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Adding the first n cubes, in an animation

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Adding the first n cubes, in an animation

If you add up the first n numbers and square the result, this produces the same answer as adding the first n cubes. This animation by Hydrodium on tumblr gives a stunningly good illustration of this identity.

The graphic shows the case n=5. In this case, the sum of the first 5 natural numbers is 1+2+3+4+5=15, which squares to 15×15=225. On the other hand, the sum of the first n cubes is (1x1x1)+(2x2x2)+(3x3x3)+(4x4x4)+(5x5x5)=1+8+27+64+125, which also adds up to 225. There is nothing special about the number 5 here: an analogous identity holds for any other positive integer, and it can be illustrated by a similar animation.

At this point, the moderators of the arXiv preprint server would rightly accuse me of “substantial text overlap” with an earlier post. Yes, I’ve posted about this before (https://plus.google.com/101584889282878921052/posts/WqimoVTZWL3) but this animation does an even better job of showing how everything fits together.

The animation comes from Hydrodium’s Graphical MathLand at http://hyrodium.tumblr.com/post/94237657514/inspired-by-this-twocubes-post-and-asked-to-make
I found it via the blog Visualizing Math (http://visualizingmath.tumblr.com/) which in turn I found via +W Younes. There are a lot of very interesting posts on Visualizing Math. However, I might be a bit biased in saying that, because I found two of my own recent posts reshared there (with attribution).

#mathematics

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A sunflower at infinity

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A sunflower at infinity

This picture by +Roice Nelson shows the ‘view at infinity’ of a honeycomb in hyperbolic space.

A honeycomb is a way of chopping space into polyhedra.  For example, we can chop ordinary 3d space into cubes.  This is called the {4,3,4} honeycomb.  Why?

• a square has 4 sides so its symbol is {4}

• a cube has 3 squares meeting at each corner so its symbol is {4,3}

• the cubical honeycomb has 4 cubes meeting at each edge so its symbol is {4,3,4}

The picture here is a view of the {3,3,7} honeycomb.  This is defined in the same sort of way, but it doesn’t fit into ordinary Euclidean space.  It fits into a curved space called hyperbolic space!   The honeycomb extends forever, and it forms this pattern where it meets the ‘plane at infinity’ of hyperbolic space.

For links to related pictures, visit my +American Mathematical Society blog Visual Insight:

http://blogs.ams.org/visualinsight/2014/09/01/intersection-of-337-honeycomb-and-the-plane-at-infinity/

#geometry  

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