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May 14

Maximum packing density of *edge connected* Regular Pentagons

The maximum packing density of Regular Pentagons according to the article is 92.131. What is the maximum packing density of Regular Pentagons if *all* Pentagons in the packing must connect from one to another in a sequence of where an edge is shared between the pentagons in each step? Naraht (talk) 03:24, 14 May 2023 (UTC)

Here is a non-trivial lower bound. Warning: I have not checked everything carefully. All edge lengths will be $1$. Take a regular dodecagon. On its ten sides we can glue ten regular pentagons, on the outside. I think there is also room on the inside to glue on three more. The combined area of these pentagons equals $13A$, where

$A={\tfrac {1}{4}}{\sqrt {5(5+2{\sqrt {5}})}}.$

The whole combination fits in a circle of radius

$R={\sqrt {5+2{\sqrt {5}}}}.$

These combinations of 13 pentagons can be packed edge to edge in the plane in a square lattice pattern with a lattice spacing of slightly less than $2R.$ (Note: none of the edges will be parallel to a lattice axis.) If I'm not mistaken, this packing leaves space to squeeze in an extra pentagon for each group of 13. All combined, this gives a packing density slightly greater than

${\frac {14A}{\,4R^{2}}}\approx 0.6357.$

This can be improved by calculating the exact lattice spacing, but since there may be a simple way of establishing a far better lower bound, the slight improvement may not be worth the effort.  --Lambiam 10:03, 15 May 2023 (UTC)

I don't think Lambian has it right here. regular dodecagon says the interior angle is 150 deg. In pentagon it says the interior angle is 108 deg. Therefore each corner has 150+108+108 deg = 366 deg which can't be done flat without overlap. -- SGBailey (talk) 11:32, 15 May 2023 (UTC)

Sorry, you're right, but I meant to write decagon, as can be seen from fitting just 10 pentagons along the outside.  --Lambiam 12:50, 15 May 2023 (UTC)

The lattice will not be rectangular but somewhat skewed. Without detailed calculation, I'm not so sure a 14th pentagon will fit in the in-between space.  --Lambiam 14:38, 15 May 2023 (UTC)

Wanted to quickly note that the Penrose tiling is constructed from edge-connected regular pentagons, though I have absolutely no idea what their density is. Also, if I'm not mistaken, some portions of it look remarkably similar to Lambiam's construction. GalacticShoe (talk) 17:34, 17 May 2023 (UTC)

Dan Mackinnon's blog has a couple posts that may also be of interest. GalacticShoe (talk) 17:45, 17 May 2023 (UTC)

There are several Penrose tilings; this must refer to his original tiling, tagged as "P1" in our article. I have made an image of my (periodic) pentagonal packing. It is clear that there is no room for a 14th pentagon.  --Lambiam 18:04, 17 May 2023 (UTC)

Whoops, my mistake on the specific kind of Penrose tiling; thanks for catching that. After some searching online for the densest P1 tiling, I found a set of lecture slides with a particularly dense tiling on slide 6 that I thought might be noteworthy.

Update: after looking at it, I'm pretty sure this tiling is composed of the central pentagon and multiple two-pentagon one-rhombus prototiles; if this is the case, then I think it's probably asymptotically just as dense as Dürer's packing, which I think has a density of roughly ${\frac {3{\sqrt {5}}-5}{2}}\approx 85.4\%$.

Update 2: A 2020 MSc thesis by Timothy Sibbald has Dürer's packing as the lower bound for pentagonal packing density, which I'm pretty sure means that it is still the best-known edge-connected regular pentagon tiling. The thesis itself finds no other better tilings, though it's possible there are more that are just very hard to find. GalacticShoe (talk) 19:01, 17 May 2023 (UTC)

I expect it should be possible to establish bounds of the nature that any edge-connected packing has at least $p$ rhombus-sized empty areas for every $q$ pentagons. Such bounds will imply upper bounds on the maximum packing density.  --Lambiam 08:19, 18 May 2023 (UTC)

Looking at my regular packing I noticed that whole chains of ten-pentagon circles could be shifted together by letting more pentagons be shared between circles if one of the ten pentagons is removed. An image is here. I expect this has a higher density than my earlier packing.  --Lambiam 17:55, 19 May 2023 (UTC)

This packing has a "fundamental tile" of 8 pentagons, 2 boats, and 1 rhombus, yielding a density of $50-22{\sqrt {5}}\approx 80.7\%$. GalacticShoe (talk) 18:25, 19 May 2023 (UTC)