# [seqfan] Re: A187781 - Number of noncongruent polygonal regions in a regular n-gon with all diagonals drawn

Neil Sloane njasloane at gmail.com
Tue Jun 6 16:40:39 CEST 2023

```PS  Hi Scott, when we are looking at the regular n-gon with all diagonals
drawn, lets call it K_n (because abstractly it is the complete graph on n
nodes), for your sequence D you mention the group D_24, of order 24, the
group of the 12-gon.  I assume you really want the group D_2n, of order 2n,
the group of the n-gon, right?

Best regards
Neil

Neil J. A. Sloane, Chairman, OEIS Foundation.
Also Visiting Scientist, Math. Dept., Rutgers University,
Email: njasloane at gmail.com

On Mon, Jun 5, 2023 at 10:22 PM Chris Scussel <scussel at kwom.com> wrote:

> Hello all,
>
> ? ? I believe I have found that the description for this sequence actually
> describes
> a different, but related, sequence. I am hoping to get a second opinion
> from SeqFans.
>
> ? ? Define an equivalence relation D on the polygonal regions, where two
> regions are
> equivalent iff they map into each other under the dihedral symmetries of
> the 12-gon.
> Define another equivalence relation C where two regions are equivalent iff
> they are
> congruent. It appears to me that the terms of A187781 count the equivalence
> classes of D, while the description specifies the count of equivalence
> classes of C.
> Certainly all of the regions in any equivalence class of D are congruent.?
> Is it possible that regions from different equivalence classes of D are
> congruent?
>
> ? ? I believe so. This first occurs at a(12), where there are 22
> equivalence classes for D,
> but only 21 for C. I have proved (by brute force analytic geometry, with
> help from
> WolframAlpha, then again later with plane geometry) the congruence of
> regions from
> two of the equivalence classes of D. Here is a description of a
> do-it-yourself illustration:
>
> ? ? ? ? Number the vertices of the 12-gon consecutively from 0 to 11,
> ? ? ? ? then draw the following chords:?
> ? ? ? ? ? ? 0 ? 5
> ? ? ? ? ? ? 1 ? 4
> ? ? ? ? ? ? 1 ? 7
> ? ? ? ? ? ? 2 ? 6
> ? ? ? ? ? ? 3 ? 10
> ? ? ? ? The resulting figure has two small, congruent triangles. It?s easy
> to see
> ? ? ? ? that they are similar, but a bit harder to prove that they are
> congruent.
> ? ? ? ? These triangles remain intact as regions in the full dissection.
> They are
> ? ? ? ? at different distances from the center of the 12-gon, and thus are
> not
> ? ? ? ? in the same equivalence class of D.
>
> An analogous pair occurs for a(16). ?Thus, a(12) and a(16) are too large
> by one if they are supposed to be counting the equivalence classes of C.
> Going beyond the currently given terms for A187781, for a(18) there are
> three such pairs, as well as a triple. Additional congruences occur for
> all even n >14 up to at least n=60. They do not occur for any odd n<=60.
>
> ? ? I discovered this while testing a tessellation-coloring program, which
> has
> since morphed into program to count these classes. The program has computed
> the equivalence class counts for D, C, and S up to a(60). It also computed
> the
> numbers of segments, vertices, regions, and polygons from triangles through
> 9-gons, each of which agree with the corresponding OEIS sequence. This and
> a several other checks give me confidence that the results are correct.
>
> ? ? It seems to me that OEIS needs an additional sequence, so that there
> is one for
> D and one for C. ?As I understand it, in situations where the terms and
> description
> conflict the terms are taken to define the sequence. ?If that is the case
> then A187781
> would be for D, and the new sequence for C. And perhaps even a third for a
> relation S,
> where two regions are equivalent iff they are similar.
>
> ? ? I am hoping someone on this list can either point out where I have
> gone wrong
> or agree that I?m making sense. ?I find it a bit hard to believe that this
> is new.
>
>
> Thanks and kind regards,
>
> ? ? ? ? Chris Scussel
>
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>
```