Uniform compound tilings

Discussion of tapertopes, uniform polytopes, and other shapes with flat hypercells.

Uniform compound tilings

Postby polychoronlover » Wed Nov 25, 2020 6:46 am

A few years ago, I found several uniform compounds of Euclidean tilings that were analogous to uniform polyhedra with cubic and dodecahedral symmetry. The faces follow trends: 4/n, 5/n, 6/n and 8/n, 10/n, 12/n. Analogs in hyperbolic space seem likely to exist as well.

Compound of two trihexagonal tilings (3.6.6/2.6):
2that.jpg
2that.jpg (18.72 KiB) Viewed 831 times

Spherical equivalents: Siid (3.6.5/2.6), Tisso (3.6.4/2.6)

Compound of three trihexagonal tilings (6/2.6.6/2.6)
3that.jpg
3that.jpg (19.78 KiB) Viewed 831 times

Spherical equivalents: Did (5.5/2.5.5/2), {4, 2}*3 (4.4/2.4.4/2)

Compound of four rhombitrihexagonal tilings (3.12/3.6.12/3)
4rothat.jpg
4rothat.jpg (15.12 KiB) Viewed 831 times

Spherical equivalents: Gidditdid (3.10/3.5.10/3), Gocco (3.8/3.4.8/3)

Compound of three rhombitrihexagonal tilings (6.4.6/2.4)
3rothat.jpg
3rothat.jpg (16.12 KiB) Viewed 831 times

Spherical equivalents: Raded (5.4.5/2.4), Rah (4.4.4/2.4)

Compound of four omnitruncated trihexagonal tilings (6.12.12/3)
4othat.jpg
4othat.jpg (24.01 KiB) Viewed 831 times

Spherical equivalents: Idtid (6.10.10/3), Cotco (6.8.8/3)

There are also some where only the "outside" faces follow the trends. In these cases, I have underlined the faces following trends.

Compound of ∞ apeirogonal prisms (12/3.12/3.∞)
inf-azip-1.jpg
inf-azip-1.jpg (45.93 KiB) Viewed 831 times

Spherical equivalents: Quit sissid (10/3.10/3.5), Quith (8/3.8/3.3)

Compound of ∞ apeirogonal prisms (4.12/3.∞)
inf-azip-2.jpg
inf-azip-2.jpg (32.24 KiB) Viewed 831 times

Spherical equivalents: Quitdid (4.10/3.10), Quitco (4.8/3.6)
Last edited by polychoronlover on Wed Nov 25, 2020 6:00 pm, edited 2 times in total.
Climbing method and elemental naming scheme are good.
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Re: Uniform compound tilings

Postby Klitzing » Wed Nov 25, 2020 11:44 am

The last pic more looks like a (12/3.12/3.∞), whereas it ought be a (4.12/3.∞) instead. That is, not ALL squares should become dodecagonal symmetric compounds.
Ah, probably you simply interchanged the last 2 pics wrt. your descriptions.
--- rk
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Re: Uniform compound tilings

Postby polychoronlover » Wed Nov 25, 2020 5:58 pm

Klitzing wrote:The last pic more looks like a (12/3.12/3.∞), whereas it ought be a (4.12/3.∞) instead. That is, not ALL squares should become dodecagonal symmetric compounds.
Ah, probably you simply interchanged the last 2 pics wrt. your descriptions.
--- rk


Whoops, thanks for spotting that. Fixed now.
Climbing method and elemental naming scheme are good.
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Re: Uniform compound tilings

Postby Klitzing » Thu Nov 26, 2020 9:08 pm

There's a further one as well: (P.6.P/3.6)
P=4 : cho
P=5 : ided
P=6 : you might want to draw - but yes, 6/3 is kind of degenerate : resulting in a Grünbaumian multi-cover of hexat
P=7 : the picture being found here: https://discord.com/channels/6772784813 ... 1297025054
Image

--- rk
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Re: Uniform compound tilings

Postby Challenger007 » Fri Dec 11, 2020 1:56 pm

This is not a very mathematical example, but such patterns are often laid out on the floor or on the walls and you can look at such patterns endlessly. Most likely, this is done on purpose. We often overlook things that subtly affect our emotional state.
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