Sum Cubes with Triangles (visual proof without words V)

This is a short, animated (wordless) visual proof demonstrating the sum of the first n positive cubes by rearranging triangular arrays into a larger triangular array.

Here are four other visual proofs of sum of cubes formulas:

To learn more about animating with manim, check out:

#mathshorts​ #mathvideo​ #math​ #calculus #mtbos​ #manim​ #animation​ #theorem​ #pww​ #proofwithoutwords​ #visualproof​ #proof​ #iteachmath #finitesums #discretemath #calculus #sum #induction
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Music in this video:
Creative Commons / Attribution 3.0 Unported License (CC BY 3.0)

[SolLederman]

This is mind blowing! Fantastic visual proof!

[Ninja]

This is always my favourite summation formula because you can troll ppl by pretending to “distribute” the power (the common algebra mistake), but change the power 2 to power 3.

Great video

[rayaneaoussar]

Thank you for all those amazing videos ❤

[GuyOnPhilippines]

Me waiting for the triangles to become cubes:

[JuliusUnique]

but in the end it's a triangle not a square, so why is it it enough to just square it?

[gordonbarnes]

I recently messed around with a convergent series in Excel. Basically, the fractions were 1/π, 1/π², 1/π³, 1/π⁴, etc.

When I added the fractions together, it converged on a value of 0.466942207.

I found that if you multiply 0.466942207 by π, you get 1.466942207. So the number multiplied by π is the same as 1 plus that number.

I then found out that this number is the same as 1/(π - 1).

I'd love to see a visualization or something of why this is the case.