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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.

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Whoever downvoted this, stand and unfold yourself... –  J. H. S. Jun 9 '11 at 3:43
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2 Answers

up vote 3 down vote accepted

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. alt text

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.

alt text alt text

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

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"In part this involves summing gemetric (sic) progressions, a natural thing to investigate..." Can you please elaborate? Anyways, thanks a lot for leaving your reply. –  J. H. S. Jun 9 '11 at 1:08
    
As I said, Triangular and Square numbers are interesting and the graphic diagrams are attractive. But is that enough for Hypsicle to say: "hey let's define pentagonal numbers!"? He may have had the philosophy that you need a problem or theorem to justify a definition. The diagrams are not that nice. However he notes that the triangular numbers are the successive partial sums of 1+2+3+4+5+... and the square numbers of 1+3+5+7+9+... so let the Petagonal numbers be the sums 1+4+7+10+13+... And then k-gonal 1+(k-1)+(2k-3)+(3k-4)+... –  Aaron Meyerowitz Jun 9 '11 at 2:08
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SO in my (pretend) history of Greek Mathematics the problem : "find a closed form expression for the nth sum of an arithmetic progression (with initial term 1)" becomes the motivation for defining these sums to be polygonal numbers. Certainly the problem of summing an arithmetic progression (as in the famous baby Gauss story: americanscientist.org/issues/pub/gausss-day-of-reckoning ) is a natural question. And that (thanks for asking) is what I meant. –  Aaron Meyerowitz Jun 9 '11 at 2:14
    
Your explanations are really good, dear Aaron. The only thing whereof I can't make heads or tails in them dwells in the part where you say "this involves summing gemetric (sic) progressions...". I don't see how those progressions are supposed to enter the scene here. Once again, heartfelt thanks for leaving your replies to this inquiry of mine. –  J. H. S. Jun 9 '11 at 3:40
    
oops, as you can see in my recent comment (and now corrected in the answer) I meant arithmetic progression but had said geometric. –  Aaron Meyerowitz Jun 9 '11 at 3:57
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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.

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Seems like my opening lines were misleading... I've rewritten them so that people who do not feel like reading the rest of the post don't jump to conclusions. Anyway, thanks for taking the time to reply. –  J. H. S. Jun 7 '11 at 5:59
    
@J.H.S., if you're implying that I didn't read the rest of your post, I can assure you that I did - I just couldn't see where the question was in the rest of the post, so I went back to the first line and answered that as best I could. I still don't know quite what you want. It seems to me that the pictures in Wikipedia let you see the approach of Hypsicles. I'm not sure what more you want. –  Gerry Myerson Jun 7 '11 at 7:27
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