Sums of Oblong Numbers III (visual proof without words)

This is a short, animated (wordless) visual proof demonstrating the sum of the first n oblong numbers by rearranging 3 copies of the sum of oblong pyramids to build a rectangular prism.

Here are two other visual proofs of this formula:

And here is a related proof for the sum of squares:

To learn more about animating with manim, check out:

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Music in this video:
Creative Commons / Attribution 3.0 Unported License (CC BY 3.0)

[RandyKing]

remember the days when we couldn’t just see the math?!? i love this one!

[magnusmcgee]

Nice. For anyone thinking 'nice' doesn't capture the beauty, please remember this is math. Less is more.

[LittleCheese-oprm]

I was wondering whether there is a visual proof of the Wallis Formula for Pi/2 ? It looks like there should be one because : = 1/1 x 2/1 x 2/3 x 4/3 x 4/5 x 6/5 x 6/7 x ... can be derived from the series : s1:= 1 / 1 x 2 / 3 x 4 / 5 x 6 / 7 x 8 .... effectively multiplied by itself (so its 'roughly s1 times s1'). So for example up to n=8, Wallis Pi/2 := (1 / 1 x 2 / 3 x 4 / 5 x 6 / 7) multiplied by (1 / 1 x 2 / 3 x 4 / 5 x 6 / 7 x 8). Each Wallis term is found by taking the nth element from s1 and multiplying it by the n+1 th element from s1. So the Wallis 2/3 is found by multiplying the 2 from s1 by the next term from s1, i.e. the / 3, giving 2/3. This is similar to the videos 1 x 2, 2 x 3, .... etc. Alternatively Wallis is the same as Pi/2 := 2 x (1-(1/3)^2) x (1-(1/5)^2) x (1-(1/7)^2) x .... so looks to have something simple going on that should be visually 'nice' ?

[hrishikeshaggrawal]

seriously went overboard with the epic music