Wednesday, March 10, 2010

polygonal wanderings

About three years ago I sent in a manuscript to Mathematics Teacher called "Triangulating Polygonal Numbers" - and it has finally made its way into the magazine's March issue. Phew!

Since writing the first draft of that article I've continued to wander along the polygonal number trail - usually recording something about them on this blog. To celebrate that old article finally getting published, I thought I would try to collect together a few of the neat things I've stumbled upon while wandering through this topic.

I can't exactly remember why I first started looking at polygonal numbers, but it may have been while trying to find examples of interesting diagrams to draw in Fathom. Using a bit of recursion, it turns out to be pretty easy to create a Fathom document that can draw nice diagrams where you can control the number of sides and the length of the sides of the polygonal numbers that are drawn.

(I haven't yet implemented the "polygonal number diagram maker" in a more open or free platform - but it would be a nice project.)

I found out later that another fun way to diagram these numbers is to put them on a quadratic number-spiral. The images below are for the triangular, pentagonal, 12-agonal, and 13-agonal numbers - the square numbers look very uninteresting when you plot them on this spiral. The image at the top of this post shows both the triangular and the hexagonal numbers plotted on the same spiral. (Update: see Mike Croucher's Mathematica and Python implementations for drawing polygonal number spirals over at Walking Randomly - the implementations are straight forward and the images look great.)

The MT article looks at generalizations of the familiar $s_n = t_n +t_{n-1}$ identity, which tells us that a square number is the sum of two triangular numbers. From a geometric point of view, its obvious that you can split a polygon into triangles, but I thought it was interesting that you could also split a polygonal number into triangular numbers. A nice outcome of this geometric point of view is that it provides some nice "proofs without words" (the diagram below illustrates a relationship between hexagonal numbers and triangular numbers, $h_n = t_{2n-1}$).

Another interesting way to generalize things is to look at "higher dimensional" polygonal numbers. If you look at three dimensional polygonal numbers (visualized as stacked pyramids of spheres with different polygon bases), the familiar $s_n = t_n +t_{n-1}$ shows up in the standard $n \times n$ multiplication table. It turns out (surprisingly) that the upward sloping diagonals of the standard multiplication table sum to tetrahedral numbers, and of course, the main downward diagonal is the sum of all the squares in the table. So, the sum of the entries in the main upward sloping diagonal and the one above it is equal to the sum of the entries in the main downward sloping diagonal (shown in the 4x4 multiplication table below).

These identities among the higher-dimensional triangular numbers come in handy when you try to sum their reciprocals.

A different way of generalizing this same idea is to to look at splitting up higher powers into higher-dimensional triangular numbers. We know that $n^2 = t_n +t_{n-1}$, but what aboutn $n^3$, and $n^4$, and so on? Exploring this question leads you to another very interesting set of numbers, the Eulerian numbers, which show up as coefficients in the equations below (the 'exponent' on the $t$ is just an index indicating its dimension - $t^d_n$ is the n-th d-dimensional triangular number).

"Higher-dimensional triangular numbers" is a bit too fancy sounding - these things are much more recognizable to most people as the diagonals in Pascal's Triangle (2-dimensional triangular numbers are highlighted in grey in the image below) - the "dimension" corresponds to the diagonal column number (starting with index zero, for the "zero dimensional triangular numbers" which are just the constant sequence of 1's).

The other polygonal numbers (square, pentagonal, etc.) also occur in (less) well-known number triangles - the Lucas and Gibonacci Triangles.

If you look at the polygonal numbers for any length of time, you begin to appreciate that there are  many formulas for them. One surprising formula for the higher dimensional triangular numbers is their ordinary power series generating function. I found this formula surprising because it illustrates an interesting relationship between the rows of Pascal's Triangle and its diagonals. It shows that if you take the reciprocal of a particular expression whose coefficients are taken from a row in Pascal's Triangle, you get a formal power series whose coefficients are the entries in a corresponding diagonal column of Pascal's Triangle. Well, I was surprised, at least.

Those are the main highlights of my tour of the polygonal numbers, for now. Mathematicians (and idlers) have been exploring them since (at least) the time of Pythagoras, so I'm confident they'll still be around when I have time to look at them again.