title

The Wallis product for pi, proved geometrically

description

A geometric proof of a famous Wallis product for pi.
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/wallis-thanks
If you want to dive into the relevant ideas required to make this proof more rigorous, the relevant search term is "dominated convergence".
https://en.wikipedia.org/wiki/Dominated_convergence_theorem
Here's a good blog post on the topic:
https://www.math3ma.com/blog/dominated-convergence-theorem
In the video, I referenced our own blog post expanding on this argument. Unfortunately, it managed to get lost during a website transition.
Another approach to this product by Johan Wästlund:
http://www.math.chalmers.se/~wastlund/monthly.pdf
With more from Donald Knuth building off this idea:
https://apetresc.wordpress.com/2010/12/28/knuths-why-pi-talk-at-stanford-part-1/
Music by Vincent Rubinetti:
https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown
------------------
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detail

{'title': 'The Wallis product for pi, proved geometrically', 'heatmap': [{'end': 465.666, 'start': 424.62, 'weight': 0.873}, {'end': 594.9, 'start': 524.691, 'weight': 0.724}, {'end': 1090.968, 'start': 1068.61, 'weight': 0.724}, {'end': 1222.571, 'start': 1182.715, 'weight': 0.82}], 'summary': 'Explores a novel proof of the wallace product for pi through unconventional thinking and its connection to the basel problem, providing a geometric perspective on the concept, while also covering the application of inverse square law, complex numbers, and arithmetic in the proof, ultimately leading to the derivation of an infinite product formula for sine of x.', 'chapters': [{'end': 94.937, 'segs': [{'end': 79.065, 'src': 'embed', 'start': 40.682, 'weight': 0, 'content': [{'end': 45.605, 'text': "And for almost all of the content on this channel, the underlying math is something that's well known in the field.", 'start': 40.682, 'duration': 4.923}, {'end': 48.767, 'text': "It's either based on general theory or some particular paper.", 'start': 45.845, 'duration': 2.922}, {'end': 52.368, 'text': 'and my hope is for the novelty to come from the communication half.', 'start': 49.347, 'duration': 3.021}, {'end': 60.212, 'text': "And with this video, the result we're discussing, a very famous infinite product for pi known as the Wallace product, is indeed well-known math.", 'start': 53.449, 'duration': 6.763}, {'end': 65.875, 'text': "However, what we'll be presenting is, to our knowledge, a more original proof of this result.", 'start': 60.752, 'duration': 5.123}, {'end': 67.496, 'text': 'For context.', 'start': 66.815, 'duration': 0.681}, {'end': 71.899, 'text': 'after watching our video on the Basel problem, Swether, the new 3b1b member,', 'start': 67.496, 'duration': 4.403}, {'end': 75.702, 'text': 'who some of you may remember from the video about color and winding numbers.', 'start': 71.899, 'duration': 3.803}, {'end': 79.065, 'text': 'well, he spent some time thinking about the approach taken in that video,', 'start': 75.702, 'duration': 3.363}], 'summary': 'The video presents a more original proof of the well-known wallace product for pi.', 'duration': 38.383, 'max_score': 40.682, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/8GPy_UMV-08/pics/8GPy_UMV-0840682.jpg'}], 'start': 4.264, 'title': 'Fractions and circles', 'summary': 'Explores a novel proof of the wallace product for pi, a famous infinite product, which came about through unconventional thinking and a connection to the basel problem, offering a fresh perspective on well-known math.', 'chapters': [{'end': 94.937, 'start': 4.264, 'title': 'Surprising connection: fractions and circles', 'summary': 'Explores a novel proof of the wallace product for pi, a famous infinite product, which came about through unconventional thinking and a connection to the basel problem, offering a fresh perspective on well-known math.', 'duration': 90.673, 'highlights': ['The video presents a more original proof of the Wallace product for pi, a well-known math result, discovered through unconventional thinking and a connection to the Basel problem.', 'The value of math presentation is attributed to both the underlying math and the choices made in communicating it, with the novelty in this case stemming from the communication half.', 'The result being discussed is the Wallace product for pi, a very famous infinite product, and the new proof of this relationship was stumbled upon by Swether, a new member of 3b1b, after thinking about the approach taken in a previous video and the connection between the Basel problem and the Wallace product.']}], 'duration': 90.673, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/8GPy_UMV-08/pics/8GPy_UMV-084264.jpg', 'highlights': ['The video presents a more original proof of the Wallace product for pi, a well-known math result, discovered through unconventional thinking and a connection to the Basel problem.', 'The result being discussed is the Wallace product for pi, a very famous infinite product, and the new proof of this relationship was stumbled upon by Swether, a new member of 3b1b, after thinking about the approach taken in a previous video and the connection between the Basel problem and the Wallace product.', 'The value of math presentation is attributed to both the underlying math and the choices made in communicating it, with the novelty in this case stemming from the communication half.']}, {'end': 215.863, 'segs': [{'end': 127.742, 'src': 'embed', 'start': 94.937, 'weight': 2, 'content': [{'end': 99.58, 'text': 'but I can at least say that it was found independently and that if it does exist out there,', 'start': 94.937, 'duration': 4.643}, {'end': 102.642, 'text': 'it has done a fantastic job hiding itself from the public view.', 'start': 99.58, 'duration': 3.062}, {'end': 105.984, 'text': "So, without further ado, let's dive into the math.", 'start': 103.562, 'duration': 2.422}, {'end': 115.148, 'text': 'Consider the product 2 over 1 times, 4 over 3 times, 6 over 5 on and on and on,', 'start': 109.181, 'duration': 5.967}, {'end': 120.094, 'text': "where what we're doing is including all the even numbers as the numerators and odd numbers as the denominators.", 'start': 115.148, 'duration': 4.946}, {'end': 127.742, 'text': 'Of course, all the factors here are bigger than 1, so as you go through the series, multiplying each new factor in one by one,', 'start': 120.834, 'duration': 6.908}], 'summary': 'The product of even numbers over consecutive odd numbers is being discussed, with all factors bigger than 1.', 'duration': 32.805, 'max_score': 94.937, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/8GPy_UMV-08/pics/8GPy_UMV-0894937.jpg'}, {'end': 163.84, 'src': 'embed', 'start': 134.526, 'weight': 0, 'content': [{'end': 136.507, 'text': "So in that sense, it's not super interesting.", 'start': 134.526, 'duration': 1.981}, {'end': 137.847, 'text': 'It just blows up to infinity.', 'start': 136.667, 'duration': 1.18}, {'end': 146.93, 'text': 'And on the other hand, if you shift things over slightly, looking at 2 divided by 3 times, 4 divided by, 5 times, 6 divided by 7, on and on,', 'start': 138.588, 'duration': 8.342}, {'end': 149.151, 'text': 'all of those factors are less than 1..', 'start': 146.93, 'duration': 2.221}, {'end': 150.951, 'text': 'So the result keeps getting smaller and smaller.', 'start': 149.151, 'duration': 1.8}, {'end': 154.532, 'text': 'And this time, the series turns out to approach 0.', 'start': 151.451, 'duration': 3.081}, {'end': 155.973, 'text': 'But what if we mix the two?', 'start': 154.532, 'duration': 1.441}, {'end': 163.84, 'text': 'If you looked at 2 over 1 times, 2 over 3 times, 4 over 3 times, 4 over 5, on and on, like this,', 'start': 157.034, 'duration': 6.806}], 'summary': 'Discussing convergence and divergence of series, approaching infinity and zero.', 'duration': 29.314, 'max_score': 134.526, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/8GPy_UMV-08/pics/8GPy_UMV-08134526.jpg'}], 'start': 94.937, 'title': 'Infinite product and pi', 'summary': 'Explores the convergence of an infinite product to pi/2 through alternating even and odd numbers, and its connection to circles, providing a geometric perspective on the concept.', 'chapters': [{'end': 215.863, 'start': 94.937, 'title': 'Infinite product and its connection to pi', 'summary': 'Explores the infinite product of alternating even and odd numbers, its divergence, convergence to a finite value of pi/2, and its connection to circles through a digression involving geometric tools.', 'duration': 120.926, 'highlights': ['The infinite product of alternating even and odd numbers diverges to infinity and then converges to a finite value of pi/2. The product of alternating even and odd numbers results in a series that gets bigger and bigger, eventually reaching infinity, but when the series is slightly shifted, it approaches 0. However, when both patterns are combined, the series converges to pi/2.', "The infinite product's value is pi/2, and its connection to circles is explored through a digression involving geometric tools. The value of the infinite product of alternating even and odd numbers is discovered to be pi/2, and its connection to circles is explained through a digression involving a circle with evenly spaced points and a special point.", 'The geometric tools introduced in the digression are useful for various mathematical problem-solving scenarios. The digression through geometric tools is described as productive, as it equips individuals with useful problem-solving tools for various mathematical scenarios.']}], 'duration': 120.926, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/8GPy_UMV-08/pics/8GPy_UMV-0894937.jpg', 'highlights': ['The infinite product of alternating even and odd numbers diverges to infinity and then converges to a finite value of pi/2.', "The infinite product's value is pi/2, and its connection to circles is explored through a digression involving geometric tools.", 'The geometric tools introduced in the digression are useful for various mathematical problem-solving scenarios.']}, {'end': 509.232, 'segs': [{'end': 264.182, 'src': 'embed', 'start': 236.476, 'weight': 7, 'content': [{'end': 241.217, 'text': 'since the inverse square law gave a really nice physical interpretation to this quantity.', 'start': 236.476, 'duration': 4.741}, {'end': 244.777, 'text': 'It was the total amount of light received by that observer.', 'start': 241.637, 'duration': 3.14}, {'end': 250.999, 'text': "But despite that nice physical interpretation, there's nothing magical about adding inverse squared distances.", 'start': 245.938, 'duration': 5.061}, {'end': 254.52, 'text': 'That just happened to be what was useful for that particular problem.', 'start': 251.479, 'duration': 3.041}, {'end': 259.241, 'text': 'Now to tackle our new problem of 2 over 1 times, 2 over 3 times, 4 over 3 times, 4 over 5 and so on.', 'start': 255.18, 'duration': 4.061}, {'end': 264.182, 'text': "We're going to do something similar, but different in the details.", 'start': 261.641, 'duration': 2.541}], 'summary': 'Inverse square law provides physical interpretation of light received. new problem requires similar approach.', 'duration': 27.706, 'max_score': 236.476, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/8GPy_UMV-08/pics/8GPy_UMV-08236476.jpg'}, {'end': 357.751, 'src': 'embed', 'start': 306.094, 'weight': 0, 'content': [{'end': 312.182, 'text': 'the thing you get by multiplying together the lengths of all these lines works out to be exactly two.', 'start': 306.094, 'duration': 6.088}, {'end': 315.386, 'text': 'No matter how many lighthouses there are.', 'start': 313.944, 'duration': 1.442}, {'end': 325.47, 'text': 'And second, if you remove one of those lighthouses and put the observer in its place,', 'start': 320.444, 'duration': 5.026}, {'end': 332.398, 'text': 'this distance product from all of the remaining lighthouses happens to equal the number of lighthouses that you started with.', 'start': 325.47, 'duration': 6.928}, {'end': 345.875, 'text': 'Again, no matter how many lighthouses there are, And if those two facts seem crazy, I agree.', 'start': 334.6, 'duration': 11.275}, {'end': 351.082, 'text': "I mean, it's not even obvious that the distance product here should work out to be an integer in either case.", 'start': 345.895, 'duration': 5.187}, {'end': 357.751, 'text': 'And also, it seems super tricky to actually compute all of the distances and then multiply them together like this.', 'start': 351.723, 'duration': 6.028}], 'summary': 'The product of distances between lighthouses equals two, and after removing one lighthouse, the product equals the original number of lighthouses.', 'duration': 51.657, 'max_score': 306.094, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/8GPy_UMV-08/pics/8GPy_UMV-08306094.jpg'}, {'end': 449.656, 'src': 'embed', 'start': 424.62, 'weight': 3, 'content': [{'end': 431.563, 'text': 'So because the angle that each one of these makes with the horizontal is an integer multiple of 1, 7th of a turn.', 'start': 424.62, 'duration': 6.943}, {'end': 438.847, 'text': 'raising any one of these numbers to the 7th power rotates you around to landing on the number 1..', 'start': 431.563, 'duration': 7.284}, {'end': 449.656, 'text': 'In other words, these are all solutions to the polynomial equation x to the 7th minus 1 equals 0.', 'start': 438.847, 'duration': 10.809}], 'summary': 'Angles are integer multiples of 1/7th turn, raising to 7th power equals 1', 'duration': 25.036, 'max_score': 424.62, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/8GPy_UMV-08/pics/8GPy_UMV-08424620.jpg'}, {'end': 465.666, 'src': 'heatmap', 'start': 424.62, 'weight': 0.873, 'content': [{'end': 431.563, 'text': 'So because the angle that each one of these makes with the horizontal is an integer multiple of 1, 7th of a turn.', 'start': 424.62, 'duration': 6.943}, {'end': 438.847, 'text': 'raising any one of these numbers to the 7th power rotates you around to landing on the number 1..', 'start': 431.563, 'duration': 7.284}, {'end': 449.656, 'text': 'In other words, these are all solutions to the polynomial equation x to the 7th minus 1 equals 0.', 'start': 438.847, 'duration': 10.809}, {'end': 458.581, 'text': 'But on the other hand, we could construct a polynomial that has these numbers as roots a totally different way, by taking x minus L0 times x minus L1,', 'start': 449.656, 'duration': 8.925}, {'end': 461.503, 'text': 'on and on and on up to x minus L6..', 'start': 458.581, 'duration': 2.922}, {'end': 465.666, 'text': 'I mean, you plug in any one of these numbers, and that product will have to equal zero.', 'start': 461.983, 'duration': 3.683}], 'summary': 'Numbers at integer multiples of 1/7 rotate to 1; solutions to x^7 - 1 = 0.', 'duration': 41.046, 'max_score': 424.62, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/8GPy_UMV-08/pics/8GPy_UMV-08424620.jpg'}, {'end': 494.739, 'src': 'embed', 'start': 466.727, 'weight': 6, 'content': [{'end': 474.997, 'text': "And because these two degree seven polynomials have the same seven distinct roots and the same leading term, it's just x to the seventh.", 'start': 466.727, 'duration': 8.27}, {'end': 477.901, 'text': 'in both cases they are in fact one and the same.', 'start': 474.997, 'duration': 2.904}, {'end': 481.586, 'text': 'Now take a moment to appreciate just what a marvelous fact that is.', 'start': 478.802, 'duration': 2.784}, {'end': 485.811, 'text': 'This right hand side looks like it would be an absolute nightmare to expand.', 'start': 482.148, 'duration': 3.663}, {'end': 488.333, 'text': 'Not only are there a lot of terms,', 'start': 486.372, 'duration': 1.961}, {'end': 494.739, 'text': 'but writing down what exactly each of those complex numbers is is going to land us in a whole mess of sines and cosines.', 'start': 488.333, 'duration': 6.406}], 'summary': 'Two degree seven polynomials with same roots and leading term are identical.', 'duration': 28.012, 'max_score': 466.727, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/8GPy_UMV-08/pics/8GPy_UMV-08466727.jpg'}], 'start': 215.863, 'title': 'Inverse square law and distance product', 'summary': 'Covers the concept of using inverse squared distances of lighthouses to measure light received and discusses the distance product, its application in the proof of the wallace product, and the properties of distance product and its algebraic representation.', 'chapters': [{'end': 254.52, 'start': 215.863, 'title': 'Inverse square law and lighthouses', 'summary': 'Discusses the concept of using inverse squared distances of lighthouses to measure the total amount of light received by an observer, providing a physical interpretation to the quantity.', 'duration': 38.657, 'highlights': ['The inverse square law provided a physical interpretation to the total amount of light received by the observer, by summing the inverse square of distances between the observer and each lighthouse.', 'The concept of adding inverse squared distances was specifically useful for measuring the total amount of light received by the observer, despite not being inherently magical.', 'The analogy of lighthouses and an observer was used to visualize the concept of using inverse squared distances to measure the total amount of light received.']}, {'end': 509.232, 'start': 255.18, 'title': 'Wallace product and distance product', 'summary': 'Discusses the concept of distance product, its application in the proof of the wallace product, and the connection between evenly spaced points around a unit circle and roots of unity, highlighting the properties of the distance product and its algebraic representation.', 'duration': 254.052, 'highlights': ['The distance product for the observer involves multiplying the lengths of all the lines, resulting in a value of exactly 2 when the observer is positioned halfway between two lighthouses on the circle, and the number of lighthouses when one lighthouse is removed, irrespective of the number of lighthouses present.', 'The geometric property of evenly spaced points around a circle corresponds to a nice algebraic property, where the points are represented as roots of unity, and the polynomial equation x to the 7th minus 1 equals 0 is discussed, highlighting the symmetry and simplification of the algebraic representation.']}], 'duration': 293.369, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/8GPy_UMV-08/pics/8GPy_UMV-08215863.jpg', 'highlights': ['The inverse square law interprets the total light received by summing inverse square distances.', 'Using inverse squared distances is specifically useful for measuring total light received.', 'The analogy of lighthouses and an observer visualizes using inverse squared distances.', 'The distance product for the observer involves multiplying the lengths of all the lines.', 'The value of the distance product is exactly 2 when the observer is positioned halfway between two lighthouses.', 'The number of lighthouses is the same when one lighthouse is removed, irrespective of the number of lighthouses present.', 'Evenly spaced points around a circle correspond to a nice algebraic property.', 'The polynomial equation x to the 7th minus 1 equals 0 is discussed, highlighting the symmetry and simplification of the algebraic representation.']}, {'end': 812.257, 'segs': [{'end': 594.9, 'src': 'heatmap', 'start': 524.691, 'weight': 0.724, 'content': [{'end': 532.538, 'text': 'If you consider the observer to be any other complex number, not necessarily on the circle, and then you plug in that number for x,', 'start': 524.691, 'duration': 7.847}, {'end': 533.539, 'text': 'that right-hand side,', 'start': 532.538, 'duration': 1.001}, {'end': 542.126, 'text': 'there is giving you some new complex number whose magnitude is the product of the distances between the observer and each lighthouse.', 'start': 533.539, 'duration': 8.587}, {'end': 544.368, 'text': 'But look at that left-hand side.', 'start': 542.987, 'duration': 1.381}, {'end': 549.973, 'text': 'It is a dramatically simpler way to understand what that product is ultimately going to simplify down to.', 'start': 544.829, 'duration': 5.144}, {'end': 559.248, 'text': "Surprisingly, this means that if our observer sits on the same circle as the lighthouses, the actual number of lighthouses won't be important.", 'start': 550.894, 'duration': 8.354}, {'end': 565.298, 'text': "It's only the fraction of the way between adjacent lighthouses that describes our observer, which will come into play.", 'start': 559.869, 'duration': 5.429}, {'end': 575.515, 'text': 'If this fraction is f, then observer to the power n lands f of the way around a full circle.', 'start': 568.433, 'duration': 7.082}, {'end': 586.258, 'text': 'So the magnitude of the complex number observer to the n minus one is the distance between the number one and a point f of the way around a full unit circle.', 'start': 576.115, 'duration': 10.143}, {'end': 594.9, 'text': 'For example, on screen right now, we have seven lighthouses, and the observer is sitting one third of the way between the first and the second.', 'start': 587.058, 'duration': 7.842}], 'summary': "The observer's position relative to lighthouses can be described using complex numbers, with the magnitude being the product of distances.", 'duration': 70.209, 'max_score': 524.691, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/8GPy_UMV-08/pics/8GPy_UMV-08524691.jpg'}, {'end': 706.463, 'src': 'embed', 'start': 664.895, 'weight': 2, 'content': [{'end': 669.219, 'text': "but sometimes it's easier to just think of it as chord of f.", 'start': 664.895, 'duration': 4.324}, {'end': 675.925, 'text': "So the result we've just shown is that, for an observer f of the way between two lighthouses, the total distance product,", 'start': 669.219, 'duration': 6.706}, {'end': 682.31, 'text': 'as complicated as that might seem, works out to be exactly chord of f, no matter how many lighthouses there are.', 'start': 675.925, 'duration': 6.385}, {'end': 685.997, 'text': 'So in particular, think about chord of one half.', 'start': 683.396, 'duration': 2.601}, {'end': 691.16, 'text': 'This is the distance between two points on the opposite ends of a unit circle, which is two.', 'start': 686.618, 'duration': 4.542}, {'end': 696.543, 'text': 'So we see that no matter how many lighthouses there are equally spread around the unit circle,', 'start': 691.901, 'duration': 4.642}, {'end': 703.187, 'text': 'putting an observer exactly halfway along the circle between two of them results in a distance product of precisely two.', 'start': 696.543, 'duration': 6.644}, {'end': 706.463, 'text': "And that's our first key fact, so just tuck that away.", 'start': 704.522, 'duration': 1.941}], 'summary': 'For an observer halfway between two lighthouses on a unit circle, the distance product is always two.', 'duration': 41.568, 'max_score': 664.895, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/8GPy_UMV-08/pics/8GPy_UMV-08664895.jpg'}, {'end': 752.468, 'src': 'embed', 'start': 728.956, 'weight': 1, 'content': [{'end': 736.219, 'text': 'Well now, instead of considering the polynomial observer to the n minus one which has a root at all of these n roots of unity,', 'start': 728.956, 'duration': 7.263}, {'end': 743.963, 'text': "we're looking at the polynomial observer to the n minus one, divided by observer minus one which has a root at all of the roots of unity,", 'start': 736.219, 'duration': 7.744}, {'end': 745.784, 'text': 'except for the number one itself.', 'start': 743.963, 'duration': 1.821}, {'end': 752.468, 'text': 'And a little algebra shows that this fraction is the same thing as 1 plus observer plus observer,', 'start': 746.784, 'duration': 5.684}], 'summary': 'The polynomial observer to the n minus one, divided by observer minus one, has roots at n roots of unity, except for the number one itself.', 'duration': 23.512, 'max_score': 728.956, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/8GPy_UMV-08/pics/8GPy_UMV-08728956.jpg'}, {'end': 820.204, 'src': 'embed', 'start': 794.422, 'weight': 0, 'content': [{'end': 799.386, 'text': 'And by the way, proving geometric facts with complex polynomials like this is pretty standard in math.', 'start': 794.422, 'duration': 4.964}, {'end': 805.391, 'text': 'And if you went up to your local mathematician and showed him or her these two facts, or other facts like these,', 'start': 799.847, 'duration': 5.544}, {'end': 810.696, 'text': "they'd quickly recognize both that these facts are true and how to prove them using the methods we just showed.", 'start': 805.391, 'duration': 5.305}, {'end': 812.257, 'text': 'And now, so can you.', 'start': 811.236, 'duration': 1.021}, {'end': 815.42, 'text': 'So next, with both these facts in our back pocket,', 'start': 812.858, 'duration': 2.562}, {'end': 820.204, 'text': "let's see how to use them to understand the product that we're interested in and how it relates to pi.", 'start': 815.42, 'duration': 4.784}], 'summary': 'Proving geometric facts with complex polynomials is standard in math, and mathematicians can recognize and prove these facts using the methods shown.', 'duration': 25.782, 'max_score': 794.422, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/8GPy_UMV-08/pics/8GPy_UMV-08794422.jpg'}], 'start': 509.613, 'title': 'Analyzing lighthouses with complex numbers', 'summary': 'Explores how complex numbers are utilized to simplify the computation of distances between an observer and lighthouses, revealing that the distance product is constant at one-third of the way between lighthouses and equals the number of lighthouses when the observer is at a lighthouse, demonstrated through complex polynomials.', 'chapters': [{'end': 615.33, 'start': 509.613, 'title': 'Complex number simplification', 'summary': 'Explains how complex numbers can be used to simplify the computation of distances between an observer and lighthouses, demonstrating that the actual number of lighthouses becomes unimportant if the observer sits on the same circle and only the fraction of the way between adjacent lighthouses is significant.', 'duration': 105.717, 'highlights': ['The magnitude of the complex number observer to the n minus one is the distance between the number one and a point f of the way around a full unit circle, showcasing the simplifying trick for computing the distance product (e.g., for seven lighthouses, the observer sitting one third of the way between the first and the second results in the magnitude of observer to the seven minus one being the length of the chord, approximately 1.73).', 'Complex numbers can be used to simplify the computation of distances between an observer and lighthouses, demonstrating that the actual number of lighthouses becomes unimportant if the observer sits on the same circle and only the fraction of the way between adjacent lighthouses is significant.']}, {'end': 812.257, 'start': 615.33, 'title': 'Lighthouse distance and observer position', 'summary': 'Discusses the relationship between the position of an observer, the number of lighthouses, and the distance product, revealing that the distance product is constant at one-third of the way between lighthouses and equals the number of lighthouses when the observer is at a lighthouse, demonstrated through complex polynomials.', 'duration': 196.927, 'highlights': ['The total distance product for an observer one-third of the way between lighthouses is constant and equals the length of the chord, independent of the number of lighthouses.', 'Placing an observer halfway between two lighthouses results in a constant distance product of two, irrespective of the number of lighthouses.', 'The distance product when the observer is at a lighthouse equals the total number of lighthouses, demonstrating a direct relationship between observer position and the number of lighthouses.']}], 'duration': 302.644, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/8GPy_UMV-08/pics/8GPy_UMV-08509613.jpg', 'highlights': ['The distance product when the observer is at a lighthouse equals the total number of lighthouses, demonstrating a direct relationship between observer position and the number of lighthouses.', 'The total distance product for an observer one-third of the way between lighthouses is constant and equals the length of the chord, independent of the number of lighthouses.', 'Complex numbers can be used to simplify the computation of distances between an observer and lighthouses, demonstrating that the actual number of lighthouses becomes unimportant if the observer sits on the same circle and only the fraction of the way between adjacent lighthouses is significant.', 'Placing an observer halfway between two lighthouses results in a constant distance product of two, irrespective of the number of lighthouses.', 'The magnitude of the complex number observer to the n minus one is the distance between the number one and a point f of the way around a full unit circle, showcasing the simplifying trick for computing the distance product (e.g., for seven lighthouses, the observer sitting one third of the way between the first and the second results in the magnitude of observer to the seven minus one being the length of the chord, approximately 1.73).']}, {'end': 1231.598, 'segs': [{'end': 929.74, 'src': 'embed', 'start': 900.059, 'weight': 4, 'content': [{'end': 901.04, 'text': 'For each lighthouse.', 'start': 900.059, 'duration': 0.981}, {'end': 906.626, 'text': "think about its contribution to the keeper's distance product, meaning its distance to the keeper,", 'start': 901.04, 'duration': 5.586}, {'end': 911.531, 'text': "divided by its contribution to the sailor's distance product, its distance to the sailor.", 'start': 906.626, 'duration': 4.905}, {'end': 916.214, 'text': 'And when we multiply all of these factors up over each lighthouse,', 'start': 912.452, 'duration': 3.762}, {'end': 922.437, 'text': 'we have to get the same ratio in the end n times the distance between the observers all divided by 2..', 'start': 916.214, 'duration': 6.223}, {'end': 929.74, 'text': 'Now that might seem like a super messy calculation, but as n gets larger, this actually gets simpler for any particular lighthouse.', 'start': 922.437, 'duration': 7.303}], 'summary': "Consider lighthouses' contributions to distances for keepers and sailors, leading to a specific ratio related to the distance between observers.", 'duration': 29.681, 'max_score': 900.059, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/8GPy_UMV-08/pics/8GPy_UMV-08900059.jpg'}, {'end': 996.997, 'src': 'embed', 'start': 939.506, 'weight': 1, 'content': [{'end': 946.091, 'text': 'Specifically, the angle from this lighthouse to the keeper is exactly twice the angle from this lighthouse to the sailor.', 'start': 939.506, 'duration': 6.585}, {'end': 951.296, 'text': "And those angles aren't exactly proportional to the straight line distances.", 'start': 947.132, 'duration': 4.164}, {'end': 955.039, 'text': 'but as n gets larger and larger, the correspondence gets better and better.', 'start': 951.296, 'duration': 3.743}, {'end': 963.026, 'text': 'And for a very large n, the distance from the lighthouse to the keeper is very nearly twice the distance from that lighthouse to the sailor.', 'start': 955.6, 'duration': 7.426}, {'end': 969.191, 'text': 'And in the same way, looking at the second lighthouse after the keeper,', 'start': 965.29, 'duration': 3.901}, {'end': 974.352, 'text': 'it has an angle-to-keeper divided by angle-to-sailor ratio of exactly 4 thirds,', 'start': 969.191, 'duration': 5.161}, {'end': 980.434, 'text': 'which is very nearly the same as the distance-to-keeper divided by distance-to-sailor ratio, as n gets large.', 'start': 974.352, 'duration': 6.082}, {'end': 988.916, 'text': 'And that third lighthouse, L3, is going to contribute a fraction that gets closer and closer to 6 fifths as n is approaching infinity.', 'start': 981.494, 'duration': 7.422}, {'end': 992.813, 'text': 'Now for this proof.', 'start': 992.233, 'duration': 0.58}, {'end': 996.997, 'text': "we're going to want to consider all the lighthouses on the bottom of the circle a little bit differently,", 'start': 992.813, 'duration': 4.184}], 'summary': 'Proportional relationships between lighthouse angles and distances improve as n increases.', 'duration': 57.491, 'max_score': 939.506, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/8GPy_UMV-08/pics/8GPy_UMV-08939506.jpg'}, {'end': 1074.775, 'src': 'embed', 'start': 1044.194, 'weight': 2, 'content': [{'end': 1050.537, 'text': 'each one of those terms reflects what the contribution for a particular lighthouse is, as n approaches infinity.', 'start': 1044.194, 'duration': 6.343}, {'end': 1058.004, 'text': "And when I say contribution, I mean the contribution to this ratio of the keeper's distance product to the sailor's distance product,", 'start': 1051.98, 'duration': 6.024}, {'end': 1063.567, 'text': 'which we know at every step has to equal n times the distance between the observers, divided by two.', 'start': 1058.004, 'duration': 5.563}, {'end': 1067.79, 'text': 'So what does that value approach as n approaches infinity?', 'start': 1064.448, 'duration': 3.342}, {'end': 1074.775, 'text': 'Well, the distance between the observers is half of 1 over n of a full turn around the circle.', 'start': 1068.61, 'duration': 6.165}], 'summary': 'The contribution for a lighthouse as n approaches infinity is discussed.', 'duration': 30.581, 'max_score': 1044.194, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/8GPy_UMV-08/pics/8GPy_UMV-081044194.jpg'}, {'end': 1101.096, 'src': 'heatmap', 'start': 1068.61, 'weight': 0.724, 'content': [{'end': 1074.775, 'text': 'Well, the distance between the observers is half of 1 over n of a full turn around the circle.', 'start': 1068.61, 'duration': 6.165}, {'end': 1083.282, 'text': 'And since this is a unit circle, its total circumference is 2 pi, so the distance between the observers approaches pi divided by n.', 'start': 1075.576, 'duration': 7.706}, {'end': 1090.968, 'text': 'And therefore, n times this distance divided by 2 approaches pi divided by 2.', 'start': 1083.282, 'duration': 7.686}, {'end': 1091.989, 'text': 'So there you have it.', 'start': 1090.968, 'duration': 1.021}, {'end': 1101.096, 'text': 'Our product, 2 over 1 times 2 over 3 times 4 over 3 times 4 over 5, on and on and on, must approach pi divided by 2.', 'start': 1092.529, 'duration': 8.567}], 'summary': 'The distance between observers approaches pi divided by n, leading to the product 2/1*2/3*4/3*4/5 approaching pi/2.', 'duration': 32.486, 'max_score': 1068.61, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/8GPy_UMV-08/pics/8GPy_UMV-081068610.jpg'}, {'end': 1117.588, 'src': 'embed', 'start': 1092.529, 'weight': 0, 'content': [{'end': 1101.096, 'text': 'Our product, 2 over 1 times 2 over 3 times 4 over 3 times 4 over 5, on and on and on, must approach pi divided by 2.', 'start': 1092.529, 'duration': 8.567}, {'end': 1107.861, 'text': "This is a truly marvelous result and it's known as the Wallace product, named after 17th century mathematician John Wallace,", 'start': 1101.096, 'duration': 6.765}, {'end': 1110.663, 'text': 'who first discovered this fact in a way more convoluted way.', 'start': 1107.861, 'duration': 2.802}, {'end': 1117.588, 'text': 'And also, little bit of trivia, this is the same guy who discovered, or, well, rather, invented, the infinity symbol.', 'start': 1111.363, 'duration': 6.225}], 'summary': 'The wallace product converges to pi/2, discovered by mathematician john wallace.', 'duration': 25.059, 'max_score': 1092.529, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/8GPy_UMV-08/pics/8GPy_UMV-081092529.jpg'}, {'end': 1222.571, 'src': 'heatmap', 'start': 1182.715, 'weight': 0.82, 'content': [{'end': 1186.819, 'text': 'So if you were to take the infinite product of each row, you just get 7 for each one of them.', 'start': 1182.715, 'duration': 4.104}, {'end': 1193.083, 'text': 'So, since every one of these products is 7, the limit of the products is also 7.', 'start': 1187.319, 'duration': 5.764}, {'end': 1194.985, 'text': 'But look at what happens if you take the limits first.', 'start': 1193.083, 'duration': 1.902}, {'end': 1201.549, 'text': "If you look at each column, the limit of a given column is going to be 1, since at some point it's nothing but 1s.", 'start': 1195.445, 'duration': 6.104}, {'end': 1212.904, 'text': "But then, if you're taking the product of those limits, you're just taking the product of a bunch of 1s, so you get a different answer, namely 1.", 'start': 1202.19, 'duration': 10.714}, {'end': 1222.571, 'text': "Luckily, mathematicians have spent a lot of time thinking about this phenomenon and they've developed tools for quickly seeing certain conditions under which this exchanging of the limits actually works.", 'start': 1212.904, 'duration': 9.667}], 'summary': 'The limit of the infinite product of each row is 7, but the limit of the product of the limits of each column is 1.', 'duration': 39.856, 'max_score': 1182.715, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/8GPy_UMV-08/pics/8GPy_UMV-081182715.jpg'}, {'end': 1222.571, 'src': 'embed', 'start': 1195.445, 'weight': 6, 'content': [{'end': 1201.549, 'text': "If you look at each column, the limit of a given column is going to be 1, since at some point it's nothing but 1s.", 'start': 1195.445, 'duration': 6.104}, {'end': 1212.904, 'text': "But then, if you're taking the product of those limits, you're just taking the product of a bunch of 1s, so you get a different answer, namely 1.", 'start': 1202.19, 'duration': 10.714}, {'end': 1222.571, 'text': "Luckily, mathematicians have spent a lot of time thinking about this phenomenon and they've developed tools for quickly seeing certain conditions under which this exchanging of the limits actually works.", 'start': 1212.904, 'duration': 9.667}], 'summary': 'The product of column limits is 1, due to the presence of 1s, leading to a different answer, 1.', 'duration': 27.126, 'max_score': 1195.445, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/8GPy_UMV-08/pics/8GPy_UMV-081195445.jpg'}], 'start': 812.858, 'title': 'Lighthouse distances and arithmetic', 'summary': 'Discusses the ratio computation for lighthouse distances around a unit circle, wallace product approaching pi/2 as n approaches infinity, and commuting limits in infinitary arithmetic with a focus on dominated convergence result.', 'chapters': [{'end': 922.437, 'start': 812.858, 'title': 'Lighthouse distance and product ratio', 'summary': 'Discusses the use of two key facts to compute the ratio between the distance product for a keeper and a sailor around a unit circle with n lighthouses, resulting in the ratio n times the distance between the observers divided by 2.', 'duration': 109.579, 'highlights': ['The total distance product for the sailor is two This key fact allows computation of the distance product ratio for the sailor.', 'The remaining distance product for the keeper, after removing a lighthouse, is n After removing a lighthouse, the remaining distance product for the keeper is quantified as n.', "The ratio between the keeper's distance product and the sailor's is n times the distance between the two observers, all divided by 2 The ratio between the keeper's and sailor's distance product is expressed as n times the distance between the observers, divided by 2."]}, {'end': 1132.558, 'start': 922.437, 'title': 'Wallace product and infinity symbol', 'summary': 'Discusses the wallace product, where the product of ratios of lighthouse distances approaches pi divided by 2 as n approaches infinity, a result discovered by 17th century mathematician john wallace.', 'duration': 210.121, 'highlights': ['The product of ratios of lighthouse distances approaches pi divided by 2 as n approaches infinity The product 2 over 1 times, 2 over 3 times, 4 over 3 times, 4 over 5, on and on, must approach pi divided by 2.', 'Distances to the keeper compared to distances to the sailor as n approaches infinity The distance from the lighthouse to the keeper is very nearly twice the distance from that lighthouse to the sailor as n gets larger.', 'Relation between angles and distances as n gets larger Specifically, the angle from a lighthouse to the keeper is exactly twice the angle from the same lighthouse to the sailor as n gets larger, and the correspondence gets better and better.']}, {'end': 1231.598, 'start': 1133.439, 'title': 'Commuting of limits in infinitary arithmetic', 'summary': "Discusses the concept of commuting limits in infinitary arithmetic, demonstrating a case where it doesn't hold and highlighting the application of the dominated convergence result in ensuring its validity.", 'duration': 98.159, 'highlights': ['The product of limits is assumed to be equal to the limit of products, even with infinitely many factors, but this does not always hold true.', 'A simple example is presented with a grid where the product of each row yields 7, but the limit of the products does not match the result obtained by taking the limits first, showcasing a case where commuting of limits fails.', 'Mathematicians have developed tools, such as the dominated convergence result, to quickly determine the conditions under which exchanging limits is valid, providing assurance for the validity of the argument demonstrated.']}], 'duration': 418.74, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/8GPy_UMV-08/pics/8GPy_UMV-08812858.jpg', 'highlights': ['The product of ratios of lighthouse distances approaches pi divided by 2 as n approaches infinity', "The ratio between the keeper's and sailor's distance product is expressed as n times the distance between the observers, divided by 2", 'The distance from the lighthouse to the keeper is very nearly twice the distance from that lighthouse to the sailor as n gets larger', 'The total distance product for the sailor is two This key fact allows computation of the distance product ratio for the sailor', 'Mathematicians have developed tools, such as the dominated convergence result, to quickly determine the conditions under which exchanging limits is valid, providing assurance for the validity of the argument demonstrated', 'The remaining distance product for the keeper, after removing a lighthouse, is n After removing a lighthouse, the remaining distance product for the keeper is quantified as n', 'The product of limits is assumed to be equal to the limit of products, even with infinitely many factors, but this does not always hold true', 'A simple example is presented with a grid where the product of each row yields 7, but the limit of the products does not match the result obtained by taking the limits first, showcasing a case where commuting of limits fails', 'Relation between angles and distances as n gets larger Specifically, the angle from a lighthouse to the keeper is exactly twice the angle from the same lighthouse to the sailor as n gets larger, and the correspondence gets better and better']}, {'end': 1504.871, 'segs': [{'end': 1395.512, 'src': 'embed', 'start': 1361.002, 'weight': 0, 'content': [{'end': 1363.486, 'text': 'where f is that fraction of the way between lighthouses?', 'start': 1361.002, 'duration': 2.484}, {'end': 1370.416, 'text': 'And if we go through the same reasoning that we just did with the sailor at this location instead and change nothing else,', 'start': 1364.335, 'duration': 6.081}, {'end': 1380.879, 'text': "what we'll find is that the ratio of the keeper's distance product to the sailor's distance product is now n times the distance between them divided by chord of f,", 'start': 1370.416, 'duration': 10.463}, {'end': 1385.52, 'text': 'which approaches f, times 2 pi divided by chord of f, as n gets larger.', 'start': 1380.879, 'duration': 4.641}, {'end': 1395.512, 'text': 'And in the same way as before, you could alternatively calculate this by considering the contributions from each individual lighthouse.', 'start': 1389.125, 'duration': 6.387}], 'summary': "The ratio of keeper's distance product to sailor's distance product is n times the distance between them divided by chord of f, which approaches f times 2 pi divided by chord of f as n gets larger.", 'duration': 34.51, 'max_score': 1361.002, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/8GPy_UMV-08/pics/8GPy_UMV-081361002.jpg'}, {'end': 1433.508, 'src': 'embed', 'start': 1406.264, 'weight': 2, 'content': [{'end': 1412.829, 'text': "and all the lighthouses before the keeper, they contribute the same thing, but you're just plugging in negative values for k.", 'start': 1406.264, 'duration': 6.565}, {'end': 1418.173, 'text': 'If you combine all those contributions over all non-zero integers k, where, in the same way as before,', 'start': 1412.829, 'duration': 5.344}, {'end': 1425.079, 'text': "you have to be careful about how you bundle the positive and negative k terms together, What you'll get is that the product of k,", 'start': 1418.173, 'duration': 6.906}, {'end': 1433.508, 'text': 'divided by k minus f over all nonzero integers k is going to equal f times 2 pi divided by the coordinate of f.', 'start': 1425.079, 'duration': 8.429}], 'summary': 'Combining contributions over non-zero integers k yields product of k/k-f = f*2pi/coordinate of f', 'duration': 27.244, 'max_score': 1406.264, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/8GPy_UMV-08/pics/8GPy_UMV-081406264.jpg'}, {'end': 1479.507, 'src': 'embed', 'start': 1455.301, 'weight': 1, 'content': [{'end': 1465.889, 'text': 'Sine of f times pi is equal to f pi times this really big product, the product of 1 minus f over k over all non-zero integers k.', 'start': 1455.301, 'duration': 10.588}, {'end': 1473.454, 'text': 'So what we found is a way to express sine of x as an infinite product, which is really cool if you think about it.', 'start': 1465.889, 'duration': 7.565}, {'end': 1479.507, 'text': 'So not only does this proof give us the Wallace product, which is incredible in its own right,', 'start': 1474.275, 'duration': 5.232}], 'summary': 'Sine of x can be expressed as an infinite product, a cool mathematical insight.', 'duration': 24.206, 'max_score': 1455.301, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/8GPy_UMV-08/pics/8GPy_UMV-081455301.jpg'}], 'start': 1232.258, 'title': 'Interpreting lighthouse contributions and generalization of distance product', 'summary': "Explores interpreting lighthouse contributions, interleaving counterclockwise and clockwise contributions to compute a product converging to pi halves, and discusses the generalization of the distance product for a sailor at a fraction f between lighthouses, leading to the derivation of an infinite product formula for sine of x, connecting to euler's solution to the basel problem.", 'chapters': [{'end': 1337.586, 'start': 1232.258, 'title': 'Interpreting lighthouse contributions', 'summary': 'Explores the delicate process of interpreting lighthouse contributions, which involves interleaving and combining counterclockwise and clockwise contributions to compute a product that converges to pi halves, with technicalities and a cool result falling out of the argument.', 'duration': 105.328, 'highlights': ['The delicate process of interpreting lighthouse contributions involves interleaving and combining counterclockwise and clockwise contributions to compute a product that converges to pi halves, with technicalities and a cool result falling out of the argument.', 'The lighthouses counterclockwise from the keeper contribute 2 over 1, 4 over 3, 6 over 5, and the ones clockwise from the keeper contribute 2 over 3, 4 over 5, 6 over 7, with the first series blowing up to infinity and the second series approaching zero.', "It's crucial to specifically combine counterclockwise and clockwise contributions in a one-for-one manner to obtain a product that converges to pi halves, as intermixing them differently can yield different results.", 'Dominated convergence justifies commuting limits in this process, and the conceptual gist of the argument remains consistent despite technicalities, leading to a neat result that falls out of the argument.']}, {'end': 1504.871, 'start': 1338.126, 'title': 'Generalization of distance product and sign formula', 'summary': "Discusses the generalization of the distance product for a sailor at a fraction f between lighthouses, leading to the derivation of an infinite product formula for sine of x, connecting to euler's solution to the basel problem.", 'duration': 166.745, 'highlights': ["The product formula for sine of x is derived as an infinite product, connecting to Euler's solution to the Basel problem, showcasing the generalization of the distance product.", 'The discussion introduces the concept of placing the sailor at any fraction f of the way between adjacent lighthouses, leading to the derivation of the distance product formula and its connection to the sign formula.', "The proof provides the Wallace product and a product formula for the sign, showcasing the generalization of the distance product and its connection to Euler's solution to the Basel problem.", "The chapter discusses the connection between the derived product formula for sine and Euler's solution to the Basel problem, emphasizing the broader implications and connections of the derived formulas."]}], 'duration': 272.613, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/8GPy_UMV-08/pics/8GPy_UMV-081232258.jpg', 'highlights': ['The delicate process of interpreting lighthouse contributions involves interleaving and combining counterclockwise and clockwise contributions to compute a product that converges to pi halves, with technicalities and a cool result falling out of the argument.', "The product formula for sine of x is derived as an infinite product, connecting to Euler's solution to the Basel problem, showcasing the generalization of the distance product.", "It's crucial to specifically combine counterclockwise and clockwise contributions in a one-for-one manner to obtain a product that converges to pi halves, as intermixing them differently can yield different results.", 'The discussion introduces the concept of placing the sailor at any fraction f of the way between adjacent lighthouses, leading to the derivation of the distance product formula and its connection to the sign formula.']}], 'highlights': ["The product formula for sine of x is derived as an infinite product, connecting to Euler's solution to the Basel problem, showcasing the generalization of the distance product.", 'The video presents a more original proof of the Wallace product for pi, a well-known math result, discovered through unconventional thinking and a connection to the Basel problem.', 'The result being discussed is the Wallace product for pi, a very famous infinite product, and the new proof of this relationship was stumbled upon by Swether, a new member of 3b1b, after thinking about the approach taken in a previous video and the connection between the Basel problem and the Wallace product.', 'The infinite product of alternating even and odd numbers diverges to infinity and then converges to a finite value of pi/2.', 'The inverse square law interprets the total light received by summing inverse square distances.', 'The distance product when the observer is at a lighthouse equals the total number of lighthouses, demonstrating a direct relationship between observer position and the number of lighthouses.', 'The delicate process of interpreting lighthouse contributions involves interleaving and combining counterclockwise and clockwise contributions to compute a product that converges to pi halves, with technicalities and a cool result falling out of the argument.', 'The value of math presentation is attributed to both the underlying math and the choices made in communicating it, with the novelty in this case stemming from the communication half.', "The infinite product's value is pi/2, and its connection to circles is explored through a digression involving geometric tools.", 'The analogy of lighthouses and an observer visualizes using inverse squared distances.', 'The product of ratios of lighthouse distances approaches pi divided by 2 as n approaches infinity', 'The distance from the lighthouse to the keeper is very nearly twice the distance from that lighthouse to the sailor as n gets larger', 'The total distance product for the sailor is two This key fact allows computation of the distance product ratio for the sailor', 'The distance product for the observer involves multiplying the lengths of all the lines.', 'The distance product for the observer involves multiplying the lengths of all the lines.', 'The distance product for the observer involves multiplying the lengths of all the lines.', 'The distance product for the observer involves multiplying the lengths of all the lines.']}