A Visual Attempt at 1 + 2 + 3 + 4 + 5 + ... = -1/12 Mathematical Visual Proofs 2024-08-04 | I often get requests to give a visual proof of the claim that 1+2+3+4+5+... = -1/12 that is, that the sum of all positive integers is -1/12. One problem with this is that it isn't true using the techniques of real analysis, so visualizing it can be challenging (not to mention that visualizing negative numbers is challenging in and of itself). Another challenge is that people get pretty angry with the various methods used to produce this claim.However, we are able to make sense of this sum using a series of visual arguments that connect three different infinite sums and choose one value to assign to them. Here I try to show visual representations for this argument. While this argument has its inconsistencies, there are more legitimate reasons out there to indicate these values make some sense. In each case, there are valid reasons to assign these values, though the ones here aren't always the best.This sum has appeared many times on YouTube and has created a lot of controversy and excitement. I am not claiming this to be a set fact; instead, my intent is to show how one standard (though with problems) argument for this claim can be made visual in some sense. If you want to know more about the intricacies involved in this argument, see the videos linked below, especially the ones from Mathologer.If you like this video, consider subscribing to the channel or consider buying me a coffee: buymeacoffee.com/visualproofs Thanks! If you want to find other nice commentaries about this infinite sum, check out the following.From @numberphile youtu.be/w-I6XTVZXww?si=w89s2GLH65oB6fUVyoutu.be/E-d9mgo8FGk?si=aB5ahicUK9z5HHbVyoutu.be/0Oazb7IWzbA?si=gnpHiIMElCF7PLwjyoutu.be/FmLIGN8ZGdw?si=RMCbT35UNDyIers1youtu.be/beakj767uG4?si=lg8h7Zw0s7PV-Fdfyoutu.be/PCu_BNNI5x4?si=ZbAqwQtYesWFB12IFrom @Mathologer : youtu.be/jcKRGpMiVTw?si=xn3aoSuYbYmiXZW6 youtu.be/YuIIjLr6vUA?si=vGP2gdB39MuYTrf4From @blackpenredpen youtu.be/6FTwMUL69u0?si=Qxh6xOhcDrNa-KMpyoutu.be/U_g_OZLt4OU?si=8V2uh0rI5iddO8LcThe alternating geometric series argument comes from a proof by The Viewpoints 2000 group from the October 2001 issue of Mathematics Magazine page 320 (jstor.org/stable/2691106 ).To learn more about animating with manim, check out:https://manim.community#manim #maths #mathematics #ramanujan #ramanujansummation #infinitesum #sumpositiveintegers #arithmeticmean #series #infiniteseries #divergentseries #divergent #convergent #geometricseries #abelsum #somepi
Happy Math Holidays! Mathematical Visual Proofs 2024-12-23 | This is a holiday tree built from a cylinder, lines on a cone, and spheres. The tree is topped off with the Koch snowflake. Thank you for your support this year. I hope everyone has a happy holiday season and a wonderful new year. #math #manim #holiday To buy me a coffee, head over to buymeacoffee.com/VisualProofs Thanks!To learn more about animating with manim, check out: https://manim.community
Approximate Square Roots with Calculus! Mathematical Visual Proofs 2024-12-22 | This is a visual technique that uses linear approximation to find the square root of a whole number by finding the equation of the tangent line to points on the square root curve. To buy me a coffee, head over to buymeacoffee.com/VisualProofs Thanks!#manim #math #mathshorts #mathvideo #algebra #calculus #mtbos #manim #animation #theorem #iteachmath #spiral #logarithm #square #root #squareroot #function To learn more about animating with manim, check out:https://manim.community
Approximate Square Roots Visually! Mathematical Visual Proofs 2024-12-20 | This is a visual technique that uses linear approximation to find the square root of a whole number by looking at nested perfect square arrays. Can you prove why this works?To buy me a coffee, head over to buymeacoffee.com/VisualProofs Thanks!#manim #math #mathshorts #mathvideo #algebra #calculus #mtbos #manim #animation #theorem #iteachmath #spiral #logarithm #square #root #squareroot #function To learn more about animating with manim, check out:https://manim.community
Logarithmic spiral | Spira Mirabilis Mathematical Visual Proofs 2024-12-17 | This is a visualization of a self-similar curve known as a logarithmic spiral or spira mirabilis. To learn more about logarithmic spirals, check out the wikipedia site or mathworld:en.wikipedia.org/wiki/Logarithmic_spiralmathworld.wolfram.com/LogarithmicSpiral.htmlTo buy me a coffee, head over to buymeacoffee.com/VisualProofs Thanks!#manim #math #mathshorts #mathvideo #construction #calculus #mtbos #manim #animation #theorem #iteachmath #spiral #logarithm #logarithmicspiral #polarequation #polarcurve #function To learn more about animating with manim, check out:https://manim.community
Nested Powers of 2 Epicycloids (synthwave enumeration) Mathematical Visual Proofs 2024-12-11 | This video shows the creation of nested epicycloids with powers of 2 cusps (starting with 2 cusps and ending with 256), generated at rates that are powers of 2. Then the image rotates 3 times before fading out.If you like this video, consider subscribing to the channel or consider buying me a coffee: buymeacoffee.com/VisualProofs. Thanks! To learn more about epicycloids, check out the wikipedia site: en.wikipedia.org/wiki/EpicycloidFor a hypocycloid version of this idea, seeyoutu.be/DC8JlortDqEFor other hypocycloid videos, check these out:youtu.be/gTsdoR2rqwkyoutu.be/5dwlLH0aqxsyoutu.be/VvX9bMUX-icyoutu.be/1ctRFgWwg5Eyoutube.com/shorts/skLikVz2JhA#manim #epicycloid #spirograph #circle #rollingcircles #rational #count #mathvideo #math #mtbos #animation #iteachmath #mathematics #enumeration #synthwaveTo learn more about animating with manim, check out:https://manim.community_________________________________________Music in this videoNeon Drive by Ghostrifter Official | soundcloud.com/ghostrifter-officialMusic promoted by free-stock-music.comCreative Commons / Attribution-NoDerivs 3.0 Unported (CC BY-ND 3.0)creativecommons.org/licenses/by-nd/3.0
Dudeney’s Hinged Dissection Mathematical Visual Proofs 2024-11-29 | A dancing version of Dudeney’s famous dissection that shows an equilateral triangle hinge into a perfect square.If you like this video, consider subscribing to the channel or consider buying me a coffee: buymeacoffee.com/VisualProofs. Thanks! #math #manim #animation #theorem #iteachmath #mathematics ##mathshorts #mathvideo #mtbos #mathematician #dissection #proof #hinge To learn more about animating with manim, check out:https://manim.community
Seven Proofs Without Words for Summing Cubes Mathematical Visual Proofs 2024-11-27 | This video is a compilation of seven different visual proofs showing the sum formula for the sum of the first n positive cubes that can be found on my channel. This is a new style of video for me inspired by @ThatsAmazing 's trick shots at various levels and @dudeperfect 's soundtracks. I hope you enjoy it. If these visual proofs are too fast to follow completely in this format, the originals are linked below.If you like this video, consider subscribing to the channel or consider buying me a coffee: buymeacoffee.com/VisualProofs. Thanks! For the original animations, see here:Level 1, Four flat stacks: youtu.be/Ye9OPNqV9FALevel 2, Triangular arrays: youtu.be/jWpyrXYZNiILevel 3, Overlapping gnomons: youtu.be/d1yM6Rq7TfwLevel 4, Cubes to gnomons: youtu.be/YQLicI8R4GsLevel 5, Triangular dissection: youtu.be/NxOcT_VKQR0Level 6, Two cuboid stacks: youtu.be/i12p4Acb3L4Level 7, Summing Odds: youtu.be/LYfKapE_g1QOriginal sources:Level 1: This animation is based on a visual proofs by Warren Lushbaugh (from an article by Solomon Golomb) in the May 1965 issue of the Mathematical Gazette (doi.org/10.2307/3612319) and, independently, Antonella Cupillari from the October 1989 issue of Mathematics Magazine page 259 (doi.org/10.2307/2689765).Level 2: This animation is based on a visual proof by Parames Laosinchai from the December 2012 issue of Mathematics Magazine (jstor.org/stable/10.4169/math.mag.85.5.360 page 360).Level 3: This animation is based on a visual proofs by Warren Lushbaugh (from an article by Solomon Golomb) in the May 1965 issue of the Mathematical Gazette (doi.org/10.2307/3612319).Level 4: This animation is based on independently discovered, separate visual proofs by J. Barry Love from the March 1977 issue of Mathematics Magazine (jstor.org/stable/2689727), page 74, and Alan L. Fry from the January 1985 issue of Mathematics Magazine (jstor.org/stable/2690228) , page 11.Level 5: This animation is based on a visual proof by Georg Schrage from the June 1992 issue of Mathematics Magazine (doi.org/10.2307/2691330 - page 185).Level 6: This animation is based on a visual proof by Tom Edgar from the September 2024 issue of Mathematics Magazine (doi.org/10.1080/0025570X.2024.2379213 page ??).Level 7: This animation is based on a visual proof by Alfinio Flores from the January 1998 issue of The College Mathematics Journal (jstor.org/stable/2687639 page 61).#mathshorts #mathvideo #math #calculus #mtbos #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #iteachmath #finitesums #discretemath #calculus #sum #induction #faulhaber #sumofcubes #cubesTo learn more about animating with manim, check out:https://manim.community
Non-differentiability Mathematical Visual Proofs 2024-11-24 | In this short, we show five different graphs of functions with a point where the function is not differentiable.If you like this video, consider subscribing to the channel or consider buying me a coffee: buymeacoffee.com/VisualProofs. Thanks! #math #manim #animation #theorem #iteachmath #mathematics ##mathshorts #mathvideo #mtbos #mathematician #calculus #differentiability #functions #derivative #discontinuity #cusp #verticalasymptote #tangentline To learn more about animating with manim, check out:https://manim.community
Cannonball Problem! Mathematical Visual Proofs 2024-11-12 | In this short, we show the only nontrivial solution to the cannonball problem, which asks for numbers that are simultaneously square pyramidal and square. The problem gets its name because cannonballs stack nicely in square pyramidal arrays. Can you prove that 0,1, and 4900 are the only such solutions to the problem?If you like this video, consider subscribing to the channel or consider buying me a coffee: buymeacoffee.com/VisualProofs. Thanks! #math #manim #animation #theorem #iteachmath #mathematics ##mathshorts #mathvideo #mtbos #mathematician #squarepyramidal #numbertheory #cannonballproblem #cannonballrun To learn more about animating with manim, check out:https://manim.community
Visualizing Derivatives on Multivariable Surface Plots of Average Rates of Change Mathematical Visual Proofs 2024-11-03 | In this video, we investigate the derivative function as the set of discontinuities on the multivariable surface plot of the multivariable function that gives the average rate of change between two points on a single-variable function.Along the way, we investigate the definition of the derivative of a function at a point and see the key idea behind differential calculus.This video arose from one of my classes after a conversations with Joe Eichholz - my thanks to Joe for his time. If you like this video, consider subscribing to the channel or consider buying me a coffee: buymeacoffee.com/VisualProofs. Thanks!#manim #calculus #derivatives #derivative #tangentline #slope #parabola #mathvideo #mathshorts #math #visualmath #graph #linearapproximation #secantline #instantrateofchange #averagerateofchange #multivariablefunction #multivariablecalculus #surfaceplot
A Mathematical Halloween Witch! Mathematical Visual Proofs 2024-10-26 | A Halloween witch drawn with mathematical objects. Happy Halloween!If you like this video, consider subscribing to the channel or consider buying me a coffee: buymeacoffee.com/VisualProofs. Thanks!#math #manim #animation #mathvideo #halloween #witch #equations To learn more about animating with manim, check out:https://manim.community
Continuous everywhere but differentiable nowhere : Weierstrass Function Visualization! Mathematical Visual Proofs 2024-10-18 | This is a visualization of an approximation of the Weierstrass function, which is a function that is continuous everywhere but differentiable nowhere. The curve is defined as an infinite sum and there are several curves like this given different parameters.To buy me a coffee, head over to buymeacoffee.com/VisualProofs Thanks!#manim #math #mathshorts #mathvideo #construction #calculus #mtbos #manim #animation #theorem #iteachmath #spiral #weierstrass #continuouseverywhere #continuity #differentiation #differentiablenowhere #function To learn more about animating with manim, check out:https://manim.community
Mean Inequalities Mathematical Visual Proofs 2024-10-10 | This short animated proof demonstrates the two variable mean inequalities for the arithmetic mean, geometric mean, harmonic mean and root mean square (or quadratic mean).If you like this video, consider subscribing to the channel or consider buying me a coffee: buymeacoffee.com/VisualProofs. Thanks! This animation of the inequalities is based on a visual proof by Roger Nelsen from the June 1987 issue of Mathematics Magazine (jstor.org/stable/2689561) p. 158.If you like visual proofs of inequalities, see these links:AM-GM inequality via rectangles/squares: youtu.be/62G9fak1vykAM-GM inequality via circles: youtu.be/Pt8IX_Q5U1AAM-GM-HM-QM via Moments: youtu.be/XJ3643WhKjY#math #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #inequality #iteachmath #mathematics #pythagoreantheorem #similartriangles #means #meaninequality #mathshorts #mathvideo #mtbosTo learn more about animating with manim, check out:https://manim.community
Pythagorean Theorem Dissection Mathematical Visual Proofs 2024-10-05 | This is a short, animated visual proof of the Pythagorean theorem (the right triangle theorem) where we show how to use Euclid's shear and rotate proof to create a dissection proof. If you like this video, consider subscribing to the channel or consider buying me a coffee: buymeacoffee.com/VisualProofs. Thanks! Check out my Pythagorean theorem playlist:youtube.com/playlist?list=PLZh9gzIvXQUvlRlnYryejWGHYQquM92rBThe dissection proof was suggested to me by Glen Whitney (studioinfinity.org), and Glen then found a similar dissection in the compendium of Pythagorean theorem proofs from Elisha Loomis (Proof 21 on p. 122 ).(I may receive a small commission at no cost to you for this affiliate link): amzn.to/3RL7rfN#math #pythagoreantheorem #pythagorean #triangle #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #mathshorts #mathvideo To learn more about animating with manim, check out:https://manim.community
Mediant Inequality I Mathematical Visual Proofs 2024-10-01 | This is a short, animated visual proof demonstrating what one might called the mediant inequality.If you like this video, consider subscribing to the channel or consider buying me a coffee: buymeacoffee.com/VisualProofs. Thanks!This animation is based on a visual proof by Richard A. Gibbs from the June 1990 issue of Mathematics Magazine (doi.org/10.2307/2691137 - page 172).#mathshorts #mathvideo #math #numbertheory #mtbos #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #iteachmath #fractions #mediant #inequality #mathematics #fareyfractions #fareyTo learn more about animating with manim, check out:https://manim.community
Pi Times Phi using a Regular Icosagon Area (visual proof) Mathematical Visual Proofs 2024-09-28 | In this video, we use a golden triangle to find the area of a regular icosagon sitting inside of the unit circle (the circle with radius 1). From this area formula, we are able to then find an interesting bound on the product of two famous constants: Pi and the golden ratio. If you like this video, consider subscribing to the channel or consider buying me a coffee: buymeacoffee.com/VisualProofs. Thanks!If you like this video, you might enjoy this compilation of four visual proofs finding the area of a regular dodecagon inscribed in the unit circle: youtu.be/I5IrNug26S8This animation is based on a visual proof by Roger B. Nelsen from a recent issue of The College Mathematics Journal (doi.org/10.1080/07468342.2024.2347160 ).#math #manim #visualproof #mathvideo #geometry #mathshorts #geometry #mtbos #animation #theorem #pww #proofwithoutwords #proof #iteachmath #icosagon #area #dissection #trigonometry #polygon #sine #pi #goldenratio #phi #goldentriangle #inequality #areaTo learn more about animating with manim, check out:https://manim.community
Can you explain the pattern? Mathematical Visual Proofs 2024-09-27 | This enumeration shows all the tilings of a circular grid divided into n pieces from n = 1 to n = 8 using tiles that cover one or two cells. Can you tell the pattern? Can you prove why or how the Lucas numbers appear? If you are looking for more information about this, I recommend the wonderful textbook Proofs that Really Count: The Art of Combinatorial Proof from Arthur Benjamin and Jennifer Quinn: bookstore.ams.org/dol-27If you like this video, consider subscribing to the channel or consider buying me a coffee: buymeacoffee.com/VisualProofs. Thanks!If you enjoy this enumeration, check out the longer form: youtu.be/4S85bHkmM_Yor check out a Fibonacci tiling video : youtu.be/AFAcKDTmYXIIf you like this video, check out my others and consider subscribing. Thanks!#dominotilings #fibonacci #lucas #circulargrid ##countingtiling #count #mathvideo #math #mtbos #manim #animation #iteachmath #mathematics #discretemath #combinatorics #enumeration #synthwave
0.bbbb... = 1 (in base b+1) | 9 geometric series dissection proofs without words Mathematical Visual Proofs 2024-09-22 | This video is a compilation of nine shorter videos I have created showing dissection proofs for infinite geometric series with ratio of the form 1/n and first term 1/n. One interpretation of these series is that 0.bbbb... = 1 in base b+1 . If you like this video, consider subscribing to the channel or consider buying me a coffee: buymeacoffee.com/VisualProofs. Thanks! For another compilation of geometric series dissections (with minimal overlaps), check out this video: youtu.be/JteQEN1XPycThis is a compilation of nine shorts (most with words). If you want to see those, the links are below (along with attribution to the original ``proof without words" author).00:06 Infinite sum of powers of 1/2 (0.111... = 1 in base 2): youtube.com/shorts/iVLE8gEJfPAThis animation is based on a visual proof by Warren Page from the September 1981 issue of Mathematics Magazine, (page 201 - jstor.org/stable/2689632 ).00:38 Infinite sum of powers of 1/3 (0.222... = 1 in base 3):youtube.com/shorts/GwIVlNRV-M0This animation is based on a proof by Elizabeth M. Markham from the October 1993 issue of Mathematics Magazine page 242 (doi.org/10.2307/2690738).01:20 Infinite sum of powers of 1/4 (0.333... = 1 in base 4): youtube.com/shorts/qk1IPvvWfr0This animation is based on a proof by Sunday A. Ajose from the June 1994 issue of Mathematics Magazine (jstor.org/stable/2690616 page 230).01:55 Infinite sum of powers of 1/5 (0.444... = 1 in base 5):youtube.com/shorts/ltKZJ8wumUIThis animation is based on a proof by Elizabeth M. Markham from the October 1993 issue of Mathematics Magazine page 242 (doi.org/10.2307/2690738).02:40 Infinite sum of powers of 1/6 (0.555... = 1 in base 6):youtube.com/shorts/T2Mi0AUIKJYThis animation is based on a proof by James Tanton from the March 2008 issue of The College Mathematics Journal page 106. (jstor.org/stable/27646594).03:20 Infinite sum of powers of 1/7 (0.666... = 1 in base 7):youtube.com/shorts/4ri5pCirFsQThis animation is based on a proof by Stephan Berendonk (2020) from the November 2020 issue of The College Mathematics Journal, (doi.org/10.1080/07468342.2020.1830649 p. 385)04:15 Infinite sum of powers of 1/8 (0.777... = 1 in base 8):youtube.com/shorts/8TDSID0BhwIThis animation is based on a visual proof from Roger B. Nelsen in the February 2006 issue of Mathematics Magazine (doi.org/10.2307/27642904 or jstor.org/stable/27642904), page 60.05:00 Infinite sum of powers of 1/9 (0.888... = 1 in base 9):youtube.com/shorts/a8YMq_q4HRsThis animation is inspired by the a proof by Elizabeth M. Markham from the October 1993 issue of Mathematics Magazine page 242 (doi.org/10.2307/2690738).05:50 Infinite sum of powers of 1/10 (0.999... = 1 in base 10):youtube.com/shorts/XJtjU_IpjnoThis animation is based on a proof by James Tanton from the March 2008 issue of The College Mathematics Journal page 106. (jstor.org/stable/27646594).To learn more about animating with manim, check out:https://manim.community___________________________________Music: Wanderer by Alexander Nakarada (CreatorChords) | creatorchords.comMusic promoted by free-stock-music.comCreative Commons / Attribution 4.0 International (CC BY 4.0)creativecommons.org/licenses/by/4.0
Dancing tilings - why this pattern? Mathematical Visual Proofs 2024-09-17 | This short shows the enumeration of all the domino tilings of the (2 x n)-grid for n=1 to n=8. Can you tell the pattern? Can you prove why or how the Fibonacci numbers (perhaps more aptly called the Virahanka numbers) appear? If you like this video, consider subscribing to the channel or consider buying me a coffee: buymeacoffee.com/VisualProofs. Thanks!If you are looking for more information about this, I recommend this fantastic video by : youtu.be/Ct7oltmdJrM or the wonderful textbook Proofs that Really Count: The Art of Combinatorial Proof from Arthur Benjamin and Jennifer Quinn: bookstore.ams.org/dol-27#dominotilings #fibonacci #counting #count #mathvideo #math #mtbos #manim #animation #iteachmath #mathematics #discretemath #combinatorics #enumeration #synthwaveTo learn more about animating with manim, check out:https://manim.community
Sums of Sums of Squares (visual proof) Mathematical Visual Proofs 2024-09-15 | This is a short, animated visual proof for the sum of the sum of squares formula using a wonderful visual proof computing the sum of squares formula. If you like this video, consider subscribing to the channel or consider buying me a coffee: buymeacoffee.com/VisualProofs. Thanks!Here is a link to the visual proof for the sum of squares:youtu.be/-tJhH_k2LaMHere are other visual proofs about the sum of squares formulas:youtu.be/a8j3sBrXchgyoutu.be/WidzHiUFWNAyoutu.be/UqVmocdLFGcyoutu.be/Q-frL0Ot2m4youtu.be/VYaEGvClg7QThis animation is based on a visual proof by C.G. Wastun according to Roger Nelsen in his second Proof without Words compendium (page 91) : bookstore.ams.org/view?ProductCode=CLRM/14This first visual proof animation is based on (independently discovered) visual proofs by Dan Kalman from the March 1991 issue of The College Mathematics Journal (jstor.org/stable/2686447) and Martin Gardner from the October 1973 Scientific American (jstor.org/stable/24923225). To learn more about animating with manim, check out:https://manim.community#math #manim #mathvideo #sumofsquares #mtbos #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #iteachmath #finitesums #discretemath #induction #sumsumsquares #partialsums
Pythagorean Theorem - Behold! Mathematical Visual Proofs 2024-09-08 | This is a short, animated visual proof of the Pythagorean theorem (the right triangle theorem) following essentially Bhāskara's proof (Behold!). This theorem states the square of the hypotenuse of a right triangle is equal to the sum of squares of the two other side lengths. Here is a bit longer version: youtu.be/I75HYC8TRNYIf you like this video, consider subscribing to the channel or consider buying me a coffee: buymeacoffee.com/VisualProofs. Thanks!This animation is based on a proof due to Bhāskara. For a static version of this proof, see Roger Nelsen's first "Proof Without Words: Exercises in Visual Thinking" compendium (page 4). You can also check out Howard Eves' "Great Moments in Mathematics (Before 1650)" page 29-32.Check out Roger's book on Amazon (affiliate link to follow; I may receive a small commission): amzn.to/3SjsqEhFor other proofs of this same fact check out: youtu.be/q28H5NY5UVUyoutu.be/yfGtbNgcrQ8youtu.be/JRjPsuJ7S-gyoutu.be/TsKTteGJpZUyoutu.be/ZlGaQdNRdqAyoutu.be/uqGjs_n-f7Uyoutu.be/wWEZ-rUlBDQ#math #manim #pythagoreantheorem #pythagorean #triangle #animation #theorem #pww #proofwithoutwords #visualproof #proof #mathshorts #mathvideo To learn more about animating with manim, check out:https://manim.community
Chaos Game finds the Mandelbrot Quintet Mathematical Visual Proofs 2024-09-03 | In this video, we show how to use a random process of iteratively applying five affine transformations in the real plane to generate a 5-rep-tile known as the Mandelbrot Quintet. What happens if you do something similar with different affine transformations?Here is a bit longer version (without words): youtu.be/rLxx6b-oJGoCheck out these videos for related constructions:youtube.com/shorts/dklWNdM9WSgyoutu.be/rLxx6b-oJGoyoutu.be/NBI_5GdvqUoIf you like this video, consider subscribing to the channel or consider buying me a coffee: buymeacoffee.com/VisualProofs. Thanks!This animation was inspired by an article written by Lorelei Koss that appeared in the Proceedings of Bridges 2015: archive.bridgesmathart.org/2015/bridges2015-423.pdf . For more information, you should also check out the related article "Fractal Tilings in the Plane" by Richard Darst, Judith Palagallo and Thomas Price from the February 1998 Mathematics Magazine: jstor.org/stable/2691339#chaos #chaosgame #quintet #mandelbrot #mathvideo #math #mtbos #manim #animation #theorem #iteachmath #mathematics #dynamicalsystems #iteratedfunctionsystem #dynamics #fractals To learn more about animating with manim, check out:https://manim.community
0.5555… = 1 (in base 6) Mathematical Visual Proofs 2024-08-30 | This is a short, animated visual proof showing the sum of the infinite geometric series with first term 5/6 and ratio 1/6, which in turn allows us to compute the sum of the series of powers of 1/6 and determine an interesting base 6 representation of 1. If you like this video, consider subscribing to the channel or consider buying me a coffee: buymeacoffee.com/VisualProofs. Thanks! For a longer, wordless version (more dramatic) of this animation seeyoutu.be/o0AyHmRV8RA?si=2fExvp1WqrmZp5SJAlso, check out my playlist on geometric sums/series: youtube.com/playlist?list=PLZh9gzIvXQUsgw8W5TUVDtF0q4jEJ3iawThis animation is based on a proof by James Tanton from the March 2008 issue of The College Mathematics Journal page 106. (jstor.org/stable/27646594).#mathshorts #mathvideo #math #calculus #mtbos #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #iteachmath #geometricsums #series #infinitesums #infiniteseries #geometric #geometricseries #pentagon #trapezoid To learn more about animating with manim, check out:https://manim.community
Two Geometric Series in an Isosceles Trapezoid (visual proof without words) Mathematical Visual Proofs 2024-08-24 | This is a short, animated visual proof demonstrating the sum of two infinite geometric series using dissection proofs in an isosceles trapezoid. In particular, we show how to find the sum of powers of 1/4 and the alternating sum of powers of 1/2 using the isosceles trapezoid. Geometric series are important for many results in calculus, discrete mathematics, and combinatorics. If you like this video, consider subscribing to the channel or consider buying me a coffee: buymeacoffee.com/VisualProofs. Thanks! Also, check out my playlist on geometric sums/series: youtube.com/playlist?list=PLZh9gzIvXQUsgw8W5TUVDtF0q4jEJ3iawThis animation is based on a proof by Roger B. Nelsen in his book Cameos for Calculus (bookstore.ams.org/clrm-49 ).#mathshorts #mathvideo #math #calculus #mtbos #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #iteachmath #geometricsums #series #infinitesums #infiniteseries #geometric #geometricseries #trapezoid #isoscelestrapezoid To learn more about animating with manim, check out:https://manim.community______________________________Music in this video:Traveling Around The World by Alex-Productions | https://onsound.eu/Music promoted by free-stock-music.comCreative Commons / Attribution 3.0 Unported License (CC BY 3.0)creativecommons.org/licenses/by/3.0/deed.en_US
Parallelogram Area from Side Angle Side Mathematical Visual Proofs 2024-08-21 | This is a short animation showing a standard fact about how to find the area of a parallelogram using the lengths of the two sides and the angle between them. This fact will be used in future videos showing the AM-GM inequality and the Cauchy Schwarz inequality (and possibly others).If you like this video, consider subscribing to the channel or consider buying me a coffee: buymeacoffee.com/VisualProofs. Thanks! #area #trigonometry #parallelogram #angle #sideangleside #geometry #visualproof #math #manimTo learn more about animating with manim, check out:https://manim.community
Top 10 Math Constants Mathematical Visual Proofs 2024-08-17 | In this video, we show visualizations of ten classic mathematical constants:1) The square root of 2 as the length of the diagonal of a 1 by 1 square.2) Pi as the area of the circle with radius 1.3) Tau as the circumference of the circle with radius 1.4) The golden ratio as the number satisfying (x+1)/x=x/1.5) The square root of 3 as the altitude or area of an equilateral triangle with side length 2.6) Euler's number e as the ending x-coordinate yielding the area under the curve of 1/x starting at x=1.7) The silver ratio as the horizontal length across the regular octagon with side length 1 or the area of the largest rectangle inside the same regular octagon.8) The natural logarithm of 2, ln(2), as the sum of the convergent alternating harmonic series.9) The plastic ratio as the limit of the ratio of side lengths in a spiral of equilateral triangles with side lengths dictated by the Padovan sequence.10) The Euler-Mascheroni constant as the difference between the harmonic series and the area under the curve y=1/x from 1 to infinity.If you like this video, consider subscribing to the channel or consider buying me a coffee: buymeacoffee.com/VisualProofs. Thanks! #mathvideo #math #numbertheory #mtbos #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #iteachmath #constants #goldenratio #silverratio #plasticratio #squareroot #squareroot2 #squareroot3 #euler #eulerconstant #eulermascheroni #ln2 #logarithm #octagon #square #pi #tau To learn more about animating with manim, check out:https://manim.community
AM-GM Inequality via negative space Mathematical Visual Proofs 2024-08-14 | This is a short, animated visual proof demonstrating the arithmetic mean geometric mean inequality using algebraic areas and the Side-Angle-Side formula for a parallelogram.If you like this video, consider subscribing to the channel or consider buying me a coffee: buymeacoffee.com/VisualProofs. Thanks! For other visual proofs of this fact see:youtu.be/Pt8IX_Q5U1Ayoutu.be/62G9fak1vykyoutu.be/HFGSsPbYc-sFor a nearly identical visual proof of another inequality (Cauchy-Schwarz) seeyoutu.be/mKg_gVagHy8This animation is based on a visual proof that appears in Icons of Mathematics by Claudi Alsina and Roger B. Nelsen (MAA, 2011): #mathshorts #mathvideo #math #amgminequality #mtbos #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #iteachmath #calculus #inequality To learn more about animating with manim, check out:https://manim.community
Sum of Cubes VII (visual proof without words) Mathematical Visual Proofs 2024-08-11 | This is a short, animated (wordless) visual proof demonstrating the sum of the first n positive cubes by rearranging two stacks of cubic arrays.If you like this video, consider subscribing to the channel or consider buying me a coffee: buymeacoffee.com/VisualProofs. Thanks!Here are six other visual proofs of sum of cubes formulas:youtu.be/LYfKapE_g1Qyoutu.be/jWpyrXYZNiIyoutu.be/YQLicI8R4Gsyoutu.be/Ye9OPNqV9FAyoutu.be/NxOcT_VKQR0youtu.be/d1yM6Rq7TfwThis animation is based on a visual proof by Tom Edgar from the September 2024 issue of Mathematics Magazine (doi.org/10.1080/0025570X.2024.2379213 page ??).To learn more about animating with manim, check out:https://manim.community#mathshorts #mathvideo #math #calculus #mtbos #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #iteachmath #finitesums #discretemath #calculus #sum #induction #faulhaber #sumofcubes #cubes___________________________________________Music in this video:Epic Cyberpunk | GLORY by Alex-Productions | https://onsound.eu/Music promoted by free-stock-music.comCreative Commons / Attribution 3.0 Unported License (CC BY 3.0)creativecommons.org/licenses/by/3.0/deed.en_US
0.8888… = 1 (in base 9) Mathematical Visual Proofs 2024-08-08 | This is a short, animated visual proof showing the sum of the infinite geometric series with first term 8/9 and ratio 1/9, which in turn allows us to compute the sum of the series of powers of 1/9 and determine an interesting base 9 representation of 1. If you like this video, consider subscribing to the channel or consider buying me a coffee: buymeacoffee.com/VisualProofs. Thanks! For a longer, wordless version (more dramatic) of this animation seeyoutu.be/C4t_ps3VKvIAlso, check out my playlist on geometric sums/series: youtube.com/playlist?list=PLZh9gzIvXQUsgw8W5TUVDtF0q4jEJ3iawThis animation is inspired by the a proof by Elizabeth M. Markham from the October 1993 issue of Mathematics Magazine page 242 (doi.org/10.2307/2690738).#mathshorts #mathvideo #math #calculus #mtbos #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #iteachmath #geometricsums #series #infinitesums #infiniteseries #geometric #geometricseries #square To learn more about animating with manim, check out:https://manim.community
Trapezoid Area Visual Proof Mathematical Visual Proofs 2024-08-06 | This is a short animation showing a standard fact about how to find the area of a trapezoid by finding a triangle with the same area. If you like this video, consider subscribing to the channel or consider buying me a coffee: buymeacoffee.com/VisualProofs. Thanks!Here is the original video in wide-format, wordless, with dramatic music:youtu.be/4ZypJJmd5ZYHere is a related visual proof for the area of a parallelogram:youtu.be/QjWbBexfJvw#area #triangle #trapezoid #geometry #visualproof #math #manimThis video was inspired by a beautiful GeoGebra applet created by Tim Brzezinski:geogebra.org/m/ZeDN7ZTWTo learn more about animating with manim, check out:https://manim.community
Trig Double Angle Formulas from Semicircle (visual proof) Mathematical Visual Proofs 2024-07-27 | This is a short, animated visual proof of the Double angle identities for sine and cosine. To get the formulas we use a semicircle diagram and rely on similarity of two right triangles formed inside.If you like this video, consider subscribing to the channel or consider buying me a coffee: buymeacoffee.com/VisualProofs. Thanks! For another visual proof of this fact using the laws of sines and cosines, check out this animation:youtu.be/0ggGQ96eaiEThis animation is based on a visual proof by Roger B. Nelsen from the January 1989 issue of The College Mathematics Journal (jstor.org/stable/2686819 - page 51).#mathshorts #mathvideo #math #trigonometry #lawofsines #lawofcosines #triangle #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #circle #obtuseangle #acuteangle #angle #trigidentities #identity #righttriangle #isoscelestriangle #doubleangle To learn more about animating with manim, check out:https://manim.community
Ten Mortal Math Constants Mathematical Visual Proofs 2024-07-22 | In this video, we show visualizations of ten classic mathematical constants:1) The square root of 2 as the length of the diagonal of a 1 by 1 square.2) Pi as the area of the circle with radius 1.3) Tau as the circumference of the circle with radius 1.4) The golden ratio as the number satisfying (x+1)/x=x/1.5) The square root of 3 as the altitude or area of an equilateral triangle with side length 2.6) Euler's number e as the ending x-coordinate yielding the area under the curve of 1/x starting at x=1.7) The silver ratio as the horizontal length across the regular octagon with side length 1 or the area of the largest rectangle inside the same regular octagon.8) The natural logarithm of 2, ln(2), as the sum of the convergent alternating harmonic series.9) The plastic ratio as the limit of the ratio of side lengths in a spiral of equilateral triangles with side lengths dictated by the Padovan sequence.10) The Euler-Mascheroni constant as the difference between the harmonic series and the area under the curve y=1/x from 1 to infinity.If you like this video, consider subscribing to the channel or consider buying me a coffee: buymeacoffee.com/VisualProofs. Thanks! #mathvideo #math #numbertheory #mtbos #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #iteachmath #constants #goldenratio #silverratio #plasticratio #squareroot #squareroot2 #squareroot3 #euler #eulerconstant #eulermascheroni #ln2 #logarithm #octagon #square #pi #tau To learn more about animating with manim, check out:https://manim.community
Inductive factorial sum visual proof Mathematical Visual Proofs 2024-07-15 | This is a short, animated visual proof demonstrating a finite sum involving products of factorials. The proof exploits the classic inductive proof of the formula in question. If you like this video, consider subscribing to the channel or consider buying me a coffee: buymeacoffee.com/VisualProofs. Thanks!This animation is based on a proofs by Tom Edgar from the December 2016 issue of Mathematics Magazine page 338 (doi.org/10.4169/math.mag.89.5.338).If you want to see a longer version, with an accompanying explanation, check out youtu.be/FE0Vl4zm20I #mathvideo #math #numbertheory #mtbos #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #iteachmath #factorial #squares #factoradic #discretemath #inductionproof #induction #factorialnumbersystem #finitesums #sum #series To learn more about animating with manim, check out:https://manim.community
Top four visual proofs? Mathematical Visual Proofs 2024-07-10 | In this short, we show animations of four of the most famous proofs without words: the formula for the sum of the first n integers; the pythagorean theorem using negative space/sliding rectangles/Chou pei suan ching; the formula for the sum of the first n odd integers; and the infinite geometric series of positive powers of 1/2 (using a rectangle/square dissection of a unit area square). We include brief justifications for these visual proofs.If you like this video, consider subscribing to the channel or consider buying me a coffee: buymeacoffee.com/VisualProofs. Thanks!For a slightly different version of this short without words, see:youtube.com/shorts/eG72HxNceEsFor longer videos with related animations, seeSumming Integers: youtu.be/eHbtc50-qXoPythagorean Theorem: youtu.be/o2mbN452ywA (and old version: youtu.be/yfGtbNgcrQ8)Summing Odds: youtu.be/I4A2T55k0cgGeometric Series: youtu.be/JteQEN1XPycThe first proof was known to the ancient greeks (cited by Martin Gardner), the second proof is adapted from the Chou pei suan ching (around 200 BCE according to Roger Nelsen), the third is attributed to Nicomachus of Gerasa by Roger Nelsen, and the final one is attributed to Warren Page (from the September 1981 issue of Mathematics Magazine, page 201 - jstor.org/stable/2689632 ). The first three can all be found in Roger Nelsen's first compendium, "Proofs Without Words: Exercises in Visual Thinking: bookstore.ams.org/view?ProductCode=CLRM/1 .#math #manim #pythagoreantheorem #pythagorean #triangle #animation #theorem #pww #proofwithoutwords #visualproof #proof #mathshorts #mathvideo #geometricseries #infiniteseries #finitesums #infinitesums #oddsumformula #sumsofintegers #sums #shorts To learn more about animating with manim, check out:https://manim.community
Two Infinite Series Sums from Regular Polygons (visual proof) Mathematical Visual Proofs 2024-07-07 | This is a short, animated visual proof computing the sums of two series - one of reciprocals of triangular numbers (i.e., certain binomial coefficients) and the other a classic series that is used to demonstrate telescoping series.If you like this video, consider subscribing to the channel or consider buying me a coffee: buymeacoffee.com/VisualProofs. Thanks! For some related videos see these videos:youtu.be/moxeqJx2YsYyoutu.be/I0juicS3SIcThis animation is based on a visual proof by Paul Stephenson from the December 2022 issue of Mathematics Magazine (doi.org/10.1080/0025570X.2022.2126173 page 572 ).#mathshorts #mathvideo #math #infiniteseries #mtbos #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #iteachmath #calculus #triangularnumbers #animation #theorem #proof #iteachmath #series #infinitesums #infiniteseries #pascaltriangle #binomialcoefficients #binomials #binomialtheorem #telescoping To learn more about animating with manim, check out:https://manim.community
Product of chords? Mathematical Visual Proofs 2024-07-01 | This short animation shows chords connecting n equally space points to one of those points on a unit circle and computes the product of the chord lengths. Do you have a conjecture based on this? Can you prove it?If you’re interested, check out the related video at youtu.be/OedLfHaj2p4?si=VFiIFOmeDCWgGBYF .This animation is based on a famous problem that has been discussed in numerous places. In particular, a great reference for this problem and related ones (with a fantastic bibliography) is the source Chords of an Ellipse, Lucas Polynomials, and Cubic Equations in Issue 8 of the 2020 American Mathematical Monthly (doi.org/10.1080/00029890.2020.1785253) by Ben Blum-Smith and Japheth Wood. You can also find the source here: arxiv.org/abs/1810.00492.#math #mathvideo #manim #circle #chords #visualproof #trigonometry #sine #cosine #tangent #rootsofunity #complexanalysis #complexnumbers #products #geometry For more about using manim, see https://www.manim.community/.
Arithmetic Mean-Geometric Mean Visual Proof Compilation Mathematical Visual Proofs 2024-06-30 | This video is a compilation of seven shorter videos that I have created showing various visual proofs without words of the famous AM-GM inequality, which states that the arithmetic mean of two numbers (a+b)/2 is greater than or equal to the geometric mean of the two numbers square root of a times b.To see the original videos (in shorter form and sometimes with words/explanations), check the links below. Of course there are many other visual proofs of this amazing fact, but these are my favorite seven.Let me know your favorite in the comments!If you like this video, consider subscribing to the channel or consider buying me a coffee: buymeacoffee.com/VisualProofs. Thanks! Here are the original series videos (along with attribution; for more detailed attribution, see the original videos):0:00 Introduction0:05 Folding Proof : youtu.be/IJGwvvQdfgg and youtube.com/shorts/YAGjrfpHq-Q (attribution: Yukio Kobayashi)0:40 Circle with Triangles : youtu.be/HFGSsPbYc-s (attribution: Roger B. Nelsen)1:07 Sliding Triangles: youtu.be/SEPkdPWKdO0 (attribution: Claudi Alsina and Roger B. Nelsen)1:52 Tangent Circles : youtu.be/jNx05gXSblE (attribution : Roland H. Eddy)2:38 Eight Kites : youtu.be/02yM5LqMWUs (attribution: Ayoub B. Ayoub)3:27 Tatami : youtu.be/62G9fak1vyk and youtube.com/shorts/if800mpoJaY(attribution: Doris Schattschneider)4:05 Semicircle : youtu.be/Pt8IX_Q5U1A and youtube.com/shorts/jB0OZYA_EI0 (attribution: Charles D. Gallant)4:49 Ending citationIf you like this inequality or others, you can check out my playlists of inequalities and AM-GM inequality:Inequalities: youtube.com/playlist?list=PLZh9gzIvXQUuKcFHdtjFkwAVELDxu9nH4&si=_cVhtnlrrAIQg9f6AM-GM : youtube.com/playlist?list=PLZh9gzIvXQUuoRR4kTd628cWdFQ1BDyS6&si=zZYGTtPAsp0jXr42#math #amgminequality #area #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #mathshorts #mathvideo #inequality #inequalities To learn more about animating with manim, check out:https://manim.community
Happy Tau Day! Mathematical Visual Proofs 2024-06-27 | In this short video, we show one way to visualize the constant tau, which is twice pi, as the area between the curves 1/(1+x^2) and its negation over the entire real line.Happy Tau Day if you celebrate!To buy me a coffee, head over to buymeacoffee.com/VisualProofs Thanks!#mathshorts #mathvideo #math #short #math #manim #radian #circle #radius #tauday #taurus #calculus To learn more about animating with manim, check out:https://manim.community
Pythagorean theorem from a (semi) circle! Mathematical Visual Proofs 2024-06-24 | This is a short, animated visual proof of the Pythagorean theorem (the right triangle theorem) using the semicircle and Thales triangle theorem. This theorem states the square of the hypotenuse of a right triangle is equal to the sum of squares of the two other side lengths. This animation is based on a proof due to Michael Hardy from the November, 1986 issue of The College Math Journal (doi.org/10.2307/2686255).To buy me a coffee, head over to buymeacoffee.com/VisualProofs Thanks!For other proofs of this same fact check out the following playlist:youtube.com/playlist?list=PLZh9gzIvXQUvlRlnYryejWGHYQquM92rB&si=o_im7-_NHp1zhkcD#mathshorts #mathvideo #math #pythagoreantheorem #pythagorean #triangle #manim #animation #theorem #pww #proofwithoutwords #visualproof #proofTo learn more about animating with manim, check out:https://manim.community
Sum Geometric Series with Ratio 1/2 in Rectangles (visual proofs) Mathematical Visual Proofs 2024-06-22 | In this video, we show an animated version of a recent proof without words that finds the sum of the geometric series with first term 1/root(2)+1/2 and ratio 1/2 using the unit square. Then we show how to find other geometric series sums with ratio 1/2 using different sizes of rectangles. Careful, the final bonus question might require a different technique than rotating :)If you like this video, consider subscribing to the channel or consider buying me a coffee: buymeacoffee.com/VisualProofs. Thanks! Also, check out my playlist on geometric sums/series: youtube.com/playlist?list=PLZh9gzIvXQUsgw8W5TUVDtF0q4jEJ3iawThis animation is based on a proof by Ángel Plaza in the June 2024 issue of Mathematics Magazine (doi.org/10.1080/0025570X.2024.2336443 page 328).#mathshorts #mathvideo #math #calculus #mtbos #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #iteachmath #geometricsums #series #infinitesums #infiniteseries #geometric #geometricseries #rectangle #silverratio #goldenratio To learn more about animating with manim, check out:https://manim.community
0.66666… = 1 (in base 7) Mathematical Visual Proofs 2024-06-17 | This is a short, animated visual proof showing the sum of the infinite geometric series with first term 6/7 and ratio 1/7, which in turn allows us to compute the sum of the series of powers of 1/7 and determine an interesting base 7 representation of 1. If you like this video, consider subscribing to the channel or consider buying me a coffee: buymeacoffee.com/VisualProofs. Thanks! For a longer, wordless version of this animation with two other proofs, see youtu.be/L-pxcCk3Xi4?si=hCnLn61vVxSOp2b4Also, check out my playlist on geometric sums/series: youtube.com/playlist?list=PLZh9gzIvXQUsgw8W5TUVDtF0q4jEJ3iawThis animation is based on a proof by Stephan Berendonk (2020) from the November 2020 issue of The College Mathematics Journal, (doi.org/10.1080/07468342.2020.1830649 p. 385)#mathshorts #mathvideo #math #calculus #mtbos #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #iteachmath #geometricsums #series #infinitesums #infiniteseries #geometric #geometricseries #equilateraltriangle To learn more about animating with manim, check out:https://manim.community
A Factorial Sum Produces the Factorial Number System (visual proof) Mathematical Visual Proofs 2024-06-15 | This is a short, animated visual proof demonstrating a finite sum involving products of factorials. The proof exploits the classic inductive proof of the formula in question. As a bonus, we discuss the factorial number system and show how the formula can be used to count up in this system.If you like this video, consider subscribing to the channel or consider buying me a coffee: buymeacoffee.com/VisualProofs. Thanks!This animation is based on a proofs by Tom Edgar from the December 2016 issue of Mathematics Magazine page 338 (doi.org/10.4169/math.mag.89.5.338).If you want to know more about the factorial number system, check out en.wikipedia.org/wiki/Factorial_number_system #mathvideo #math #numbertheory #mtbos #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #iteachmath #factorial #squares #factoradic #discretemath #inductionproof #induction #factorialnumbersystem #finitesums #sum #series To learn more about animating with manim, check out:https://manim.community
Alternating Geometric Series Sum Mathematical Visual Proofs 2024-06-13 | This is a short, animated visual proof demonstrating the infinite alternating geometric series formula for any positive ratio r with r less than 1 and with positive first term a. This series is important for many results in calculus, discrete mathematics, and combinatorics.To buy me a coffee, head over to buymeacoffee.com/VisualProofs Thanks!For a slower, wide format version of this video (without words), see youtu.be/Qa0bOHYjnW0Also, check out my playlist on geometric sums/series: youtube.com/playlist?list=PLZh9gzIvXQUsgw8W5TUVDtF0q4jEJ3iawThis animation is based on a proof by The Viewpoints 2000 group from the October 2001 issue of Mathematics Magazine page 320 (jstor.org/stable/2691106 ). #mathshorts #mathvideo #math #calculus #mtbos #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #iteachmath #geometricsums #series #infinitesums #infiniteseries #geometric #geometricseriesTo learn more about animating with manim, check out:https://manim.community
Spiral of Theodorus Mathematical Visual Proofs 2024-06-10 | This is a short, animated visual proof demonstrating how to construct square roots of any positive integer using the Spiral of TheodorusTo buy me a coffee, head over to buymeacoffee.com/VisualProofs Thanks!#manim #math #mathshorts #mathvideo #construction #geometry #mtbos #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #iteachmath #spiral #theodorus #squarerootTo learn more about animating with manim, check out:https://manim.community
Terdragon from the Chaos Game (visual fractal construction from iterated function system) Mathematical Visual Proofs 2024-06-08 | In this video, we show how to use a random process of iteratively applying three affine transformations in the real plane to generate a 3-rep-tile known as the Terragon. What happens if you do something similar with different affine linear transformations?Check out these videos for related constructions:youtu.be/rLxx6b-oJGoyoutu.be/NBI_5GdvqUoyoutu.be/XITi6Bnn5Ccyoutu.be/fFIyqMiycs4If you like this video, consider subscribing to the channel or consider buying me a coffee: buymeacoffee.com/VisualProofs. Thanks!This animation was inspired by articles written by Lorelei Koss, one that appeared in the Proceedings of Bridges 2015: archive.bridgesmathart.org/2015/bridges2015-423.pdf and one that appeared in the September 2022 issue of Math Horizons:doi.org/10.1080/10724117.2022.2088009 For more information, you should also check out the related article "Fractal Tilings in the Plane" by Richard Darst, Judith Palagallo and Thomas Price from the February 1998 Mathematics Magazine: jstor.org/stable/2691339Or you can check out Larry Riddle's fantastic site with lots about this fractal and others: larryriddle.agnesscott.org/ifs/heighway/terdragon.htm#chaos #chaosgame #terdragon #dragonfractal #mathvideo #math #mtbos #manim #animation #theorem #iteachmath #mathematics #dynamicalsystems #iteratedfunctionsystem #dynamics #fractals To learn more about animating with manim, check out:https://manim.community_________________________________________Music in this video:The Sun Will Rise by Vlad Gluschenko | soundcloud.com/vgl9Music promoted by free-stock-music.comCreative Commons / Attribution 3.0 Unported License (CC BY 3.0)creativecommons.org/licenses/by/3.0/deed.en_US
Sum of a Positive Number and its Reciprocal from Calculus Mathematical Visual Proofs 2024-06-06 | This is a short, animated visual proof demonstrating that sum of a positive real number and its reciprocal is always greater than or equal to 2. To buy me a coffee, head over to buymeacoffee.com/VisualProofs Thanks!It turns out that this theorem is equivalent to the Arithmetic Mean-Geometric Mean inequality. The equivalence is implied by the following two proofs: youtu.be/62G9fak1vykyoutu.be/IghOHBl0Do8And here are two alternate ways to prove this fact via the Pythagorean Theorem and triangle areas:youtu.be/zBUK8C6wSqsyoutu.be/BkQvnriVblYThis animation is based on a proof by Roger B. Nelsen from the December 1994 issue of Mathematics Magazine, page 374 (doi.org/10.2307/2690999). You can find this here too: maa.org/sites/default/files/Nelsen99959975.pdf#math #manim #visualproof #mathvideo #geometry #mathshorts #algebra #mtbos #animation #theorem #pww #proofwithoutwords #proof #iteachmath #inequality #calculus #derivative #tangentline #slope #amgmTo learn more about animating with manim, check out:https://manim.community
Garfield’s Pythagorean Proof Mathematical Visual Proofs 2024-06-03 | This is a short, animated visual proof of the Pythagorean theorem (the right triangle theorem) using the trapezoid that is now attributed to President James Garfield. This theorem states the square of the hypotenuse of a right triangle is equal to the sum of squares of the two other side lengths.To buy me a coffee, head over to buymeacoffee.com/VisualProofs Thanks!This animation is based on a proof due to James A. Garfield. For a static version of this proof, see Roger Nelsen's first "Proof Without Words: Exercises in Visual Thinking" compendium (page 7) : bookstore.ams.org/clrm-1. You can also check out Howard Eves' "Great Moments in Mathematics (Before 1650)" page 34-36.You can also see the original on MAA Convergence: maa.org/press/periodicals/convergence/mathematical-treasure-james-a-garfields-proof-of-the-pythagorean-theoremFor other proofs of this same fact check out the following playlist:youtube.com/playlist?list=PLZh9gzIvXQUvlRlnYryejWGHYQquM92rB&si=o_im7-_NHp1zhkcD#mathshorts #mathvideo #math #pythagoreantheorem #pythagorean #triangle #manim #animation #theorem #pww #proofwithoutwords #visualproof #proofTo learn more about animating with manim, check out:https://manim.community
Root 2 is Irrational from Isosceles Triangle (visual proof) Mathematical Visual Proofs 2024-06-01 | In this short, we use a famous argument by Tom Apostol to prove that the square root of two is irrational by infinite descent using a right isosceles triangle. We also go a bit further and show how this proof hints at the number theoretic construction of the convergents of the square root of two, which are the best rational approximations of root 2. If you like this video, consider subscribing to the channel or consider buying me a coffee: buymeacoffee.com/VisualProofs. Thanks!For an alternate visual proof of this fact, see this video: youtu.be/-dbzvN4jnfYThis animation is based on an argument due to Tom Apostol from issue 9 of the 2000 American Mathematical Monthly: doi.org/10.1080/00029890.2000.12005280To learn more about the convergents argument and the relationship between this proof and convergents, see this wonderful article by Doron Zeilberger:https://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/sqrt2.pdfand here you can learn more about convergents:en.wikipedia.org/wiki/Continued_fraction#Convergents#irrationalnumbers #realnumbers #manim #math #mtbos #animation #theorem #visualproof #proof #iteachmath #mathematics #irrational #triangle #righttriangle #isoscelestriangle #proofbycontradiction #root2 #algebra #infinitedescent #numbertheory #convergents To learn more about animating with manim, check out:https://manim.community
Sums of Fibonacci Squares Mathematical Visual Proofs 2024-05-30 | This is a short, animated visual proof demonstrating the sum of the squares of the first n consecutive Fibonacci numbers. To buy me a coffee, head over to buymeacoffee.com/VisualProofs Thanks!To see a slower, wider high definition version of this visual identity (without words), check out this video: youtu.be/2EsoiBdnIAM?si=3bW8M_zxxx-ueAhCAlso, check out my playlist on other finite sums:youtube.com/playlist?list=PLZh9gzIvXQUubr38YfIlul9j7_54hXZy_This animation is based on a proof by Alfred Brouusseau from "A Primer for the Fibonacci Numbers by M. Bicknell and V.E. Hoggatt Jr (eds.). You can also find this proof in Roger Nelsen's first Proof Without Words compendium (page 83).Get Nelsen's Amazing Book (Affiliate link so I may receive a commission): amzn.to/3QF6Zvz#fibonacci #math #manim #mathvideo #calculus #mtbos #mathshorts #animation #theorem #pww #proofwithoutwords #visualproof #proof #iteachmath #series #finitesums #finiteseries #geometric #squaresTo learn more about animating with manim, check out:https://manim.community
