List of uniform honeycombs

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

Re: List of uniform honeycombs

Postby polychoronlover » Wed Oct 21, 2020 12:23 am

My name is Eric#0289. I am also EricABQ on the Miraheze wiki.
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Re: List of uniform honeycombs

Postby polychoronlover » Sat Jan 02, 2021 6:59 am

It's been 6 years since I posted my first of uniform honeycombs. To celebrate, I've published a website about uniform honeycombs, in the style of Bowers' uniform polychora website: https://ericbinnendyk.github.io/website/honeycombs.html
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Re: List of uniform honeycombs

Postby polychoronlover » Sun Jan 10, 2021 6:03 am

Over the past few months, I've been sharing my new honeycomb discoveries with Username5243 over discord. I will be posting these discoveries here for reference.

I found several new members of the gyrate and elongate regiments, by viewing them as stacks of slab honeycombs and faceting the slabs.
  • Octet is a stack of trataps (triangular tiling antiprisms). Tratap has two other regiment members (not counting subdivisions of trats into azaps): ditathahap and ditathatap (see the spreadsheet for details). A stack of ditathahaps is ohtrah (cho verf) and a stack of ditathataps is tehtrah (oho verf). But if you alternate between two of these three slabs on alternate layers, it gives three new uniform honeycombs: tratap + ditathahap, tratap + ditathatap, and ditathahap + ditathatap. (The last one actually comes in two forms, one with stacks of azaps and one with trats, but I'm excluding the former from the main list because it has coplanar faces in its verf.)
  • Gytoh is also a stack of trataps, so it gives five new uniform honeycombs: ditathatap+ditathatap, ditathahap+ditathahap, tratap+ditathatap, tratap+ditathahap, and ditathatap+ditathahap.
  • Gyetoh is a stack of alternating trataps and trattips (triangular tiling prisms). The trattips have o3o3o3*a prism symmetry so they act like they have three regiment members: trattip, ditathap1 (has trips touching octs), and ditathap2 (has trips touching tets). All the combinations of trattip members alternating with tratap members lead to eight new uniform honeycombs in addition to gyetoh.
  • Etoh is also a stack of alternating trataps and trattips, but unlike gyetoh, etoh's symmetries allow us to map trips onto adjacent trips by turning the layer upside down. If we replace trattip by ditathap1 or ditathap2, the result is no longer uniform. Thus, the etoh regiment contains two new uniform honeycombs: ditathatap+trattip and ditathahap+trattip.
  • Regyetoh (retroelongated gytoh) has eight other regiment members analogous to gyetoh.
  • Retoh (retroelongated octet) has two other regiment members analogous to etoh.
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Re: List of uniform honeycombs

Postby polychoronlover » Sat Jan 23, 2021 6:35 am

On the Discord, _Geometer mentioned finding a uniform member of the squatap (square tiling antiprism) regiment. (Squatap itself is scaliform). It has tets and azaps. The verf is the blend of two wedges with square bases; a faceting of a square antipodium. It is also half of an oho. I propose the name "Spinosquare tiling hemiantiprism" for this, similar to the polyteron nophap (spinopentachoric hemiantiprism).

In fact, _Geometer described three members of the squatap regiment, whose verfs are half of co, cho, and oho. The other two are non-uniform scaliforms. I call them "Square tiling antiprism", "Square tiling hemiantiprism" (I used to give this name to o~x o~x || o~x x~o, but I now call that "apeirogonal antiprism pseudoprismatic honeycomb"), and "Spinosquare tiling hemiantiprism" respectively.
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