60
$\begingroup$

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. $$\begin{array}{c|c|c} \begin{array}{cc} {\color{red} \bullet} & {\color{red} \bullet}\\ {\color{red} \bullet} & {\color{red} \bullet} \end{array} & \begin{array}{cc} {\color{red} \bullet} & {\color{red} \bullet}\\ {\color{red} \bullet} & {\color{red} \bullet} \end{array} & \begin{array}{cc} {\color{blue} \bullet} & {\color{blue} \bullet}\\ {\color{blue} \bullet} & {\color{blue} \bullet} \end{array} \\\hline \begin{array}{cc} {\color{green} \bullet} & {\color{green} \bullet}\\ {\color{green} \bullet} & {\color{green} \bullet} \end{array} & \begin{array}{cc} {\color{red} \bullet} & {\color{blue} \bullet}\\ {\color{green} \bullet} & {\color{magenta} \bullet} \end{array} & \begin{array}{cc} {\color{blue} \bullet} & {\color{blue} \bullet}\\ {\color{blue} \bullet} & {\color{blue} \bullet} \end{array} \\\hline \begin{array}{cc} {\color{green} \bullet} & {\color{green} \bullet}\\ {\color{green} \bullet} & {\color{green} \bullet} \end{array} & \begin{array}{cc} {\color{magenta} \bullet} & {\color{magenta} \bullet}\\ {\color{magenta} \bullet} & {\color{magenta} \bullet} \end{array} & \begin{array}{cc} {\color{magenta} \bullet} & {\color{magenta} \bullet}\\ {\color{magenta} \bullet} & {\color{magenta} \bullet} \end{array} \\ \end{array}$$ This is a $2\times2\times3\times3$ rectangular hypersolid made up of four copies of $1\cdot1^3+1\cdot2^3$.


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.

$\endgroup$
11
  • 31
    $\begingroup$ +1. But I want no part in this madness... $\endgroup$ Commented Nov 21, 2012 at 10:46
  • $\begingroup$ 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? $\endgroup$ Commented Nov 21, 2012 at 17:55
  • 1
    $\begingroup$ 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. $\endgroup$
    – Amzoti
    Commented Nov 21, 2012 at 18:28
  • 1
    $\begingroup$ 30 comes from the fact that the Bernoulli number $B_4=-1/30$. See "Faulhaber's formula" in Wikipedia. $\endgroup$
    – pharmine
    Commented Apr 23, 2013 at 0:26
  • 1
    $\begingroup$ @pharmine, Yes and so does $6$ in the sum of squares formula. But perhaps coincidentally there is a way to configure 6 shapes together as seen in the link. I'm asking if coincidentally this can be done with thirty 5-dimensional shapes. (I doubt that it can, but I'd love to see it if I'm wrong.) $\endgroup$
    – 2'5 9'2
    Commented Apr 23, 2013 at 4:16

2 Answers 2

26
$\begingroup$

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...

$\endgroup$
1
  • 4
    $\begingroup$ +1 This is cool, but I'm holding out for five-dimensional blocks :) $\endgroup$
    – 2'5 9'2
    Commented Oct 21, 2013 at 20:26
4
$\begingroup$

Your construction sounds like that of the Ehrhart polynomials. Snooping around Wikipedia a bit gives us confirmation. From the article square pyramidal number:

In modern mathematics, figurate numbers are formalized by the Ehrhart polynomials.

The Ehrhart polynomial $L(P,t)$ of a polyhedron $P$ is a polynomial that counts the number of integer points in a copy of $P$ that is expanded by multiplying all its coordinates by the number $t$. The Ehrhart polynomial of a pyramid whose base is the unit square $[0,1]^2 \times 0$, and whose apex is an integer point at height one above the base plane $(0,0,1)$ has Ehrhart polynomial $ \frac{1}{6}(t+1)(t+2)(2t+3)$.

In your problem the choice of convex polytope is pretty clear. It is the convex hull of the hypercube $[0,1]^4 \times 0$ and the point $(0,0,0,0,1)$.


Reading the article Ehrhart Polynomial we get some definitions:

Let $P$ be a polytope with vertices in a lattice $L$ (e.g. $L = \mathbb{Z}^d$ then) $L(P,t) = \# (tP \cap L)$ is a polynomial in $t$, called the Erhart polynomial of degree $d = \dim L$.

The Ehrhart polynomial of the interior (those strictly inside $P$) is $L(\text{int}(P), t)= (-1)^d L(P,-t)$.

$\endgroup$
8
  • $\begingroup$ But a convex polytope is not an analogy to blocks. With blocks in the smaller dimensional cases, we can see directly the connection to $1^k+2^k+\cdots+n^k$. The polytope answer to this question for summing $1+2+\cdots+n$ would be two triangles glued together instead of the familiar interlocking stairsteps. $\endgroup$
    – 2'5 9'2
    Commented Mar 12, 2015 at 20:33
  • $\begingroup$ Well, yes, $1^k+2^k+...+n^k$ is the value of an Ehrhart polynomial for the 'hybercubic pyramid'. But does this help to compute the sum? $\endgroup$
    – Grigory M
    Commented Mar 12, 2015 at 21:06
  • 1
    $\begingroup$ @alex.jordan Did you notice the sum of squares divides sum of 4th powers? $$ \sum k^2 =\frac{1}{6}n(n+1)(2n+1) \Bigg| \sum k^4 = \frac{1}{30}n(n+1)(2n+1)(3n^2+3n-1)$$ The blog you mentioned has two solutions. One uses a triangular lattice. $$ .$$ The volume of a pyramid is $\frac{1}{3}Ah$ where $A$ is the base and $h$ is the height. By Cavalieri principle we don't worry if the Pyramid is slanted or not. $\endgroup$
    – cactus314
    Commented Mar 12, 2015 at 21:13
  • 1
    $\begingroup$ @alex.jordan Well no; $\sum k^2$ doesn't divide $\sum k^{2a}$ for all $n$, but it does indeed divide $5\sum k^4, 7\sum k^6, 5\sum k^8, 11\sum k^{10}, 455\sum k^{12}, \sum k^{14}, 85\sum k^{16}$. $\endgroup$
    – Rosie F
    Commented May 15, 2017 at 19:08
  • 1
    $\begingroup$ @RosieF I was speaking with polynomial divisibility, referring to the polynomial factors $n$, $(n+1)$, and $(2n+1)\sim(n+1/2)$. So divisibility in $\mathbb{Q}[n]$, not $\mathbb{Z}[n]$. But your point is well taken. $\endgroup$
    – 2'5 9'2
    Commented May 15, 2017 at 19:51

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .