On figurate numbers - MathOverflow

On figurate numbers

2011-06-07T04:53:42Z

Do you know a text where I can find a definition of polygonal number that is both geometrically and operationally sound?

I've basically seen two ways in which this topic is approached in the literature. For instance, in the celebrated historical opus by Dickson (volume II, chapter 1), the ancient treatment by a Hypsicles is mentioned:

"If there are as many numbers as we please beginning with one and increasing by the same common difference, then when the common difference is 1, the sum of all the terms is a triangular number; when 2, a square..."

Clearly enough, the above definition suits just right the needs of those persons interested in determining explicitly the $k$-element of the sequence of $n$-gonal numbers, but it gives no clue about the possibility of representing geometrically all those sequences.

That it is actually an issue becomes apparent when one notices that just about every presentation of this topic begins by providing diagrams that illustrate, to some extent, the process by which the first $n$-gonal numbers are built (they will typically focus their attention in the cases $n=3$, $n=4$, and $n=5$) and then, rush to introduce the general formulas without mentioning the way in which the corresponding patterns are supposed to be preserved by them (e.g., M. B. Nathanson. A short proof of Cauchy' polygonal number theorem. Proc. Amer. Math. Soc. 99 (1987) no. 1, 22-24.)... As it turns out, what it's at stake here is the possibility of a definition that reconciles the geometry inherent to these sequences and the ease of manipulation offered by an approach akin to that of Hypsicles.

Answer by Gerry Myerson for On figurate numbers

2011-06-07T05:17:26Z

Is there something geometrically unsound about the definition given at Wikipedia, http://en.wikipedia.org/wiki/Polygonal_number ? "By convention, 1 is the first polygonal number for any number of sides. The rule for enlarging the polygon to the next size is to extend two adjacent arms by one point and to then add the required extra sides between those points. In the following diagrams, each extra layer is shown as in red."

EDIT: I'll take advantage of the first row of diagrams in Aaron's answer to try to provide a Hypsiclean interpretation of the geometry (although I'm still not sure if this is what OP wants).

To go from each pentagonal to the next, you add three lines of red dots. More precisely, to go from $n-1$ to $n$, you add three lines of $n$ dots (minus 2, since you don't want to count the two corner dots twice). To go from $n$ to $n+1$, you'll add 1 more dot on each of the three sides than you added going from $n-1$ to $n$, which amounts to adding 3 more dots than you added previously, which, a la Hypsicles, is beginning with one and increasing by the common difference 3.

So Aaron's diagrams (not to mention Wikipedia's) seem to me to illustrate the geometry inherent to the sequences and at the same time to follow the treatment by Hypsicles.

Answer by Aaron Meyerowitz for On figurate numbers

2011-06-07T09:02:18Z

The triangular and square numbers arise naturally as figures one might make in the plane and occurred early in the history of mathematics. They generalize easily to higher dimension and have myriad connections to Pascals triangle, convergents to $\sqrt{2}$ and many other things. There are results with attractive visual proofs. This being so fruitful, it is natural to generalize to pentagonal hexagonal and other planar figurate numbers. The Greeks did so. In part this involves summing arithmetic progressions, a natural thing to investigate. However (to get to your point) the actual representation as a pattern in the plane is definable but not nearly as nice. (centered polygonal numbers are nicer, but that is another topic).

Let $P(s,n)$ be the $n$th $s$-gonal number. Then $P(s,1)=1$ and , as Gerry notes, one can build up by adding gnomons so $P(s,n+1)=P(s,n)+(1+n(s-2))=\sum_0^n(1+n(s-2)).$ The resulting figures are just not as attractive for $S \gt 4$ as for $s=3$ and $4$. There is not rotational or reflective symmetry of the resulting diagram.

One can complement this picture by looking instead at the transition from $P(s,n)$ to $P(s+1,n).$ Then $P(s+1,n)=P(s,N)+P(3,n-1)=P(3,n)+(s-3)P(3,n-1).$ this corresponds to our view of an $s+1$-gon split into $s-2$ triangles by the diagonals from a selected vertex. If we want them totally disjoint then one triangle is one level larger then the others. This is a generalization of the famous graphical proof for the fact that a square is the sum of two consecutive triangular numbers.

It is reasonable to define $P(2,n)=n$ and then we can also write $P(s+1,n)=P(2,n)+(s-2)P(3,n-1).$