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"Generalized" polygons?

Posted: Wed Aug 29, 2018 12:45 pm UTC
Hi all,

In my copious free time I often enjoy wandering around the OEIS learning about various sequences. There's one set that has caught my eye recently, but I'm having trouble understanding it: the concept of a "generalized polygon" - the "generalized pentagonal numbers", the "generalized heptagonal numbers", etc.

I haven't managed to find a primer on the subject for a math-enthusiastic layman. Anybody here know of one, or, barring that, can be bothered to provide it themselves?

Thanks!

Re: "Generalized" polygons?

Posted: Wed Aug 29, 2018 2:27 pm UTC
It's probably better to parse it as "generalized (pentagonal number)" rather than "(generalized pentagon)al number". They seem to be created by taking the formula for pentagonal numbers, and plugging in negative numbers for the length of a side.

Re: "Generalized" polygons?

Posted: Wed Aug 29, 2018 4:18 pm UTC
Yup, that's all this is - a generalization of the polygonal number sequences to allow for 0 and negative sides.

Re: "Generalized" polygons?

Posted: Wed Aug 29, 2018 6:41 pm UTC
Interesting! Is there any physical meaning to, or representation of, a polygon with negative sides?

Re: "Generalized" polygons?

Posted: Wed Aug 29, 2018 7:13 pm UTC
Hmm.

A square with negative sides would none-the-less have a positive area.

Meaningless or mind-blowing..? You decide...

Re: "Generalized" polygons?

Posted: Wed Aug 29, 2018 8:41 pm UTC
I'd think a polygon with negative-length sides would be identical to the same polygon with positive-length sides, rotated 180 degrees.

And perhaps a polygon with imaginary-length sides would be identical to the same polygon with real-length sides, rotated 90 degrees.

Re: "Generalized" polygons?

Posted: Wed Aug 29, 2018 9:08 pm UTC
Can't be a simple rotation; if it were, there wouldn't be a separate term in the sequence for it.

Re: "Generalized" polygons?

Posted: Wed Aug 29, 2018 9:26 pm UTC
Possibly the generalized sequence no longer corresponds to the geometry that the non-generalized sequence was based on? I was commenting on the geometry, not the number sequence derived from it.

This seems to shed some light on the meaning of the negative-indexed members of the sequence, though.

Re: "Generalized" polygons?

Posted: Thu Aug 30, 2018 12:27 pm UTC
There is a really neat history of Euler's work on this:
https://arxiv.org/abs/math/0510054

Euler was a beast.

Re: "Generalized" polygons?

Posted: Thu Aug 30, 2018 4:35 pm UTC
The sequence of polygonal numbers isn't (at least as I see it) as simple as a number n for f(x) where f(x)=f(-x). It relies upon the union of f(x-1) and the (unfilled) polygon g(x) that is the perimeter extant of f(x), and the special case of f(0) being 1, so that for squares length 1 sides is 4x1 minus L0's 1 (added to L1's 1), length 2 is 4x2 minus the three coincident L2 spots (added to L2's total of four), length 3 is 12-5(+9), etc.

(Noting that I'm using length of side x that is actually one less than the real number used, which is "fenceposts, not fence-bars", just to save fuss in this segment.)

The transition up the negatives (towards zero) is therefore different in principle from backtracking down the positives (towards zero) and just reversing the sign of both. Indeed (even after readjusting for my adjust-by-1 'error') it can be seen that a positive output number comes out of the negative square. The simpler, in this case, y=x² analogue to the more complicated sum of 1,3,5,7,9,… makes it obvious (much as adding -n,-n+2,-n+4 works on the flip-side, and …-1,+1,… nicely bridges across the central pillar of zero), but for the less trivial y=NPolygonOf(N,x) for N≠4 it'd be a funny Union¹ of old full-polygon and new edge-polygon to make the set deplete.

What I'd find interesting is the efficacy of negative sides (or zero, if not 1 or 2 also), then fractional and eventually complex+ numbers in either/both the N and x positions (for N=4, 'otherworldly' values of x can be compared against the existing known treatment of 'just squaring them', as a first test for reality). Assuming we aren't going to disallow them (as an incalculable singularity/undefined value) or lazily neuter them into just counting numbers (as per the negatives in the lazy way of handling them) into accepted order.

But minds more learned than mine² have no doubt applied themselves to these issues, even if I'd spent more than just the half an hour on my own analysis/reinventing-the-wheel!

¹ Probably just means that one should not think of it in terms of sets, but I'm not entirely convinced we should just use |x| in the equation and disavow the handling of negative numbers in the series by glossing over them.
² I think I would be no false modesty to count Euler amongst that number. Re: "Generalized" polygons?

Posted: Thu Aug 30, 2018 7:30 pm UTC
They describe exactly how the generalization works. You just take the general formula for n-gonal numbers and plug in negative values. For instance, the very first link is called "Generalized pentagonal numbers: m*(3*m-1)/2, m=0, +-1, +-2, +-3, .... ."