[seqfan] Pentagonal tiling coordination sequence?
Allan Wechsler
acwacw at gmail.com
Thu Nov 17 03:23:49 CET 2022
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 Schläfli
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.
Oh, apparently coordination sequences of this kind are called layer
sequences. I find it odd that we have two different terms for concepts so
closely related. For any layer sequence, there is clearly a dual graph that
has the same numbers as its coordination sequence. But hey, there's lots of
odd mathematical nomenclature.
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