[seqfan] Re: Pentagonal tiling coordination sequence?
Peter Munn
techsubs at pearceneptune.co.uk
Mon Nov 21 19:51:52 CET 2022
The orientation of the pentagons depends only on the parity of the level,
so we need only record this parity and the centre of each pentagon. Using
a spreadsheet, I get 1, 5, 20, 50, 110. This sequence is not in the
database. My value of 50 matches my visual count of pentagons in Richard
Mather's recently posted illustration.
Best regards,
Peter
On Thu, November 17, 2022 2:23 am, Allan Wechsler wrote:
> Start with a single regular pentagon in the Euclidean plane. The first
> entry in my proposed sequence is 1, because we start with 1 pentagon.
>
> On the other side of each of the edges of this seed pentagon, draw a new
> pentagon. This creates 5 new pentagons, so the second entry is 5.
>
> On the other side of each of the *unoccupied *edges of the new pentagons,
> draw a new pentagon. Some of them will intersect with each other, but for
> this sequence we consider that to be okay. This creates 20 distinct new
> pentagons, so the third entry is 20.
>
> On the other side of each of the unoccupied edges of these second-tier
> pentagons, draw a third-tier pentagon. You would think there would be 80
> of
> these, but in fact they include ten coincident pairs, five very easy to
> see, and five a bit more obscure, so there are only 70 new pentagons at
> level 3.
>
> I did this all in my head, and my visualization skills are not up to doing
> level 4. There are fifteen sequences in OEIS starting with 1, 5, 20, 70,
> and none of them are obviously this pseudo-tiling coordination sequence,
> although some of them might turn out to be. One of them is the
> coordination
> sequence of a closely related tiling, a hyperbolic tiling with SchlÃ¤fli
> symbol {5,5}. It's barely plausible that there is an isomorphism of some
> sort between these two tilings, in which case https://oeis.org/A054889 is
> indeed the sequence I'm looking for, and we should just add a comment that
> explains this.
>
[...]
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