Visual Sum of Cubes

08 Aug 2022

This article discusses a pattern I noticed in ‘visual’ derivations of the formulas for and , which led me to a similar derivation for .

using two lines

There’s a well-known trick for adding . Leaving out a bunch of symbols, it looks like this:

In other words, we arrange the sum as a line, then add a flipped copy of the line and multiply by to keep the total the same.

Since consecutive entries increase by in the first line but decrease by in the flipped line, the combined entries all have the same value.

So we have entries all of value , and can simply multiply to get the total.

using three triangles

Triangle arrangement of

I recently encountered a similar trick for the sum of squares , but this time using three triangles instead of two lines!

In other words, we arrange the sum as a triangle—one one (), followed by two twos (), and so on, to the last row of ’s ().

We then add two rotated copies of the triangle so we have all three orientations (i.e. the gets to be at each of the three corners) and multiply by to keep the total the same.

Every entry in the resulting triangle has the same value of , and there are (as derived above!) entries, so we simply multiply to get the total.

using four tetrahedra

Since this trick worked for using lines and using triangles, I wanted to see if any shape would work for the sum of cubes .

Pyramids?

The simplest way to arrange is as a pyramid, where the top layer is one one (), the second layer is two-by-two twos (), and so on, to the last layer of -by- ’s (). For example, for :

Pyramid arrangement of

But pyramids aren’t very symmetrical—the sides are triangles but the base is a square, so every symmetry leaves the at the top and doesn’t actually change the entries at all, meaning we can’t combine copies in a helpful way.

Octahedra?

If you double the pyramid, you get a much more symmetrical object—the octahedron. It represents (two pyramids, minus one th layer since it isn’t doubled). For example, for :

Octahedron arrangement of

This looks promising, since we can combine rotated copies of it as we did with lines and triangles. But this only helps if the combined entries all have the same value, and it turns out they don’t. For , for example:

Combining the three unique rotations of the octahedron

The combined entries aren’t all equal—for example, the top is but the center is .

Tetrahedra!

Finally I tried a tetrahedron:

This tetrahedron doesn’t sum as conveniently as the lines, triangles, and pyramids, so we have to rearrange things a bit to get our desired .

Since the th layer is a triangle with (as derived above!) entries all of value , its sum is just , so the sum of all the layers is:

We can rearrange this to get on the left-hand side, and then use the rotated copies trick again to get to the final formula:

In other words, we add three rotated copies of the tetrahedron so we have all four orientations (i.e. the gets to be at each of the four corners) and multiply by to keep the total the same.

Every entry in the resulting tetrahedron has the same value of , and there are entries, so we simply multiply to get the total.

And lastly we just substitute the formula for (derived above!) and simplify the polynomial.

Summary

So there we have it, all the ‘visual’ sums of powers before you need more than three dimensions, which isn’t very visual for humans.

In the context of simplices, going from 2 line segments to 3 triangles to 4 tetrahedra is a nice pattern—line segments are 1-simplices, triangles are 2-simplices, and tetrahedra are 3-simplices.

The pattern can continue, using 5 four-dimensional 4-simplices to derive the formula for , and so on in increasingly high dimensions. But that might defeat the point of it being a ‘visual’ derivation.

…but also

If we write the formula for in terms of the formula for , an interesting identity emerges, known as Nicomachus’s Theorem:

We can express this visually, as in this image by Wikipedia user cmglee:

Visual proof that [source]

This is a much easier way to visually derive the formula for , but don’t worry I still had fun figuring out the tetrahedron way—the more, the merrier!

from Hacker News https://ift.tt/1JyrOtq