List of uniform honeycombs

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

Re: List of uniform honeycombs

Postby Klitzing » Mon Apr 17, 2017 5:41 pm

username5243 wrote:
Klitzing wrote:
Code: Select all
      _3-  |  -4_
   o<      3      >o
      -3_  |  4/3


... i think you have x3x3o4/3x4*a3*c and o3x3x4/3x4*a3*c wrong ...

Oh yes, misplaced that single name. :sweatdrop: Thanx for pointing that out. (At least the used cells all where correct.)
So here - for reference - is the corrected list, now including the then missing names:

  • x3x3x4/3x4*a3*c - dicroch = dicubatirhombated cubihexagonal HC, cells are: ^ girco, <| hexat, v quitco, |> cotco
  • o3x3x4/3x4*a3*c - skivcadach = small skewverted cubatiapeiroducubatic HC, cells are: ^ sirco, <| that, v quitco, |> gocco
  • x3o3x4/3x4*a3*c - dichac = dicubatihexacubatic HC, cells are: ^ tic, <| that, v quith, |> cotco
  • x3x3o4/3x4*a3*c - gikkivcadach = great skewverted cubatiapeiroducubatic HC, cells are: ^ girco, <| that, v querco, |> socco
  • o3o3x4/3x4*a3*c - stut cadoca = small tritrigonary cubatidicubatiapeiratic HC, cells are: ^ cube, <| trat, v quith, |> gocco
  • x3o3o4/3x4*a3*c - getit cadoca = great tritrigonary cubatidicubatiapeiratic HC, cells are: ^ tic, <| trat, v cube, |> socco
--- rk
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Re: List of uniform honeycombs

Postby polychoronlover » Wed Apr 19, 2017 3:26 am

username5243 wrote:
polychoronlover wrote:x4/3x4x3/2x3*a *a~c *b4d

wait, this one contains x3/2x and is degenerate. Not sure what you meant to write here...

When I saw this, I first thought of digging up my old papers from 2015 that first described these, but then I realized that I would probably be doing a more exhaustive search soon, so there wasn't really any need. But before I completed that search I suddenly remembered that one of the first uniform honeycombs I discovered after making my list had the symbol x4x~x4x3*a4/3*c *b4/3*d. It may well be that this is the one I originally wrote down, but I didn't notice that it was in a different family from the second-to-last one when I put it online (both have tetrahedral diagrams).

So I guess that my plan should have x4/3x4x3/2x3*a *a~c *b4d replaced with x4x~x4x3*a4/3*c *b4/3*d. I also missed x4o3/2x~x3*b, a member of the skivcadach regiment, on my 2015 list.

Klitzing wrote:First of all: the linearisation symbol of the above symmetry group you gave as "o4o3o3o4/3*a *b3d" is wrong, as the letter "d" likewise isn't a real node, rather it is a virtaul one too, which re-reffers to the d-th so far already provided real node from the left. Accordingly it ought be prefixed by an asterisk as well. So you'd write rather "o4o3o3o4/3*a *b3*d".

Thanks, I didn't realize asterisks before virtual nodes were mandatory. I guess I haven't learned all the ins and outs of the linearization rules yet.

username5243 wrote:
polychoronlover wrote:x4x3x3x4/3*a *b3d

Possible name: dicroch (dicubatirhombated cubihexagonal honeycomb)

polychoronlover wrote:x4x3o3x4/3*a *b3d

Possible name: Dichac (dicubatihexacubatic honeycomb)

I don't like the idea of using "hexa-" when the hexagonal-tiling symmetry of the cell doesn't carry over into the entire pattern. In these, as in almost every other uniform honeycomb, the hexagons act like truncated triangles. So I would prefer something like "tri-", or "triati-" to reflect the o3o3o3*a symmetry, or maybe just "apeirati-" to be consistent.

username5243 wrote:I have some ideas for names:

polychoronlover wrote:o3x4x~x4/3*b

Possible name: Dacta (dicubatitruncated apeiratic honeycomb)

polychoronlover wrote:x3x4x~x4/3*b

Possible name: Dactapa (dicubiatitruncated prismato-apeiratic honeybomc)

I'm not entirely sure how the "x-truncated y" nomenclature emerged, or what, if anything, are the exact rules for using it, but I have a feeling these should be called apeiratitruncated dicubatic honeycomb and apeiratitruncated prismatodicubatic honeycomb. Maybe it's best to leave these ones alone for now.
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