Sum of Cubes IV (Nicomachus's Theorem)

This is a short, animated (wordless) visual proof demonstrating the sum of the first n positive cubes by rearranging cubes into a flat square. #mathshorts​ #mathvideo​ #math​ #calculus #mtbos​ #manim​ #animation​ #theorem​ #pww​ #proofwithoutwords​ #visualproof​ #proof​ #iteachmath #finitesums #discretemath #calculus #sum #induction
Here are three other visual proofs of sum of cubes formulas:
• Visual Sum of Cubes II
• Sum Of Cubes I (visual...
• Visual Sum of Cubes III
This animation is based on independently discovered, separate visual proofs by J. Barry Love from the March 1977 issue of Mathematics Magazine (www.jstor.org/..., page 74, and Alan L. Fry from the January 1985 issue of Mathematics Magazine (www.jstor.org/...) , page 11.
To learn more about animating with manim, check out:
manim.community
This is my version of a classic visual proof. Here are other renditions:
• Nicomachus's theorem |... by ‪@ThinkTwiceLtu‬
• Nicomachus's theorem |... by ‪@Mathocube‬
Edit: and thanks to a viewer I learned that
‪@Mathologer‬ posted a similar video a day ago : • Animating Nicomachus's...
___________________________________________
Music in this video:
Rain On The Window by Alex-Productions | onsound.eu/
Music promoted by www.free-stock...
Creative Commons Attribution 3.0 Unported License
creativecommon...

[@gegebenein.gaussprozess7539]

This is such a beautiful theorem, beautifully demonstrated and animated. Together with this well-chosen music track, I must confess that it moves me to tears every time I see it. Thank you so much for giving us this wonderful gift.

[@MathVisualProofs]

Glad you liked it. Thanks!

[@tacemus]

I could not put it better myself. I agree with every word! A big thank you from Australia.

[@MathPhysicsEngineering]

I wanted to make this video as a follow up for my sum of squares video!
Well done, great job!

[@MathVisualProofs]

Do it anyway!

[@Mr_Happy_Face]

nice

[@MathVisualProofs]

Thanks :)

[@halosandhugs8129]

thank u so much for this incredible visual so helpful!!!!

[@jakobthomsen1595]

Very good visualization of a wonderful formula!

[@MathVisualProofs]

Thanks!

[@verudoralast4890]

Lovely

[@pengin6035]

One comment: I wouldn't really call this a complete proof because it's not entirely obvious that this method of tiling can be done in general

[@MathVisualProofs]

Better take it up with the published versions :) kidding of course. Can you convince yourself that the even/odd steps will always work?

[@spikypichu]

A quick proof:
To prove that each cube completes a square with side length equal to the next triangular number, we simply must prove that (1 + 2 + ... + n)^2 - (1 + 2 + [n-1])^2 = n^3. Expanding the LHS with difference of squares, it's n * (2(1 + 2 + ... + n) - n) - n = n * (n(n + 1) - n) = n * n^2 = n^3. :)

[@QuantumHistorian]

@@MathVisualProofs I can convince myself of many things, that doesn't mean I have proofs of them.

[@kavithavelu8282]

Fantastic

[@MathVisualProofs]

Thanks!

[@Inspirator_AG112]

I have seen this on r/math before.

[@MathVisualProofs]

Yes. This is a famous one. I linked to two other animations that I like and I linked to the original two published proofs without words that the visualization is based on. I am just finally learning to do some more 3D animations so this one was high on my list to try 😀

[@AnglandAlamehnaSwedish]

Beautiful

[@MathVisualProofs]

Thanks!

[@AnglandAlamehnaSwedish]

Y don't kids need protractors today in schools???? I was required to have one