title

Why is pi here? And why is it squared? A geometric answer to the Basel problem

description

A most beautiful proof of the Basel problem, using light.
Help fund future projects: https://www.patreon.com/3blue1brown
An equally valuable form of support is to simply share some of the videos.
Special thanks to these supporters: http://3b1b.co/basel-thanks
This video was sponsored by Brilliant: https://brilliant.org/3b1b
Brilliant's principles list that I referenced:
https://brilliant.org/principles/
Get early access and more through Patreon:
https://www.patreon.com/3blue1brown
The content here was based on a paper by Johan Wästlund
http://www.math.chalmers.se/~wastlund/Cosmic.pdf
Check out Mathologer's video on the many cousins of the Pythagorean theorem:
https://youtu.be/p-0SOWbzUYI
On the topic of Mathologer, he also has a nice video about the Basel problem:
https://youtu.be/yPl64xi_ZZA
A simple Geogebra to play around with the Inverse Pythagorean Theorem argument shown here.
https://ggbm.at/yPExUf7b
Some of you may be concerned about the final step here where we said the circle approaches a line. What about all the lighthouses on the far end? Well, a more careful calculation will show that the contributions from those lights become more negligible. In fact, the contributions from almost all lights become negligible. For the ambitious among you, see this paper for full details.
If you want to contribute translated subtitles or to help review those that have already been made by others and need approval, you can click the gear icon in the video and go to subtitles/cc, then "add subtitles/cc". I really appreciate those who do this, as it helps make the lessons accessible to more people.
Music by Vincent Rubinetti:
https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown
------------------
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detail

{'title': 'Why is pi here? And why is it squared? A geometric answer to the Basel problem', 'heatmap': [{'end': 534.753, 'start': 516.806, 'weight': 1}, {'end': 915.489, 'start': 874.191, 'weight': 0.783}], 'summary': 'Delves into the basel problem, solved by euler, showing the sum of inverse square numbers approach pi squared divided by 6, and illustrates the connection between pi and circles in mathematics through lighthouse demonstrations and geometric transformations.', 'chapters': [{'end': 84.316, 'segs': [{'end': 26.6, 'src': 'embed', 'start': 0.37, 'weight': 0, 'content': [{'end': 8.572, 'text': "Take 1 plus 1 4th plus 1 9th plus 1 16th and so on, where you're adding the inverses of the next square number.", 'start': 0.37, 'duration': 8.202}, {'end': 14.632, 'text': 'What does this sum approach as you keep adding on more and more terms?', 'start': 11.229, 'duration': 3.403}, {'end': 23.598, 'text': 'Now, this is a challenge that remained unsolved for 90 years after it was initially posed, until finally, it was Euler who found the answer,', 'start': 15.712, 'duration': 7.886}, {'end': 26.6, 'text': 'super surprisingly, to be pi squared, divided by 6..', 'start': 23.598, 'duration': 3.002}], 'summary': 'The sum of inverses of square numbers approaches pi squared divided by 6.', 'duration': 26.23, 'max_score': 0.37, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d-o3eB9sfls/pics/d-o3eB9sfls370.jpg'}, {'end': 88.36, 'src': 'embed', 'start': 63.88, 'weight': 1, 'content': [{'end': 71.403, 'text': "And that's all well and good, but whatever your own perspective holds as fundamental, the fact is, pi is very much tied to circles.", 'start': 63.88, 'duration': 7.523}, {'end': 79.991, 'text': 'So if you see it show up, there will be a path somewhere in the massive interconnected web of mathematics leading you back to circles and geometry.', 'start': 71.843, 'duration': 8.148}, {'end': 84.316, 'text': 'The question is just how long and convoluted that path might be.', 'start': 80.852, 'duration': 3.464}, {'end': 88.36, 'text': "And in the case of the Basel problem, it's a lot shorter than you might first think.", 'start': 84.856, 'duration': 3.504}], 'summary': 'Pi is tied to circles and geometry, leading to a shorter path in the basel problem.', 'duration': 24.48, 'max_score': 63.88, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d-o3eB9sfls/pics/d-o3eB9sfls63880.jpg'}], 'start': 0.37, 'title': 'The basel problem and its connection to pi', 'summary': 'Explores the basel problem, which remained unsolved for 90 years, until euler found the sum of inverse square numbers to approach pi squared divided by 6, highlighting the fundamental connection between pi and circles in mathematics.', 'chapters': [{'end': 84.316, 'start': 0.37, 'title': 'The basel problem and its connection to pi', 'summary': 'Explores the basel problem, which remained unsolved for 90 years, until euler found the sum of inverse square numbers to approach pi squared divided by 6, highlighting the fundamental connection between pi and circles in mathematics.', 'duration': 83.946, 'highlights': ["Euler's solution to the Basel problem Euler found the sum of inverse square numbers to approach pi squared divided by 6 after 90 years of it remaining unsolved.", 'Fundamental connection between pi and circles The chapter emphasizes the fundamental connection between pi and circles in mathematics, indicating that whenever pi shows up, there will be a path leading back to circles and geometry in the interconnected web of mathematics.']}], 'duration': 83.946, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d-o3eB9sfls/pics/d-o3eB9sfls370.jpg', 'highlights': ['Euler found the sum of inverse square numbers to approach pi squared divided by 6 after 90 years of it remaining unsolved.', 'The chapter emphasizes the fundamental connection between pi and circles in mathematics, indicating that whenever pi shows up, there will be a path leading back to circles and geometry in the interconnected web of mathematics.']}, {'end': 511.757, 'segs': [{'end': 112.217, 'src': 'embed', 'start': 84.856, 'weight': 1, 'content': [{'end': 88.36, 'text': "And in the case of the Basel problem, it's a lot shorter than you might first think.", 'start': 84.856, 'duration': 3.504}, {'end': 90.562, 'text': 'And it all starts with light.', 'start': 89.06, 'duration': 1.502}, {'end': 93.6, 'text': "Here's the basic idea.", 'start': 92.759, 'duration': 0.841}, {'end': 100.286, 'text': 'Imagine standing at the origin of a positive number line and putting a little lighthouse on all of the positive integers.', 'start': 94.18, 'duration': 6.106}, {'end': 102.789, 'text': '1, 2, 3, 4, and so on.', 'start': 100.787, 'duration': 2.002}, {'end': 107.853, 'text': 'That first lighthouse has some apparent brightness from your point of view.', 'start': 103.649, 'duration': 4.204}, {'end': 112.217, 'text': 'Some amount of energy that your eye is receiving from the light per unit time.', 'start': 108.354, 'duration': 3.863}], 'summary': 'Basel problem relates to lighthouses on positive integers.', 'duration': 27.361, 'max_score': 84.856, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d-o3eB9sfls/pics/d-o3eB9sfls84856.jpg'}, {'end': 157.96, 'src': 'embed', 'start': 129.75, 'weight': 0, 'content': [{'end': 132.312, 'text': 'And you can probably see why this is useful for the Basel problem.', 'start': 129.75, 'duration': 2.562}, {'end': 136.014, 'text': "It gives us a physical representation of what's being asked.", 'start': 132.772, 'duration': 3.242}, {'end': 143.599, 'text': 'Since the brightness received from the whole infinite line of lighthouses is going to be 1 plus 1 4th plus 1 9th plus 1 16th and so on.', 'start': 136.935, 'duration': 6.664}, {'end': 151.41, 'text': 'So the result that we are aiming to show is that this total brightness is equal to pi squared,', 'start': 146.342, 'duration': 5.068}, {'end': 154.815, 'text': 'divided by 6 times the brightness of that first lighthouse.', 'start': 151.41, 'duration': 3.405}, {'end': 157.96, 'text': 'And at first, that might seem useless.', 'start': 156.038, 'duration': 1.922}], 'summary': 'Physical representation of basel problem, brightness equals pi squared divided by 6.', 'duration': 28.21, 'max_score': 129.75, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d-o3eB9sfls/pics/d-o3eB9sfls129750.jpg'}, {'end': 302.053, 'src': 'embed', 'start': 274.234, 'weight': 3, 'content': [{'end': 278.719, 'text': 'the brightness of that light decreases by the inverse square of that distance.', 'start': 274.234, 'duration': 4.485}, {'end': 284.022, 'text': "And as I'm sure many of you know, this inverse square law is not at all special to light.", 'start': 280.14, 'duration': 3.882}, {'end': 292.207, 'text': "It pops up whenever you have some kind of quantity that spreads out evenly from a point source, whether that's sound or heat or a radio signal,", 'start': 284.443, 'duration': 7.764}, {'end': 292.807, 'text': 'things like that.', 'start': 292.207, 'duration': 0.6}, {'end': 302.053, 'text': "And remember, it's because of this inverse square law that an infinite array of evenly spaced lighthouses physically implements the Basel problem.", 'start': 293.788, 'duration': 8.265}], 'summary': 'Inverse square law applies to light, sound, heat, and radio signals spreading from a point source.', 'duration': 27.819, 'max_score': 274.234, 'thumbnail': ''}, {'end': 365.587, 'src': 'embed', 'start': 337.028, 'weight': 2, 'content': [{'end': 341.672, 'text': 'Now place two lighthouses where this new line intersects the coordinate axes,', 'start': 337.028, 'duration': 4.644}, {'end': 346.676, 'text': "which I'll go ahead and call Lighthouse A over here on the left and Lighthouse B on the upper side.", 'start': 341.672, 'duration': 5.004}, {'end': 359.665, 'text': "It turns out and you'll see why this is true in just a minute the brightness that the observer experiences from that first lighthouse is equal to the combined brightness experienced from lighthouses A and B together.", 'start': 347.797, 'duration': 11.868}, {'end': 365.587, 'text': 'And I should say, by the way, that the standing assumption throughout this video is that all lighthouses are equivalent.', 'start': 360.703, 'duration': 4.884}], 'summary': 'Placing two lighthouses at intersecting axes results in equal combined brightness.', 'duration': 28.559, 'max_score': 337.028, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d-o3eB9sfls/pics/d-o3eB9sfls337028.jpg'}, {'end': 403.335, 'src': 'embed', 'start': 380.778, 'weight': 4, 'content': [{'end': 390.725, 'text': 'and the distance to the first lighthouse h, we have the relation 1 over a squared plus 1 over b squared equals 1 over h squared.', 'start': 380.778, 'duration': 9.947}, {'end': 399.792, 'text': "This is the much less well-known inverse Pythagorean theorem, which some of you may recognize from Mathologer's most recent and, I'll say,", 'start': 391.785, 'duration': 8.007}, {'end': 403.335, 'text': 'most excellent video on the many cousins of the Pythagorean theorem.', 'start': 399.792, 'duration': 3.543}], 'summary': 'Inverse pythagorean theorem relates a, b, and h in 1/a^2 + 1/b^2 = 1/h^2', 'duration': 22.557, 'max_score': 380.778, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d-o3eB9sfls/pics/d-o3eB9sfls380778.jpg'}], 'start': 84.856, 'title': 'Basel problem and inverse square law', 'summary': 'Introduces the basel problem using lighthouses, demonstrating the total brightness from an infinite line of lighthouses is pi squared divided by 6 times the brightness of the first lighthouse. it also discusses the inverse square law and a method to transform a single lighthouse into two while maintaining total brightness for the observer.', 'chapters': [{'end': 170.974, 'start': 84.856, 'title': 'The basel problem and lighthouses', 'summary': 'Introduces the basel problem with a physical representation using lighthouses and aims to demonstrate that the total brightness from the infinite line of lighthouses is equal to pi squared divided by 6 times the brightness of the first lighthouse.', 'duration': 86.118, 'highlights': ['The chapter introduces the Basel problem with a physical representation using lighthouses. This sets the stage for a visual understanding of the problem.', 'The total brightness from the infinite line of lighthouses is equal to pi squared divided by 6 times the brightness of the first lighthouse. This is the key result that the chapter aims to demonstrate.']}, {'end': 511.757, 'start': 171.715, 'title': 'Inverse square law and light sources', 'summary': 'Discusses the inverse square law, which describes the decrease in brightness of a light source as the distance from the source increases, and introduces a method to transform a single lighthouse into two, maintaining the total brightness for the observer.', 'duration': 340.042, 'highlights': ['The inverse square law describes how the brightness of light decreases by the square of the distance from the source, resulting in the observed light appearing one fourth as bright at two times the distance away and one ninth as bright at three times the distance away. The brightness of light decreases by the inverse square of the distance; at two times the distance away, the light appears one fourth as bright, and at three times the distance away, it appears one ninth as bright.', 'A method is introduced to transform a single lighthouse into two, where the combined brightness experienced from the two new lighthouses is equal to the brightness experienced from the original lighthouse. The method involves placing two lighthouses at points where a line from the original lighthouse intersects the coordinate axes, resulting in the combined brightness from the two new lighthouses being equal to the brightness from the original lighthouse.', 'The relation 1/a^2 + 1/b^2 = 1/h^2, known as the inverse Pythagorean theorem, is derived, with a, b, and h representing the distances from the observer to the lighthouses. The relation 1/a^2 + 1/b^2 = 1/h^2 is derived, representing the distances from the observer to the lighthouses, and is referred to as the inverse Pythagorean theorem.']}], 'duration': 426.901, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d-o3eB9sfls/pics/d-o3eB9sfls84856.jpg', 'highlights': ['The total brightness from the infinite line of lighthouses is equal to pi squared divided by 6 times the brightness of the first lighthouse. This is the key result that the chapter aims to demonstrate.', 'The chapter introduces the Basel problem with a physical representation using lighthouses. This sets the stage for a visual understanding of the problem.', 'A method is introduced to transform a single lighthouse into two, where the combined brightness experienced from the two new lighthouses is equal to the brightness experienced from the original lighthouse. The method involves placing two lighthouses at points where a line from the original lighthouse intersects the coordinate axes, resulting in the combined brightness from the two new lighthouses being equal to the brightness from the original lighthouse.', 'The inverse square law describes how the brightness of light decreases by the square of the distance from the source, resulting in the observed light appearing one fourth as bright at two times the distance away and one ninth as bright at three times the distance away. The brightness of light decreases by the inverse square of the distance; at two times the distance away, the light appears one fourth as bright, and at three times the distance away, it appears one ninth as bright.', 'The relation 1/a^2 + 1/b^2 = 1/h^2, known as the inverse Pythagorean theorem, is derived, with a, b, and h representing the distances from the observer to the lighthouses. The relation 1/a^2 + 1/b^2 = 1/h^2 is derived, representing the distances from the observer to the lighthouses, and is referred to as the inverse Pythagorean theorem.']}, {'end': 1012.221, 'segs': [{'end': 548.14, 'src': 'heatmap', 'start': 516.806, 'weight': 0, 'content': [{'end': 519.147, 'text': "Alright, buckle up now, because here's where things get good.", 'start': 516.806, 'duration': 2.341}, {'end': 521.587, 'text': "We've got this inverse Pythagorean theorem right?", 'start': 519.667, 'duration': 1.92}, {'end': 528.011, 'text': "And that's going to let us transform a single lighthouse into two others without changing the brightness experienced by the observer.", 'start': 521.828, 'duration': 6.183}, {'end': 534.753, 'text': 'With that in hand, and no small amount of cleverness, we can use this to build up the infinite array that we need.', 'start': 528.791, 'duration': 5.962}, {'end': 540.336, 'text': 'Picture yourself at the edge of a circular lake directly opposite a lighthouse.', 'start': 535.774, 'duration': 4.562}, {'end': 548.14, 'text': "We're going to want it to be the case that the distance between you and the lighthouse along the border of the lake is 1.", 'start': 541.277, 'duration': 6.863}], 'summary': 'Using inverse pythagorean theorem to transform a lighthouse, maintaining brightness for observer, and building an infinite array.', 'duration': 31.334, 'max_score': 516.806, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d-o3eB9sfls/pics/d-o3eB9sfls516806.jpg'}, {'end': 694.864, 'src': 'embed', 'start': 662.539, 'weight': 2, 'content': [{'end': 665.763, 'text': 'And of course, you can do that same thing over on the other side,', 'start': 662.539, 'duration': 3.224}, {'end': 670.768, 'text': 'drawing a line through the top of the smaller circle and getting two new lighthouses on the larger circle.', 'start': 665.763, 'duration': 5.005}, {'end': 676.595, 'text': 'And even nicer, these four lighthouses are all going to be evenly spaced around the lake.', 'start': 671.77, 'duration': 4.825}, {'end': 684.156, 'text': 'Why? Well, the lines from those lighthouses to the center are at 90 degree angles with each other.', 'start': 678.311, 'duration': 5.845}, {'end': 694.864, 'text': 'So, since things are symmetric left to right, that means that the distances along the circumference are 1, 2, 2, 2, and 1.', 'start': 685.116, 'duration': 9.748}], 'summary': 'Drawing a line through circles creates evenly spaced lighthouses with 1, 2, 2, 2, and 1 distances.', 'duration': 32.325, 'max_score': 662.539, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d-o3eB9sfls/pics/d-o3eB9sfls662539.jpg'}, {'end': 842.663, 'src': 'embed', 'start': 813.245, 'weight': 1, 'content': [{'end': 820.511, 'text': 'doubling the size of each circle and transforming each lighthouse into two new ones along a line drawn through the center of the larger circle.', 'start': 813.245, 'duration': 7.266}, {'end': 825.635, 'text': 'At every step, the apparent brightness to the observer remains the same, pi squared over four.', 'start': 821.512, 'duration': 4.123}, {'end': 832.36, 'text': 'And at every step, the lighthouses remain evenly spaced with a distance two between each one of them on the circumference.', 'start': 826.355, 'duration': 6.005}, {'end': 842.663, 'text': "And in the limit, what we're getting here is a flat horizontal line with an infinite number of lighthouses evenly spaced in both directions.", 'start': 834.494, 'duration': 8.169}], 'summary': 'Lighthouses double in size, become two, remain evenly spaced, forming an infinite line.', 'duration': 29.418, 'max_score': 813.245, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d-o3eB9sfls/pics/d-o3eB9sfls813245.jpg'}, {'end': 915.489, 'src': 'heatmap', 'start': 874.191, 'weight': 0.783, 'content': [{'end': 877.353, 'text': 'And just take a step back and think about how unreal this seems.', 'start': 874.191, 'duration': 3.162}, {'end': 883.695, 'text': 'The sum of simple fractions that at first sight have nothing to do with geometry, nothing to do with circles at all,', 'start': 877.933, 'duration': 5.762}, {'end': 886.777, 'text': "apparently gives us this result that's related to pi.", 'start': 883.695, 'duration': 3.082}, {'end': 890.812, 'text': 'Except now you can actually see what it has to do with geometry.', 'start': 887.751, 'duration': 3.061}, {'end': 895.133, 'text': 'The number line is kind of like a limit of ever-growing circles.', 'start': 891.472, 'duration': 3.661}, {'end': 902.295, 'text': 'And as you sum across that number line, making sure to sum all the way to infinity on either side.', 'start': 895.933, 'duration': 6.362}, {'end': 909.797, 'text': "it's sort of like you're adding up along the boundary of an infinitely large circle in a very loose but very fun way of speaking.", 'start': 902.295, 'duration': 7.502}, {'end': 915.489, 'text': 'But wait, you might say, this is not the sum that you promised us at the start of the video.', 'start': 911.107, 'duration': 4.382}], 'summary': 'Summing fractions leads to surprising connection with pi and circles.', 'duration': 41.298, 'max_score': 874.191, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d-o3eB9sfls/pics/d-o3eB9sfls874191.jpg'}, {'end': 1009.62, 'src': 'embed', 'start': 985.206, 'weight': 4, 'content': [{'end': 992.57, 'text': 'that means going from the sum over the odd numbers to the sum over all positive integers requires multiplying by 4 thirds.', 'start': 985.206, 'duration': 7.364}, {'end': 1000.214, 'text': "So, taking that pi squared over 8, multiplying by 4 thirds, bada boom bada bing, we've got ourselves a solution to the Basel problem.", 'start': 993.711, 'duration': 6.503}, {'end': 1009.62, 'text': 'Now, this video that you just watched was primarily written and animated by one of the new 3Blue1Brown team members, Ben Hambrecht.', 'start': 1002.676, 'duration': 6.944}], 'summary': 'Summing odd numbers to all positive integers involves multiplying by 4/3, solving the basel problem.', 'duration': 24.414, 'max_score': 985.206, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d-o3eB9sfls/pics/d-o3eB9sfls985206.jpg'}], 'start': 516.806, 'title': 'Lighthouse geometry & mathematical transformations', 'summary': 'Explores using the inverse pythagorean theorem to transform a single lighthouse into an infinite array of lighthouses, maintaining consistent brightness for the observer, and demonstrates the process through geometric constructions. it also delves into transforming lighthouses on concentric circles, leading to evenly spaced lighthouses and deriving the basel problem solution, showing a sum of inverse squares equal to pi squared over four.', 'chapters': [{'end': 704.586, 'start': 516.806, 'title': 'Infinite lighthouses & inverse pythagorean theorem', 'summary': 'Explores using the inverse pythagorean theorem to transform a single lighthouse into an infinite array of lighthouses, maintaining consistent brightness for the observer, demonstrating the process through geometric constructions and explaining the symmetric arrangement of evenly spaced lighthouses around a circular lake.', 'duration': 187.78, 'highlights': ["The inverse Pythagorean theorem allows transforming a single lighthouse into two others without changing the observer's brightness, enabling the construction of an infinite array of lighthouses. This theorem is used to create two new lighthouses from the original one, maintaining consistent brightness for the observer.", 'Geometric constructions involving circles and triangles are used to demonstrate the process of expanding the array of lighthouses while maintaining consistent apparent brightness. The speaker explains the process of drawing new circles and constructing right triangles to achieve the desired transformations while preserving brightness.', 'The symmetric arrangement of evenly spaced lighthouses around the circular lake is achieved by drawing circles of increasing circumference and creating new lighthouses at specific intersections, resulting in a sequence of distances along the circumference. The process of drawing circles of increasing size and creating new lighthouses at specific intersections results in evenly spaced lighthouses around the lake, with distances along the circumference following a specific sequence.']}, {'end': 1012.221, 'start': 704.586, 'title': 'Lighthouse geometry and the basel problem', 'summary': 'Demonstrates how transforming lighthouses on concentric circles leads to evenly spaced lighthouses and the derivation of the basel problem solution, showing a sum of inverse squares equal to pi squared over four, with a final solution of pi squared over 8 obtained by multiplying by 4 thirds.', 'duration': 307.635, 'highlights': ['The transformation of lighthouses on concentric circles leads to evenly spaced lighthouses on the larger lake, with a distance of two between each one of them on the circumference. The process of transforming lighthouses results in evenly spaced lighthouses on the larger lake, with a distance of two between each one, creating a visual pattern.', 'The solution to the Basel problem shows a sum of inverse squares equal to pi squared over four, derived from adding all the odd integers and their negative counterparts. The derivation of the Basel problem solution involves adding all the odd integers and their negative counterparts, resulting in a sum of inverse squares equal to pi squared over four.', 'Obtaining the final solution of pi squared over 8 involves multiplying the pi squared over 4 solution by 4 thirds, resulting in a solution to the Basel problem. The final solution to the Basel problem is obtained by multiplying the pi squared over 4 solution by 4 thirds, leading to a solution of pi squared over 8.']}], 'duration': 495.415, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d-o3eB9sfls/pics/d-o3eB9sfls516806.jpg', 'highlights': ['The inverse Pythagorean theorem enables transforming a single lighthouse into an infinite array, maintaining consistent brightness for the observer.', 'Geometric constructions demonstrate expanding the array of lighthouses while preserving brightness.', 'Drawing circles of increasing size and creating new lighthouses at specific intersections results in evenly spaced lighthouses around the lake.', 'The transformation of lighthouses on concentric circles leads to evenly spaced lighthouses on the larger lake, creating a visual pattern.', 'The solution to the Basel problem shows a sum of inverse squares equal to pi squared over four, derived from adding all the odd integers and their negative counterparts.', 'Obtaining the final solution of pi squared over 8 involves multiplying the pi squared over 4 solution by 4 thirds, resulting in a solution to the Basel problem.']}], 'highlights': ['Euler found the sum of inverse square numbers to approach pi squared divided by 6 after 90 years of it remaining unsolved.', 'The total brightness from the infinite line of lighthouses is equal to pi squared divided by 6 times the brightness of the first lighthouse.', 'The inverse Pythagorean theorem enables transforming a single lighthouse into an infinite array, maintaining consistent brightness for the observer.', 'The chapter emphasizes the fundamental connection between pi and circles in mathematics, indicating that whenever pi shows up, there will be a path leading back to circles and geometry in the interconnected web of mathematics.', 'The solution to the Basel problem shows a sum of inverse squares equal to pi squared over four, derived from adding all the odd integers and their negative counterparts.']}