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Summing the first $n$ first powers of natural numbers: $$\sum_{k=1}^nk=\frac12n(n+1)$$ and there is a geometric proof involving two copies of a 2D representation of $(1+2+\cdots+n)$ that form a $n\times(n+1)$-rectangle.


Similarly, $$\sum_{k=1}^nk^2=\frac16n(n+1)(2n+1)$$ has a geometric proof (scroll down just a bit til you see the wooden blocks) involving six copies of a 3D representation of $(1^2+2^2+\cdots+n^2)$ that form a $n\times(n+1)\times(2n+1)$-rectangular solid.


And similarly, $$\sum_{k=1}^nk^3=\frac14n^2(n+1)^2$$ has a proof involving four copies of a 4D representation of $(1^3+2^3+\cdots+n^3)$ that form a $n\times n\times(n+1)\times(n+1)$-rectangular hypersolid in four-space. I can't find a resource to demonstrate this last one better, but I've sketched it out for $n=3$, sketching the four dimensions using a $4\times4$ grid for two dimensions, and within each cell a $3\times 3$ grid for the other two. (Try it - it's fun!) Here's a smaller $n=2$ version. Note that $1^3+2^3$ is represented by some 4D blocks $1\times1^3+1\times2^3$ that are configured in a way that makes use of all four dimensions. $$\begin{array}{c|c|c} \begin{array}{cc} {\color{red} \bullet} & {\color{red} \bullet}\\ {\color{red} \bullet} & {\color{red} \bullet} \end{array} & \begin{array}{cc} {\color{red} \bullet} & {\color{red} \bullet}\\ {\color{red} \bullet} & {\color{red} \bullet} \end{array} & \begin{array}{cc} {\color{blue} \bullet} & {\color{blue} \bullet}\\ {\color{blue} \bullet} & {\color{blue} \bullet} \end{array} \\\hline \begin{array}{cc} {\color{green} \bullet} & {\color{green} \bullet}\\ {\color{green} \bullet} & {\color{green} \bullet} \end{array} & \begin{array}{cc} {\color{red} \bullet} & {\color{blue} \bullet}\\ {\color{green} \bullet} & {\color{magenta} \bullet} \end{array} & \begin{array}{cc} {\color{blue} \bullet} & {\color{blue} \bullet}\\ {\color{blue} \bullet} & {\color{blue} \bullet} \end{array} \\\hline \begin{array}{cc} {\color{green} \bullet} & {\color{green} \bullet}\\ {\color{green} \bullet} & {\color{green} \bullet} \end{array} & \begin{array}{cc} {\color{magenta} \bullet} & {\color{magenta} \bullet}\\ {\color{magenta} \bullet} & {\color{magenta} \bullet} \end{array} & \begin{array}{cc} {\color{magenta} \bullet} & {\color{magenta} \bullet}\\ {\color{magenta} \bullet} & {\color{magenta} \bullet} \end{array} \\ \end{array}$$


So we move on to fourth powers: $$\sum_{k=1}^nk^4=\frac1{30}n(n+1)(2n+1)(3n^2+3n-1)$$ This time the polynomial does not completely factor. I would like to know if anyone can find a similar geometric proof. It would involve 30 copies of a 5D representation of $(1^4+2^4+\cdots+n^4)$ forming a 5D hypersolid, one of whose 2-faces somehow works out to have area $3n^2+3n-1$, with the orthogonal edge lengths being $n$, $n+1$, and $2n+1$. I know it seems hopeless...30 copies? $3n^2+3n-1$? But it would make for some nice art.

It would be a nice start if there were some sort of connection between $30$ and a symmetry group of $\mathbb{R}^5$. For instance, if there were a subgroup of $\operatorname{SO}_5$ whose order was some large divisor of $30$, then that might help place the $30$ blocks of volume $1^5$ into their configuration.

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+1. But I want no part in this madness... –  Benjamin Dickman Nov 21 '12 at 10:46
    
In the same way that there are nice 'visual' proofs of $\sum_kk^3$ that instead go through its representation as $\left(\sum_kk\right)^2$, maybe you can wedge 5 copies of $\sum_kk^4$ together and find $(3n^2+3n-1)$ copies of $\sum_kk^2$ in the result? –  Steven Stadnicki Nov 21 '12 at 17:55
    
@StevenStadnicki Interesting! –  alex.jordan Nov 21 '12 at 17:57
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Are you familiar with the book: Charming Proofs: A Journey into Elegant Mathematics (Dolciani Mathematical Expositions)? I believe there are others also for visualizing proofs and they are very interesting indeed. –  Amzoti Nov 21 '12 at 18:28
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30 comes from the fact that the Bernoulli number $B_4=-1/30$. See "Faulhaber's formula" in Wikipedia. –  pharmine Apr 23 '13 at 0:26
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On this book: http://www.amazon.com/Proofs-without-Words-Exercises-Classroom/dp/0883857006

Is given the following solution:

Proofs Without Words

I guess that with a bit of calculation it is possible to get to the formula for $$\sum_{i=1}^ni^4$$

I know that we miss all the stuff about the fifth dimension... but in this way the proof is much easier to visualize...

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+1 This is cool, but I'm holding out for five-dimensional blocks :) –  alex.jordan Oct 21 '13 at 20:26
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