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 ( but this animation does an even better job of showing how everything fits together.

The animation comes from Hydrodium’s Graphical MathLand at
I found it via the blog Visualizing Math ( 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).


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