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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Re: List of uniform honeycombs

Postby polychoronlover » Wed Mar 10, 2021 6:36 am

I also found some more members of the prismatic honeycombs' regiments. The ones I found involve blending the honeycombs with polygon-apeirogon duoprisms, but I suspect some prismatic honeycombs have many other regiment members yet to be discovered.

First, prismatic honeycombs made from tilings containing n-gons can be blended with n-gonal apeirogonal duoprisms. However, no more than one of them can blend around each lateral edge of a prism (corresponding to a vertex in the original tiling). To find blendings that are uniform honeycombs, we need to find ways to color one type of face of a uniform tiling so that exactly one face around each vertex is colored and the tiling is still vertex transitive under the coloring.

I know of several ways to do this:

Blend 1/4 of the squares in squat (already listed as quabassiph)
Image

Blend 1/4 of the squares in squat, but shifted on alternate layers (a new member of the chon regiment!):
Image

Blend 1/6 of the triangles in trat (e.g. the blue triangles in this picture of s3s3s3*a: https://commons.wikimedia.org/wiki/File ... 121314.png):
Image

Blend 1/3 of the hexagons in hexat

Blend half the triangles in that

Blend the squares in tosquat

Blend half the octagons in tosquat

Blend the squares in quitsquat

Blend half the octagons in quitsquat

Blend the triangles in toxat

Blend the triangles in quothat

Blend the triangles in rothat

Blend the hexagons in rothat

Blend the triangles in shothat

Blend the hexagons in shothat

Blend the triangles in ghothat

Blend the hexagons in ghothat

Blend the triangles in qrothat

Blend the hexagons in qrothat

Blend the squares in sossa

Note: This and the next one are wild honeycombs because the pseudo-apeirogons in the middle of squats in the sossa pseudoprismatic honeycomb are intercepted by the ridge between two vertical azips.

Also, if we try to blend the apeirogons in any of these tilings' pseudoprismatic honeycombs, we end up with subdivisions with vertical rows of apeirogons instead of squats, so they would go in Category B (subdivisions) , not here.

Blend the squares in gossa

Blend the hexagons in shaha

Blend the hexagons in ghaha

Blend the squares in grothat

Blend the hexagons in grothat

Blend the dodecagons in grothat

Blend the octagons in satsa

Blend the octagrams in satsa

(If we blend the apeirogons it will just lead to a subdivided version of the satsa prismatic honeycomb, which will go into category B as before)

Blend the dodecagons in hatha

Blend the dodecagrams in hatha

Blend the octagons in qrasquit

Blend the octagrams in qrasquit

Blend the squares in qrasquit

Blend the dodecagons in thotithit

Blend the dodecagrams in thotithit

Blend the hexagons in thotithit

Blend the squares in quitothit

Blend the hexagons in quitothit

Blend the dodecagrams in quitothit

Blend half the squares in snasquat

Blend half the squares in rasisquat

Blend half the squares in snassa

Blend the non-snub triangles in snathat

Blend the hexagons in snathat

There may well be other blendings with snubs that I haven't investigated yet.

Each of these 43 selections of polygons from tilings corresponds to two honeycombs: one where the n-gonal apeirogonal duoprisms blend with the tiling's prismatic honeycomb, and another where the duoprisms are blended with tiling prisms which take up half the layers. Thus we have a total of 86 new uniform honeycombs.
Attachments
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Climbing method and elemental naming scheme are good.
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