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Mathematical Visual Proofs
United States
Приєднався 22 бер 2021
I animate and provide some explanation for classic and newer "proofs without words," which are typically diagrams without any words that indicate how a theorem could be proved.
I often pay homage to Proofs Without Words by including some animations without narration (only dramatic music :) ). I have also noticed that while there are many wonderful visual proofs available on UA-cam, most of the creators do not credit the original visual proofs. I will do my best to include links and citations for each visual proof so that you can track down the original static images, which have a beauty and a wonder all their own.
I create my animations using the amazing software Manimgl from 3Blue1Brown.
If you want to buy me a coffee, you can do that here:
www.buymeacoffee.com/VisualProofs
I often pay homage to Proofs Without Words by including some animations without narration (only dramatic music :) ). I have also noticed that while there are many wonderful visual proofs available on UA-cam, most of the creators do not credit the original visual proofs. I will do my best to include links and citations for each visual proof so that you can track down the original static images, which have a beauty and a wonder all their own.
I create my animations using the amazing software Manimgl from 3Blue1Brown.
If you want to buy me a coffee, you can do that here:
www.buymeacoffee.com/VisualProofs
Arithmetic Mean-Geometric Mean Visual Proof Compilation
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: www.buymeacoffee.com/VisualProofs. Thanks!
Here are the original series videos (along with attribution; for more detailed attribution, see the original videos):
0:00 Introduction
0:05 Folding Proof : ua-cam.com/video/IJGwvvQdfgg/v-deo.html and ua-cam.com/users/shortsYAGjrfpHq-Q (attribution: Yukio Kobayashi)
0:40 Circle with Triangles : ua-cam.com/video/HFGSsPbYc-s/v-deo.html (attribution: Roger B. Nelsen)
1:07 Sliding Triangles: ua-cam.com/video/SEPkdPWKdO0/v-deo.html (attribution: Claudi Alsina and Roger B. Nelsen)
1:52 Tangent Circles : ua-cam.com/video/jNx05gXSblE/v-deo.html (attribution : Roland H. Eddy)
2:38 Eight Kites : ua-cam.com/video/02yM5LqMWUs/v-deo.html (attribution: Ayoub B. Ayoub)
3:27 Tatami : ua-cam.com/video/62G9fak1vyk/v-deo.html and ua-cam.com/users/shortsif800mpoJaY(attribution: Doris Schattschneider)
4:05 Semicircle : ua-cam.com/video/Pt8IX_Q5U1A/v-deo.html and ua-cam.com/users/shortsjB0OZYA_EI0 (attribution: Charles D. Gallant)
4:49 Ending citation
If you like this inequality or others, you can check out my playlists of inequalities and AM-GM inequality:
Inequalities: ua-cam.com/play/PLZh9gzIvXQUuKcFHdtjFkwAVELDxu9nH4.html&si=_cVhtnlrrAIQg9f6
AM-GM : ua-cam.com/play/PLZh9gzIvXQUuoRR4kTd628cWdFQ1BDyS6.html&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:
manim.community
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: www.buymeacoffee.com/VisualProofs. Thanks!
Here are the original series videos (along with attribution; for more detailed attribution, see the original videos):
0:00 Introduction
0:05 Folding Proof : ua-cam.com/video/IJGwvvQdfgg/v-deo.html and ua-cam.com/users/shortsYAGjrfpHq-Q (attribution: Yukio Kobayashi)
0:40 Circle with Triangles : ua-cam.com/video/HFGSsPbYc-s/v-deo.html (attribution: Roger B. Nelsen)
1:07 Sliding Triangles: ua-cam.com/video/SEPkdPWKdO0/v-deo.html (attribution: Claudi Alsina and Roger B. Nelsen)
1:52 Tangent Circles : ua-cam.com/video/jNx05gXSblE/v-deo.html (attribution : Roland H. Eddy)
2:38 Eight Kites : ua-cam.com/video/02yM5LqMWUs/v-deo.html (attribution: Ayoub B. Ayoub)
3:27 Tatami : ua-cam.com/video/62G9fak1vyk/v-deo.html and ua-cam.com/users/shortsif800mpoJaY(attribution: Doris Schattschneider)
4:05 Semicircle : ua-cam.com/video/Pt8IX_Q5U1A/v-deo.html and ua-cam.com/users/shortsjB0OZYA_EI0 (attribution: Charles D. Gallant)
4:49 Ending citation
If you like this inequality or others, you can check out my playlists of inequalities and AM-GM inequality:
Inequalities: ua-cam.com/play/PLZh9gzIvXQUuKcFHdtjFkwAVELDxu9nH4.html&si=_cVhtnlrrAIQg9f6
AM-GM : ua-cam.com/play/PLZh9gzIvXQUuoRR4kTd628cWdFQ1BDyS6.html&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:
manim.community
Переглядів: 2 515
Відео
Sum Geometric Series with Ratio 1/2 in Rectangles (visual proofs)
Переглядів 3,2 тис.День тому
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 vid...
A Factorial Sum Produces the Factorial Number System (visual proof)
Переглядів 10 тис.14 днів тому
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: www.buymeacoff...
Terdragon from the Chaos Game (visual fractal construction from iterated function system)
Переглядів 2,2 тис.21 день тому
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: ua-cam.com/video/rLxx6b-oJGo/v-deo.html ua-cam.com/video/NBI_5GdvqUo/v-deo.html ua-cam.com...
Root 2 is Irrational from Isosceles Triangle (visual proof)
Переглядів 19 тис.Місяць тому
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 ...
Three Geometric Series in an Equilateral Triangle (visual proof without words)
Переглядів 12 тис.Місяць тому
Three Geometric Series in an Equilateral Triangle (visual proof without words)
Which is larger: A^ B or B^A? (two visual proofs)
Переглядів 35 тис.Місяць тому
Which is larger: A^ B or B^A? (two visual proofs)
Levy Dragon L-System Visualization (65535 steps!)
Переглядів 2,3 тис.Місяць тому
Levy Dragon L-System Visualization (65535 steps!)
Viviani's Theorem III (visual proof without words via area)
Переглядів 5 тис.Місяць тому
Viviani's Theorem III (visual proof without words via area)
Exact Trig Values for Multiples of 18 Degrees Geometrically (visual proof)
Переглядів 8 тис.2 місяці тому
Exact Trig Values for Multiples of 18 Degrees Geometrically (visual proof)
Infinitely Many Infinite Sums Geometrically (visual proof)
Переглядів 3,6 тис.2 місяці тому
Infinitely Many Infinite Sums Geometrically (visual proof)
Sum of Odd Cubes is Triangular (visual proof without words)
Переглядів 3,8 тис.2 місяці тому
Sum of Odd Cubes is Triangular (visual proof without words)
Area of a Regular Dodecagon | Proofs without Words Levels 1 to 4
Переглядів 3,6 тис.3 місяці тому
Area of a Regular Dodecagon | Proofs without Words Levels 1 to 4
Five Pi Formulas in "Pi/2 minutes" (Pi Day visual proofs)
Переглядів 6 тис.3 місяці тому
Five Pi Formulas in "Pi/2 minutes" (Pi Day visual proofs)
Consecutive Squares as Up/Down Sum of Odds
Переглядів 2,8 тис.3 місяці тому
Consecutive Squares as Up/Down Sum of Odds
Sum of Cubes as Sum of Odds and more (visual proof VI)
Переглядів 3 тис.3 місяці тому
Sum of Cubes as Sum of Odds and more (visual proof VI)
Infinite Geometric Series in a Square (visual proof without words) III
Переглядів 2,9 тис.4 місяці тому
Infinite Geometric Series in a Square (visual proof without words) III
Integrating by Weierstrass Substitution (visual proof)
Переглядів 7 тис.4 місяці тому
Integrating by Weierstrass Substitution (visual proof)
Unfolding the Year of the Dragon: 2024 Lunar New Year Special
Переглядів 2 тис.4 місяці тому
Unfolding the Year of the Dragon: 2024 Lunar New Year Special
Picture me Differentiating (visual calculus)
Переглядів 47 тис.4 місяці тому
Picture me Differentiating (visual calculus)
3D Sum of Triangular Numbers (visual proof without words III)
Переглядів 2,6 тис.5 місяців тому
3D Sum of Triangular Numbers (visual proof without words III)
Infinite Sum of Powers of -1/7 in a Square (visual proof without words)
Переглядів 3,2 тис.5 місяців тому
Infinite Sum of Powers of -1/7 in a Square (visual proof without words)
A proof to remember: Double Angle Formulas I (visual proof)
Переглядів 13 тис.5 місяців тому
A proof to remember: Double Angle Formulas I (visual proof)
A Factorial Series Geometric Dissection (visual proof)
Переглядів 4,7 тис.5 місяців тому
A Factorial Series Geometric Dissection (visual proof)
Sum of Squares Visual Proofs: From Level 1 to Level 5
Переглядів 5 тис.6 місяців тому
Sum of Squares Visual Proofs: From Level 1 to Level 5
Recreational Math Christmas Tree Problem (with visual proof)
Переглядів 3 тис.6 місяців тому
Recreational Math Christmas Tree Problem (with visual proof)
Geometry of the Twelve Days of X-mas (visual proof)
Переглядів 2,5 тис.6 місяців тому
Geometry of the Twelve Days of X-mas (visual proof)
Pythagorean Theorem XII (visual proof)
Переглядів 7 тис.6 місяців тому
Pythagorean Theorem XII (visual proof)
Much more quickly: since (x-1)^2 ≥ 0 for all x, x^2-2x+1 ≥ 0 -> x^2+1 ≥ 2x If you divide both sides by x the inequality is preserved as long as x>0 and you end up with x+1/x > 2
How ironic that the dude who died on monday has the same name as someone who despises mondays
🥐
I dont see a Dragon
7 is very weard
Me: tree centimeter times for centimeter, divide by 2 since its a treeangle, not a reltangle, area is 6²
Basically Summation to infinity n=0 of A×(-r)^n § is stand in for summation because no symbol in keypad :( Or just Aק-r^n
Wait so 4 points has 2×(√2)² soo maybe which us 4 chords So if we back track The chord would be a triangle with a base of 1+|cos120| or 3/2 and a height of √3/2 so Pythagorean theorem it comes out as 9/4+3/4 which comes down to 3sq Then √3 Then realization kicks in I'm not smart enough to conclude anything with this Google is my trusted navigator
_n_ th roots of unity
I did , its hell
Maybe I could, if it were physically possible to draw a perfect circle.
ua-cam.com/video/8kYkciD9VjU/v-deo.html ?
The Chanel name covers it, but for anyone who’s wondering the bottom number says “chord length product”
Huh???
I wonder if there is a proof for this using roots of unity! Each chord would have length of abs(1-[root]).
Yes!
Best content creator 😎❤
Grant's liitle theorem!
😎🆒🆒
I’d love to see an explanation/proof for this!
ua-cam.com/video/8GPy_UMV-08/v-deo.htmlsi=8I89MK0V8FCePpXx
See linked video. 😀
Boo
I hadn't seen this proof!
Why this is so satisfying and interesting 😊😢
😀
sec^2(theta)= 1^2 +tan^2(theta) by the pythagorean theorem. same for cosec and cot
How is the smaller circles 1 and also the inital circle 1?
Yeah this is definitely useful to know and has infinite practical applications in the real world.
Looks like a nautilus shell.
This is how the Greeks invented bikini bottoms and SpongeBob…
Average calamity mod boss attack:
the roblox stairs
Amazing video, cool music is a bonus too!
😎🎵
0 be like: Am I joke to you?
So...... 6+1=7?
What is the real-life use case for this?
This sounds like alien talk to me bro
I don’t understand some of it but it looks like it makes sense
Amazing, great explanation. Thank you. ❤
Bro please Give Your video Source code 😢😢
If you have a^b and b^a, the one with the base closer to e will be bigger. 3^2 > 2^3, 10^100 > 100^10
It fails because even though it approaches the size, it never reaches it.
This is beautifu! and delightful to watch! Pls what software/tool do you use for your animations?
I use manim for these (python library from 3blue1brown)
Za mene je ovo mnogo komplikovano, nista ne razumem. Svaka cast vama koji razumete!
Hey, diddle diddle, the median is middle... 😜
How about using the geometrical multiplying as a basis of proof? That is, if we have a,b > 0, we can construct the product a*b.
SNAIL
Many of them are cool. However, some of them are just proving that (a+b)^2=(a-b)^2+4ab then simplify.
Yes. That is essentially the proof of am-gm so you expect visual proofs to reflect that but geometrically.
Really nice proofs 👌🏻
Elegant ❣️
😎👍
Can you make a video on this concept on why x +1/x range. Is( -infinite,-2] and [2, infinit)
These essentially prove that. I have four shorts investigating that variation of these.
@@MathVisualProofs 😊okay , I am going to watch those
I love this concept, it is going on now in my maths coaching
👍😀
You be square. Too cool for school.