing the ones place to reset to 0. The rhythm of counting repeats like this, counting up 9, rolling over to the tens place, counting up 9 more, rolling over to the tens place, etc. Until, after repeating that process 9 times, you roll over to a hundreds place,",
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"text": "You say that 1 is in the tens place, since it's meant to encapsulate the group of 10"
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"text": "that you've already counted up to so far, while freeing the ones place to reset to 0."
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"text": "The rhythm of counting repeats like this, counting up 9,"
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"text": "rolling over to the tens place, counting up 9 more, rolling over to the tens place, etc."
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"text": "Until, after repeating that process 9 times, you roll over to a hundreds place,"
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],
"title": "Binary, Hanoi and Sierpinski, part 1",
"url": "https://youtube.com/watch?v=2SUvWfNJSsM",
"video_id": "2SUvWfNJSsM"
}
|
2SUvWfNJSsM_43-47
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
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{
"caption": "a digit that keeps track of how many groups of 100 you've hit, freeing up the other two digits to reset to 0. In this way, the rhythm of counting is kind of self-similar. Even if you zoom out to a larger scale, the process looks like doing something, rolling over, doing that same thing, rolling over, and repeating 9 times before an even larger rollover.",
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"text": "a digit that keeps track of how many groups of 100 you've hit,"
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"text": "freeing up the other two digits to reset to 0."
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"text": "In this way, the rhythm of counting is kind of self-similar."
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"text": "Even if you zoom out to a larger scale, the process looks like doing something,"
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"text": "rolling over, doing that same thing, rolling over,"
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"text": "and repeating 9 times before an even larger rollover."
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],
"title": "Binary, Hanoi and Sierpinski, part 1",
"url": "https://youtube.com/watch?v=2SUvWfNJSsM",
"video_id": "2SUvWfNJSsM"
}
|
2SUvWfNJSsM_48-53
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
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{
"caption": "In binary, also known as base-2, you limit yourself to two digits, 0 and 1, commonly called bits, which is short for binary digits. The result is that when you're counting, you have to roll over all the time. After counting 01, you've already run out of bits, so you need to roll over to a twos place, writing 10, and resisting every urge in your base-10 trained brain to read this as 10,",
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"text": "In binary, also known as base-2, you limit yourself to two digits,"
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"text": "0 and 1, commonly called bits, which is short for binary digits."
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"text": "The result is that when you're counting, you have to roll over all the time."
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"text": "After counting 01, you've already run out of bits,"
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"text": "so you need to roll over to a twos place, writing 10,"
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"end_time": "00:04:15.401000",
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"start_time": "00:04:10.542000",
"text": "and resisting every urge in your base-10 trained brain to read this as 10,"
}
],
"title": "Binary, Hanoi and Sierpinski, part 1",
"url": "https://youtube.com/watch?v=2SUvWfNJSsM",
"video_id": "2SUvWfNJSsM"
}
|
2SUvWfNJSsM_54-59
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
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{
"caption": "and instead understand it to mean 1 group of 2 plus 0. Then increment up to 11, which represents 3, and already you have to roll over again, and since there's a 1 in that twos place, that has to roll over as well, giving you 100, which represents 1 group of 4 plus 0 groups of 2 plus 0. In the same way that digits in base-10 represent powers of 10,",
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"text": "and instead understand it to mean 1 group of 2 plus 0."
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"text": "Then increment up to 11, which represents 3, and already you have to roll over again,"
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"text": "and since there's a 1 in that twos place, that has to roll over as well,"
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"text": "giving you 100, which represents 1 group of 4 plus 0 groups of 2 plus 0."
},
{
"end_time": "00:04:41.490000",
"segment_index": 64,
"start_time": "00:04:36.860000",
"text": "In the same way that digits in base-10 represent powers of 10,"
}
],
"title": "Binary, Hanoi and Sierpinski, part 1",
"url": "https://youtube.com/watch?v=2SUvWfNJSsM",
"video_id": "2SUvWfNJSsM"
}
|
2SUvWfNJSsM_60-64
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
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{
"caption": "bits in base-2 represent different powers of 2, so instead of a tens place, a hundreds place, a thousands place, you talk about a twos place, a fours place, and an eights place. The rhythm of counting is now a lot faster, but that almost makes it more noticeable.",
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"text": "bits in base-2 represent different powers of 2, so instead of a tens place,"
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"text": "a hundreds place, a thousands place, you talk about a twos place, a fours place,"
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"text": "and an eights place."
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"text": "The rhythm of counting is now a lot faster, but that almost makes it more noticeable."
}
],
"title": "Binary, Hanoi and Sierpinski, part 1",
"url": "https://youtube.com/watch?v=2SUvWfNJSsM",
"video_id": "2SUvWfNJSsM"
}
|
2SUvWfNJSsM_65-68
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
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{
"caption": "Again, there's a certain self-similarity to this pattern. At every scale, the process is to do something, roll over, then do that same thing again. At the small scale, say counting up to 3, which is 11 in binary, this means flip the last bit, roll over to the twos, then flip the last bit.",
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"text": "Again, there's a certain self-similarity to this pattern."
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"text": "At every scale, the process is to do something, roll over, then do that same thing again."
},
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"text": "At the small scale, say counting up to 3, which is 11 in binary,"
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"text": "this means flip the last bit, roll over to the twos, then flip the last bit."
}
],
"title": "Binary, Hanoi and Sierpinski, part 1",
"url": "https://youtube.com/watch?v=2SUvWfNJSsM",
"video_id": "2SUvWfNJSsM"
}
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2SUvWfNJSsM_69-72
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
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{
"caption": "At a larger scale, like counting up to 15, which is 1111 in binary, the process is to let the last 3 count up to 7, roll over to the eights place, then let the last 3 bits count up again. Counting up to 255, which is 8 successive ones, this looks like letting the last 7 bits count up till they're full, rolling over to the 128th place, then letting the last 7 bits count up again.",
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"text": "At a larger scale, like counting up to 15, which is 1111 in binary,"
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"text": "the process is to let the last 3 count up to 7,"
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"text": "roll over to the eights place, then let the last 3 bits count up again."
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"text": "Counting up to 255, which is 8 successive ones,"
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"text": "this looks like letting the last 7 bits count up till they're full,"
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"start_time": "00:05:54.496000",
"text": "rolling over to the 128th place, then letting the last 7 bits count up again."
}
],
"title": "Binary, Hanoi and Sierpinski, part 1",
"url": "https://youtube.com/watch?v=2SUvWfNJSsM",
"video_id": "2SUvWfNJSsM"
}
|
2SUvWfNJSsM_73-78
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
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{
"caption": "Alright, so with that mini-introduction, the surprising fact that Keith showed me is that we can use this rhythm to solve the towers of Hanoi. You start by counting from 0. Whenever you're only flipping that last bit, from a 0 to a 1, move disk 0 one peg to the right. If it was already on the right-most peg, you just loop it back to the first peg.",
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"text": "Alright, so with that mini-introduction, the surprising fact that Keith"
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"text": "showed me is that we can use this rhythm to solve the towers of Hanoi."
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"text": "You start by counting from 0."
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"end_time": "00:06:16.797000",
"segment_index": 82,
"start_time": "00:06:12.660000",
"text": "Whenever you're only flipping that last bit, from a 0 to a 1,"
},
{
"end_time": "00:06:19",
"segment_index": 83,
"start_time": "00:06:16.797000",
"text": "move disk 0 one peg to the right."
},
{
"end_time": "00:06:26.020000",
"segment_index": 84,
"start_time": "00:06:22.020000",
"text": "If it was already on the right-most peg, you just loop it back to the first peg."
}
],
"title": "Binary, Hanoi and Sierpinski, part 1",
"url": "https://youtube.com/watch?v=2SUvWfNJSsM",
"video_id": "2SUvWfNJSsM"
}
|
2SUvWfNJSsM_79-84
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
|
|
{
"caption": "In case you're unfamiliar, let's just lay down what the Towers of Hanoi puzzle actually is. So you have a collection of three pegs, and you have these disks of descending size. You think of these disks as having a hole in the middle so that you can fit them onto a peg. The setup pictured here has five disks, which I'll label 0, 1, 2, 3, 4, but in principle, you could have as many disks as you want.",
"caption_code": "",
"channel_id": "3blue1brown",
"clip_duration": 29.340000000000003,
"clip_end": 67.26,
"clip_start": 37.92,
"full_duration": 838.124263,
"segments": [
{
"end_time": "00:00:43.181000",
"segment_index": 8,
"start_time": "00:00:38.420000",
"text": "In case you're unfamiliar, let's just lay down"
},
{
"end_time": "00:00:47.640000",
"segment_index": 9,
"start_time": "00:00:43.181000",
"text": "what the Towers of Hanoi puzzle actually is."
},
{
"end_time": "00:00:54",
"segment_index": 10,
"start_time": "00:00:47.640000",
"text": "So you have a collection of three pegs, and you have these disks of descending size."
},
{
"end_time": "00:00:56.519000",
"segment_index": 11,
"start_time": "00:00:54.400000",
"text": "You think of these disks as having a hole in the"
},
{
"end_time": "00:00:58.380000",
"segment_index": 12,
"start_time": "00:00:56.519000",
"text": "middle so that you can fit them onto a peg."
},
{
"end_time": "00:01:03.008000",
"segment_index": 13,
"start_time": "00:00:59.200000",
"text": "The setup pictured here has five disks, which I'll label 0, 1, 2,"
},
{
"end_time": "00:01:06.760000",
"segment_index": 14,
"start_time": "00:01:03.008000",
"text": "3, 4, but in principle, you could have as many disks as you want."
}
],
"title": "Binary, Hanoi and Sierpinski, part 1",
"url": "https://youtube.com/watch?v=2SUvWfNJSsM",
"video_id": "2SUvWfNJSsM"
}
|
2SUvWfNJSsM_8-14
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
|
|
{
"caption": "If, in your binary counting, you roll over once to the twos place, meaning you flip the last two bits, you move disk number 1. Where do you move it, you might ask? Well, you have no choice. You can't put it on top of disk 0, and there's only one other peg, so you move it where you're forced to move it. So after this, counting up to 1,1, that involves just flipping the last bit, so you move disk 0 again. Then when your binary counting rolls over twice to the fours place,",
"caption_code": "",
"channel_id": "3blue1brown",
"clip_duration": 29.435000000000002,
"clip_end": 417.735,
"clip_start": 388.3,
"full_duration": 838.124263,
"segments": [
{
"end_time": "00:06:32.990000",
"segment_index": 85,
"start_time": "00:06:28.800000",
"text": "If, in your binary counting, you roll over once to the twos place,"
},
{
"end_time": "00:06:36.680000",
"segment_index": 86,
"start_time": "00:06:32.990000",
"text": "meaning you flip the last two bits, you move disk number 1."
},
{
"end_time": "00:06:38.980000",
"segment_index": 87,
"start_time": "00:06:37.620000",
"text": "Where do you move it, you might ask?"
},
{
"end_time": "00:06:40.400000",
"segment_index": 88,
"start_time": "00:06:39.300000",
"text": "Well, you have no choice."
},
{
"end_time": "00:06:43.821000",
"segment_index": 89,
"start_time": "00:06:40.620000",
"text": "You can't put it on top of disk 0, and there's only one other peg,"
},
{
"end_time": "00:06:46.020000",
"segment_index": 90,
"start_time": "00:06:43.821000",
"text": "so you move it where you're forced to move it."
},
{
"end_time": "00:06:50.676000",
"segment_index": 91,
"start_time": "00:06:46.659000",
"text": "So after this, counting up to 1,1, that involves just flipping the last bit,"
},
{
"end_time": "00:06:51.980000",
"segment_index": 92,
"start_time": "00:06:50.676000",
"text": "so you move disk 0 again."
},
{
"end_time": "00:06:57.235000",
"segment_index": 93,
"start_time": "00:06:52.640000",
"text": "Then when your binary counting rolls over twice to the fours place,"
}
],
"title": "Binary, Hanoi and Sierpinski, part 1",
"url": "https://youtube.com/watch?v=2SUvWfNJSsM",
"video_id": "2SUvWfNJSsM"
}
|
2SUvWfNJSsM_85-93
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
|
|
{
"caption": "move disk number 2, and the pattern continues like this. Flip the last, move disk 0. Flip the last two, move disk 1. Flip the last, move disk 0. And here, we're going to have to roll over three times to the eights place, and that corresponds to moving disk number 3. There's something magical about it. When I first saw this, I was like, this can't work. I don't know how this works, I don't know why this works. Now I know, but it's just magical when you see it.",
"caption_code": "",
"channel_id": "3blue1brown",
"clip_duration": 28.04499999999996,
"clip_end": 444.78,
"clip_start": 416.735,
"full_duration": 838.124263,
"segments": [
{
"end_time": "00:07:01.020000",
"segment_index": 94,
"start_time": "00:06:57.235000",
"text": "move disk number 2, and the pattern continues like this."
},
{
"end_time": "00:07:02.880000",
"segment_index": 95,
"start_time": "00:07:01.320000",
"text": "Flip the last, move disk 0."
},
{
"end_time": "00:07:04.900000",
"segment_index": 96,
"start_time": "00:07:03.260000",
"text": "Flip the last two, move disk 1."
},
{
"end_time": "00:07:07.200000",
"segment_index": 97,
"start_time": "00:07:05.760000",
"text": "Flip the last, move disk 0."
},
{
"end_time": "00:07:11.660000",
"segment_index": 98,
"start_time": "00:07:07.980000",
"text": "And here, we're going to have to roll over three times to the eights place,"
},
{
"end_time": "00:07:13.840000",
"segment_index": 99,
"start_time": "00:07:11.660000",
"text": "and that corresponds to moving disk number 3."
},
{
"end_time": "00:07:16.180000",
"segment_index": 100,
"start_time": "00:07:14.800000",
"text": "There's something magical about it."
},
{
"end_time": "00:07:17.920000",
"segment_index": 101,
"start_time": "00:07:16.300000",
"text": "When I first saw this, I was like, this can't work."
},
{
"end_time": "00:07:21.080000",
"segment_index": 102,
"start_time": "00:07:18.540000",
"text": "I don't know how this works, I don't know why this works."
},
{
"end_time": "00:07:24.280000",
"segment_index": 103,
"start_time": "00:07:21.200000",
"text": "Now I know, but it's just magical when you see it."
}
],
"title": "Binary, Hanoi and Sierpinski, part 1",
"url": "https://youtube.com/watch?v=2SUvWfNJSsM",
"video_id": "2SUvWfNJSsM"
}
|
2SUvWfNJSsM_94-103
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
|
|
{
"caption": "When I first learned about Taylor series, I definitely didn't appreciate just how important they are. But time and time again they come up in math, physics, and many fields of engineering because they're one of the most powerful tools that math has to offer for approximating functions. I think one of the first times this clicked for me as a student was not in a calculus class but a physics class. We were studying a certain problem that had to do with the potential energy of a",
"caption_code": "",
"channel_id": "3blue1brown",
"clip_duration": 26.463,
"clip_end": 40.603,
"clip_start": 14.14,
"full_duration": 1339.094785,
"segments": [
{
"end_time": "00:00:17.395000",
"segment_index": 1,
"start_time": "00:00:14.640000",
"text": "When I first learned about Taylor series, I definitely"
},
{
"end_time": "00:00:19.700000",
"segment_index": 2,
"start_time": "00:00:17.395000",
"text": "didn't appreciate just how important they are."
},
{
"end_time": "00:00:22.828000",
"segment_index": 3,
"start_time": "00:00:20.120000",
"text": "But time and time again they come up in math, physics,"
},
{
"end_time": "00:00:25.930000",
"segment_index": 4,
"start_time": "00:00:22.828000",
"text": "and many fields of engineering because they're one of the most"
},
{
"end_time": "00:00:29.180000",
"segment_index": 5,
"start_time": "00:00:25.930000",
"text": "powerful tools that math has to offer for approximating functions."
},
{
"end_time": "00:00:32.710000",
"segment_index": 6,
"start_time": "00:00:30",
"text": "I think one of the first times this clicked for me as a"
},
{
"end_time": "00:00:35.420000",
"segment_index": 7,
"start_time": "00:00:32.710000",
"text": "student was not in a calculus class but a physics class."
},
{
"end_time": "00:00:40.103000",
"segment_index": 8,
"start_time": "00:00:35.840000",
"text": "We were studying a certain problem that had to do with the potential energy of a"
}
],
"title": "Taylor series | Chapter 11, Essence of calculus",
"url": "https://youtube.com/watch?v=3d6DsjIBzJ4",
"video_id": "3d6DsjIBzJ4"
}
|
3d6DsjIBzJ4_1-8
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
|
|
{
"caption": "which is 1, c4 has to be 1 over 24. And indeed, the polynomial 1 minus ½ x2 plus 1 24 times x to the fourth, which looks like this, is a very close approximation for cosine x around x equals 0. In any physics problem involving the cosine of a small angle, for example, predictions would be almost unnoticeably different if you substituted this polynomial for cosine of x.",
"caption_code": "",
"channel_id": "3blue1brown",
"clip_duration": 28.92900000000003,
"clip_end": 504.56,
"clip_start": 475.631,
"full_duration": 1339.094785,
"segments": [
{
"end_time": "00:07:58.760000",
"segment_index": 105,
"start_time": "00:07:56.131000",
"text": "which is 1, c4 has to be 1 over 24."
},
{
"end_time": "00:08:05.873000",
"segment_index": 106,
"start_time": "00:07:59.820000",
"text": "And indeed, the polynomial 1 minus ½ x2 plus 1 24 times x to the fourth,"
},
{
"end_time": "00:08:12.840000",
"segment_index": 107,
"start_time": "00:08:05.873000",
"text": "which looks like this, is a very close approximation for cosine x around x equals 0."
},
{
"end_time": "00:08:18.112000",
"segment_index": 108,
"start_time": "00:08:13.740000",
"text": "In any physics problem involving the cosine of a small angle, for example,"
},
{
"end_time": "00:08:23.127000",
"segment_index": 109,
"start_time": "00:08:18.112000",
"text": "predictions would be almost unnoticeably different if you substituted this polynomial"
},
{
"end_time": "00:08:24.060000",
"segment_index": 110,
"start_time": "00:08:23.127000",
"text": "for cosine of x."
}
],
"title": "Taylor series | Chapter 11, Essence of calculus",
"url": "https://youtube.com/watch?v=3d6DsjIBzJ4",
"video_id": "3d6DsjIBzJ4"
}
|
3d6DsjIBzJ4_105-110
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
|
|
{
"caption": "Take a step back and notice a few things happening with this process. First of all, factorial terms come up very naturally in this process. When you take n successive derivatives of the function x to the n, letting the power rule keep cascading on down, what you'll be left with is 1 times 2 times 3 on and on up to whatever n is. So you don't simply set the coefficients of the polynomial equal to whatever derivative",
"caption_code": "",
"channel_id": "3blue1brown",
"clip_duration": 28.888000000000034,
"clip_end": 534.488,
"clip_start": 505.6,
"full_duration": 1339.094785,
"segments": [
{
"end_time": "00:08:29.760000",
"segment_index": 111,
"start_time": "00:08:26.100000",
"text": "Take a step back and notice a few things happening with this process."
},
{
"end_time": "00:08:34.200000",
"segment_index": 112,
"start_time": "00:08:30.520000",
"text": "First of all, factorial terms come up very naturally in this process."
},
{
"end_time": "00:08:39.801000",
"segment_index": 113,
"start_time": "00:08:35.020000",
"text": "When you take n successive derivatives of the function x to the n,"
},
{
"end_time": "00:08:43.156000",
"segment_index": 114,
"start_time": "00:08:39.801000",
"text": "letting the power rule keep cascading on down,"
},
{
"end_time": "00:08:48.580000",
"segment_index": 115,
"start_time": "00:08:43.156000",
"text": "what you'll be left with is 1 times 2 times 3 on and on up to whatever n is."
},
{
"end_time": "00:08:53.988000",
"segment_index": 116,
"start_time": "00:08:49.220000",
"text": "So you don't simply set the coefficients of the polynomial equal to whatever derivative"
}
],
"title": "Taylor series | Chapter 11, Essence of calculus",
"url": "https://youtube.com/watch?v=3d6DsjIBzJ4",
"video_id": "3d6DsjIBzJ4"
}
|
3d6DsjIBzJ4_111-116
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
|
|
{
"caption": "you want, you have to divide by the appropriate factorial to cancel out this effect. For example, that x to the fourth coefficient was the fourth derivative of cosine, 1, but divided by 4 factorial, 24. The second thing to notice is that adding on new terms, like this c4 times x to the fourth, doesn't mess up what the old terms should be, and that's really important.",
"caption_code": "",
"channel_id": "3blue1brown",
"clip_duration": 26.311999999999898,
"clip_end": 559.8,
"clip_start": 533.488,
"full_duration": 1339.094785,
"segments": [
{
"end_time": "00:08:58.540000",
"segment_index": 117,
"start_time": "00:08:53.988000",
"text": "you want, you have to divide by the appropriate factorial to cancel out this effect."
},
{
"end_time": "00:09:05.344000",
"segment_index": 118,
"start_time": "00:08:59.400000",
"text": "For example, that x to the fourth coefficient was the fourth derivative of cosine,"
},
{
"end_time": "00:09:07.780000",
"segment_index": 119,
"start_time": "00:09:05.344000",
"text": "1, but divided by 4 factorial, 24."
},
{
"end_time": "00:09:12.739000",
"segment_index": 120,
"start_time": "00:09:09.400000",
"text": "The second thing to notice is that adding on new terms,"
},
{
"end_time": "00:09:17.630000",
"segment_index": 121,
"start_time": "00:09:12.739000",
"text": "like this c4 times x to the fourth, doesn't mess up what the old terms should be,"
},
{
"end_time": "00:09:19.300000",
"segment_index": 122,
"start_time": "00:09:17.630000",
"text": "and that's really important."
}
],
"title": "Taylor series | Chapter 11, Essence of calculus",
"url": "https://youtube.com/watch?v=3d6DsjIBzJ4",
"video_id": "3d6DsjIBzJ4"
}
|
3d6DsjIBzJ4_117-122
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
|
|
{
"caption": "For example, the second derivative of this polynomial at x equals 0 is still equal to 2 times the second coefficient, even after you introduce higher order terms. And it's because we're plugging in x equals 0, so the second derivative of any higher order term, which all include an x, will just wash away. And the same goes for any other derivative, which is why each derivative of a",
"caption_code": "",
"channel_id": "3blue1brown",
"clip_duration": 26.37900000000002,
"clip_end": 585.979,
"clip_start": 559.6,
"full_duration": 1339.094785,
"segments": [
{
"end_time": "00:09:25.213000",
"segment_index": 123,
"start_time": "00:09:20.100000",
"text": "For example, the second derivative of this polynomial at x equals 0 is still equal"
},
{
"end_time": "00:09:30.080000",
"segment_index": 124,
"start_time": "00:09:25.213000",
"text": "to 2 times the second coefficient, even after you introduce higher order terms."
},
{
"end_time": "00:09:33.879000",
"segment_index": 125,
"start_time": "00:09:30.960000",
"text": "And it's because we're plugging in x equals 0,"
},
{
"end_time": "00:09:38.537000",
"segment_index": 126,
"start_time": "00:09:33.879000",
"text": "so the second derivative of any higher order term, which all include an x,"
},
{
"end_time": "00:09:39.780000",
"segment_index": 127,
"start_time": "00:09:38.537000",
"text": "will just wash away."
},
{
"end_time": "00:09:45.479000",
"segment_index": 128,
"start_time": "00:09:40.740000",
"text": "And the same goes for any other derivative, which is why each derivative of a"
}
],
"title": "Taylor series | Chapter 11, Essence of calculus",
"url": "https://youtube.com/watch?v=3d6DsjIBzJ4",
"video_id": "3d6DsjIBzJ4"
}
|
3d6DsjIBzJ4_123-128
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
|
|
{
"caption": "polynomial at x equals 0 is controlled by one and only one of the coefficients. If instead you were approximating near an input other than 0, like x equals pi, in order to get the same effect you would have to write your polynomial in terms of powers of x minus pi, or whatever input you're looking at. This makes it look noticeably more complicated, but all we're doing is making sure that the point pi looks and behaves like 0,",
"caption_code": "",
"channel_id": "3blue1brown",
"clip_duration": 29.48199999999997,
"clip_end": 614.461,
"clip_start": 584.979,
"full_duration": 1339.094785,
"segments": [
{
"end_time": "00:09:50.280000",
"segment_index": 129,
"start_time": "00:09:45.479000",
"text": "polynomial at x equals 0 is controlled by one and only one of the coefficients."
},
{
"end_time": "00:09:57.353000",
"segment_index": 130,
"start_time": "00:09:52.640000",
"text": "If instead you were approximating near an input other than 0, like x equals pi,"
},
{
"end_time": "00:10:01.772000",
"segment_index": 131,
"start_time": "00:09:57.353000",
"text": "in order to get the same effect you would have to write your polynomial in"
},
{
"end_time": "00:10:05.720000",
"segment_index": 132,
"start_time": "00:10:01.772000",
"text": "terms of powers of x minus pi, or whatever input you're looking at."
},
{
"end_time": "00:10:09.208000",
"segment_index": 133,
"start_time": "00:10:06.320000",
"text": "This makes it look noticeably more complicated,"
},
{
"end_time": "00:10:13.961000",
"segment_index": 134,
"start_time": "00:10:09.208000",
"text": "but all we're doing is making sure that the point pi looks and behaves like 0,"
}
],
"title": "Taylor series | Chapter 11, Essence of calculus",
"url": "https://youtube.com/watch?v=3d6DsjIBzJ4",
"video_id": "3d6DsjIBzJ4"
}
|
3d6DsjIBzJ4_129-134
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
|
|
{
"caption": "so that plugging in x equals pi will result in a lot of nice cancellation that leaves only one constant. And finally, on a more philosophical level, notice how what we're doing here is basically taking information about higher order derivatives of a function at a single point, and translating that into information about the value of the function near that point.",
"caption_code": "",
"channel_id": "3blue1brown",
"clip_duration": 24.81899999999996,
"clip_end": 638.28,
"clip_start": 613.461,
"full_duration": 1339.094785,
"segments": [
{
"end_time": "00:10:18.715000",
"segment_index": 135,
"start_time": "00:10:13.961000",
"text": "so that plugging in x equals pi will result in a lot of nice cancellation that"
},
{
"end_time": "00:10:20.220000",
"segment_index": 136,
"start_time": "00:10:18.715000",
"text": "leaves only one constant."
},
{
"end_time": "00:10:27.731000",
"segment_index": 137,
"start_time": "00:10:22.380000",
"text": "And finally, on a more philosophical level, notice how what we're doing here is basically"
},
{
"end_time": "00:10:32.666000",
"segment_index": 138,
"start_time": "00:10:27.731000",
"text": "taking information about higher order derivatives of a function at a single point,"
},
{
"end_time": "00:10:37.780000",
"segment_index": 139,
"start_time": "00:10:32.666000",
"text": "and translating that into information about the value of the function near that point."
}
],
"title": "Taylor series | Chapter 11, Essence of calculus",
"url": "https://youtube.com/watch?v=3d6DsjIBzJ4",
"video_id": "3d6DsjIBzJ4"
}
|
3d6DsjIBzJ4_135-139
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hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
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{
"caption": "You can take as many derivatives of cosine as you want. It follows this nice cyclic pattern, cosine of x, negative sine of x, negative cosine, sine, and then repeat. And the value of each one of these is easy to compute at x equals 0. It gives this cyclic pattern 1, 0, negative 1, 0, and then repeat. And knowing the values of all those higher order derivatives is a lot of information",
"caption_code": "",
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"text": "You can take as many derivatives of cosine as you want."
},
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"text": "It follows this nice cyclic pattern, cosine of x,"
},
{
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"text": "negative sine of x, negative cosine, sine, and then repeat."
},
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"segment_index": 143,
"start_time": "00:10:52.320000",
"text": "And the value of each one of these is easy to compute at x equals 0."
},
{
"end_time": "00:11:01.100000",
"segment_index": 144,
"start_time": "00:10:56.100000",
"text": "It gives this cyclic pattern 1, 0, negative 1, 0, and then repeat."
},
{
"end_time": "00:11:07.149000",
"segment_index": 145,
"start_time": "00:11:02",
"text": "And knowing the values of all those higher order derivatives is a lot of information"
}
],
"title": "Taylor series | Chapter 11, Essence of calculus",
"url": "https://youtube.com/watch?v=3d6DsjIBzJ4",
"video_id": "3d6DsjIBzJ4"
}
|
3d6DsjIBzJ4_140-145
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hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
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{
"caption": "about cosine of x, even though it only involves plugging in a single number, x equals 0. So what we're doing is leveraging that information to get an approximation around this input, and you do it by creating a polynomial whose higher order derivatives are designed to match up with those of cosine, following this same 1, 0, negative 1, 0, cyclic pattern. And to do that, you just make each coefficient of the polynomial follow that",
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"text": "about cosine of x, even though it only involves plugging in a single number, x equals 0."
},
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"text": "So what we're doing is leveraging that information to get an approximation around this"
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"text": "input, and you do it by creating a polynomial whose higher order derivatives are designed"
},
{
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"start_time": "00:11:25.131000",
"text": "to match up with those of cosine, following this same 1, 0, negative 1, 0, cyclic pattern."
},
{
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"segment_index": 150,
"start_time": "00:11:31.420000",
"text": "And to do that, you just make each coefficient of the polynomial follow that"
}
],
"title": "Taylor series | Chapter 11, Essence of calculus",
"url": "https://youtube.com/watch?v=3d6DsjIBzJ4",
"video_id": "3d6DsjIBzJ4"
}
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3d6DsjIBzJ4_146-150
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hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
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{
"caption": "But if you approximate cosine of theta as 1 minus theta squared over 2, everything just fell into place much more easily. If you've never seen anything like this before, an approximation like that might seem completely out of left field. If you graph cosine of theta along with this function, 1 minus theta squared over 2, they do seem rather close to each other, at least for small angles near 0,",
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"text": "But if you approximate cosine of theta as 1 minus theta squared over 2,"
},
{
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"text": "everything just fell into place much more easily."
},
{
"end_time": "00:01:19.214000",
"segment_index": 17,
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"text": "If you've never seen anything like this before,"
},
{
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"segment_index": 18,
"start_time": "00:01:19.214000",
"text": "an approximation like that might seem completely out of left field."
},
{
"end_time": "00:01:28.804000",
"segment_index": 19,
"start_time": "00:01:23.820000",
"text": "If you graph cosine of theta along with this function, 1 minus theta squared over 2,"
},
{
"end_time": "00:01:33.203000",
"segment_index": 20,
"start_time": "00:01:28.804000",
"text": "they do seem rather close to each other, at least for small angles near 0,"
}
],
"title": "Taylor series | Chapter 11, Essence of calculus",
"url": "https://youtube.com/watch?v=3d6DsjIBzJ4",
"video_id": "3d6DsjIBzJ4"
}
|
3d6DsjIBzJ4_15-20
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hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
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{
"caption": "same pattern, but you have to divide each one by the appropriate factorial. Like I mentioned before, this is what cancels out the cascading effect of many power rule applications. The polynomials you get by stopping this process at any point are called Taylor polynomials for cosine of x. More generally, and hence more abstractly, if we were dealing with some other function other than cosine, you would compute its derivative, its second derivative, and so on,",
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"text": "same pattern, but you have to divide each one by the appropriate factorial."
},
{
"end_time": "00:11:42.615000",
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"text": "Like I mentioned before, this is what cancels out"
},
{
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"segment_index": 153,
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"text": "the cascading effect of many power rule applications."
},
{
"end_time": "00:11:50.111000",
"segment_index": 154,
"start_time": "00:11:47.280000",
"text": "The polynomials you get by stopping this process at"
},
{
"end_time": "00:11:53.160000",
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"start_time": "00:11:50.111000",
"text": "any point are called Taylor polynomials for cosine of x."
},
{
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"segment_index": 156,
"start_time": "00:11:53.900000",
"text": "More generally, and hence more abstractly, if we were dealing with some other function"
},
{
"end_time": "00:12:03.420000",
"segment_index": 157,
"start_time": "00:11:58.660000",
"text": "other than cosine, you would compute its derivative, its second derivative, and so on,"
}
],
"title": "Taylor series | Chapter 11, Essence of calculus",
"url": "https://youtube.com/watch?v=3d6DsjIBzJ4",
"video_id": "3d6DsjIBzJ4"
}
|
3d6DsjIBzJ4_151-157
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
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{
"caption": "getting as many terms as you'd like, and you would evaluate each one of them at x equals 0. Then for the polynomial approximation, the coefficient of each x to the n term should be the value of the nth derivative of the function evaluated at 0, but divided by n factorial. This whole rather abstract formula is something you'll likely see in any text or course that touches on Taylor polynomials.",
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"text": "getting as many terms as you'd like, and you would evaluate each one of them at x equals"
},
{
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"segment_index": 159,
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"text": "0."
},
{
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"start_time": "00:12:09.580000",
"text": "Then for the polynomial approximation, the coefficient of each x to the n term should be"
},
{
"end_time": "00:12:20.511000",
"segment_index": 161,
"start_time": "00:12:15.938000",
"text": "the value of the nth derivative of the function evaluated at 0,"
},
{
"end_time": "00:12:22.440000",
"segment_index": 162,
"start_time": "00:12:20.511000",
"text": "but divided by n factorial."
},
{
"end_time": "00:12:27.371000",
"segment_index": 163,
"start_time": "00:12:23.480000",
"text": "This whole rather abstract formula is something you'll likely"
},
{
"end_time": "00:12:31.200000",
"segment_index": 164,
"start_time": "00:12:27.371000",
"text": "see in any text or course that touches on Taylor polynomials."
}
],
"title": "Taylor series | Chapter 11, Essence of calculus",
"url": "https://youtube.com/watch?v=3d6DsjIBzJ4",
"video_id": "3d6DsjIBzJ4"
}
|
3d6DsjIBzJ4_158-164
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hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
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{
"caption": "And when you see it, think to yourself that the constant term ensures that the value of the polynomial matches with the value of f, the next term ensures that the slope of the polynomial matches the slope of the function at x equals 0, the next term ensures that the rate at which the slope changes is the same at that point, and so on, depending on how many terms you want. And the more terms you choose, the closer the approximation,",
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"text": "And when you see it, think to yourself that the constant term ensures that"
},
{
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"start_time": "00:12:36.139000",
"text": "the value of the polynomial matches with the value of f,"
},
{
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"start_time": "00:12:39.452000",
"text": "the next term ensures that the slope of the polynomial matches the slope"
},
{
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"start_time": "00:12:43.696000",
"text": "of the function at x equals 0, the next term ensures that the rate at which"
},
{
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"start_time": "00:12:48.114000",
"text": "the slope changes is the same at that point, and so on,"
},
{
"end_time": "00:12:53.520000",
"segment_index": 170,
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"text": "depending on how many terms you want."
},
{
"end_time": "00:12:57.451000",
"segment_index": 171,
"start_time": "00:12:54.620000",
"text": "And the more terms you choose, the closer the approximation,"
}
],
"title": "Taylor series | Chapter 11, Essence of calculus",
"url": "https://youtube.com/watch?v=3d6DsjIBzJ4",
"video_id": "3d6DsjIBzJ4"
}
|
3d6DsjIBzJ4_165-171
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
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{
"caption": "but the tradeoff is that the polynomial you'd get would be more complicated. And to make things even more general, if you wanted to approximate near some input other than 0, which we'll call a, you would write this polynomial in terms of powers of x minus a, and you would evaluate all the derivatives of f at that input, a. This is what Taylor polynomials look like in their fullest generality.",
"caption_code": "",
"channel_id": "3blue1brown",
"clip_duration": 26.668999999999983,
"clip_end": 803.62,
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"segments": [
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"text": "but the tradeoff is that the polynomial you'd get would be more complicated."
},
{
"end_time": "00:13:07.727000",
"segment_index": 173,
"start_time": "00:13:02.640000",
"text": "And to make things even more general, if you wanted to approximate near some input"
},
{
"end_time": "00:13:12.937000",
"segment_index": 174,
"start_time": "00:13:07.727000",
"text": "other than 0, which we'll call a, you would write this polynomial in terms of powers"
},
{
"end_time": "00:13:17.780000",
"segment_index": 175,
"start_time": "00:13:12.937000",
"text": "of x minus a, and you would evaluate all the derivatives of f at that input, a."
},
{
"end_time": "00:13:23.120000",
"segment_index": 176,
"start_time": "00:13:18.680000",
"text": "This is what Taylor polynomials look like in their fullest generality."
}
],
"title": "Taylor series | Chapter 11, Essence of calculus",
"url": "https://youtube.com/watch?v=3d6DsjIBzJ4",
"video_id": "3d6DsjIBzJ4"
}
|
3d6DsjIBzJ4_172-176
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
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{
"caption": "Changing the value of a changes where this approximation is hugging the original function, where its higher order derivatives will be equal to those of the original function. One of the simplest meaningful examples of this is the function e to the x around the input x equals 0. Computing the derivatives is super nice, as nice as it gets, because the derivative of e to the x is itself,",
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"text": "Changing the value of a changes where this approximation is hugging the original"
},
{
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"segment_index": 178,
"start_time": "00:13:28.534000",
"text": "function, where its higher order derivatives will be equal to those of the original"
},
{
"end_time": "00:13:33.740000",
"segment_index": 179,
"start_time": "00:13:33.236000",
"text": "function."
},
{
"end_time": "00:13:38.860000",
"segment_index": 180,
"start_time": "00:13:35.880000",
"text": "One of the simplest meaningful examples of this is"
},
{
"end_time": "00:13:41.900000",
"segment_index": 181,
"start_time": "00:13:38.860000",
"text": "the function e to the x around the input x equals 0."
},
{
"end_time": "00:13:46.406000",
"segment_index": 182,
"start_time": "00:13:42.760000",
"text": "Computing the derivatives is super nice, as nice as it gets,"
},
{
"end_time": "00:13:49.275000",
"segment_index": 183,
"start_time": "00:13:46.406000",
"text": "because the derivative of e to the x is itself,"
}
],
"title": "Taylor series | Chapter 11, Essence of calculus",
"url": "https://youtube.com/watch?v=3d6DsjIBzJ4",
"video_id": "3d6DsjIBzJ4"
}
|
3d6DsjIBzJ4_177-183
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
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|
{
"caption": "so the second derivative is also e to the x, as is its third, and so on. So at the point x equals 0, all of these are equal to 1. And what that means is our polynomial approximation should look like 1 plus 1 times x plus 1 over 2 times x squared plus 1 over 3 factorial times x cubed,",
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"clip_end": 854.448,
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"start_time": "00:13:49.275000",
"text": "so the second derivative is also e to the x, as is its third, and so on."
},
{
"end_time": "00:13:58.240000",
"segment_index": 185,
"start_time": "00:13:54.340000",
"text": "So at the point x equals 0, all of these are equal to 1."
},
{
"end_time": "00:14:05.720000",
"segment_index": 186,
"start_time": "00:13:59.120000",
"text": "And what that means is our polynomial approximation should look like"
},
{
"end_time": "00:14:13.948000",
"segment_index": 187,
"start_time": "00:14:05.720000",
"text": "1 plus 1 times x plus 1 over 2 times x squared plus 1 over 3 factorial times x cubed,"
}
],
"title": "Taylor series | Chapter 11, Essence of calculus",
"url": "https://youtube.com/watch?v=3d6DsjIBzJ4",
"video_id": "3d6DsjIBzJ4"
}
|
3d6DsjIBzJ4_184-187
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
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{
"caption": "and so on, depending on how many terms you want. These are the Taylor polynomials for e to the x. Ok, so with that as a foundation, in the spirit of showing you just how connected all the topics of calculus are, let me turn to something kind of fun, a completely different way to understand this second order term of the Taylor polynomials, but geometrically.",
"caption_code": "",
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"clip_duration": 27.572000000000003,
"clip_end": 881.02,
"clip_start": 853.448,
"full_duration": 1339.094785,
"segments": [
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"text": "and so on, depending on how many terms you want."
},
{
"end_time": "00:14:22.700000",
"segment_index": 189,
"start_time": "00:14:19.400000",
"text": "These are the Taylor polynomials for e to the x."
},
{
"end_time": "00:14:31.039000",
"segment_index": 190,
"start_time": "00:14:26.380000",
"text": "Ok, so with that as a foundation, in the spirit of showing you just how connected all"
},
{
"end_time": "00:14:34.614000",
"segment_index": 191,
"start_time": "00:14:31.039000",
"text": "the topics of calculus are, let me turn to something kind of fun,"
},
{
"end_time": "00:14:38.840000",
"segment_index": 192,
"start_time": "00:14:34.614000",
"text": "a completely different way to understand this second order term of the Taylor"
},
{
"end_time": "00:14:40.520000",
"segment_index": 193,
"start_time": "00:14:38.840000",
"text": "polynomials, but geometrically."
}
],
"title": "Taylor series | Chapter 11, Essence of calculus",
"url": "https://youtube.com/watch?v=3d6DsjIBzJ4",
"video_id": "3d6DsjIBzJ4"
}
|
3d6DsjIBzJ4_188-193
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
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{
"caption": "It's related to the fundamental theorem of calculus, which I talked about in chapters 1 and 8 if you need a quick refresher. Like we did in those videos, consider a function that gives the area under some graph between a fixed left point and a variable right point. What we're going to do here is think about how to approximate this area function, not the function for the graph itself, like we've been doing before. Focusing on that area is what's going to make the second order term pop out.",
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"text": "It's related to the fundamental theorem of calculus,"
},
{
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"text": "which I talked about in chapters 1 and 8 if you need a quick refresher."
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{
"end_time": "00:14:52.001000",
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"text": "Like we did in those videos, consider a function that gives the area"
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"segment_index": 197,
"start_time": "00:14:52.001000",
"text": "under some graph between a fixed left point and a variable right point."
},
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"segment_index": 198,
"start_time": "00:14:56.980000",
"text": "What we're going to do here is think about how to approximate this area function,"
},
{
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"segment_index": 199,
"start_time": "00:15:00.915000",
"text": "not the function for the graph itself, like we've been doing before."
},
{
"end_time": "00:15:09.440000",
"segment_index": 200,
"start_time": "00:15:04.900000",
"text": "Focusing on that area is what's going to make the second order term pop out."
}
],
"title": "Taylor series | Chapter 11, Essence of calculus",
"url": "https://youtube.com/watch?v=3d6DsjIBzJ4",
"video_id": "3d6DsjIBzJ4"
}
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3d6DsjIBzJ4_194-200
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hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
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{
"caption": "Remember, the fundamental theorem of calculus is that this graph itself represents the derivative of the area function, and it's because a slight nudge dx to the right bound of the area gives a new bit of area approximately equal to the height of the graph times dx. And that approximation is increasingly accurate for smaller and smaller choices of dx.",
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"text": "Remember, the fundamental theorem of calculus is that this graph itself represents the"
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"text": "derivative of the area function, and it's because a slight nudge dx to the right bound"
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"text": "of the area gives a new bit of area approximately equal to the height of the graph times"
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{
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"text": "dx."
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{
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"segment_index": 205,
"start_time": "00:15:30.040000",
"text": "And that approximation is increasingly accurate for smaller and smaller choices of dx."
}
],
"title": "Taylor series | Chapter 11, Essence of calculus",
"url": "https://youtube.com/watch?v=3d6DsjIBzJ4",
"video_id": "3d6DsjIBzJ4"
}
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3d6DsjIBzJ4_201-205
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hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
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{
"caption": "But if you wanted to be more accurate about this change in area, given some change in x that isn't meant to approach 0, you would have to take into account this portion right here, which is approximately a triangle. Let's name the starting input a, and the nudged input above it x, so that change is x-a. The base of that little triangle is that change, x-a,",
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"text": "But if you wanted to be more accurate about this change in area,"
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"text": "given some change in x that isn't meant to approach 0,"
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"text": "you would have to take into account this portion right here,"
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"text": "which is approximately a triangle."
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"text": "Let's name the starting input a, and the nudged input above it x, so that change is x-a."
},
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"start_time": "00:15:58.100000",
"text": "The base of that little triangle is that change, x-a,"
}
],
"title": "Taylor series | Chapter 11, Essence of calculus",
"url": "https://youtube.com/watch?v=3d6DsjIBzJ4",
"video_id": "3d6DsjIBzJ4"
}
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3d6DsjIBzJ4_206-211
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hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
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{
"caption": "but how would you even think to make this approximation, and how would you find that particular quadratic? The study of Taylor series is largely about taking non-polynomial functions and finding polynomials that approximate them near some input. The motive here is that polynomials tend to be much easier to deal with than other functions, they're easier to compute, easier to take derivatives, easier to integrate, just all around more friendly.",
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"text": "but how would you even think to make this approximation,"
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"text": "and how would you find that particular quadratic?"
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"text": "The study of Taylor series is largely about taking non-polynomial"
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"text": "functions and finding polynomials that approximate them near some input."
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"text": "The motive here is that polynomials tend to be much easier to deal"
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"text": "with than other functions, they're easier to compute,"
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"text": "easier to take derivatives, easier to integrate, just all around more friendly."
}
],
"title": "Taylor series | Chapter 11, Essence of calculus",
"url": "https://youtube.com/watch?v=3d6DsjIBzJ4",
"video_id": "3d6DsjIBzJ4"
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3d6DsjIBzJ4_21-27
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hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
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{
"caption": "and its height is the slope of the graph times x-a. Since this graph is the derivative of the area function, its slope is the second derivative of the area function, evaluated at the input a. So the area of this triangle, 1 half base times height, is 1 half times the second derivative of this area function, evaluated at a, multiplied by x-a2.",
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"text": "and its height is the slope of the graph times x-a."
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"text": "Since this graph is the derivative of the area function,"
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"start_time": "00:16:11.987000",
"text": "its slope is the second derivative of the area function, evaluated at the input a."
},
{
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"text": "So the area of this triangle, 1 half base times height,"
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"text": "is 1 half times the second derivative of this area function, evaluated at a,"
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{
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"start_time": "00:16:28.467000",
"text": "multiplied by x-a2."
}
],
"title": "Taylor series | Chapter 11, Essence of calculus",
"url": "https://youtube.com/watch?v=3d6DsjIBzJ4",
"video_id": "3d6DsjIBzJ4"
}
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3d6DsjIBzJ4_212-217
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hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
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{
"caption": "And this is exactly what you would see with a Taylor polynomial. If you knew the various derivative information about this area function at the point a, how would you approximate the area at the point x? Well you have to include all that area up to a, f of a, plus the area of this rectangle here, which is the first derivative, times x-a,",
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"text": "And this is exactly what you would see with a Taylor polynomial."
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"text": "If you knew the various derivative information about this area function at the point a,"
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"text": "how would you approximate the area at the point x?"
},
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"text": "Well you have to include all that area up to a, f of a,"
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{
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"start_time": "00:16:49.316000",
"text": "plus the area of this rectangle here, which is the first derivative, times x-a,"
}
],
"title": "Taylor series | Chapter 11, Essence of calculus",
"url": "https://youtube.com/watch?v=3d6DsjIBzJ4",
"video_id": "3d6DsjIBzJ4"
}
|
3d6DsjIBzJ4_218-222
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hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
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{
"caption": "plus the area of that little triangle, which is 1 half times the second derivative, times x-a2. I really like this, because even though it looks a bit messy all written out, each one of the terms has a very clear meaning that you can just point to on the diagram. If you wanted, we could call it an end here, and you would have a phenomenally useful tool for approximating these Taylor polynomials.",
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"text": "plus the area of that little triangle, which is 1 half times the second derivative,"
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"text": "times x-a2."
},
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"text": "I really like this, because even though it looks a bit messy all written out,"
},
{
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"start_time": "00:17:06.539000",
"text": "each one of the terms has a very clear meaning that you can just point to on the diagram."
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"end_time": "00:17:16.877000",
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"start_time": "00:17:13.400000",
"text": "If you wanted, we could call it an end here, and you would have a"
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{
"end_time": "00:17:20.460000",
"segment_index": 228,
"start_time": "00:17:16.877000",
"text": "phenomenally useful tool for approximating these Taylor polynomials."
}
],
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"url": "https://youtube.com/watch?v=3d6DsjIBzJ4",
"video_id": "3d6DsjIBzJ4"
}
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3d6DsjIBzJ4_223-228
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hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
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{
"caption": "But if you're thinking like a mathematician, one question you might ask is whether or not it makes sense to never stop and just add infinitely many terms. In math, an infinite sum is called a series, so even though one of these approximations with finitely many terms is called a Taylor polynomial, adding all infinitely many terms gives what's called a Taylor series. You have to be really careful with the idea of an infinite series,",
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"text": "But if you're thinking like a mathematician, one question you might ask is"
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"text": "whether or not it makes sense to never stop and just add infinitely many terms."
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"text": "In math, an infinite sum is called a series, so even though one of these"
},
{
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"text": "approximations with finitely many terms is called a Taylor polynomial,"
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"start_time": "00:17:40.263000",
"text": "adding all infinitely many terms gives what's called a Taylor series."
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"text": "You have to be really careful with the idea of an infinite series,"
}
],
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"url": "https://youtube.com/watch?v=3d6DsjIBzJ4",
"video_id": "3d6DsjIBzJ4"
}
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3d6DsjIBzJ4_229-234
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hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
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{
"caption": "because it doesn't actually make sense to add infinitely many things, you can only hit the plus button on the calculator so many times. But if you have a series where adding more and more of the terms, which makes sense at each step, gets you increasingly close to some specific value, what you say is that the series converges to that value. Or, if you're comfortable extending the definition of equality to",
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"text": "because it doesn't actually make sense to add infinitely many things,"
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"text": "you can only hit the plus button on the calculator so many times."
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"text": "But if you have a series where adding more and more of the terms,"
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{
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"text": "which makes sense at each step, gets you increasingly close to some specific value,"
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{
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"text": "what you say is that the series converges to that value."
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"text": "Or, if you're comfortable extending the definition of equality to"
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"url": "https://youtube.com/watch?v=3d6DsjIBzJ4",
"video_id": "3d6DsjIBzJ4"
}
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3d6DsjIBzJ4_235-240
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
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{
"caption": "include this kind of series convergence, you'd say that the series as a whole, this infinite sum, equals the value it's converging to. For example, look at the Taylor polynomial for e to the x, and plug in some input, like x equals 1. As you add more and more polynomial terms, the total sum gets closer and closer to the value e, so you say that this infinite series converges to the number e,",
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"text": "include this kind of series convergence, you'd say that the series as a whole,"
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"text": "this infinite sum, equals the value it's converging to."
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"text": "For example, look at the Taylor polynomial for e to the x,"
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"start_time": "00:18:27.452000",
"text": "and plug in some input, like x equals 1."
},
{
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"text": "As you add more and more polynomial terms, the total sum gets closer and"
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{
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"text": "closer to the value e, so you say that this infinite series converges to the number e,"
}
],
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"url": "https://youtube.com/watch?v=3d6DsjIBzJ4",
"video_id": "3d6DsjIBzJ4"
}
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3d6DsjIBzJ4_241-246
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hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
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{
"caption": "or what's saying the same thing, that it equals the number e. In fact, it turns out that if you plug in any other value of x, like x equals 2, and look at the value of the higher and higher order Taylor polynomials at this value, they will converge towards e to the x, which is e squared. This is true for any input, no matter how far away from 0 it is,",
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"text": "or what's saying the same thing, that it equals the number e."
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"text": "In fact, it turns out that if you plug in any other value of x, like x equals 2,"
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"text": "and look at the value of the higher and higher order Taylor polynomials at this value,"
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"text": "they will converge towards e to the x, which is e squared."
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"text": "This is true for any input, no matter how far away from 0 it is,"
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],
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"video_id": "3d6DsjIBzJ4"
}
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3d6DsjIBzJ4_247-251
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hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
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{
"caption": "even though these Taylor polynomials are constructed only from derivative information gathered at the input 0. In a case like this, we say that e to the x equals its own Taylor series at all inputs x, which is kind of a magical thing to have happen. Even though this is also true for a couple other important functions, like sine and cosine, sometimes these series only converge within a",
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"text": "even though these Taylor polynomials are constructed only from derivative information"
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"text": "gathered at the input 0."
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"text": "In a case like this, we say that e to the x equals its own Taylor series at all inputs x,"
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"text": "which is kind of a magical thing to have happen."
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"text": "Even though this is also true for a couple other important functions,"
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"text": "like sine and cosine, sometimes these series only converge within a"
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],
"title": "Taylor series | Chapter 11, Essence of calculus",
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3d6DsjIBzJ4_252-257
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hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
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{
"caption": "certain range around the input whose derivative information you're using. If you work out the Taylor series for the natural log of x around the input x equals 1, which is built by evaluating the higher order derivatives of the natural log of x at x equals 1, this is what it would look like. When you plug in an input between 0 and 2, adding more and more terms of this",
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"text": "certain range around the input whose derivative information you're using."
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"text": "If you work out the Taylor series for the natural log of x around the input x equals 1,"
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"text": "which is built by evaluating the higher order derivatives of the natural"
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"text": "log of x at x equals 1, this is what it would look like."
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"text": "When you plug in an input between 0 and 2, adding more and more terms of this"
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],
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3d6DsjIBzJ4_258-262
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hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
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{
"caption": "series will indeed get you closer and closer to the natural log of that input. But outside of that range, even by just a little bit, the series fails to approach anything. As you add on more and more terms, the sum bounces back and forth wildly. It does not, as you might expect, approach the natural log of that value, even though the natural log of x is perfectly well defined for inputs above 2.",
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"text": "series will indeed get you closer and closer to the natural log of that input."
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"text": "But outside of that range, even by just a little bit,"
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"text": "the series fails to approach anything."
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"text": "As you add on more and more terms, the sum bounces back and forth wildly."
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"text": "It does not, as you might expect, approach the natural log of that value,"
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"text": "even though the natural log of x is perfectly well defined for inputs above 2."
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"title": "Taylor series | Chapter 11, Essence of calculus",
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3d6DsjIBzJ4_263-268
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hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
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"caption": "In some sense, the derivative information of ln of x at x equals 1 doesn't propagate out that far. In a case like this, where adding more terms of the series doesn't approach anything, you say that the series diverges. And that maximum distance between the input you're approximating near and points where the outputs of these polynomials actually converge is called the radius of convergence for the Taylor series.",
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"text": "In some sense, the derivative information of ln"
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"text": "of x at x equals 1 doesn't propagate out that far."
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"text": "In a case like this, where adding more terms of the series doesn't approach anything,"
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"text": "you say that the series diverges."
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"text": "And that maximum distance between the input you're approximating"
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"text": "near and points where the outputs of these polynomials actually"
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"text": "converge is called the radius of convergence for the Taylor series."
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3d6DsjIBzJ4_269-275
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"caption": "There remains more to learn about Taylor series. There are many use cases, tactics for placing bounds on the error of these approximations, tests for understanding when series do and don't converge, and for that matter, there remains more to learn about calculus as a whole, and the countless topics not touched by this series. The goal with these videos is to give you the fundamental intuitions that make you feel confident and efficient in learning more on your own,",
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"text": "There remains more to learn about Taylor series."
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"text": "There are many use cases, tactics for placing bounds on the error of"
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"text": "these approximations, tests for understanding when series do and don't converge,"
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"text": "and for that matter, there remains more to learn about calculus as a whole,"
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"start_time": "00:21:11.759000",
"text": "and the countless topics not touched by this series."
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"text": "The goal with these videos is to give you the fundamental intuitions"
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"text": "that make you feel confident and efficient in learning more on your own,"
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"url": "https://youtube.com/watch?v=3d6DsjIBzJ4",
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3d6DsjIBzJ4_276-282
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hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
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"caption": "So let's take a look at that function, cosine of x, and really take a moment to think about how you might construct a quadratic approximation near x equals 0. That is, among all of the possible polynomials that look like c0 plus c1 times x plus c2 times x squared, for some choice of these constants, c0, c1, and c2, find the one that most resembles cosine of x near x equals 0,",
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"text": "So let's take a look at that function, cosine of x,"
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"text": "and really take a moment to think about how you might construct a quadratic"
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"text": "approximation near x equals 0."
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"text": "That is, among all of the possible polynomials that look like c0 plus c1"
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"text": "times x plus c2 times x squared, for some choice of these constants, c0,"
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"text": "c1, and c2, find the one that most resembles cosine of x near x equals 0,"
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],
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3d6DsjIBzJ4_28-33
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hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
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{
"caption": "and potentially even rediscovering more of the topic for yourself. In the case of Taylor series, the fundamental intuition to keep in mind as you explore more of what there is, is that they translate derivative information at a single point to approximation information around that point. Thank you once again to everybody who supported this series. The next series like it will be on probability,",
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"text": "and potentially even rediscovering more of the topic for yourself."
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"end_time": "00:21:32.327000",
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"start_time": "00:21:28.060000",
"text": "In the case of Taylor series, the fundamental intuition to keep in mind"
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{
"end_time": "00:21:36.595000",
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"start_time": "00:21:32.327000",
"text": "as you explore more of what there is, is that they translate derivative"
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"text": "information at a single point to approximation information around that point."
},
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"start_time": "00:21:43.920000",
"text": "Thank you once again to everybody who supported this series."
},
{
"end_time": "00:21:49.529000",
"segment_index": 288,
"start_time": "00:21:47.300000",
"text": "The next series like it will be on probability,"
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],
"title": "Taylor series | Chapter 11, Essence of calculus",
"url": "https://youtube.com/watch?v=3d6DsjIBzJ4",
"video_id": "3d6DsjIBzJ4"
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3d6DsjIBzJ4_283-288
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hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
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"caption": "and if you want early access as those videos are made, you know where to go.",
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"text": "and if you want early access as those videos are made, you know where to go."
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3d6DsjIBzJ4_289
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"caption": "Thank you.",
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"text": "Thank you."
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"title": "Taylor series | Chapter 11, Essence of calculus",
"url": "https://youtube.com/watch?v=3d6DsjIBzJ4",
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3d6DsjIBzJ4_290
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hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
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"caption": "whose graph kind of spoons with the graph of cosine x at that point. Well, first of all, at the input 0, the value of cosine of x is 1, so if our approximation is going to be any good at all, it should also equal 1 at the input x equals 0. Plugging in 0 just results in whatever c0 is, so we can set that equal to 1.",
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"text": "whose graph kind of spoons with the graph of cosine x at that point."
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"text": "Well, first of all, at the input 0, the value of cosine of x is 1,"
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"text": "so if our approximation is going to be any good at all,"
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"text": "it should also equal 1 at the input x equals 0."
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"text": "Plugging in 0 just results in whatever c0 is, so we can set that equal to 1."
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],
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3d6DsjIBzJ4_34-38
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hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
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"caption": "This leaves us free to choose constants c1 and c2 to make this approximation as good as we can, but nothing we do with them is going to change the fact that the polynomial equals 1 at x equals 0. It would also be good if our approximation had the same tangent slope as cosine x at this point of interest. Otherwise the approximation drifts away from the cosine graph much faster than it needs to.",
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"text": "This leaves us free to choose constants c1 and c2 to make this"
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"text": "going to change the fact that the polynomial equals 1 at x equals 0."
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"text": "It would also be good if our approximation had the same"
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"text": "tangent slope as cosine x at this point of interest."
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"text": "Otherwise the approximation drifts away from the"
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"text": "cosine graph much faster than it needs to."
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3d6DsjIBzJ4_39-45
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hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
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"caption": "The derivative of cosine is negative sine, and at x equals 0, that equals 0, meaning the tangent line is perfectly flat. On the other hand, when you work out the derivative of our quadratic, you get c1 plus 2 times c2 times x. At x equals 0, this just equals whatever we choose for c1. So this constant c1 has complete control over the derivative of our approximation around x equals 0.",
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"text": "The derivative of cosine is negative sine, and at x equals 0,"
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"text": "that equals 0, meaning the tangent line is perfectly flat."
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"text": "On the other hand, when you work out the derivative of our quadratic,"
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"text": "you get c1 plus 2 times c2 times x."
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"text": "At x equals 0, this just equals whatever we choose for c1."
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"text": "So this constant c1 has complete control over the"
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"text": "derivative of our approximation around x equals 0."
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3d6DsjIBzJ4_46-52
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hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
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"caption": "Setting it equal to 0 ensures that our approximation also has a flat tangent line at this point. This leaves us free to change c2, but the value and the slope of our polynomial at x equals 0 are locked in place to match that of cosine. The final thing to take advantage of is the fact that the cosine graph curves downward above x equals 0, it has a negative second derivative.",
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"text": "Setting it equal to 0 ensures that our approximation"
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"text": "This leaves us free to change c2, but the value and the slope of our"
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"text": "polynomial at x equals 0 are locked in place to match that of cosine."
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"end_time": "00:04:08.379000",
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"start_time": "00:04:04.260000",
"text": "The final thing to take advantage of is the fact that the cosine graph"
},
{
"end_time": "00:04:12.440000",
"segment_index": 58,
"start_time": "00:04:08.379000",
"text": "curves downward above x equals 0, it has a negative second derivative."
}
],
"title": "Taylor series | Chapter 11, Essence of calculus",
"url": "https://youtube.com/watch?v=3d6DsjIBzJ4",
"video_id": "3d6DsjIBzJ4"
}
|
3d6DsjIBzJ4_53-58
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
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|
{
"caption": "Or in other words, even though the rate of change is 0 at that point, the rate of change itself is decreasing around that point. Specifically, since its derivative is negative sine of x, its second derivative is negative cosine of x, and at x equals 0, that equals negative 1. Now in the same way that we wanted the derivative of our approximation to match that of the cosine so that their values wouldn't drift apart needlessly quickly,",
"caption_code": "",
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"text": "Or in other words, even though the rate of change is 0 at that point,"
},
{
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"text": "the rate of change itself is decreasing around that point."
},
{
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"segment_index": 61,
"start_time": "00:04:21.279000",
"text": "Specifically, since its derivative is negative sine of x,"
},
{
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"segment_index": 62,
"start_time": "00:04:25.446000",
"text": "its second derivative is negative cosine of x, and at x equals 0, that equals negative 1."
},
{
"end_time": "00:04:37.150000",
"segment_index": 63,
"start_time": "00:04:33.080000",
"text": "Now in the same way that we wanted the derivative of our approximation to"
},
{
"end_time": "00:04:41.934000",
"segment_index": 64,
"start_time": "00:04:37.150000",
"text": "match that of the cosine so that their values wouldn't drift apart needlessly quickly,"
}
],
"title": "Taylor series | Chapter 11, Essence of calculus",
"url": "https://youtube.com/watch?v=3d6DsjIBzJ4",
"video_id": "3d6DsjIBzJ4"
}
|
3d6DsjIBzJ4_59-64
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
|
|
{
"caption": "making sure that their second derivatives match will ensure that they curve at the same rate, that the slope of our polynomial doesn't drift away from the slope of cosine x any more quickly than it needs to. Pulling up the same derivative we had before, and then taking its derivative, we see that the second derivative of this polynomial is exactly 2 times c2. So to make sure that this second derivative also equals negative 1 at x equals 0,",
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"text": "making sure that their second derivatives match will ensure that they"
},
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"text": "curve at the same rate, that the slope of our polynomial doesn't drift"
},
{
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"segment_index": 67,
"start_time": "00:04:49.690000",
"text": "away from the slope of cosine x any more quickly than it needs to."
},
{
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"segment_index": 68,
"start_time": "00:04:54.220000",
"text": "Pulling up the same derivative we had before, and then taking its derivative,"
},
{
"end_time": "00:05:04.040000",
"segment_index": 69,
"start_time": "00:04:59.226000",
"text": "we see that the second derivative of this polynomial is exactly 2 times c2."
},
{
"end_time": "00:05:10.436000",
"segment_index": 70,
"start_time": "00:05:04.960000",
"text": "So to make sure that this second derivative also equals negative 1 at x equals 0,"
}
],
"title": "Taylor series | Chapter 11, Essence of calculus",
"url": "https://youtube.com/watch?v=3d6DsjIBzJ4",
"video_id": "3d6DsjIBzJ4"
}
|
3d6DsjIBzJ4_65-70
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
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|
{
"caption": "2 times c2 has to be negative 1, meaning c2 itself should be negative 1 half. This gives us the approximation 1 plus 0x minus 1 half x squared. To get a feel for how good it is, if you estimate cosine of 0.1 using this polynomial, you'd estimate it to be 0.995, and this is the true value of cosine of 0.1. It's a really good approximation!",
"caption_code": "",
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"text": "2 times c2 has to be negative 1, meaning c2 itself should be negative 1 half."
},
{
"end_time": "00:05:22.140000",
"segment_index": 72,
"start_time": "00:05:16.380000",
"text": "This gives us the approximation 1 plus 0x minus 1 half x squared."
},
{
"end_time": "00:05:29.977000",
"segment_index": 73,
"start_time": "00:05:23.200000",
"text": "To get a feel for how good it is, if you estimate cosine of 0.1 using this polynomial,"
},
{
"end_time": "00:05:35.820000",
"segment_index": 74,
"start_time": "00:05:29.977000",
"text": "you'd estimate it to be 0.995, and this is the true value of cosine of 0.1."
},
{
"end_time": "00:05:38.440000",
"segment_index": 75,
"start_time": "00:05:36.640000",
"text": "It's a really good approximation!"
}
],
"title": "Taylor series | Chapter 11, Essence of calculus",
"url": "https://youtube.com/watch?v=3d6DsjIBzJ4",
"video_id": "3d6DsjIBzJ4"
}
|
3d6DsjIBzJ4_71-75
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
|
|
{
"caption": "Take a moment to reflect on what just happened. You had 3 degrees of freedom with this quadratic approximation, the constants c0, c1, and c2. c0 was responsible for making sure that the output of the approximation matches that of cosine x at x equals 0, c1 was in charge of making sure that the derivatives match at that point, and c2 was responsible for making sure that the second derivatives match up.",
"caption_code": "",
"channel_id": "3blue1brown",
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"text": "Take a moment to reflect on what just happened."
},
{
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"text": "You had 3 degrees of freedom with this quadratic approximation,"
},
{
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"start_time": "00:05:46.993000",
"text": "the constants c0, c1, and c2."
},
{
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"segment_index": 79,
"start_time": "00:05:49.520000",
"text": "c0 was responsible for making sure that the output of the approximation matches that of"
},
{
"end_time": "00:06:01.952000",
"segment_index": 80,
"start_time": "00:05:55.807000",
"text": "cosine x at x equals 0, c1 was in charge of making sure that the derivatives match at"
},
{
"end_time": "00:06:08.240000",
"segment_index": 81,
"start_time": "00:06:01.952000",
"text": "that point, and c2 was responsible for making sure that the second derivatives match up."
}
],
"title": "Taylor series | Chapter 11, Essence of calculus",
"url": "https://youtube.com/watch?v=3d6DsjIBzJ4",
"video_id": "3d6DsjIBzJ4"
}
|
3d6DsjIBzJ4_76-81
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
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|
{
"caption": "This ensures that the way your approximation changes as you move away from x equals 0, and the way that the rate of change itself changes, is as similar as possible to the behaviour of cosine x, given the amount of control you have. You could give yourself more control by allowing more terms in your polynomial and matching higher order derivatives. For example, let's say you added on the term c3 times x cubed for some constant c3.",
"caption_code": "",
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"text": "This ensures that the way your approximation changes as you move away from x equals 0,"
},
{
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"start_time": "00:06:14.272000",
"text": "and the way that the rate of change itself changes,"
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{
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"segment_index": 84,
"start_time": "00:06:17.459000",
"text": "is as similar as possible to the behaviour of cosine x,"
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{
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"segment_index": 85,
"start_time": "00:06:20.892000",
"text": "given the amount of control you have."
},
{
"end_time": "00:06:27.187000",
"segment_index": 86,
"start_time": "00:06:24.080000",
"text": "You could give yourself more control by allowing more terms"
},
{
"end_time": "00:06:30.140000",
"segment_index": 87,
"start_time": "00:06:27.187000",
"text": "in your polynomial and matching higher order derivatives."
},
{
"end_time": "00:06:36.580000",
"segment_index": 88,
"start_time": "00:06:30.840000",
"text": "For example, let's say you added on the term c3 times x cubed for some constant c3."
}
],
"title": "Taylor series | Chapter 11, Essence of calculus",
"url": "https://youtube.com/watch?v=3d6DsjIBzJ4",
"video_id": "3d6DsjIBzJ4"
}
|
3d6DsjIBzJ4_82-88
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hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
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|
{
"caption": "In that case, if you take the third derivative of a cubic polynomial, anything quadratic or smaller goes to 0. As for that last term, after 3 iterations of the power rule, it looks like 1 times 2 times 3 times c3. On the other hand, the third derivative of cosine x comes out to sine x, which equals 0 at x equals 0.",
"caption_code": "",
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"text": "In that case, if you take the third derivative of a cubic polynomial,"
},
{
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"segment_index": 90,
"start_time": "00:06:41.480000",
"text": "anything quadratic or smaller goes to 0."
},
{
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"start_time": "00:06:45.560000",
"text": "As for that last term, after 3 iterations of the power rule,"
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{
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"segment_index": 92,
"start_time": "00:06:50.882000",
"text": "it looks like 1 times 2 times 3 times c3."
},
{
"end_time": "00:07:01.340000",
"segment_index": 93,
"start_time": "00:06:56.460000",
"text": "On the other hand, the third derivative of cosine x comes out to sine x,"
},
{
"end_time": "00:07:03.280000",
"segment_index": 94,
"start_time": "00:07:01.340000",
"text": "which equals 0 at x equals 0."
}
],
"title": "Taylor series | Chapter 11, Essence of calculus",
"url": "https://youtube.com/watch?v=3d6DsjIBzJ4",
"video_id": "3d6DsjIBzJ4"
}
|
3d6DsjIBzJ4_89-94
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
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{
"caption": "pendulum, and for that you need an expression for how high the weight of the pendulum is above its lowest point, and when you work that out it comes out to be proportional to 1 minus the cosine of the angle between the pendulum and the vertical. The specifics of the problem we were trying to solve are beyond the point here, but what I'll say is that this cosine function made the problem awkward and unwieldy, and made it less clear how pendulums relate to other oscillating phenomena.",
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"text": "pendulum, and for that you need an expression for how high the weight of the"
},
{
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"text": "pendulum is above its lowest point, and when you work that out it comes out to be"
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{
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"text": "proportional to 1 minus the cosine of the angle between the pendulum and the vertical."
},
{
"end_time": "00:00:57.875000",
"segment_index": 12,
"start_time": "00:00:53.580000",
"text": "The specifics of the problem we were trying to solve are beyond the point here,"
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{
"end_time": "00:01:02.493000",
"segment_index": 13,
"start_time": "00:00:57.875000",
"text": "but what I'll say is that this cosine function made the problem awkward and unwieldy,"
},
{
"end_time": "00:01:06.520000",
"segment_index": 14,
"start_time": "00:01:02.493000",
"text": "and made it less clear how pendulums relate to other oscillating phenomena."
}
],
"title": "Taylor series | Chapter 11, Essence of calculus",
"url": "https://youtube.com/watch?v=3d6DsjIBzJ4",
"video_id": "3d6DsjIBzJ4"
}
|
3d6DsjIBzJ4_9-14
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hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
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|
{
"caption": "So to make sure that the third derivatives match, the constant c3 should be 0. Or in other words, not only is 1 minus ½ x2 the best possible quadratic approximation of cosine, it's also the best possible cubic approximation. You can make an improvement by adding on a fourth order term, c4 times x to the fourth.",
"caption_code": "",
"channel_id": "3blue1brown",
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"text": "So to make sure that the third derivatives match, the constant c3 should be 0."
},
{
"end_time": "00:07:14.696000",
"segment_index": 96,
"start_time": "00:07:09.880000",
"text": "Or in other words, not only is 1 minus ½ x2 the best possible quadratic"
},
{
"end_time": "00:07:19.580000",
"segment_index": 97,
"start_time": "00:07:14.696000",
"text": "approximation of cosine, it's also the best possible cubic approximation."
},
{
"end_time": "00:07:27.060000",
"segment_index": 98,
"start_time": "00:07:21.280000",
"text": "You can make an improvement by adding on a fourth order term, c4 times x to the fourth."
}
],
"title": "Taylor series | Chapter 11, Essence of calculus",
"url": "https://youtube.com/watch?v=3d6DsjIBzJ4",
"video_id": "3d6DsjIBzJ4"
}
|
3d6DsjIBzJ4_95-98
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
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|
{
"caption": "The fourth derivative of cosine is itself, which equals 1 at x equals 0. And what's the fourth derivative of our polynomial with this new term? Well, when you keep applying the power rule over and over, with those exponents all hopping down in front, you end up with 1 times 2 times 3 times 4 times c4, which is 24 times c4. So if we want this to match the fourth derivative of cosine x,",
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"text": "The fourth derivative of cosine is itself, which equals 1 at x equals 0."
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{
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"start_time": "00:07:34.300000",
"text": "And what's the fourth derivative of our polynomial with this new term?"
},
{
"end_time": "00:07:42.677000",
"segment_index": 101,
"start_time": "00:07:38.620000",
"text": "Well, when you keep applying the power rule over and over,"
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"start_time": "00:07:42.677000",
"text": "with those exponents all hopping down in front,"
},
{
"end_time": "00:07:51",
"segment_index": 103,
"start_time": "00:07:45.979000",
"text": "you end up with 1 times 2 times 3 times 4 times c4, which is 24 times c4."
},
{
"end_time": "00:07:56.131000",
"segment_index": 104,
"start_time": "00:07:51.400000",
"text": "So if we want this to match the fourth derivative of cosine x,"
}
],
"title": "Taylor series | Chapter 11, Essence of calculus",
"url": "https://youtube.com/watch?v=3d6DsjIBzJ4",
"video_id": "3d6DsjIBzJ4"
}
|
3d6DsjIBzJ4_99-104
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
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{
"caption": "Let's talk about space-filling curves. They are incredibly fun to animate, and they also give a chance to address a certain philosophical question. Math often deals with infinite quantities, sometimes so intimately that the very substance of a result only actually makes sense in an infinite world. So the question is, how can these results ever be useful in a finite context? As with all philosophizing, this is best left to discuss until after we look at the concrete case and the real math.",
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"clip_start": 3.6399999999999997,
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"text": "Let's talk about space-filling curves."
},
{
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"segment_index": 2,
"start_time": "00:00:06.420000",
"text": "They are incredibly fun to animate, and they also give"
},
{
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"segment_index": 3,
"start_time": "00:00:08.864000",
"text": "a chance to address a certain philosophical question."
},
{
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"start_time": "00:00:11.820000",
"text": "Math often deals with infinite quantities, sometimes so intimately that the"
},
{
"end_time": "00:00:20.180000",
"segment_index": 5,
"start_time": "00:00:16.055000",
"text": "very substance of a result only actually makes sense in an infinite world."
},
{
"end_time": "00:00:25.680000",
"segment_index": 6,
"start_time": "00:00:20.940000",
"text": "So the question is, how can these results ever be useful in a finite context?"
},
{
"end_time": "00:00:29.598000",
"segment_index": 7,
"start_time": "00:00:26.660000",
"text": "As with all philosophizing, this is best left to discuss"
},
{
"end_time": "00:00:32.640000",
"segment_index": 8,
"start_time": "00:00:29.598000",
"text": "until after we look at the concrete case and the real math."
}
],
"title": "Hilbert's Curve: Is infinite math useful?",
"url": "https://youtube.com/watch?v=3s7h2MHQtxc",
"video_id": "3s7h2MHQtxc"
}
|
3s7h2MHQtxc_1-8
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
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{
"caption": "curve approach turns out to be exactly what you want. In fact, given how specific the goal is, it seems almost weirdly perfect. So you go back to your mathematician friend and ask her, what was the original motivation for defining one of these curves? She explains that near the end of the 19th century, in the aftershock of Cantor's research on infinity, mathematicians were interested in finding a mapping from a one-dimensional",
"caption_code": "",
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"text": "curve approach turns out to be exactly what you want."
},
{
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"start_time": "00:07:26.220000",
"text": "In fact, given how specific the goal is, it seems almost weirdly perfect."
},
{
"end_time": "00:07:35.148000",
"segment_index": 107,
"start_time": "00:07:32.220000",
"text": "So you go back to your mathematician friend and ask her,"
},
{
"end_time": "00:07:38.540000",
"segment_index": 108,
"start_time": "00:07:35.148000",
"text": "what was the original motivation for defining one of these curves?"
},
{
"end_time": "00:07:42.618000",
"segment_index": 109,
"start_time": "00:07:39.740000",
"text": "She explains that near the end of the 19th century,"
},
{
"end_time": "00:07:45.497000",
"segment_index": 110,
"start_time": "00:07:42.618000",
"text": "in the aftershock of Cantor's research on infinity,"
},
{
"end_time": "00:07:49.648000",
"segment_index": 111,
"start_time": "00:07:45.497000",
"text": "mathematicians were interested in finding a mapping from a one-dimensional"
}
],
"title": "Hilbert's Curve: Is infinite math useful?",
"url": "https://youtube.com/watch?v=3s7h2MHQtxc",
"video_id": "3s7h2MHQtxc"
}
|
3s7h2MHQtxc_105-111
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
|
|
{
"caption": "line into two-dimensional space in such a way that the line runs through every single point in space. To be clear, we're not talking about a finite bounded grid of pixels, like we had in the sound-to-sight application. This is continuous space, which is very infinite, and the goal is to have a line which is as thin as can be and has zero area, somehow pass through every single one of those infinitely many points that makes up the infinite area of space.",
"caption_code": "",
"channel_id": "3blue1brown",
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"start_time": "00:07:49.648000",
"text": "line into two-dimensional space in such a way that the line runs through"
},
{
"end_time": "00:07:55.240000",
"segment_index": 113,
"start_time": "00:07:53.690000",
"text": "every single point in space."
},
{
"end_time": "00:07:59.703000",
"segment_index": 114,
"start_time": "00:07:56.240000",
"text": "To be clear, we're not talking about a finite bounded grid of pixels,"
},
{
"end_time": "00:08:01.980000",
"segment_index": 115,
"start_time": "00:07:59.703000",
"text": "like we had in the sound-to-sight application."
},
{
"end_time": "00:08:05.978000",
"segment_index": 116,
"start_time": "00:08:02.680000",
"text": "This is continuous space, which is very infinite,"
},
{
"end_time": "00:08:11.057000",
"segment_index": 117,
"start_time": "00:08:05.978000",
"text": "and the goal is to have a line which is as thin as can be and has zero area,"
},
{
"end_time": "00:08:16.401000",
"segment_index": 118,
"start_time": "00:08:11.057000",
"text": "somehow pass through every single one of those infinitely many points that makes"
},
{
"end_time": "00:08:18.380000",
"segment_index": 119,
"start_time": "00:08:16.401000",
"text": "up the infinite area of space."
}
],
"title": "Hilbert's Curve: Is infinite math useful?",
"url": "https://youtube.com/watch?v=3s7h2MHQtxc",
"video_id": "3s7h2MHQtxc"
}
|
3s7h2MHQtxc_112-119
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
|
|
{
"caption": "Before 1890, a lot of people thought this was obviously impossible, but then Peano discovered the first of what would come to be known as space-filling curves. In 1891, Hilbert followed with his own slightly simpler space-filling curve. Technically, each one fills a square, not all of space, but I'll show you later on how once you filled a square with a line, filling all of space is not an issue. By the way, mathematicians use the word curve to talk about",
"caption_code": "",
"channel_id": "3blue1brown",
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"text": "Before 1890, a lot of people thought this was obviously impossible,"
},
{
"end_time": "00:08:28.819000",
"segment_index": 121,
"start_time": "00:08:23.768000",
"text": "but then Peano discovered the first of what would come to be known as space-filling"
},
{
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"segment_index": 122,
"start_time": "00:08:28.819000",
"text": "curves."
},
{
"end_time": "00:08:34.400000",
"segment_index": 123,
"start_time": "00:08:30.180000",
"text": "In 1891, Hilbert followed with his own slightly simpler space-filling curve."
},
{
"end_time": "00:08:38.206000",
"segment_index": 124,
"start_time": "00:08:35.400000",
"text": "Technically, each one fills a square, not all of space,"
},
{
"end_time": "00:08:41.665000",
"segment_index": 125,
"start_time": "00:08:38.206000",
"text": "but I'll show you later on how once you filled a square with a line,"
},
{
"end_time": "00:08:43.520000",
"segment_index": 126,
"start_time": "00:08:41.665000",
"text": "filling all of space is not an issue."
},
{
"end_time": "00:08:48.038000",
"segment_index": 127,
"start_time": "00:08:44.620000",
"text": "By the way, mathematicians use the word curve to talk about"
}
],
"title": "Hilbert's Curve: Is infinite math useful?",
"url": "https://youtube.com/watch?v=3s7h2MHQtxc",
"video_id": "3s7h2MHQtxc"
}
|
3s7h2MHQtxc_120-127
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
|
|
{
"caption": "a line running through space even if it has jagged corners. This is especially counterintuitive terminology in the context of a space-filling curve, which in a sense consists of nothing but sharp corners. A better name might be something like space-filling fractal, which some people do use, but hey, it's math, so we live with bad terminology. None of the pseudo-Hilbert curves that you use to fill pixelated space",
"caption_code": "",
"channel_id": "3blue1brown",
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"clip_end": 554.564,
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"text": "a line running through space even if it has jagged corners."
},
{
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"segment_index": 129,
"start_time": "00:08:52.200000",
"text": "This is especially counterintuitive terminology in the context of a space-filling curve,"
},
{
"end_time": "00:09:00.320000",
"segment_index": 130,
"start_time": "00:08:57.218000",
"text": "which in a sense consists of nothing but sharp corners."
},
{
"end_time": "00:09:04.362000",
"segment_index": 131,
"start_time": "00:09:00.860000",
"text": "A better name might be something like space-filling fractal,"
},
{
"end_time": "00:09:08.840000",
"segment_index": 132,
"start_time": "00:09:04.362000",
"text": "which some people do use, but hey, it's math, so we live with bad terminology."
},
{
"end_time": "00:09:14.064000",
"segment_index": 133,
"start_time": "00:09:10.360000",
"text": "None of the pseudo-Hilbert curves that you use to fill pixelated space"
}
],
"title": "Hilbert's Curve: Is infinite math useful?",
"url": "https://youtube.com/watch?v=3s7h2MHQtxc",
"video_id": "3s7h2MHQtxc"
}
|
3s7h2MHQtxc_128-133
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
|
|
{
"caption": "would count as a space-filling curve, no matter how high the order. Just zoom in on one of the pixels. When this pixel is considered part of infinite, continuous space, the curve only passes through the tiniest zero-area slice of it, and it certainly doesn't hit every point. Your mathematician friend explains that an actual bonafide Hilbert curve is not any one of these pseudo-Hilbert curves. Instead it's the limit of all of them.",
"caption_code": "",
"channel_id": "3blue1brown",
"clip_duration": 29.49599999999998,
"clip_end": 583.06,
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"segments": [
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"start_time": "00:09:14.064000",
"text": "would count as a space-filling curve, no matter how high the order."
},
{
"end_time": "00:09:20.200000",
"segment_index": 135,
"start_time": "00:09:18.480000",
"text": "Just zoom in on one of the pixels."
},
{
"end_time": "00:09:25.076000",
"segment_index": 136,
"start_time": "00:09:20.940000",
"text": "When this pixel is considered part of infinite, continuous space,"
},
{
"end_time": "00:09:29.150000",
"segment_index": 137,
"start_time": "00:09:25.076000",
"text": "the curve only passes through the tiniest zero-area slice of it,"
},
{
"end_time": "00:09:31.720000",
"segment_index": 138,
"start_time": "00:09:29.150000",
"text": "and it certainly doesn't hit every point."
},
{
"end_time": "00:09:36.751000",
"segment_index": 139,
"start_time": "00:09:33.420000",
"text": "Your mathematician friend explains that an actual bonafide"
},
{
"end_time": "00:09:40.140000",
"segment_index": 140,
"start_time": "00:09:36.751000",
"text": "Hilbert curve is not any one of these pseudo-Hilbert curves."
},
{
"end_time": "00:09:42.560000",
"segment_index": 141,
"start_time": "00:09:40.820000",
"text": "Instead it's the limit of all of them."
}
],
"title": "Hilbert's Curve: Is infinite math useful?",
"url": "https://youtube.com/watch?v=3s7h2MHQtxc",
"video_id": "3s7h2MHQtxc"
}
|
3s7h2MHQtxc_134-141
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
|
|
{
"caption": "Defining this limit rigorously is delicate. You first have to formalize what these curves are as functions, specifically functions which take in a single number somewhere between 0 and 1 as their input, and output a pair of numbers. This input can be thought of as a point on the line, and the output can be thought of as coordinates in 2D space. But in principle it's just an association between a single number and pairs of numbers.",
"caption_code": "",
"channel_id": "3blue1brown",
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"clip_end": 610.82,
"clip_start": 583.2,
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"start_time": "00:09:43.700000",
"text": "Defining this limit rigorously is delicate."
},
{
"end_time": "00:09:51.266000",
"segment_index": 143,
"start_time": "00:09:47.420000",
"text": "You first have to formalize what these curves are as functions,"
},
{
"end_time": "00:09:55.053000",
"segment_index": 144,
"start_time": "00:09:51.266000",
"text": "specifically functions which take in a single number somewhere"
},
{
"end_time": "00:09:58.720000",
"segment_index": 145,
"start_time": "00:09:55.053000",
"text": "between 0 and 1 as their input, and output a pair of numbers."
},
{
"end_time": "00:10:02.160000",
"segment_index": 146,
"start_time": "00:09:59.600000",
"text": "This input can be thought of as a point on the line,"
},
{
"end_time": "00:10:05.060000",
"segment_index": 147,
"start_time": "00:10:02.160000",
"text": "and the output can be thought of as coordinates in 2D space."
},
{
"end_time": "00:10:10.320000",
"segment_index": 148,
"start_time": "00:10:05.480000",
"text": "But in principle it's just an association between a single number and pairs of numbers."
}
],
"title": "Hilbert's Curve: Is infinite math useful?",
"url": "https://youtube.com/watch?v=3s7h2MHQtxc",
"video_id": "3s7h2MHQtxc"
}
|
3s7h2MHQtxc_142-148
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
|
|
{
"caption": "For example, an order-2 pseudo-Hilbert curve as a function maps the input 0.3 to the output pair 0.125, 0.75. An order-3 pseudo-Hilbert curve maps that same input 0.3 to the output pair 0.0758, 0.6875. Now the core property that makes a function like this a curve,",
"caption_code": "",
"channel_id": "3blue1brown",
"clip_duration": 26.583000000000084,
"clip_end": 637.363,
"clip_start": 610.78,
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"segments": [
{
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"segment_index": 149,
"start_time": "00:10:11.280000",
"text": "For example, an order-2 pseudo-Hilbert curve as a"
},
{
"end_time": "00:10:21.640000",
"segment_index": 150,
"start_time": "00:10:16.032000",
"text": "function maps the input 0.3 to the output pair 0.125, 0.75."
},
{
"end_time": "00:10:31.109000",
"segment_index": 151,
"start_time": "00:10:22.580000",
"text": "An order-3 pseudo-Hilbert curve maps that same input 0.3 to the output pair 0.0758,"
},
{
"end_time": "00:10:31.820000",
"segment_index": 152,
"start_time": "00:10:31.109000",
"text": "0.6875."
},
{
"end_time": "00:10:36.863000",
"segment_index": 153,
"start_time": "00:10:33.140000",
"text": "Now the core property that makes a function like this a curve,"
}
],
"title": "Hilbert's Curve: Is infinite math useful?",
"url": "https://youtube.com/watch?v=3s7h2MHQtxc",
"video_id": "3s7h2MHQtxc"
}
|
3s7h2MHQtxc_149-153
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
|
|
{
"caption": "and not just any ol' association between single numbers and pairs of numbers, is continuity. The intuition behind continuity is that you don't want the output of your function to suddenly jump at any point when the input is only changing smoothly. And the way this is made rigorous in math is actually pretty clever, and fully appreciating space-filling curves requires digesting the formal idea",
"caption_code": "",
"channel_id": "3blue1brown",
"clip_duration": 26.284999999999968,
"clip_end": 662.648,
"clip_start": 636.363,
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"segments": [
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"start_time": "00:10:36.863000",
"text": "and not just any ol' association between single numbers and pairs of numbers,"
},
{
"end_time": "00:10:42.300000",
"segment_index": 155,
"start_time": "00:10:41.472000",
"text": "is continuity."
},
{
"end_time": "00:10:47.667000",
"segment_index": 156,
"start_time": "00:10:43.660000",
"text": "The intuition behind continuity is that you don't want the output of your"
},
{
"end_time": "00:10:52",
"segment_index": 157,
"start_time": "00:10:47.667000",
"text": "function to suddenly jump at any point when the input is only changing smoothly."
},
{
"end_time": "00:10:57.169000",
"segment_index": 158,
"start_time": "00:10:52.820000",
"text": "And the way this is made rigorous in math is actually pretty clever,"
},
{
"end_time": "00:11:02.148000",
"segment_index": 159,
"start_time": "00:10:57.169000",
"text": "and fully appreciating space-filling curves requires digesting the formal idea"
}
],
"title": "Hilbert's Curve: Is infinite math useful?",
"url": "https://youtube.com/watch?v=3s7h2MHQtxc",
"video_id": "3s7h2MHQtxc"
}
|
3s7h2MHQtxc_154-159
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
|
|
{
"caption": "from sight even when the raw data is scrambled into a different format. I've left a few links in the description to studies to this effect. To make initial experiments easier, you might start by treating incoming images with a low resolution, maybe 256 by 256 pixels. And to make my own animation efforts easier, let's represent one of these images with a square grid, each cell corresponding with a pixel. One approach to this sound-to-sight software would be to find a nice",
"caption_code": "",
"channel_id": "3blue1brown",
"clip_duration": 29.974000000000004,
"clip_end": 90.013,
"clip_start": 60.039,
"full_duration": 1097.862676,
"segments": [
{
"end_time": "00:01:04.080000",
"segment_index": 16,
"start_time": "00:01:00.539000",
"text": "from sight even when the raw data is scrambled into a different format."
},
{
"end_time": "00:01:07.680000",
"segment_index": 17,
"start_time": "00:01:04.800000",
"text": "I've left a few links in the description to studies to this effect."
},
{
"end_time": "00:01:12.422000",
"segment_index": 18,
"start_time": "00:01:08.300000",
"text": "To make initial experiments easier, you might start by treating"
},
{
"end_time": "00:01:16.480000",
"segment_index": 19,
"start_time": "00:01:12.422000",
"text": "incoming images with a low resolution, maybe 256 by 256 pixels."
},
{
"end_time": "00:01:20.740000",
"segment_index": 20,
"start_time": "00:01:17.340000",
"text": "And to make my own animation efforts easier, let's represent one of"
},
{
"end_time": "00:01:24.240000",
"segment_index": 21,
"start_time": "00:01:20.740000",
"text": "these images with a square grid, each cell corresponding with a pixel."
},
{
"end_time": "00:01:29.513000",
"segment_index": 22,
"start_time": "00:01:25.080000",
"text": "One approach to this sound-to-sight software would be to find a nice"
}
],
"title": "Hilbert's Curve: Is infinite math useful?",
"url": "https://youtube.com/watch?v=3s7h2MHQtxc",
"video_id": "3s7h2MHQtxc"
}
|
3s7h2MHQtxc_16-22
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
|
|
{
"caption": "of continuity, so it's definitely worth taking a brief side-step to go over it now. Consider a particular input point, a, and the corresponding output of the function, b. Draw a circle centered around a, and look at all the other input points inside that circle, and consider where the function takes all those points in the output space.",
"caption_code": "",
"channel_id": "3blue1brown",
"clip_duration": 24.91199999999992,
"clip_end": 686.56,
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"full_duration": 1097.862676,
"segments": [
{
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"segment_index": 160,
"start_time": "00:11:02.148000",
"text": "of continuity, so it's definitely worth taking a brief side-step to go over it now."
},
{
"end_time": "00:11:14.160000",
"segment_index": 161,
"start_time": "00:11:08.339000",
"text": "Consider a particular input point, a, and the corresponding output of the function, b."
},
{
"end_time": "00:11:20.632000",
"segment_index": 162,
"start_time": "00:11:15.140000",
"text": "Draw a circle centered around a, and look at all the other input points inside that"
},
{
"end_time": "00:11:26.060000",
"segment_index": 163,
"start_time": "00:11:20.632000",
"text": "circle, and consider where the function takes all those points in the output space."
}
],
"title": "Hilbert's Curve: Is infinite math useful?",
"url": "https://youtube.com/watch?v=3s7h2MHQtxc",
"video_id": "3s7h2MHQtxc"
}
|
3s7h2MHQtxc_160-163
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
|
|
{
"caption": "Now draw the smallest circle you can centered at b that contains those outputs. Different choices for the size of the input circle might result in larger or smaller circles in the output space. But notice what happens when we go through this process at a point where the function jumps, drawing a circle around a, and looking at the input points within the circle, seeing where they map,",
"caption_code": "",
"channel_id": "3blue1brown",
"clip_duration": 26.66300000000001,
"clip_end": 713.223,
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"segments": [
{
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"segment_index": 164,
"start_time": "00:11:27.060000",
"text": "Now draw the smallest circle you can centered at b that contains those outputs."
},
{
"end_time": "00:11:36.609000",
"segment_index": 165,
"start_time": "00:11:33.240000",
"text": "Different choices for the size of the input circle might"
},
{
"end_time": "00:11:39.920000",
"segment_index": 166,
"start_time": "00:11:36.609000",
"text": "result in larger or smaller circles in the output space."
},
{
"end_time": "00:11:44.852000",
"segment_index": 167,
"start_time": "00:11:40.700000",
"text": "But notice what happens when we go through this process at a point"
},
{
"end_time": "00:11:48.137000",
"segment_index": 168,
"start_time": "00:11:44.852000",
"text": "where the function jumps, drawing a circle around a,"
},
{
"end_time": "00:11:52.723000",
"segment_index": 169,
"start_time": "00:11:48.137000",
"text": "and looking at the input points within the circle, seeing where they map,"
}
],
"title": "Hilbert's Curve: Is infinite math useful?",
"url": "https://youtube.com/watch?v=3s7h2MHQtxc",
"video_id": "3s7h2MHQtxc"
}
|
3s7h2MHQtxc_164-169
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
|
|
{
"caption": "and drawing the smallest possible circle centered at b containing those points. No matter how small the circle around a, the corresponding circle around b just cannot be smaller than that jump. For this reason, we say that the function is discontinuous at a if there's any lower bound on the size of this circle that surrounds b. If the circle around b can be made as small as you want,",
"caption_code": "",
"channel_id": "3blue1brown",
"clip_duration": 29.047000000000025,
"clip_end": 741.27,
"clip_start": 712.223,
"full_duration": 1097.862676,
"segments": [
{
"end_time": "00:11:57.620000",
"segment_index": 170,
"start_time": "00:11:52.723000",
"text": "and drawing the smallest possible circle centered at b containing those points."
},
{
"end_time": "00:12:02.403000",
"segment_index": 171,
"start_time": "00:11:58.540000",
"text": "No matter how small the circle around a, the corresponding"
},
{
"end_time": "00:12:05.940000",
"segment_index": 172,
"start_time": "00:12:02.403000",
"text": "circle around b just cannot be smaller than that jump."
},
{
"end_time": "00:12:11.727000",
"segment_index": 173,
"start_time": "00:12:07.340000",
"text": "For this reason, we say that the function is discontinuous at a if"
},
{
"end_time": "00:12:16.180000",
"segment_index": 174,
"start_time": "00:12:11.727000",
"text": "there's any lower bound on the size of this circle that surrounds b."
},
{
"end_time": "00:12:20.770000",
"segment_index": 175,
"start_time": "00:12:17.460000",
"text": "If the circle around b can be made as small as you want,"
}
],
"title": "Hilbert's Curve: Is infinite math useful?",
"url": "https://youtube.com/watch?v=3s7h2MHQtxc",
"video_id": "3s7h2MHQtxc"
}
|
3s7h2MHQtxc_170-175
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
|
|
{
"caption": "with sufficiently small choices for circles around a, you say that the function is continuous at a. A function as a whole is called continuous if it's continuous at every possible input point. Now with that as a formal definition of curves, you're ready to define what an actual Hilbert curve is. Doing this relies on a wonderful property of the sequence of pseudo-Hilbert curves, which should feel familiar.",
"caption_code": "",
"channel_id": "3blue1brown",
"clip_duration": 26.629999999999995,
"clip_end": 766.9,
"clip_start": 740.27,
"full_duration": 1097.862676,
"segments": [
{
"end_time": "00:12:23.906000",
"segment_index": 176,
"start_time": "00:12:20.770000",
"text": "with sufficiently small choices for circles around a,"
},
{
"end_time": "00:12:26.520000",
"segment_index": 177,
"start_time": "00:12:23.906000",
"text": "you say that the function is continuous at a."
},
{
"end_time": "00:12:29.750000",
"segment_index": 178,
"start_time": "00:12:27.340000",
"text": "A function as a whole is called continuous if"
},
{
"end_time": "00:12:32.160000",
"segment_index": 179,
"start_time": "00:12:29.750000",
"text": "it's continuous at every possible input point."
},
{
"end_time": "00:12:35.813000",
"segment_index": 180,
"start_time": "00:12:32.980000",
"text": "Now with that as a formal definition of curves,"
},
{
"end_time": "00:12:39.060000",
"segment_index": 181,
"start_time": "00:12:35.813000",
"text": "you're ready to define what an actual Hilbert curve is."
},
{
"end_time": "00:12:44.848000",
"segment_index": 182,
"start_time": "00:12:40.020000",
"text": "Doing this relies on a wonderful property of the sequence of pseudo-Hilbert curves,"
},
{
"end_time": "00:12:46.400000",
"segment_index": 183,
"start_time": "00:12:44.848000",
"text": "which should feel familiar."
}
],
"title": "Hilbert's Curve: Is infinite math useful?",
"url": "https://youtube.com/watch?v=3s7h2MHQtxc",
"video_id": "3s7h2MHQtxc"
}
|
3s7h2MHQtxc_176-183
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
|
|
{
"caption": "Take a given input point, like 0.3, and apply each successive pseudo-Hilbert curve function to this point. The corresponding outputs, as we increase the order of the curve, approaches some particular point in space. It doesn't matter what input you start with, this sequence of outputs you get by applying each successive pseudo-Hilbert curve to this point always stabilizes and approaches some particular point in 2D space.",
"caption_code": "",
"channel_id": "3blue1brown",
"clip_duration": 27.659999999999968,
"clip_end": 794.56,
"clip_start": 766.9,
"full_duration": 1097.862676,
"segments": [
{
"end_time": "00:12:50.681000",
"segment_index": 184,
"start_time": "00:12:47.400000",
"text": "Take a given input point, like 0.3, and apply each"
},
{
"end_time": "00:12:54.220000",
"segment_index": 185,
"start_time": "00:12:50.681000",
"text": "successive pseudo-Hilbert curve function to this point."
},
{
"end_time": "00:12:58.885000",
"segment_index": 186,
"start_time": "00:12:55.060000",
"text": "The corresponding outputs, as we increase the order of the curve,"
},
{
"end_time": "00:13:01.320000",
"segment_index": 187,
"start_time": "00:12:58.885000",
"text": "approaches some particular point in space."
},
{
"end_time": "00:13:06.284000",
"segment_index": 188,
"start_time": "00:13:02.340000",
"text": "It doesn't matter what input you start with, this sequence of outputs"
},
{
"end_time": "00:13:10.284000",
"segment_index": 189,
"start_time": "00:13:06.284000",
"text": "you get by applying each successive pseudo-Hilbert curve to this point"
},
{
"end_time": "00:13:14.060000",
"segment_index": 190,
"start_time": "00:13:10.284000",
"text": "always stabilizes and approaches some particular point in 2D space."
}
],
"title": "Hilbert's Curve: Is infinite math useful?",
"url": "https://youtube.com/watch?v=3s7h2MHQtxc",
"video_id": "3s7h2MHQtxc"
}
|
3s7h2MHQtxc_184-190
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
|
|
{
"caption": "This is absolutely not true, by the way, for snake curves, or for that matter most sequences of curves filling pixelated space of higher and higher resolutions. The outputs associated with a given input become wildly erratic as the resolution increases, always jumping from left to right, and never actually approaching anything. Now because of this property, we can define a Hilbert curve function like this.",
"caption_code": "",
"channel_id": "3blue1brown",
"clip_duration": 26.039999999999964,
"clip_end": 820.88,
"clip_start": 794.84,
"full_duration": 1097.862676,
"segments": [
{
"end_time": "00:13:18.444000",
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"start_time": "00:13:15.340000",
"text": "This is absolutely not true, by the way, for snake curves,"
},
{
"end_time": "00:13:22.549000",
"segment_index": 192,
"start_time": "00:13:18.444000",
"text": "or for that matter most sequences of curves filling pixelated space of higher"
},
{
"end_time": "00:13:23.760000",
"segment_index": 193,
"start_time": "00:13:22.549000",
"text": "and higher resolutions."
},
{
"end_time": "00:13:29.382000",
"segment_index": 194,
"start_time": "00:13:24.370000",
"text": "The outputs associated with a given input become wildly erratic as the resolution"
},
{
"end_time": "00:13:34.640000",
"segment_index": 195,
"start_time": "00:13:29.382000",
"text": "increases, always jumping from left to right, and never actually approaching anything."
},
{
"end_time": "00:13:40.380000",
"segment_index": 196,
"start_time": "00:13:35.900000",
"text": "Now because of this property, we can define a Hilbert curve function like this."
}
],
"title": "Hilbert's Curve: Is infinite math useful?",
"url": "https://youtube.com/watch?v=3s7h2MHQtxc",
"video_id": "3s7h2MHQtxc"
}
|
3s7h2MHQtxc_191-196
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
|
|
{
"caption": "For a given input value between 0 and 1, consider the sequence of points in 2D space you get by applying each successive pseudo-Hilbert curve function at that point. The output of the Hilbert curve function evaluated on this input is just defined to be the limit of those points. Because the sequence of pseudo-Hilbert curve outputs always converges no matter what input you start with, this is actually a well-defined",
"caption_code": "",
"channel_id": "3blue1brown",
"clip_duration": 27.918999999999983,
"clip_end": 848.459,
"clip_start": 820.54,
"full_duration": 1097.862676,
"segments": [
{
"end_time": "00:13:45.751000",
"segment_index": 197,
"start_time": "00:13:41.040000",
"text": "For a given input value between 0 and 1, consider the sequence of points in 2D"
},
{
"end_time": "00:13:50.880000",
"segment_index": 198,
"start_time": "00:13:45.751000",
"text": "space you get by applying each successive pseudo-Hilbert curve function at that point."
},
{
"end_time": "00:13:55.042000",
"segment_index": 199,
"start_time": "00:13:51.420000",
"text": "The output of the Hilbert curve function evaluated on"
},
{
"end_time": "00:13:59",
"segment_index": 200,
"start_time": "00:13:55.042000",
"text": "this input is just defined to be the limit of those points."
},
{
"end_time": "00:14:04.196000",
"segment_index": 201,
"start_time": "00:14:00.380000",
"text": "Because the sequence of pseudo-Hilbert curve outputs always converges"
},
{
"end_time": "00:14:07.959000",
"segment_index": 202,
"start_time": "00:14:04.196000",
"text": "no matter what input you start with, this is actually a well-defined"
}
],
"title": "Hilbert's Curve: Is infinite math useful?",
"url": "https://youtube.com/watch?v=3s7h2MHQtxc",
"video_id": "3s7h2MHQtxc"
}
|
3s7h2MHQtxc_197-202
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
|
|
{
"caption": "function in a way that it never could have been had we used snake curves. Now I'm not going to go through the proof for why this gives a space-filling curve, but let's at least see what needs to be proved. First, verify that this is a well-defined function by proving that the outputs of the pseudo-Hilbert curve functions really do converge the way I'm telling you they do. Second, show that this function gives a curve, meaning it's continuous.",
"caption_code": "",
"channel_id": "3blue1brown",
"clip_duration": 27.021000000000072,
"clip_end": 874.48,
"clip_start": 847.459,
"full_duration": 1097.862676,
"segments": [
{
"end_time": "00:14:11.940000",
"segment_index": 203,
"start_time": "00:14:07.959000",
"text": "function in a way that it never could have been had we used snake curves."
},
{
"end_time": "00:14:17.223000",
"segment_index": 204,
"start_time": "00:14:13.440000",
"text": "Now I'm not going to go through the proof for why this gives a space-filling curve,"
},
{
"end_time": "00:14:19.340000",
"segment_index": 205,
"start_time": "00:14:17.223000",
"text": "but let's at least see what needs to be proved."
},
{
"end_time": "00:14:23.986000",
"segment_index": 206,
"start_time": "00:14:19.340000",
"text": "First, verify that this is a well-defined function by proving that the outputs of"
},
{
"end_time": "00:14:28.860000",
"segment_index": 207,
"start_time": "00:14:23.986000",
"text": "the pseudo-Hilbert curve functions really do converge the way I'm telling you they do."
},
{
"end_time": "00:14:33.980000",
"segment_index": 208,
"start_time": "00:14:29.400000",
"text": "Second, show that this function gives a curve, meaning it's continuous."
}
],
"title": "Hilbert's Curve: Is infinite math useful?",
"url": "https://youtube.com/watch?v=3s7h2MHQtxc",
"video_id": "3s7h2MHQtxc"
}
|
3s7h2MHQtxc_203-208
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
|
|
{
"caption": "Third, and most important, show that it fills space, in the sense that every single point in the unit square is an output of this function. I really do encourage anyone watching this to take a stab at each one of these. Spoiler alert, all three of these facts turn out to be true. You can extend this to a curve that fills all of space just by tiling space with squares and then chaining a bunch of Hilbert curves together",
"caption_code": "",
"channel_id": "3blue1brown",
"clip_duration": 26.942999999999984,
"clip_end": 901.583,
"clip_start": 874.64,
"full_duration": 1097.862676,
"segments": [
{
"end_time": "00:14:38.388000",
"segment_index": 209,
"start_time": "00:14:35.140000",
"text": "Third, and most important, show that it fills space,"
},
{
"end_time": "00:14:43.660000",
"segment_index": 210,
"start_time": "00:14:38.388000",
"text": "in the sense that every single point in the unit square is an output of this function."
},
{
"end_time": "00:14:48.360000",
"segment_index": 211,
"start_time": "00:14:44.580000",
"text": "I really do encourage anyone watching this to take a stab at each one of these."
},
{
"end_time": "00:14:51.860000",
"segment_index": 212,
"start_time": "00:14:48.880000",
"text": "Spoiler alert, all three of these facts turn out to be true."
},
{
"end_time": "00:14:57.319000",
"segment_index": 213,
"start_time": "00:14:53.660000",
"text": "You can extend this to a curve that fills all of space just by tiling"
},
{
"end_time": "00:15:01.083000",
"segment_index": 214,
"start_time": "00:14:57.319000",
"text": "space with squares and then chaining a bunch of Hilbert curves together"
}
],
"title": "Hilbert's Curve: Is infinite math useful?",
"url": "https://youtube.com/watch?v=3s7h2MHQtxc",
"video_id": "3s7h2MHQtxc"
}
|
3s7h2MHQtxc_209-214
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
|
|
{
"caption": "in a spiraling pattern of tiles, connecting the end of one tile to the start of a new tile with an added little stretch of line if you need to. You can think of the first tile as coming from the interval from 0 to 1, the second tile as coming from the interval from 1 to 2, and so on, so the entire positive real number line is getting mapped into all of 2D space. Take a moment to let that fact sink in.",
"caption_code": "",
"channel_id": "3blue1brown",
"clip_duration": 27.23700000000008,
"clip_end": 927.82,
"clip_start": 900.583,
"full_duration": 1097.862676,
"segments": [
{
"end_time": "00:15:04.795000",
"segment_index": 215,
"start_time": "00:15:01.083000",
"text": "in a spiraling pattern of tiles, connecting the end of one tile to the"
},
{
"end_time": "00:15:08.560000",
"segment_index": 216,
"start_time": "00:15:04.795000",
"text": "start of a new tile with an added little stretch of line if you need to."
},
{
"end_time": "00:15:14.624000",
"segment_index": 217,
"start_time": "00:15:09.660000",
"text": "You can think of the first tile as coming from the interval from 0 to 1,"
},
{
"end_time": "00:15:19.248000",
"segment_index": 218,
"start_time": "00:15:14.624000",
"text": "the second tile as coming from the interval from 1 to 2, and so on,"
},
{
"end_time": "00:15:24.620000",
"segment_index": 219,
"start_time": "00:15:19.248000",
"text": "so the entire positive real number line is getting mapped into all of 2D space."
},
{
"end_time": "00:15:27.320000",
"segment_index": 220,
"start_time": "00:15:25.420000",
"text": "Take a moment to let that fact sink in."
}
],
"title": "Hilbert's Curve: Is infinite math useful?",
"url": "https://youtube.com/watch?v=3s7h2MHQtxc",
"video_id": "3s7h2MHQtxc"
}
|
3s7h2MHQtxc_215-220
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
|
|
{
"caption": "A line, the platonic form of thinness itself, can wander through an infinitely extending and richly dense space and hit every single point. Notice, the core property that made pseudo-Hilbert curves useful in both the sound-to-sight application and in their infinite origins is that points on",
"caption_code": "",
"channel_id": "3blue1brown",
"clip_duration": 26.240999999999985,
"clip_end": 953.401,
"clip_start": 927.16,
"full_duration": 1097.862676,
"segments": [
{
"end_time": "00:15:32.816000",
"segment_index": 221,
"start_time": "00:15:27.660000",
"text": "A line, the platonic form of thinness itself, can wander through an"
},
{
"end_time": "00:15:38.200000",
"segment_index": 222,
"start_time": "00:15:32.816000",
"text": "infinitely extending and richly dense space and hit every single point."
},
{
"end_time": "00:15:48.134000",
"segment_index": 223,
"start_time": "00:15:43.240000",
"text": "Notice, the core property that made pseudo-Hilbert curves useful in both the"
},
{
"end_time": "00:15:52.901000",
"segment_index": 224,
"start_time": "00:15:48.134000",
"text": "sound-to-sight application and in their infinite origins is that points on"
}
],
"title": "Hilbert's Curve: Is infinite math useful?",
"url": "https://youtube.com/watch?v=3s7h2MHQtxc",
"video_id": "3s7h2MHQtxc"
}
|
3s7h2MHQtxc_221-224
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
|
|
{
"caption": "the curve move around less and less as you increase the order of those curves. While translating images to sound, this was useful because it means upgrading to higher resolutions doesn't require retraining your senses all over again. For mathematicians interested in filling continuous space, this property is what ensured that talking about the limit of a sequence of curves was a meaningful thing to do.",
"caption_code": "",
"channel_id": "3blue1brown",
"clip_duration": 26.278999999999996,
"clip_end": 978.68,
"clip_start": 952.401,
"full_duration": 1097.862676,
"segments": [
{
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"segment_index": 225,
"start_time": "00:15:52.901000",
"text": "the curve move around less and less as you increase the order of those curves."
},
{
"end_time": "00:16:02.912000",
"segment_index": 226,
"start_time": "00:15:58.780000",
"text": "While translating images to sound, this was useful because it means upgrading"
},
{
"end_time": "00:16:06.940000",
"segment_index": 227,
"start_time": "00:16:02.912000",
"text": "to higher resolutions doesn't require retraining your senses all over again."
},
{
"end_time": "00:16:11.158000",
"segment_index": 228,
"start_time": "00:16:07.460000",
"text": "For mathematicians interested in filling continuous space,"
},
{
"end_time": "00:16:16.361000",
"segment_index": 229,
"start_time": "00:16:11.158000",
"text": "this property is what ensured that talking about the limit of a sequence of curves"
},
{
"end_time": "00:16:18.180000",
"segment_index": 230,
"start_time": "00:16:16.361000",
"text": "was a meaningful thing to do."
}
],
"title": "Hilbert's Curve: Is infinite math useful?",
"url": "https://youtube.com/watch?v=3s7h2MHQtxc",
"video_id": "3s7h2MHQtxc"
}
|
3s7h2MHQtxc_225-230
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
|
|
{
"caption": "way to associate each one of those pixels with a unique frequency value. Then when that pixel is brighter, the frequency associated with it would be played louder, and if the pixel were darker, the frequency would be quiet. Listening to all of the pixels all at once would then sound like a bunch of frequencies overlaid on top of one another, with dominant frequencies corresponding to the brighter regions of the image sounding like some",
"caption_code": "",
"channel_id": "3blue1brown",
"clip_duration": 27.126999999999995,
"clip_end": 116.14,
"clip_start": 89.013,
"full_duration": 1097.862676,
"segments": [
{
"end_time": "00:01:34.140000",
"segment_index": 23,
"start_time": "00:01:29.513000",
"text": "way to associate each one of those pixels with a unique frequency value."
},
{
"end_time": "00:01:38.759000",
"segment_index": 24,
"start_time": "00:01:35.020000",
"text": "Then when that pixel is brighter, the frequency associated with it would be"
},
{
"end_time": "00:01:42.400000",
"segment_index": 25,
"start_time": "00:01:38.759000",
"text": "played louder, and if the pixel were darker, the frequency would be quiet."
},
{
"end_time": "00:01:47.706000",
"segment_index": 26,
"start_time": "00:01:43.400000",
"text": "Listening to all of the pixels all at once would then sound like a bunch of"
},
{
"end_time": "00:01:51.673000",
"segment_index": 27,
"start_time": "00:01:47.706000",
"text": "frequencies overlaid on top of one another, with dominant frequencies"
},
{
"end_time": "00:01:55.640000",
"segment_index": 28,
"start_time": "00:01:51.673000",
"text": "corresponding to the brighter regions of the image sounding like some"
}
],
"title": "Hilbert's Curve: Is infinite math useful?",
"url": "https://youtube.com/watch?v=3s7h2MHQtxc",
"video_id": "3s7h2MHQtxc"
}
|
3s7h2MHQtxc_23-28
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
|
|
{
"caption": "And this connection here between the infinite and finite worlds seems to be more of a rule in math than an exception. Another example that several astute commenters on the Inventing Math video pointed out is the connection between the divergent sum of all powers of 2 and the way that the number of 1 is represented in computers with bits. It's not so much that the infinite result is directly useful,",
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"text": "And this connection here between the infinite and finite"
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"text": "worlds seems to be more of a rule in math than an exception."
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"start_time": "00:16:26.020000",
"text": "Another example that several astute commenters on the Inventing Math video"
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{
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"text": "pointed out is the connection between the divergent sum of all powers of"
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"segment_index": 235,
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"text": "2 and the way that the number of 1 is represented in computers with bits."
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{
"end_time": "00:16:43.640000",
"segment_index": 236,
"start_time": "00:16:39.580000",
"text": "It's not so much that the infinite result is directly useful,"
}
],
"title": "Hilbert's Curve: Is infinite math useful?",
"url": "https://youtube.com/watch?v=3s7h2MHQtxc",
"video_id": "3s7h2MHQtxc"
}
|
3s7h2MHQtxc_231-236
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
|
|
{
"caption": "but instead the same patterns and constructs that are used to define and prove infinite facts have finite analogs, and these finite analogs are directly useful. But the connection is often deeper than a mere analogy. Many theorems about an infinite object are often equivalent to some theorem regarding a family of finite objects. For example, if during your sound-to-sight project you were to sit down",
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"text": "but instead the same patterns and constructs that are used to define and"
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{
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"text": "prove infinite facts have finite analogs, and these finite analogs are directly useful."
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{
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"segment_index": 239,
"start_time": "00:16:55.100000",
"text": "But the connection is often deeper than a mere analogy."
},
{
"end_time": "00:17:02.049000",
"segment_index": 240,
"start_time": "00:16:58.280000",
"text": "Many theorems about an infinite object are often equivalent"
},
{
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"segment_index": 241,
"start_time": "00:17:02.049000",
"text": "to some theorem regarding a family of finite objects."
},
{
"end_time": "00:17:10.411000",
"segment_index": 242,
"start_time": "00:17:06.280000",
"text": "For example, if during your sound-to-sight project you were to sit down"
}
],
"title": "Hilbert's Curve: Is infinite math useful?",
"url": "https://youtube.com/watch?v=3s7h2MHQtxc",
"video_id": "3s7h2MHQtxc"
}
|
3s7h2MHQtxc_237-242
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
|
|
{
"caption": "and really formalize what it means for your curve to stay stable as you increase camera resolution, you would end up effectively writing the definition of what it means for a sequence of curves to have a limit. In fact, a statement about some infinite object, whether that's a sequence or a fractal, can usually be viewed as a particularly clean way to encapsulate a truth about a family of finite objects.",
"caption_code": "",
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"text": "and really formalize what it means for your curve to stay stable as"
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{
"end_time": "00:17:18.271000",
"segment_index": 244,
"start_time": "00:17:14.312000",
"text": "you increase camera resolution, you would end up effectively writing"
},
{
"end_time": "00:17:22.460000",
"segment_index": 245,
"start_time": "00:17:18.271000",
"text": "the definition of what it means for a sequence of curves to have a limit."
},
{
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"segment_index": 246,
"start_time": "00:17:23.400000",
"text": "In fact, a statement about some infinite object,"
},
{
"end_time": "00:17:30.929000",
"segment_index": 247,
"start_time": "00:17:26.636000",
"text": "whether that's a sequence or a fractal, can usually be viewed as"
},
{
"end_time": "00:17:36.280000",
"segment_index": 248,
"start_time": "00:17:30.929000",
"text": "a particularly clean way to encapsulate a truth about a family of finite objects."
}
],
"title": "Hilbert's Curve: Is infinite math useful?",
"url": "https://youtube.com/watch?v=3s7h2MHQtxc",
"video_id": "3s7h2MHQtxc"
}
|
3s7h2MHQtxc_243-248
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
|
|
{
"caption": "The lesson to take away here is that even when a statement seems very far removed from reality, you should always be willing to look under the hood and at the nuts and bolts of what's really being said. Who knows, you might find insights for representing numbers from divergent sums, or for seeing with your ears from filling space.",
"caption_code": "",
"channel_id": "3blue1brown",
"clip_duration": 18.420000000000073,
"clip_end": 1075.4,
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"start_time": "00:17:37.480000",
"text": "The lesson to take away here is that even when a statement seems"
},
{
"end_time": "00:17:44.235000",
"segment_index": 250,
"start_time": "00:17:40.781000",
"text": "very far removed from reality, you should always be willing to look"
},
{
"end_time": "00:17:47.740000",
"segment_index": 251,
"start_time": "00:17:44.235000",
"text": "under the hood and at the nuts and bolts of what's really being said."
},
{
"end_time": "00:17:52.511000",
"segment_index": 252,
"start_time": "00:17:48.480000",
"text": "Who knows, you might find insights for representing numbers from divergent sums,"
},
{
"end_time": "00:17:54.900000",
"segment_index": 253,
"start_time": "00:17:52.511000",
"text": "or for seeing with your ears from filling space."
}
],
"title": "Hilbert's Curve: Is infinite math useful?",
"url": "https://youtube.com/watch?v=3s7h2MHQtxc",
"video_id": "3s7h2MHQtxc"
}
|
3s7h2MHQtxc_249-253
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
|
|
{
"caption": "cacophonous mess until your brain learns to make sense out of the information it contains. Let's temporarily set aside worries about whether or not this would actually work, and instead think about what function, from pixel space down to frequency space, gives this software the best chance of working. The tricky part is that pixel space is two-dimensional, but frequency space is one-dimensional.",
"caption_code": "",
"channel_id": "3blue1brown",
"clip_duration": 25.64,
"clip_end": 140.78,
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{
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"start_time": "00:01:55.640000",
"text": "cacophonous mess until your brain learns to make sense out of the information it contains."
},
{
"end_time": "00:02:06.455000",
"segment_index": 30,
"start_time": "00:02:01.900000",
"text": "Let's temporarily set aside worries about whether or not this would actually work,"
},
{
"end_time": "00:02:10.900000",
"segment_index": 31,
"start_time": "00:02:06.455000",
"text": "and instead think about what function, from pixel space down to frequency space,"
},
{
"end_time": "00:02:13.480000",
"segment_index": 32,
"start_time": "00:02:10.900000",
"text": "gives this software the best chance of working."
},
{
"end_time": "00:02:17.907000",
"segment_index": 33,
"start_time": "00:02:14.500000",
"text": "The tricky part is that pixel space is two-dimensional,"
},
{
"end_time": "00:02:20.280000",
"segment_index": 34,
"start_time": "00:02:17.907000",
"text": "but frequency space is one-dimensional."
}
],
"title": "Hilbert's Curve: Is infinite math useful?",
"url": "https://youtube.com/watch?v=3s7h2MHQtxc",
"video_id": "3s7h2MHQtxc"
}
|
3s7h2MHQtxc_29-34
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
|
|
{
"caption": "You could, of course, try doing this with a random mapping. After all, we're hoping that people's brains make sense out of pretty wonky data anyway. However, it might be nice to leverage some of the intuitions that a given human brain already has about sound. For example, if we think in terms of the reverse mapping from frequency space to pixel space, frequencies that are close together should stay close together in the pixel space.",
"caption_code": "",
"channel_id": "3blue1brown",
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"text": "You could, of course, try doing this with a random mapping."
},
{
"end_time": "00:02:29.600000",
"segment_index": 36,
"start_time": "00:02:25.700000",
"text": "After all, we're hoping that people's brains make sense out of pretty wonky data anyway."
},
{
"end_time": "00:02:33.081000",
"segment_index": 37,
"start_time": "00:02:30.400000",
"text": "However, it might be nice to leverage some of the"
},
{
"end_time": "00:02:36.300000",
"segment_index": 38,
"start_time": "00:02:33.081000",
"text": "intuitions that a given human brain already has about sound."
},
{
"end_time": "00:02:42.051000",
"segment_index": 39,
"start_time": "00:02:36.960000",
"text": "For example, if we think in terms of the reverse mapping from frequency space to pixel"
},
{
"end_time": "00:02:47.260000",
"segment_index": 40,
"start_time": "00:02:42.051000",
"text": "space, frequencies that are close together should stay close together in the pixel space."
}
],
"title": "Hilbert's Curve: Is infinite math useful?",
"url": "https://youtube.com/watch?v=3s7h2MHQtxc",
"video_id": "3s7h2MHQtxc"
}
|
3s7h2MHQtxc_35-40
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
|
|
{
"caption": "That way, even if an ear has a hard time distinguishing between two nearby frequencies, they will at least refer to the same basic point in space. To ensure this happens, you could first describe a way to weave a line through each one of these pixels. Then if you fix each pixel to a spot on that line and unravel the whole thread to make it straight, you could interpret this line as a frequency space,",
"caption_code": "",
"channel_id": "3blue1brown",
"clip_duration": 27.594000000000023,
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"text": "That way, even if an ear has a hard time distinguishing between two nearby frequencies,"
},
{
"end_time": "00:02:56.320000",
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"start_time": "00:02:52.895000",
"text": "they will at least refer to the same basic point in space."
},
{
"end_time": "00:03:00.254000",
"segment_index": 43,
"start_time": "00:02:57.400000",
"text": "To ensure this happens, you could first describe a"
},
{
"end_time": "00:03:03.220000",
"segment_index": 44,
"start_time": "00:03:00.254000",
"text": "way to weave a line through each one of these pixels."
},
{
"end_time": "00:03:08.594000",
"segment_index": 45,
"start_time": "00:03:04.220000",
"text": "Then if you fix each pixel to a spot on that line and unravel the"
},
{
"end_time": "00:03:14.294000",
"segment_index": 46,
"start_time": "00:03:08.594000",
"text": "whole thread to make it straight, you could interpret this line as a frequency space,"
}
],
"title": "Hilbert's Curve: Is infinite math useful?",
"url": "https://youtube.com/watch?v=3s7h2MHQtxc",
"video_id": "3s7h2MHQtxc"
}
|
3s7h2MHQtxc_41-46
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
|
|
{
"caption": "and you have an association from pixels to frequencies. One weaving method would be to just go one row at a time, alternating between left and right as it moves up that pixel space. This is like a well-played game of Snake, so let's call this a Snake Curve. When you tell your mathematician friend about this idea, she says, why not use a Hilbert curve? When you ask her what that is, she stumbles for a moment.",
"caption_code": "",
"channel_id": "3blue1brown",
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"start_time": "00:03:14.294000",
"text": "and you have an association from pixels to frequencies."
},
{
"end_time": "00:03:23.152000",
"segment_index": 48,
"start_time": "00:03:19.840000",
"text": "One weaving method would be to just go one row at a time,"
},
{
"end_time": "00:03:26.980000",
"segment_index": 49,
"start_time": "00:03:23.152000",
"text": "alternating between left and right as it moves up that pixel space."
},
{
"end_time": "00:03:31.400000",
"segment_index": 50,
"start_time": "00:03:27.780000",
"text": "This is like a well-played game of Snake, so let's call this a Snake Curve."
},
{
"end_time": "00:03:35.516000",
"segment_index": 51,
"start_time": "00:03:32.600000",
"text": "When you tell your mathematician friend about this idea,"
},
{
"end_time": "00:03:37.460000",
"segment_index": 52,
"start_time": "00:03:35.516000",
"text": "she says, why not use a Hilbert curve?"
},
{
"end_time": "00:03:40.600000",
"segment_index": 53,
"start_time": "00:03:38.220000",
"text": "When you ask her what that is, she stumbles for a moment."
}
],
"title": "Hilbert's Curve: Is infinite math useful?",
"url": "https://youtube.com/watch?v=3s7h2MHQtxc",
"video_id": "3s7h2MHQtxc"
}
|
3s7h2MHQtxc_47-53
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
|
|
{
"caption": "So it's not a curve, but an infinite family of curves. She starts, well no, it's just one thing, but I need to tell you about a certain infinite family first. She pulls out a piece of paper and starts explaining what she decides to call pseudo-Hilbert curves, for lack of a better term. For an order-one pseudo-Hilbert curve, you divide a square into a 2x2 grid, and connect the center of the lower left quadrant to the center of the upper left,",
"caption_code": "",
"channel_id": "3blue1brown",
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"text": "So it's not a curve, but an infinite family of curves."
},
{
"end_time": "00:03:47.549000",
"segment_index": 55,
"start_time": "00:03:44.380000",
"text": "She starts, well no, it's just one thing, but I need"
},
{
"end_time": "00:03:50.540000",
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"start_time": "00:03:47.549000",
"text": "to tell you about a certain infinite family first."
},
{
"end_time": "00:03:54.351000",
"segment_index": 57,
"start_time": "00:03:51.120000",
"text": "She pulls out a piece of paper and starts explaining what she"
},
{
"end_time": "00:03:57.740000",
"segment_index": 58,
"start_time": "00:03:54.351000",
"text": "decides to call pseudo-Hilbert curves, for lack of a better term."
},
{
"end_time": "00:04:03.132000",
"segment_index": 59,
"start_time": "00:03:58.320000",
"text": "For an order-one pseudo-Hilbert curve, you divide a square into a 2x2 grid,"
},
{
"end_time": "00:04:08.387000",
"segment_index": 60,
"start_time": "00:04:03.132000",
"text": "and connect the center of the lower left quadrant to the center of the upper left,"
}
],
"title": "Hilbert's Curve: Is infinite math useful?",
"url": "https://youtube.com/watch?v=3s7h2MHQtxc",
"video_id": "3s7h2MHQtxc"
}
|
3s7h2MHQtxc_54-60
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
|
|
{
"caption": "over to the upper right, and then down in the lower right. For an order-two pseudo-Hilbert curve, rather than just going straight from one quadrant to another, we let our curve do a little work to fill out each quadrant while it does so. Specifically, subdivide the square further into a 4x4 grid, and we have our curve trace out a miniature order-one pseudo-Hilbert curve inside each quadrant before it moves on to the next.",
"caption_code": "",
"channel_id": "3blue1brown",
"clip_duration": 27.252999999999986,
"clip_end": 275.14,
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"segments": [
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"text": "over to the upper right, and then down in the lower right."
},
{
"end_time": "00:04:17.580000",
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"start_time": "00:04:12.620000",
"text": "For an order-two pseudo-Hilbert curve, rather than just going straight from one quadrant"
},
{
"end_time": "00:04:22.540000",
"segment_index": 63,
"start_time": "00:04:17.580000",
"text": "to another, we let our curve do a little work to fill out each quadrant while it does so."
},
{
"end_time": "00:04:26.775000",
"segment_index": 64,
"start_time": "00:04:23.060000",
"text": "Specifically, subdivide the square further into a 4x4 grid,"
},
{
"end_time": "00:04:31.048000",
"segment_index": 65,
"start_time": "00:04:26.775000",
"text": "and we have our curve trace out a miniature order-one pseudo-Hilbert"
},
{
"end_time": "00:04:34.640000",
"segment_index": 66,
"start_time": "00:04:31.048000",
"text": "curve inside each quadrant before it moves on to the next."
}
],
"title": "Hilbert's Curve: Is infinite math useful?",
"url": "https://youtube.com/watch?v=3s7h2MHQtxc",
"video_id": "3s7h2MHQtxc"
}
|
3s7h2MHQtxc_61-66
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
|
|
{
"caption": "If we left those mini-curves oriented as they are, going from the end of the mini-curve in the lower left to the start of the mini-curve in the upper left requires an awkward jump, same deal with going from the upper right down to the lower right, so we flip the curves in the lower left and lower right to make that connection shorter. Going from an order-two to an order-three pseudo-Hilbert curve is similar. You divide the square into an 8x8 grid, then put an order-two",
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"channel_id": "3blue1brown",
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"text": "If we left those mini-curves oriented as they are,"
},
{
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"start_time": "00:04:38.261000",
"text": "going from the end of the mini-curve in the lower left to the start of the mini-curve"
},
{
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"start_time": "00:04:42.883000",
"text": "in the upper left requires an awkward jump, same deal with going from the upper right"
},
{
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"segment_index": 70,
"start_time": "00:04:47.506000",
"text": "down to the lower right, so we flip the curves in the lower left and lower right to"
},
{
"end_time": "00:04:53.580000",
"segment_index": 71,
"start_time": "00:04:52.021000",
"text": "make that connection shorter."
},
{
"end_time": "00:04:58.780000",
"segment_index": 72,
"start_time": "00:04:54.780000",
"text": "Going from an order-two to an order-three pseudo-Hilbert curve is similar."
},
{
"end_time": "00:05:03.401000",
"segment_index": 73,
"start_time": "00:04:59.460000",
"text": "You divide the square into an 8x8 grid, then put an order-two"
}
],
"title": "Hilbert's Curve: Is infinite math useful?",
"url": "https://youtube.com/watch?v=3s7h2MHQtxc",
"video_id": "3s7h2MHQtxc"
}
|
3s7h2MHQtxc_67-73
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
|
|
{
"caption": "pseudo-Hilbert curve in each quadrant, flip the lower left and lower right appropriately, and connect them all tip to tail. And the pattern continues like that for higher orders. For the 256x256 pixel array, your mathematician friend explains, you would use an order-eight pseudo-Hilbert curve.",
"caption_code": "",
"channel_id": "3blue1brown",
"clip_duration": 27.738999999999976,
"clip_end": 330.64,
"clip_start": 302.901,
"full_duration": 1097.862676,
"segments": [
{
"end_time": "00:05:09.122000",
"segment_index": 74,
"start_time": "00:05:03.401000",
"text": "pseudo-Hilbert curve in each quadrant, flip the lower left and lower right appropriately,"
},
{
"end_time": "00:05:11.220000",
"segment_index": 75,
"start_time": "00:05:09.122000",
"text": "and connect them all tip to tail."
},
{
"end_time": "00:05:14.780000",
"segment_index": 76,
"start_time": "00:05:12.100000",
"text": "And the pattern continues like that for higher orders."
},
{
"end_time": "00:05:26.609000",
"segment_index": 77,
"start_time": "00:05:22.020000",
"text": "For the 256x256 pixel array, your mathematician friend explains,"
},
{
"end_time": "00:05:30.140000",
"segment_index": 78,
"start_time": "00:05:26.609000",
"text": "you would use an order-eight pseudo-Hilbert curve."
}
],
"title": "Hilbert's Curve: Is infinite math useful?",
"url": "https://youtube.com/watch?v=3s7h2MHQtxc",
"video_id": "3s7h2MHQtxc"
}
|
3s7h2MHQtxc_74-78
|
hf://datasets/OrcinusOrca/YouTube-English@4b3d19ba4c84ead7b8dc368b9258f1bb0de39d9b/3blue1brown/3blue1brown_000.tar
|
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