title
All possible pythagorean triples, visualized

description
To understand all pythagorean triples like (3, 4, 5), (5, 12, 13), etc. look to complex numbers. This video was sponsored by Remix: https://www.remix.com/jobs 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/triples-thanks Home page: https://www.3blue1brown.com/ Regarding the brief reference to Fermat's Last Theorem, what should be emphasized is that it refers to *positive* integers. You can of course have things like 0^3 + 2^3 = 2^3, or (-3)^3 + 3^3 = 0^3. Music by Vincent Rubinetti: https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted about new videos, subscribe, and click the bell to receive notifications (if you're into that). If you are new to this channel and want to see more, a good place to start is this playlist: http://3b1b.co/recommended Various social media stuffs: Website: https://www.3blue1brown.com Twitter: https://twitter.com/3Blue1Brown Patreon: https://patreon.com/3blue1brown Facebook: https://www.facebook.com/3blue1brown Reddit: https://www.reddit.com/r/3Blue1Brown

detail
{'title': 'All possible pythagorean triples, visualized', 'heatmap': [{'end': 51.203, 'start': 30.199, 'weight': 0.839}, {'end': 463.551, 'start': 426.641, 'weight': 0.703}, {'end': 495.58, 'start': 467.273, 'weight': 0.904}, {'end': 519.932, 'start': 509.488, 'weight': 0.824}], 'summary': "Delves into pythagorean triples and fermat's last theorem, discussing the relationships between squares and triangles, demonstrating the area of a big square minus four times the area of the triangle, and exploring rational points on a unit circle and complex numbers to hit every possible pythagorean triple.", 'chapters': [{'end': 84.743, 'segs': [{'end': 84.743, 'src': 'heatmap', 'start': 30.199, 'weight': 0, 'content': [{'end': 36.425, 'text': 'But keep in mind for comparison if you were to change that exponent to any whole number bigger than two,', 'start': 30.199, 'duration': 6.226}, {'end': 40.349, 'text': 'you go from having many integer solutions to no solutions whatsoever.', 'start': 36.425, 'duration': 3.924}, {'end': 42.912, 'text': "This is Fermat's famous last theorem.", 'start': 41.07, 'duration': 1.842}, {'end': 51.203, 'text': "Now there's a special name for any triplet of whole numbers, a, b, c, where a squared plus b squared equals c squared.", 'start': 44.881, 'duration': 6.322}, {'end': 53.363, 'text': "It's called a Pythagorean triple.", 'start': 51.763, 'duration': 1.6}, {'end': 57.765, 'text': "And what we're going to do here is find every single possible example.", 'start': 54.064, 'duration': 3.701}, {'end': 62.886, 'text': "And moreover, we'll do so in a way where you can visualize how all of these triples fit together.", 'start': 58.325, 'duration': 4.561}, {'end': 67.209, 'text': 'This is an old question, pretty much as old as they come in math.', 'start': 64.144, 'duration': 3.065}, {'end': 75.804, 'text': 'There are some Babylonian clay tablets from 1800 BC, more than a millennium before Pythagoras himself, that just list these triples.', 'start': 67.77, 'duration': 8.034}, {'end': 80.679, 'text': "And, by the way, while we're talking about the Pythagorean theorem,", 'start': 77.335, 'duration': 3.344}, {'end': 84.743, 'text': "it would be a shame not to share my favorite proof for anyone who hasn't already seen this", 'start': 80.679, 'duration': 4.064}], 'summary': "Exploring pythagorean triples and fermat's last theorem in math history.", 'duration': 63.514, 'max_score': 30.199, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/QJYmyhnaaek/pics/QJYmyhnaaek30199.jpg'}], 'start': 4.315, 'title': "Pythagorean triples and fermat's last theorem", 'summary': "Explores pythagorean triples, which are special whole number triplets where a squared plus b squared equals c squared, and delves into fermat's last theorem, stating no solutions for exponents larger than two.", 'chapters': [{'end': 84.743, 'start': 4.315, 'title': "Pythagorean triples and fermat's last theorem", 'summary': "Explores the concept of pythagorean triples, highlighting the special triple of whole numbers where a squared plus b squared equals c squared, and delves into fermat's famous last theorem, which states that changing the exponent to any whole number bigger than two results in no solutions.", 'duration': 80.428, 'highlights': ["Fermat's famous last theorem states that changing the exponent to any whole number bigger than two results in no solutions whatsoever, contrasting the existence of examples such as the 3-4-5 triangle or the 5-12-13 triangle.", 'Pythagorean triple refers to any triplet of whole numbers, a, b, c, where a squared plus b squared equals c squared, and the chapter aims to find every single possible example and visualize how all of these triples fit together.', 'The chapter mentions Babylonian clay tablets from 1800 BC, which list these triples, emphasizing the antiquity and historical significance of the concept.']}], 'duration': 80.428, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/QJYmyhnaaek/pics/QJYmyhnaaek4315.jpg', 'highlights': ["Fermat's last theorem states no solutions for exponents larger than two", 'Pythagorean triple: a^2 + b^2 = c^2, aims to find every possible example', 'Babylonian clay tablets from 1800 BC list these triples']}, {'end': 420.816, 'segs': [{'end': 128.46, 'src': 'embed', 'start': 85.885, 'weight': 1, 'content': [{'end': 94.635, 'text': 'You start off by drawing a square on each side of the triangle, and if you take that C square and add four copies of the original triangle around it,', 'start': 85.885, 'duration': 8.75}, {'end': 97.518, 'text': 'you can get a big square whose side lengths are a plus b.', 'start': 94.635, 'duration': 2.883}, {'end': 107.928, 'text': 'but you can also arrange the a square and the b square together with four copies of the original triangle to get a big square whose side lengths are a plus b.', 'start': 98.579, 'duration': 9.349}, {'end': 115.775, 'text': 'What this means is that the negative space in each of these diagrams, the area of that big square minus four times the area of the triangle,', 'start': 107.928, 'duration': 7.847}, {'end': 121.14, 'text': "is from one perspective clearly a squared plus b squared, but from another perspective it's c squared.", 'start': 115.775, 'duration': 5.365}, {'end': 125.657, 'text': 'Anyway, back to the question of finding whole number solutions.', 'start': 122.614, 'duration': 3.043}, {'end': 128.46, 'text': 'Start by reframing the question slightly.', 'start': 126.458, 'duration': 2.002}], 'summary': 'Using triangles and squares to find whole number solutions for a^2 + b^2 = c^2.', 'duration': 42.575, 'max_score': 85.885, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/QJYmyhnaaek/pics/QJYmyhnaaek85885.jpg'}, {'end': 171.709, 'src': 'embed', 'start': 149.34, 'weight': 0, 'content': [{'end': 157.708, 'text': 'The question of finding Pythagorean triples is completely equivalent to finding lattice points which are a whole number distance away from the origin.', 'start': 149.34, 'duration': 8.368}, {'end': 167.558, 'text': 'Of course, for most points, like the distance from the origin is not a whole number, but it is at least the square root of a whole number.', 'start': 158.589, 'duration': 8.969}, {'end': 171.709, 'text': 'In this case, 2 squared plus 1 squared is 5.', 'start': 168.5, 'duration': 3.209}], 'summary': 'Finding pythagorean triples is equivalent to finding whole number distance lattice points.', 'duration': 22.369, 'max_score': 149.34, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/QJYmyhnaaek/pics/QJYmyhnaaek149340.jpg'}, {'end': 217.271, 'src': 'embed', 'start': 192.966, 'weight': 3, 'content': [{'end': 200.788, 'text': 'What this gives is a surprisingly simple way to modify it to get a new point whose distance away from the origin is guaranteed to be a whole number.', 'start': 192.966, 'duration': 7.822}, {'end': 202.349, 'text': 'Just square it.', 'start': 201.469, 'duration': 0.88}, {'end': 210.488, 'text': 'Algebraically when you square a complex number, expanding out this product and matching up all of the like terms,', 'start': 203.706, 'duration': 6.782}, {'end': 214.55, 'text': 'because everything here just involves multiplying and adding integers.', 'start': 210.488, 'duration': 4.062}, {'end': 217.271, 'text': 'each component of the result is guaranteed to be an integer.', 'start': 214.55, 'duration': 2.721}], 'summary': 'Squaring a complex number guarantees integer components.', 'duration': 24.305, 'max_score': 192.966, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/QJYmyhnaaek/pics/QJYmyhnaaek192966.jpg'}, {'end': 384.397, 'src': 'embed', 'start': 354.752, 'weight': 5, 'content': [{'end': 362.354, 'text': "the triple that it corresponds to is 0 squared plus 8 squared equals 8 squared, which isn't exactly something to write home about.", 'start': 354.752, 'duration': 7.602}, {'end': 370.836, 'text': 'But for the most part, this method of squaring complex numbers is a surprisingly simple way to generate non-trivial Pythagorean triples.', 'start': 363.254, 'duration': 7.582}, {'end': 374.168, 'text': 'You can even generalize it to get a nice formula.', 'start': 372.026, 'duration': 2.142}, {'end': 384.397, 'text': 'If you write the coordinates of your initial point as u and v, then when you work out u plus vi squared, the real part is u squared,', 'start': 374.929, 'duration': 9.468}], 'summary': 'Squaring complex numbers generates non-trivial pythagorean triples.', 'duration': 29.645, 'max_score': 354.752, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/QJYmyhnaaek/pics/QJYmyhnaaek354752.jpg'}], 'start': 85.885, 'title': 'Triangle geometry and pythagorean triples', 'summary': 'Discusses the relationships between squares and triangles, demonstrating the area of a big square minus four times the area of the triangle and reframes the question of finding whole number solutions. it also explains how complex numbers can be used to generate pythagorean triples from lattice points, providing a surprisingly simple method to obtain non-trivial triples and a general formula for the process.', 'chapters': [{'end': 128.46, 'start': 85.885, 'title': 'Triangle geometry and whole numbers', 'summary': 'Discusses the relationships between squares and triangles, demonstrating that the area of a big square minus four times the area of the triangle can be expressed as a^2 + b^2 or as c^2, and then reframes the question of finding whole number solutions.', 'duration': 42.575, 'highlights': ['By arranging the a square and the b square together with four copies of the original triangle, a big square whose side lengths are a plus b can be obtained, revealing a relationship between the areas of the squares and the triangle.', 'The area of the big square minus four times the area of the triangle can be expressed as a^2 + b^2, demonstrating a geometric relationship between the squares and the triangle.', 'Reframing the question of finding whole number solutions is suggested, indicating a shift in approach to solving the problem.']}, {'end': 420.816, 'start': 129.16, 'title': 'Pythagorean triples from lattice points', 'summary': 'Explains how complex numbers can be used to generate pythagorean triples from lattice points, providing a surprisingly simple method to obtain non-trivial triples, and even a general formula for the process.', 'duration': 291.656, 'highlights': ['Complex numbers can be squared to generate new points whose distance from the origin is a whole number, providing a simple method to obtain Pythagorean triples.', 'A general formula to obtain Pythagorean triples from lattice points using complex numbers is provided, allowing any pair of integers to produce a Pythagorean triple.', 'The process of squaring complex numbers is a surprisingly simple method to generate non-trivial Pythagorean triples from lattice points.']}], 'duration': 334.931, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/QJYmyhnaaek/pics/QJYmyhnaaek85885.jpg', 'highlights': ['A general formula to obtain Pythagorean triples from lattice points using complex numbers is provided, allowing any pair of integers to produce a Pythagorean triple.', 'By arranging the a square and the b square together with four copies of the original triangle, a big square whose side lengths are a plus b can be obtained, revealing a relationship between the areas of the squares and the triangle.', 'The area of the big square minus four times the area of the triangle can be expressed as a^2 + b^2, demonstrating a geometric relationship between the squares and the triangle.', 'Complex numbers can be squared to generate new points whose distance from the origin is a whole number, providing a simple method to obtain Pythagorean triples.', 'Reframing the question of finding whole number solutions is suggested, indicating a shift in approach to solving the problem.', 'The process of squaring complex numbers is a surprisingly simple method to generate non-trivial Pythagorean triples from lattice points.']}, {'end': 874.766, 'segs': [{'end': 463.551, 'src': 'heatmap', 'start': 426.641, 'weight': 0.703, 'content': [{'end': 438.371, 'text': 'So for example, the point is going to move over to The point i is going to rotate 90 degrees to its square, negative 1.', 'start': 426.641, 'duration': 11.73}, {'end': 441.915, 'text': 'The point negative 1 is going to move over to 1, and so on.', 'start': 438.371, 'duration': 3.544}, {'end': 449.582, 'text': "Now, when you do this to every single point on the plane, including the grid lines, which I'll make more colorful so they're easier to follow.", 'start': 442.876, 'duration': 6.706}, {'end': 450.363, 'text': "here's what it looks like.", 'start': 449.582, 'duration': 0.781}, {'end': 463.551, 'text': 'So the gridlines all get turned into these parabolic arcs, and every point where these arcs intersect is a place where a lattice point landed.', 'start': 455.768, 'duration': 7.783}], 'summary': 'Transforming points on a plane creates parabolic arcs intersecting at lattice points.', 'duration': 36.91, 'max_score': 426.641, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/QJYmyhnaaek/pics/QJYmyhnaaek426641.jpg'}, {'end': 519.932, 'src': 'heatmap', 'start': 467.273, 'weight': 1, 'content': [{'end': 475.656, 'text': 'That is, if you draw a triangle whose hypotenuse is the line between any one of these points and the origin and whose legs are parallel to the axes,', 'start': 467.273, 'duration': 8.383}, {'end': 478.377, 'text': 'all three side lengths of that triangle will be whole numbers.', 'start': 475.656, 'duration': 2.721}, {'end': 486.278, 'text': 'What I love about this is that usually, when you view Pythagorean triples just on their own, they seem completely random and unconnected,', 'start': 479.417, 'duration': 6.861}, {'end': 487.938, 'text': "and you'd be tempted to say there's no pattern.", 'start': 486.278, 'duration': 1.66}, {'end': 495.58, 'text': 'But here, we have a lot of them sitting together really organized, just sitting on the intersections of these nicely spaced curves.', 'start': 488.519, 'duration': 7.061}, {'end': 508.462, 'text': 'Now you might ask if this accounts for every possible Pythagorean triple.', 'start': 505.021, 'duration': 3.441}, {'end': 510.668, 'text': 'Sadly, it does not.', 'start': 509.488, 'duration': 1.18}, {'end': 519.932, 'text': 'For example, you will never get the point 6 plus 8i using this method, even though 6, 8, 10 is a perfectly valid Pythagorean triple.', 'start': 511.489, 'duration': 8.443}], 'summary': 'Pythagorean triples form organized patterns in the triangle, but not all possible triples are accounted for.', 'duration': 31.413, 'max_score': 467.273, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/QJYmyhnaaek/pics/QJYmyhnaaek467273.jpg'}, {'end': 618.161, 'src': 'embed', 'start': 593.754, 'weight': 2, 'content': [{'end': 599.8, 'text': 'Marking all of the lattice points that this line hits will account for any multiples of these points that we might have missed.', 'start': 593.754, 'duration': 6.046}, {'end': 609.238, 'text': "Doing this for all possible points, you'll account for every possible Pythagorean triple.", 'start': 604.096, 'duration': 5.142}, {'end': 618.161, 'text': 'Every right triangle that you ever have seen or ever will see that has whole number side lengths is accounted for somewhere in this diagram.', 'start': 609.998, 'duration': 8.163}], 'summary': 'Marking lattice points accounts for all pythagorean triples.', 'duration': 24.407, 'max_score': 593.754, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/QJYmyhnaaek/pics/QJYmyhnaaek593754.jpg'}, {'end': 670.379, 'src': 'embed', 'start': 645.42, 'weight': 3, 'content': [{'end': 651.905, 'text': 'This gives us some point on the unit circle x squared plus y squared equals 1 whose coordinates are each rational numbers.', 'start': 645.42, 'duration': 6.485}, {'end': 655.627, 'text': 'This is what we call a rational point of the unit circle.', 'start': 652.645, 'duration': 2.982}, {'end': 658.089, 'text': 'And going the other way around.', 'start': 656.568, 'duration': 1.521}, {'end': 664.855, 'text': 'if you find some rational point on the unit circle, when you multiply out by a common denominator for each of those coordinates,', 'start': 658.089, 'duration': 6.766}, {'end': 670.379, 'text': "what you'll land on is a point that has integer coordinates and whose distance from the origin is also an integer.", 'start': 664.855, 'duration': 5.524}], 'summary': 'Rational points on unit circle have integer coordinates and distance from origin.', 'duration': 24.959, 'max_score': 645.42, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/QJYmyhnaaek/pics/QJYmyhnaaek645420.jpg'}, {'end': 705.884, 'src': 'embed', 'start': 682.256, 'weight': 4, 'content': [{'end': 688.918, 'text': 'If you project all of these points onto the unit circle, each one moving along its corresponding radial line,', 'start': 682.256, 'duration': 6.662}, {'end': 692.179, 'text': "what you'll end up with is a whole bunch of rational points on that circle.", 'start': 688.918, 'duration': 3.261}, {'end': 697.981, 'text': "And keep in mind, by the way, I'm drawing only finitely many of these dots and lines,", 'start': 693.42, 'duration': 4.561}, {'end': 703.543, 'text': 'but if I drew all infinitely many lines corresponding to every possible squared lattice point,', 'start': 697.981, 'duration': 5.562}, {'end': 705.884, 'text': 'it would actually fill every single pixel of the screen.', 'start': 703.543, 'duration': 2.341}], 'summary': 'Projecting points onto unit circle yields rational points, filling every pixel on the screen.', 'duration': 23.628, 'max_score': 682.256, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/QJYmyhnaaek/pics/QJYmyhnaaek682256.jpg'}, {'end': 748.808, 'src': 'embed', 'start': 723.223, 'weight': 0, 'content': [{'end': 729.684, 'text': 'Take any one of those rational points and draw a line between it and the point at.', 'start': 723.223, 'duration': 6.461}, {'end': 738.706, 'text': 'When you compute the rise over run slope of this line, the rise between the two points is rational and the run is also rational.', 'start': 729.684, 'duration': 9.022}, {'end': 741.667, 'text': 'so the slope itself is just going to be some rational number.', 'start': 738.706, 'duration': 2.961}, {'end': 748.808, 'text': 'So if we can show that our method of squaring complex numbers accounts for every possible rational slope here,', 'start': 742.707, 'duration': 6.101}], 'summary': 'Rational points form rational slopes in squaring complex numbers.', 'duration': 25.585, 'max_score': 723.223, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/QJYmyhnaaek/pics/QJYmyhnaaek723223.jpg'}, {'end': 826.249, 'src': 'embed', 'start': 797.005, 'weight': 6, 'content': [{'end': 804.126, 'text': "that turns out to be exactly two times the angle made by those same points and any other point on the circle's circumference,", 'start': 797.005, 'duration': 7.121}, {'end': 807.026, 'text': "provided that that other point isn't between the original two points.", 'start': 804.126, 'duration': 2.9}, {'end': 817.348, 'text': 'What this means for our situation is that the line between and the rational point on the circle must make an angle theta with the horizontal.', 'start': 808.346, 'duration': 9.002}, {'end': 826.249, 'text': 'In other words, that line has the same slope as the line between the origin and our initial complex number, u plus vi.', 'start': 818.806, 'duration': 7.443}], 'summary': 'Line between rational point and circle has same slope as the line between origin and initial complex number.', 'duration': 29.244, 'max_score': 797.005, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/QJYmyhnaaek/pics/QJYmyhnaaek797005.jpg'}, {'end': 874.766, 'src': 'embed', 'start': 845.782, 'weight': 5, 'content': [{'end': 846.383, 'text': 'So there you go!', 'start': 845.782, 'duration': 0.601}, {'end': 855.79, 'text': 'The radial lines from our method, determined by all possible choices of u and v, must pass through every rational point on this circle.', 'start': 847.043, 'duration': 8.747}, {'end': 860.174, 'text': 'And that means our method must hit every possible Pythagorean triple.', 'start': 856.491, 'duration': 3.683}, {'end': 874.766, 'text': "If you haven't already watched the video about pi hiding in prime regularities, the topics there are highly related to the ones here.", 'start': 868.08, 'duration': 6.686}], 'summary': 'Method hits every pythagorean triple via radial lines.', 'duration': 28.984, 'max_score': 845.782, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/QJYmyhnaaek/pics/QJYmyhnaaek845782.jpg'}], 'start': 420.816, 'title': 'Pythagorean triples and rational points', 'summary': 'Explores the squaring method and pythagorean triples, rational points on a unit circle, and complex numbers, guaranteeing the coverage of every rational point and hitting every possible pythagorean triple.', 'chapters': [{'end': 592.733, 'start': 420.816, 'title': 'Squaring method and pythagorean triples', 'summary': 'Explores the squaring method transforming points on a plane to their squares, creating organized intersections of parabolic arcs that correspond to pythagorean triples, although not accounting for every possible triple.', 'duration': 171.917, 'highlights': ['The squaring method transforms points on a plane to their squares, creating organized intersections of parabolic arcs that correspond to Pythagorean triples.', 'This method does not account for every possible Pythagorean triple, as there are points it misses, such as those with odd imaginary components, and only hits multiples of the triples it does reach.', 'The organized intersections of parabolic arcs provide a structured visualization of Pythagorean triples, demonstrating a pattern within what usually seems random and unconnected.']}, {'end': 721.975, 'start': 593.754, 'title': 'Pythagorean triples and rational points', 'summary': 'Discusses how a diagram with squared lattice points and radial lines accounts for every possible pythagorean triple, and how rational points on a unit circle correspond to integer points with integer distance from the origin.', 'duration': 128.221, 'highlights': ['Every right triangle with whole number side lengths is accounted for by marking lattice points and drawing radial lines.', 'Finding rational points on a unit circle corresponds to finding integer points with integer distance from the origin.', 'The method of squaring lattice points and drawing radial lines results in a diagram with rational points filling the unit circle.']}, {'end': 874.766, 'start': 723.223, 'title': 'Squaring complex numbers and rational points', 'summary': 'Discusses how squaring complex numbers accounts for every possible rational slope, guaranteeing the coverage of every rational point on the unit circle, and consequently, hitting every possible pythagorean triple.', 'duration': 151.543, 'highlights': ['The method of squaring complex numbers accounts for every possible rational slope, guaranteeing the coverage of every rational point on the unit circle.', 'Radial lines from the method, determined by all possible choices of u and v, must pass through every rational point on the circle.', "Any time you have an angle between two points on the circumference of a circle and its center, that turns out to be exactly two times the angle made by those same points and any other point on the circle's circumference."]}], 'duration': 453.95, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/QJYmyhnaaek/pics/QJYmyhnaaek420816.jpg', 'highlights': ['The method of squaring complex numbers accounts for every possible rational slope, guaranteeing the coverage of every rational point on the unit circle.', 'The organized intersections of parabolic arcs provide a structured visualization of Pythagorean triples, demonstrating a pattern within what usually seems random and unconnected.', 'Every right triangle with whole number side lengths is accounted for by marking lattice points and drawing radial lines.', 'Finding rational points on a unit circle corresponds to finding integer points with integer distance from the origin.', 'The method of squaring lattice points and drawing radial lines results in a diagram with rational points filling the unit circle.', 'Radial lines from the method, determined by all possible choices of u and v, must pass through every rational point on the circle.', "Any time you have an angle between two points on the circumference of a circle and its center, that turns out to be exactly two times the angle made by those same points and any other point on the circle's circumference.", 'This method does not account for every possible Pythagorean triple, as there are points it misses, such as those with odd imaginary components, and only hits multiples of the triples it does reach.']}], 'highlights': ['A general formula to obtain Pythagorean triples from lattice points using complex numbers is provided, allowing any pair of integers to produce a Pythagorean triple.', 'The method of squaring complex numbers accounts for every possible rational slope, guaranteeing the coverage of every rational point on the unit circle.', 'By arranging the a square and the b square together with four copies of the original triangle, a big square whose side lengths are a plus b can be obtained, revealing a relationship between the areas of the squares and the triangle.', 'The area of the big square minus four times the area of the triangle can be expressed as a^2 + b^2, demonstrating a geometric relationship between the squares and the triangle.', 'Every right triangle with whole number side lengths is accounted for by marking lattice points and drawing radial lines.', 'The organized intersections of parabolic arcs provide a structured visualization of Pythagorean triples, demonstrating a pattern within what usually seems random and unconnected.', 'Babylonian clay tablets from 1800 BC list these triples', "Fermat's last theorem states no solutions for exponents larger than two", 'Pythagorean triple: a^2 + b^2 = c^2, aims to find every possible example', 'The process of squaring complex numbers is a surprisingly simple method to generate non-trivial Pythagorean triples from lattice points.', 'Finding rational points on a unit circle corresponds to finding integer points with integer distance from the origin.', 'Radial lines from the method, determined by all possible choices of u and v, must pass through every rational point on the circle.', "Any time you have an angle between two points on the circumference of a circle and its center, that turns out to be exactly two times the angle made by those same points and any other point on the circle's circumference.", 'This method does not account for every possible Pythagorean triple, as there are points it misses, such as those with odd imaginary components, and only hits multiples of the triples it does reach.', 'Reframing the question of finding whole number solutions is suggested, indicating a shift in approach to solving the problem.']}