List of uniform honeycombs

[Klitzing]

Code: Select all

          _o_
      _3-  |  -4_
   o<      3      >o
      -3_  |  4/3
          -o-


...

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

Oh yes, misplaced that single name. 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

[polychoronlover]

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.

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.

x4x3x3x4/3*a *b3d

Possible name: dicroch (dicubatirhombated cubihexagonal honeycomb)

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.

I have some ideas for names:

o3x4x~x4/3*b

Possible name: Dacta (dicubatitruncated apeiratic honeycomb)

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.

[polychoronlover]

I just did an alternating family-regiment search, starting with the cubic honeycomb. For each honeycomb, I identified the head of its regiment. Here are the results:

o4o3o4x (chon)
o4o3x4o (rich)
o4o3x4x (tich)
o4x3o4x (srich)
o4x3x4o (batch)
o4x3x4x (grich)
x4o3x4x (prich)
x4x3x4x (otch)

(o3o4x4/3o3 *a)/2 (chon r)
o3o4x~x4/3 *b (chon r)
o4o3x~x3/2 *b (rich r)
(o3x4o4x3/2 *a)/2 (rich r)
x3o3x3o3/2 *a3*c (rich r)
(x3o3x4o4/3 *a3*c)/2 (rich r)
(x3o3/2x4/3o4 *a~*c)/2 (rich r)
(o3x4x4x3/2 *a)/2 (tich r)
o4x3o4/3x4 *b (srich r)
x3x3/2o4/3x4 *a3*c (srich r)
x4x3o4/3x4 *b (prich r)

o3o4x4/3x3 *a (getdoca)
o3x4x4/3o3 *a (ratoh r)
o3x4x4/3x3 *a (new?)
x3o4x4/3o3 *a (gacoco r)
x3o4x4/3x3 *a (new?)
x3x4x4/3o3 *a (x3x4x4o3 *a4/3*c r) (new?)
x3x4x4/3x3 *a (new?)
o3x4o~x4/3 *b (wavicac)
o3x4x~o4/3 *b (wavicac r)
o3x4x~x4/3 *b (dacta)
x3o4x~x4/3 *b (skivpacoca r)
x3x4o~x4/3 *b (sacpaca)
x3x4x~o4/3 *b (sacpaca r)
x3x4x~x4/3 *b (dactapa)
x4o3o~x3/2 *b (stut cadoca r)
x4o3x~o3/2 *b (stut cadoca r)
x4o3x~x3/2 *b (skivcadach r)
o3o3x3x3/2 *a3*c (This one overlaps even though it has a Bowers-style acronym)
o3x3o3o3/2 *a3*c (octet r)
x3x3x3o3/2 *a3*c (tatoh r)
o3o3x4x4/3 *a3*c (stut cadoca r)
o3x3x4x4/3 *a3*c (skivcadach r)
(o3x3o4o4/3 *a3*c)/2 (octet r)
x3o3o4x4/3 *a3*c (stut cadoca)
x3o3x4x4/3 *a3*c (dichac)
x3x3o4x4/3 *a3*c (skivcadach)
(x3x3x4o4/3 *a3*c)/2 (tatoh r)
x3x3x4x4/3 *a3*c (dicroch)
o3o3/2x4/3x4 *a~*c (stut cadoca r)
x3o3/2o4/3x4 *a~*c (stut cadoca r)
x3o3/2x4/3o4 *a~*c (dichac r)
o4o3x4/3x4 *b (gacoco)
o4x3x4/3x4 *b (cuteca)
x4o3x4/3x4 *b (skivpacoca)
x4x3x4/3x4 *b (cutpica)
x3o3/2x4/3x4 *a3*c (o3x4x4/3x3 *a r) (new?)

x3o4/3x *b3o (getdoca r)
x3o4x *b3o (ratoh)
o4o3x4x4/3 *b (gacoco r)
x3o4/3x4/3x3 *a4*c (x3o4x4/3x3 *a r) (new?)
x3x4x4o3 *a4/3*c (new?)
x4/3o3x4x4/3 *b (skivpacoca r)
x4x4/3o4x4/3 *a3*c *b~*d (skivpacoca r)
x4/3x4x3/2o3 *a~*c *b4*d (skivcadach r)
x3o4o *b3o (octet)
o3o3o~x3/2 *b (octet r)
x3x4o *b3o (tatoh)

o4o3x4/3x (quitch)
x3x4x *b3o (gaqrahch)
o4x3o4/3x (wavicoca r)
o4x3x4/3x (gaqrich)
x3x4x *b3o (gratoh)
o4x3o4x4/3 *b (wavicoca)
o4x3x4x4/3 *b (caquiteca)
x4x3o4x4/3 *b (gepdica)
x4x3x4x4/3 *b (caquitpica)
x4x4/3x4x4/3 *a3*c *b~*d (unnamed but known)
o3o3x~x3/2 *b (batatoh r)

(o3x4/3x4/3x3/2 *a)/2 (quitch r)
x3x3/2o4x4/3 *a3*c (wavicoca r)
x4x3o4/3x (gepdica r)
o3o3x3x3 *a (batatoh)

x4x3x4/3x (gaquapech)

As you can see, there are 4 new regiments in this list, including a lone operative and three pairs of prismatorhombates. I'm not sure whether I encountered these before -- it could be that I wrote them down on a piece of paper somewhere after I posted my September 2015 update. I also can't get rid of the feeling that some of these are actually fissary and their verfs can be cut from larger verfs in ways I haven't noticed, like how the verfs of o5/2x5x5/2o and x3o3o5/4x can be cut from sidtaxhi's verf.

[username5243]

Interesting. Can't seem to find any degenerate ones right away... Guess it just means more work on the list for you.

Would you like me to offer up possible names for the new regiment colonels here?

I'd also be interested in seeing how the technique you used would work if you started from one of the regular polychora. Maybe I'll investigate this sometime...

At first I was surprised you managed to get all symmetries in one go - but the three main convex honeycomb families (cubic, demicubic, cyclotetrahedral) are all related, so it makes sense. (That probably means if I did one for the uniform polychora, I'd have the tessics and icoics would all be in one list.)

[polychoronlover]

I divided the honeycombs up into categories, and enumerated the typical nonprismatic regiments to the best of my knowledge. I have the new listings for the octet, gacoco, and rich regiments on an unpublished file on my computer but I'm still not 100% sure I listed them correctly so I put question marks next to the counts.

Also, a side note: o4o3x4/3x4*b is called gacoco on the incmats site, but gacoca on the OBSA list.

Code: Select all

Category 1:
    chon regiment members w/ A(3) or C(3) verfs (1-2)
    octet regiment members w/ C(3) verfs (3-13?)
Category 2:
    tich regiment (14-15)
    tatoh regiment (16-18)
    quitch regiment (19-20)
Category 3:
    rich regiment members w/ C(2)*A(1) or A(1)^3 verfs (21-27? and 28-52?)
Category 4:
    srich regiment members w/ A(1)^2 verfs (53-59)
    wavicoca regiment members w/ A(1)^2 verfs (60-66)
    wavicac regiment (67-73)
Category 5:
    batch (74)
    grich, gratoh, cuteca, dacta, gaqrich, gaqrahch, caquiteca (75-81)
    otch, cutpica, gactipeca, dactapa, x3x4x4/3x3 *a, dicroch, x4x4/3x4x4/3 *a3*c *b~*d, gaquapech, quequapech, caquitpica, gacquitpica (82-92)
Category 6:
    prich regiment (93-95)
    quiprich regiment (96-98)
    gepdica regiment (99-101)
    gapdica regiment (102-104)
    sacpaca regiment (105-107)
    dichac regiment (108-110)
    o3x4x4/3x3 *a regiment (111-113)
    x3o4x4/3x3 *a regiment (114-116)
    x3o4/3x4/3x3 *a4*c regiment (117-119)
Category 7:
    chon regiment members w/ A(2) verfs (120-122)
    batatoh regiment (123-127)
    'gotactictosquath' regiment (128-134)
    ratoh regiment (135-141)
Category 8:
    rich regiment members w/ C(2) verfs (142-147?)
    gacoco regiment members w/ C(2) verfs (148-166?)
Category 9:
    srich regiment members w/ A(1) verfs (167-170)
    skivpacoca regiment (171-185)
    wavicoca regiment members w/ A(1) verfs (186-189)
    skivcadach regiment (190-204)
Category 10:
    rich regiment members w/ A(1)^2 verfs (205-216?)
    gacoca regiment members w/ A(1)^2 verfs (217-263?)
Category 11:
    octet regiment members w/ A(3) verfs (264-285?)
Category 12:
    stut cadoca regiment members (286-354)
Category 13:
    prismatics and their regiments
Category 14:
    prisms
    regiment members of prisms
    other slabs and their regiments
Category 15:
    gyrates and elongates


[wendy]

Have you looked at the laminate honeycombs?

There are beside the LPA2, LPB2, and LPC2, there are the 'stary' versions of LQA2, LQB2, and LQC2. These do not have wythoff symbols. The LQ forms have a pleated form, but still are uniform.

[username5243]

Hi, if you don't mind I will make a list of Wythoffian honeycombs and the names I coined for them (many of these are not official, and I have changed a few official names in what follows). I will list by the categories given above.

Code: Select all

Category 1:
x4o3o4o = chon (cubic honeycomb)
o3o3o4x4/3*a/2 = cuhsquah (cubihemisquare honeycomb)
x3o3o *b4o = octet (octetahedral tetrahedral honyecomb)
o3x3o3o3/2*a3*c = tehtrah (tetrahemitriangular honeycomb)
o3x3o4o4/3*a3*c/2 = ohtrah (octahemitriangular honeycomb)

Category 2:
x4x3o4o = tich (truncated chon)
x3/2o3x4x4*a/2 = ticuhsquah (truncated cuhsquah)
x3x3o *b4o = tatoh (truncated octet)
x3x3x3o3/2*a3*c = tutehtrah (truncated tehtrah)
x3x3x4o4/3*a3*c/2 = tohtrah (truncated ohtrah)
x4/3x3o4o = quitch (quasitruncated chon)
x3/2o3x4/3x4/3*a/2 = quitcuhsquah (quasitruncated cuhsquah)

Category 3:
o4x3o4o = rich (rectified chon)
o4o3x~x3/2*b = doha (disoctahemiapeiratic honeycomb)
x3/2o3x4o4*a/2 = ricuhsquah (rectified cuhsquah)
x3o3x3o3/2 *a3*c = retehtrah (rectified tehtrah)
x3o3x4o4/3*a3*c/2 = rohtrah (rectified ohtrah)
x3o3/2x4/3o4*a~*c/2 = cuhada (cubatidihemiapeiratic honeycomb)

Category 4;
x4o3x4o = srich (small rhombated chon)
o4x3o4/3x4*b = rawvicoca (retrosphenoverted cubaticubatiapeiratic honeycomb
o4x3o4x4/3*b = wavicoca (sphenoverted cubaticubatiapeiratic honeycomb)
x4/3o3x4o = querch (quasirhombated chon)
o3x4o~x4/3*b = wavicac (sphenoverted cubatiapeirocubatic honeycomb)
o3x4x~o4/3*b = rawvicac (retrosphenoverted cubatiapeirocubatic honeycomb)

Category 5:
o4x3x4o = batch (bitruncated chon)
x4x3x4o = grich (great rhombated chon)
x3x3o *b4x = gratoh (great rhombated octet)
o4x3x4/3x4*b = cuteca (cubatitruncated cubatiapeiratic honeycomb)
o3x4x~x4/3*b = dacta (dicubatitruncated apeiratic honeycomb)
x4/3x3x4o = gaqrich (great quasirhombated chon)
x3x3o *b4/3x = gaqrahch (great quasirhombated demicubic honeycomb; I prefer to call this one gaqratoh = great quasirhombated octet)
o4x3x4x4/3*b = caquiteca (cubatiquasitruncated cubatiapeiratic honeycomb)
x4x3x4x = otch (omnitruncated chon; I sometimes call this one gippich = great prismated chon)
x4x3x4/3x4*b = cutpica (cubatitruncated prismatocubatiapeiratic honeycomb)
x4/3x3x4/3x4*b = gactipeca (great cubatitruncated prismatocubatiapeiratic honeycomb)
x3x4x~x4/3*b = dactapa (dicubatitruncated prismatapeiratic honeycomb)
x3x3x4x4/3*a = cadca (cubatidicubatiapeiratic honeycomb)
x3x3x4/3x4*a3*c = dicroch (dicubatirhombated cubatihexagonal honeycomb)
x4x4/3x4x4/3*a3*c *b~*d = dacda (dicubatidiapeiratic honeycomb)
x4/3x3x4x = gaquapech (great quasiprismated chon)
x4/3x3x4/3x = quequapech (quasiquasiprismated chon)
x4x3x4x4/3*b = caquitpica (cubatiquasitruncated prismatocubatiapeiratic honeycomb)
x4/3x3x4x4/3*b = gacquitpica (great cubatiquasitruncated prismatocubatiapeiratic honeycomb)

Category 6;
x4x3o4x = prich (prismatorhombated chon)
x4x3o4/3x4*b = sepdica (small prismatodicubatiapeiratic honeycomb)
x4/3x3o4x = quiprich (quasiprismatorhombated chon)
x4/3x3o4/3x4*b = mipdica (medial prismatodicubatiapeiratic honeycomb)
x4x3o4x4/3*b = gepdica (great prismatodicubatiapeiratic honeycomb)
x4x3o4/3x = paqrich (prismatoquasirhombated chon)
x4/3x3o4x4/3*b = gapdica (grand prismatodicubatiapeiratic honeycomb)
x4/3x3o4/3x = quipqrich (quasiprismatoquasirhombated chon)
x3x4o~x4/3*b = sacpaca (small cubatiprismatocubatiapeiratic honeycomb)
x3x4x~o4/3*b = gacpaca (great cubatiprismatocubatiapeiratic honeycomb)
x3o3x4/3x4*a3*c = dicac (dicubatiapeirocubatic honeycomb)
x3o3/2x4/3x4*a~*c = cadic (cubatiapeirodicubatic honeycomb)
x3o3x4x4/3*a = tica (tricubatiapeiratic honeycomb)
x3o3/2x4/3x4 *a3*c = dacac (dicubatiapeirocubatic honeycomb)
x3x3o4x4/3*a = setcac (small tetracubatiapeirocubatic honeycomb)
x3o4/3x4/3x3*a4*c = sadcata (small dicubatitetrapeiratic honeycomb)
x3o4x4x3*a4/3*c = gadcata (great dicubatitetrapeiratic honeycomb)
o3x3x4x4/3*a = getcac (great tetracubatiapeirocubatic honeycomb)

Category 7:
o3o4x~x4/3*a = dacha (dicubatihemiapeiratic honeycomb)
x3x3o3o3*a = batatoh (bitruncated octet)
o3o3x~x3/2*b = datha (ditetrahemiapeiratic honeycomb)
x3o3o4x4/3*a = getdoca (great tetradicubatiapeiratic honeycomb)
x3o3o *b4/3x = qratoh (quasirhombated octet)
x3o3o *b4x = ratoh (runcinated octet, I sometimes call it sratoh = small rhombated octet)
o3o3x4x4/3*a = stedoca (small tetradicubatiapeiratic honeycomb)

Category 8:
o4o3x4/3x4*b = gacoca (great cubaticubatiapeiratic honeycomb)
o4o3x4x4/3*b = scoca (small cubaticubatiapeiratic honeycomb)

Category 9:
x3x3/2o4/3x4*a3*c = skivdacdic (small skewverted dicubatidicubatic honeycomb)
x4o3x4/3x4*b = skivpacoca (small skewverted prismatocubaticubatiapeiratic honeycomb)
x4x4/3o4x4/3*a3*c *b~*d = kavidacda (skewverted dicubatidiapeiratic honeycomb)
x4o3x4x4/3*b = gikkivpacoca (great skewverted prismatocubaticubatiapeiratic honeycomb)
x3o4x~x4/3*b = kevpadoca (skewverted prismatodicubatiapeiratic honeycomb)
x3x3/2o4x4/3*a3*c = gikvidacdic (great skewverted dicubatidicubatic honeycomb)
x3x3o4x4/3*a3*c = skivcadach (small skewverted cubatiapeirodicubatic honeycomb)
x4/3x4x3/2o3*a~*c *b4*d = kevcaca (skewverted cubatiapeirocubatiapeiratic honeycomb)
o3x3x4x4/3*a3*c = gikkivcadach (great skewverted cubatiapeirodicubatic honeycomb)
x4o3x~x3/2*b = kivpacoch (skewverted prismatocubaticubatiapeiratic honeycomb)

Category 10:
o3x3o4x4/3*a = odaca (octidicubatiapeiratic honeycomb)

Category 11:
o3o3o~x3/2*b = teha (tetrahemiapeiratic honeycomb)

Category 12:
x3o3o4x4/3*a3*c = stut cadoca (small tritrigonary cubatidicubatiapeiratic honeycomb)
o3o3/2x4/3x4*a~*c = stut dacda (small tritrigonary dicubatidiapeiratic honeycomb)
o3o3x4x4/3*a3*c = getit cadoca (great tritrigonary cubatidicubatiapeiratic honeycomb)
o3o3/2x4x4/3*a~*c = getit dacda (great tritrigonary dicubatidiapeiratic honeycomb)
x4o3o~x3/2*b = stut padoca (small tritrigonary prismatodicubatiapeiratic honeycomb)
x4o3x~o3/2*b = getit padoca (great tritrigonary prismatodicubatiapeiratic honeycomb)


I think that's all of the WYthoffians. I also remember having done short names for a few of the others in smaller regiments if you're interested.

Also, there seems to be a typo in your list: rich is listed as up to 52, but srich starts at 25, which makes no sense.

Also Klitzing, on your website it says x4o3o4/3x = chon, but this isn't true. It's verf is a qo3/2oq&#q, which I think is a degenerate triangle, so that is just a infinite covered cube instead.

[polychoronlover]

Also, there seems to be a typo in your list: rich is listed as up to 52, but srich starts at 25, which makes no sense.

Thanks for that -- I was pretty sleepy when i made that list and was comparing it to my existing uniform honeycomb list. I will change this as soon as I can, but in the meantime just add 28 to all the numbers from the 4th category up.

I'll also be discussing categories 13, 14, and 15 as soon as I can. I have found a few honeycombs in those categories still not listed. For example, the triangular tiling prism has two copycats where the trats come apart into apeirogonal antiprisms (azaps); one has the azaps going parallel on each side, and the other has one set turned 60 degrees relative to the other set. I call these "apeirogonal prism hemiantiprismatic honeycomb" and "apeirogonal prism gyrohemiantiprismatic honeycomb". The other notable discovery was that you can remove half the trips from the triangular prismatic honeycomb and use triangular and square tilings to fill the gaps -- similar to chon turning into cuhsquah, as you call it. A similar thing can also be done to the trihexagonal prismatic honeycomb.

Edit:This is now fixed.

[polychoronlover]

I'm pretty sure the triangular tiling antiprism does have some regiment members besides the 3 I listed, though. I just need to figure out what they are.

I think I found two of these. Do you know sidtidap, the small ditrigonal icosidodecahedral antiprism? It has a triangular cupolaic verf and two other regiment members, didtidap and gidtidap. The verf of the triangular tiling antiprism (let's call it tratap) can be faceted in the same way, leading to one honeycomb with ditathas, octs, and azaps and another with ditathas, tets, and azaps.

EDIT: I don't have any real preferences for the names of these honeycombs, but if one of them is to be the ditatha antiprism, I'll suggest it'd be the one with tets, not octs.

[polychoronlover]

Here is what I got for the members of the gacoco (gacoca?) regiment:

Code: Select all

Chon symmetry:
Great cubaticubatiapeiratic honeycomb. Symbol is o'(o'x"x). Cells are goccoes, octs, and squats. Verf is a flattish square podium.
Great cubicuboctahedral rhombicuboctahedral cubic honeycomb. Cells are goccoes, sircoes, and cubes. Verf is a polar faceted square podium.
Great cubicuboctahedral rhombicuboctahedral apeiroprismatic honeycomb. Cells are goccoes, sircoes, and azips (apeirogonal prisms). Verf is a polar faceted square podium.
Discubicuboctahedral honeycomb. Cells are soccoes and goccoes. Verf is an inverted square podium.
Great rhombihexahedral cubic honeycomb. Cells are grohs and cubes. Verf is a spiky faceted square podium.
Great rhombihexahedral apeiroprismatic honeycomb. Cells are grohs and azips. Verf is a spiky faceted square podium.
Small rhombicuboctahedral rhombihexahedral octahedral square tiling honeycomb. Cells are sircoes, grohs, octs, and squats. Verf is a square podium w/ the trapezoids replaced by butterfly shapes.
Dirhombihexahedral honeycomb. Cells are srohs and grohs. Verf is a spiky faceting of a square podium.
Great cubintercepted rhombicuboctahedral octahedral square tiling honeycomb. Cells are cubes, quercoes, octs, and squats. Verf is a square podium w/ trapezoids replaced by crossed trapezoids.
Great apeiroprism intercepted rhombicuboctahedral octahedral square tiling honeycomb. Cells are azips, quercoes, octs, and squats. Verf is a square podium w/ trapezoids replaced by crossed trapezoids.
Great rhombicuboctahedral rhombihexahedral octahedral square tiling honeycomb. Cells are quercoes, srohs, octs, and squats. Verf is a square podium w/ trapezoids replaced by crossed trapezoids.
Dirhombicuboctahedral honeycomb. Cells are sircoes and quercoes. Verf is a square podium w/ squares removed and trapezoids crossed.
Small cubicuboctahedral rhombicuboctahedral cubic honeycomb. Cells are soccoes, quercoes, and cubes. Verf is a polar faceted square podium w/ trapezoids crossed.
Small cubicuboctahedral rhombicuboctahedral apeiroprismatic honeycomb. Cells are soccoes, quercoes, and azips. Verf is a polar faceted square podium w/ trapezoids crossed.
Small cubintercepted rhombicuboctahedral octahedral square tiling honeycomb. Cells are cubes, sircoes, octs, and squats. Verf is a circumfaceted square podium.
Small apeiroprism intercepted rhombicuboctahedral octahedral square tiling honeycomb. Cells are azips, sircoes, octs, and squats. Verf is a circumfaceted square podium.
Small cubaticubatiapeiratic honeycomb. Symbol is o'(o"x'x). Cells are soccoes, octs, and squats. Verf is a crossed square podium.
Small rhombihexahedral cubic honeycomb. Cells are srohs and cubes. Verf is a spiky faceted square podium.
Small rhombihexahedral apeiroprismatic honeycomb. Cells are srohs and azips. Verf is a spiky faceted square podium.

Octet symmetry:
Small octahedral rhombicuboctahedral cubicuboctahedral rhombihexahedral cubisquare tiling honeycomb. Cells are octs, goccoes, grohs, sircoes, cubes, and squats.
Small octahedral rhombicuboctahedral cubicuboctahedral rhombihexahedral apeiroprismatosquare tiling honeycomb. Cells are octs goccoes, grohs, sircoes, azips, and squats.
Cubicuboctahedral rhombihexahedral cubic honeycomb. Cells are soccoes, goccoes, srohs, grohs, and cubes.
Cubicuboctahedral rhombihexahedral apeiroprismatic honeycomb. Cells are soccoes, goccoes, srohs, grohs, and azips.
Great octahedral rhombicuboctahedral cubicuboctahedral rhombihexahedral cubic squarisquare tiling square tiling honeycomb. Cells are octs, quercoes, goccoes, srohs, cubes, sosts, and squats.
Great octahedral rhombicuboctahedral cubicuboctahedral rhombihexahedral apeiroprismatic squarisquare tiling square tiling honeycomb. Cells are octs, quercoes, goccoes, srohs, azips, sosts, and squats.
Small octahedral rhombicuboctahedral cubicuboctahedral rhombihexahedral cubic squarisquare tiling square tiling honeycomb. Cells are octs, sircoes, goccoes, srohs, cubes, sosts, and squats.
Small octahedral rhombicuboctahedral cubicuboctahedral rhombihexahedral apeiroprismatic squarisquare tiling square tiling honeycomb. Cells are octs, sircoes, goccoes, srohs, azips, sosts, and squats.
Small rhombicuboctahedral cubicuboctahedral rhombihexahedral cubic squarisquare tiling honeycomb. Cells are sircoes, goccoes, srohs, cubes, and sosts.
Small rhombicuboctahedral cubicuboctahedral rhombihexahedral apeiroprismatic squarisquare tiling honeycomb. Cells are sircoes, goccoes, srohs, azips, and sosts.
Great octahedral rhombicuboctahedral cubicuboctahedral rhombihexahedral cubic squarisquare tiling square tiling honeycomb. Cells are octs, quercoes, soccoes, grohs, cubes, sosts, and squats.
Great octahedral rhombicuboctahedral cubicuboctahedral rhombihexahedral apeiroprismatic squarisquare tiling square tiling honeycomb. Cells are octs, quercoes, soccoes, grohs, azips, sosts, and squats.
Great rhombicuboctahedral cubicuboctahedral rhombihexahedral cubic squarisquare tiling honeycomb. Cells are quercoes, soccoes, grohs, cubes, and sosts.
Great rhombicuboctahedral cubicuboctahedral rhombihexahedral apeiroprismatic squarisquare tiling honeycomb. Cells are quercoes, soccoes, grohs, azips, and sosts.
Small octahedral rhombicuboctahedral cubicuboctahedral rhombihexahedral cubic squarisquare tiling square tiling honeycomb. Cells are octs, sircoes, soccoes, grohs, cubes, sosts, and squats.
Small octahedral rhombicuboctahedral cubicuboctahedral rhombihexahedral apeiorprismatic squarisquare tiling square tiling honeycomb. Cells are octs, sircoes, soccoes, grohs, azips, sosts, and squats.
Dirhombihexahedral cubic squarisquare tiling honeycomb. Cells are srohs, grohs, cubes, and sosts.
Dirhombihexahedral apeiroprismatic squarisquare tiling honeycomb. Cells are srohs, grohs, azips, and sosts.
Great octahedral rhombicuboctahedral cubicuboctahedral rhombihexahedral cubisquare tiling honeycomb. Cells are octs, soccoes, srohs, quercoes, cubes, and squats.
Great octahedral rhombicuboctahedral cubicuboctahedral rhombihexahedral apeiroprismatosquare tiling honeycomb. Cells are octs soccoes, srohs, quercoes, azips, and squats.
Dirhombicuboctahedral cubic honeycomb. Cells are sircoes, quercoes, and cubes.
Dirhombicuboctahedral apeiroprismatic honeycomb. Cells are sircoes, quercoes, and azips.
Great hemihexahedral triscubicuboctahedral squarisquariapeirogonal tiling alternate square tiling honeycomb. Cells are thahs, soccoes, goccoes in two different orientations, gossas, and shas.
Octahedral dicubicuboctahedral dirhombihexahedral square tiling honeycomb. Cells are octs, soccoes, goccoes, srohs, grohs, and squats.
Great hemihexahedral cubicuboctahedral dirhombihexahedral squarisquariapeirogonal tiling alternate square tiling honeycomb. Cells are thahs, goccoes, srohs, grohs, gossas, and shas.
Small rhombicuboctahedral cubicuboctahedral rhombihexahedral honeycomb. Cells are sircoes, goccoes, and grohs.
Great hemihexahedral dirhombicuboctahedral cubicuboctahedral squarisquariapeirogonal tiling hemialternate square tiling honeycomb. Cells are thahs, sircoes, quercoes, goccoes, sossas, and shas.
Small hemihexahedral triscubicuboctahedral squarisquariapeirogonal tiling alternate square tiling honeycomb. Cells are thahs, soccoes in two different orientations, goccoes, sossas, and shas.
Great hemihexahedral cubicuboctahedral squarisquariapeirogonal tiling alternate square tiling honeycomb. Cells are thahs, goccoes, sossas and shas.
Great octahedral rhombicuboctahedral cubicuboctahedral rhombicuboctahedral squarisquare tiling square tiling honeycomb. Cells are octs, quercoes, goccoes, srohs, sosts, and squats.
Octahedral dicubicuboctahedral squarisquare tiling square tiling honeycomb. Cells are octs, soccoes, goccoes, sosts, and squats.
Dirhombicuboctahedral dicubicuboctahedral squarisquare tiling honeycomb. Cells are sircoes, quercoes, soccoes, goccoes, and sosts.
Dicubicuboctahedral squarisquare tiling honeycomb. Cells are soccoes, goccoes, and sosts.
Small hemihexahedral rhombicuboctahedral dirhombihexahedral squarisquariapeirogonal tiling alternate square tiling honeycomb. Cells are thahs, sircoes, srohs, grohs in two different orientations, gossas, and shas.
Great hemihexahedral rhombicuboctahedral dirhombihexahedral squarisquariapeirogonal tiling alternate square tiling honeycomb. Cells are thahs, quercoes, srohs in two different orientations, grohs, sossas, and shas.
Small hemihexahedral rhombicuboctahedral rhombihexahedral squarisquariapeirogonal tiling square tiling honeycomb. Cells are thahs, sircoes, grohs, sossas, and shas.
Small hemihexahedral trisrhombicuboctahedral rhombihexahedral squarisquariapeirogonal tiling alternate square tiling honeycomb. Cells are thahs, sircoes in two different orientations, quercoes, grohs, sossas, and shas.
Small hemihexahedral cubicuboctahedral dirhombihexahedral squarisquariapeirogonal tiling alternate square tiling honeycomb. Cells are thahs, soccoes, srohs, grohs, sossas, and shas.
Dirhombicuboctahedral dirhombihexahedral squarisquare tiling honeycomb. Cells are sircoes, quercoes, srohs, grohs, and sosts.
Small octahedral rhombicuboctahedral cubicuboctahedral rhombicuboctahedral squarisquare tiling square tiling honeycomb. Cells are octs, sircoes, soccoes, grohs, sosts, and squats.
Dirhombihexahedral squarisquare tiling honeycomb. Cells are srohs, grohs, and sosts.
Octahedral dirhombicuboctahedral square tiling honeycomb. Cells are octs, sircoes, quercoes, and squats.
Great hemihexahedral trisrhombicuboctahedral rhombihexahedral squarisquariapeirogonal tiling alternate square tiling honeycomb. Cells are thahs, sircoes, quercoes in two different orientations, srohs, gossas, and shas.
Small hemihexahedral dirhombicuboctahedral cubicuboctahedral squarisquariapeirogonal tiling hemialternate square tiling honeycomb. Cells are thahs, sircoes, quercoes, soccoes, gossas, and shas.
Great rhombicuboctahedral cubicuboctahedral rhombihexahedral honeycomb. Cells are quercoes, soccoes, and srohs.
Great hemihexahedral rhombicuboctahedral rhombihexahedral squarisquariapeirogonal tiling square tiling honeycomb. Cells are thahs, quercoes, srohs, gossas, and shas.
Small hemihexahedral cubicuboctahedral squarisquariapeirogonal tiling alternate square tiling honeycomb. Cells are thahs, soccoes, gossas and shas.


EDIT: I just fixed some errors in the cell listings.

[polychoronlover]

And here is the updated list of octet regiment members.

The names and descriptions are from my latest (unpublished) list of uniform honeycombs; only the numbers (which were messy and nonconsecutive) were removed. The goal is to publish the content of regiments when it isn't immediately apparent, without having to publish a whole list of honeycombs (which might contain errors). I already checked this regiment over a year ago but I just double-checked and also checked for fissaries. I checked for fissaries by comparing my list to Jonathan Bowers' list of ditetrahedronary polychora, which means that I didn't check the honeycombs with squats in them, but it doesn't seem likely that these would be fissary. The 16th honeycomb on the list wasn't listed as having hexats in the file but I fixed this too.

Here are the 32 members of the octet regiment.

Code: Select all

Tetrahedral-octahedral honeycomb, also known as alternated cubic honeycomb or octet. Symbol is o6'o, can also be (ooox). Cells are tets and octs. Verf is a co. This one has hemicubic honeycomb symmetry.
Tetrahedral hemitriangular tiling honeycomb. Cells are tets and trats (triangular tilings). Verf is an oho. Symbol is (x(oo),o). This has hemicubic honeycomb symmetry, but is only wythoffian if it has quarter cubic honeycomb symmetry, (oooo), also known as cyclotetrahedral symmetry.
Octahedral hemitriangular tiling honeycomb. This is the blend of the first two members of the regiment. Cells are octs and trats. Verf is a cho. Numbers 3, 4, and 5 are all in the same company.
Tetrahedral cuboctahedral honeycomb. Cells are tets and coes. Verf is a co w/ the squares replaced by K4 complete graphs (omnifaceted). I was kind of suprised by this one.
Tetrahedral cubohemioctahedral hemihexagonal tiling honeycomb. Cells are tets, ohoes, and hexats in compounds of three. Verf is a co w/ the squares indented to the middle, like an oho, plus pockets around the edges.
Octahedral truncated-tetrahedral honeycomb. Cells are octs and tuts. Verf is a co w/ the triangles drilled in.
Tomotetrahedral cuboctahedral honeycomb. Cells are tuts and coes, exactly what it says on the tin. Verf is a co w/ the triangles buckled in lightly and the squares omnifaceted (and removed).
Cuboctahedral hemitriangular tiling honeycomb. Cells are coes and trats. Verf is a co w/ the triangles buckled in lightly exactly the way they were before, and squares omnifaceted except this time all the way to the center.
Cubohemioctahedral hemihexagonal tiling honeycomb. Cells are choes and hexats in compounds of three. Verf is a co faceting w/ all original edges removed, squares replaced by 'X' configurations of edges, and dents all the way to the center.
Tomotetrahedral hemitriangular tiling honeycomb. Cells are tuts and trats. Verf is a co w/ squares indented into four-pseudofaced configurations, and triangles indented into six-pseudofaced orientations.
Octahedral octahemioctahedral hemihexagonal tiling honeycomb. Cells are octs, ohoes, and hexats in compounds of three. Verf is a co w/ lightly inverted pseudo-triangles.
Ditetrahedronary distetrahedral octahedral hemioctahedral honeycomb. Cells are tets, octs, tuts, and choes. Verf is a co w/ four of the triangles drilled in. This has cyclotetrahderal symmetry.
Ditetrahedronary distetrahedral octahedral hemihexagonal tiling honeycomb. Cells are tets, octs, tuts, and hexats (hexagonal tilings) in compounds of three. Verf is a co w/ four of the triangles drilled in. This has cyclotetrahedral symmetry.
Ditetrahedronary tetrahedral octahedral hemiditrigonal triangular tiling honeycomb. Cells are tets, octs, and ditathas (ditrigonal triangular hemiapeirogonal tilings). Verf is a co w/ four of the triangles inverted to the center. This has cyclotetrahedral symmetry, and seems halfway between numbers 3 and 5.
Ditetrahedronary distetrahedral hemihexahedral hemisquare tiling honeycomb. Cells are tets, thahs, and squats. Verf is a co w/ four of the tetrahedra replaced by cube shaped dents. This has cyclotetrahedral symmetry.
Ditetrahedronary distetrahedral cuboctahedral hemihexagonal tiling honeycomb. Cells are tets, tuts, coes, and hexats in compounds of three. Verf is a co w/ the squares omnifaceted and half of the triangles indented. This has cyclotetrahedral symmetry and is just the last honeycomb w/ half the tets replaced by tuts.
Ditetrahedronary tetrahedral cuboctahedral hemiditrigonal triangular tiling honeycomb. Cells are tets, coes, and ditathas. Verf is a co w/ the squares omnifaceted and triangles indented and looks like a deeper version of the previous verf. This has cyclotetrahedral symmetry.
Ditetrahedronary hemisquare tiling intercepted tetrahedral hemihexahedral hemioctahedral hemihexagonal tiling honeycomb. Cells are tets, thahs, ohoes, squats, and hexats in compounds of three. Verf is a co w/ half of the triangles removed and the squares replaced by crossed quadrilaterals. Has cyclotetrahedral symmetry.
Ditetrahedronary distetrahedral hemioctahedral honeycomb. Cells are tets, tuts, and ohoes. Verf is a co w/ half of the triangles removed and the squares buckled in w/ "pockets" on the two sides touching the triangles. Has cyclotetrahedral symmetry.
Toroidoditetrahedronary distetrahedral hemioctahedral honeycomb. Cells are tets, tuts, and choes. Verf is a co w/ half of the triangles' edges removed and changed into holes, and the squares replaced by pseudo-crossed quadrilaterals. Has cyclotetrahedral symmetry.
Ditetrahedronary hemisquare tiling intercepted tristetrahedral hemihexahedral honeycomb. Cells are tets, tuts in two different orientations, thahs, and squats. Verf is a co w/ half of the triangles buckled w/ pockets and squares replaced by crossed quadrilaterals. Has cyclotetrahedral symmetry.
Ditetrahedronary distetrahedral hemihexagonal tiling honeycomb. Cells are tets, tuts, and hexats in compounds of three. Verf is a co w/ the squares buckled in and half the triangles drilled to the center. Has cyclotetrahedral symmetry.
Ditetrahedronary hemisquare tiling intercepted tetrahedral hemihexahedral hemitriangular tiling honeycomb. Cells are tets, thahs, trats, and squats. Verf is a co w/ half of the triangles drilled to the center, and squares replaced by crossed quadrilaterals, which don't touch the triangles except at the vertices. Has cyclotetrahedral symmetry.
Ditetrahedronary distetrahedral hemioctahedral hemitriangular tiling honeycomb. Cells are tets, tuts, choes, and trats. Verf is a co w/ omnifaceted squares and half the triangles removed. Has cyclotetrahedral symmetry.
Ditetrahedronary tetrahedral hemiditrigonal triangular tiling honeycomb. Symbol is o(o,o~x). Cells are tets and ditathas. Verf is an inflectoverted tet, which looks like four tets attached at a point. Has cyclotetrahedral symmetry.
Ditetrahedronary tetrahedral octahedral hemioctahedral hemiditrigonal triangular tiling honeycomb. Cells are tuts, octs, choes, and ditathas. Verf is a co w/ half the triangles drilled to the center, half just buckled in w/ pockets. Has cyclotetrahedral symmetry.
Ditetrahedronary hemisquare tiling intercepted tetrahedral hemihexahedral hemioctahedral honeycomb. Cells are tuts, thahs, ohoes, and squats. Verf is a co w/ squares replaced by crossed quadrilaterals, and triangles buckled in lightly. Has cyclotetrahedral symmetry.
Ditetrahedronary hemisquare tiling intercepted distetrahedral hemihexahedral ditrigonal hemitriangular tiling honeycomb. Cells are tuts in two different orientations, thahs, squats, and ditathas. Verf is a co w/ the triangles buckled in, half w/ pockets, and squares replaced by crossed quadrilaterals. Has cyclotetrahedral symmetry.
Ditetrahedronary tetrahedral hemihexahedral hemihexagonal tiling hemisquare tiling honeycomb. Cells are tuts, thahs, hexats in compounds of three, and squats. Verf is a co w/ squares replaced by crossed quadrilaterals and triangles drilled in in two different ways, in one case all the way to the center. Note that despite having squats, this is not an "intercepted" honeycomb. Has cyclotetrahedral symmetry.
Ditetrahedronary hemihexahedral hemiditrigonal triangular tiling hemisquare tiling honeycomb. Cells are thahs, ditathas, and squats. Verf is a co w/ four of the triangles replaced by cube shaped dents, and the other four drilled to the center. Has cyclotetrahedral symmetry.
Ditetrahedronary tetrahedral hemioctahedral hemiditrigonal triangular tiling honeycomb. Cells are tuts, ohoes, and ditathas. Verf is a co w/ the squares and triangles slightly indented. Has cyclotetrahedral symmetry.
Toroidoditetrahedronary tetrahedral hemioctahedral hemiditrigonal tiling honeycomb. Cells are tuts, choes, and ditathas. Verf is a co faceting w/ holes and a center that looks like four tetrahedra stuck together. Has cyclotetrahedral symmetry.


And here is the one edge-fissary case:

Code: Select all

Ditetrahedronary tristetrahedral hemiditrigonal triangular tiling honeycomb. Cells are tets, tuts in two different orientations, and ditathas. Verf is a co w/ half of the triangles drilled in and squares dented slightly. Has cyclotetrahedral symmetry.

The regiment also contains compounds of batatohs, batches, and riches.

[username5243]

Tetrahedral cubohemioctahedral hemihexagonal tiling honeycomb. Cells are tets, ohoes, and hexats in compounds of three. Verf is a co w/ the squares indented to the middle, like an oho, plus pockets around the edges.

Why does this one have "cubohemioctahedral" in the name, but ohoes for cells? I'm pretty sure the analogue in the gadtaxady regiment is gadtacaxady, which has qrids for cells.

Octahedral octahemioctahedral hemihexagonal tiling honeycomb. Cells are octs, ohoes, and hexats in compounds of three. Verf is a co w/ lightly inverted pseudo-triangles.

is this one new? (It wasn't on your last list, though its analogues in the ditetrahedronary polychora were not - I think this one corresponds to gadtacady in the gadtaxacy regiment.)

Ditetrahedronary distetrahedral hemihexahedral hemisquare tiling honeycomb. Cells are tets, thahs, and squats. Verf is a co w/ four of the tetrahedra replaced by cube shaped dents. This has cyclotetrahedral symmetry.

Is this one's verf the rectified thah, by any chance? It would make sense, given that the rectified thah should have triangles, squares, and bowties, which is what this one's verf has.

This list is interesting. The squat containing ones are unique, as they don't seem to have any analogues among the ditetrahedronary polychora. (I think it has to do with the fact that octet's verf hass all unit edges, while in, say, gadtaxady's verf, half the edges are shorter thena the other half.)

It might be a good idea to "group" these in a similar way to the ditetreahedronary polychora - ie: primary ((contains no tuts, co members, or squats), secondary (contains tuts, but no co members or squats), coic, ohoic, choic, tetracelli-choic (all these groups still lacking squats), and squatic).

What regiment will you post next? The only larger regiments you haven't uploaded here yet are stut cadoca (trigonal cupolivert) and rich (which should be easy enough to derive from the earlier list of gacoca members).

[wendy]

What are you using '~' for. I'm curious.

[Klitzing]

What are you using '~' for. I'm curious.

When I got him right,
'~' = '∞'.
Just a bit more typewriter friendly.
--- rk

[polychoronlover]

Tetrahedral cubohemioctahedral hemihexagonal tiling honeycomb. Cells are tets, ohoes, and hexats in compounds of three. Verf is a co w/ the squares indented to the middle, like an oho, plus pockets around the edges.

Why does this one have "cubohemioctahedral" in the name, but ohoes for cells? I'm pretty sure the analogue in the gadtaxady regiment is gadtacaxady, which has qrids for cells.

Yes, it does. In general, when the name says one thing, and the cell types say something else, the cell types are right.

Octahedral octahemioctahedral hemihexagonal tiling honeycomb. Cells are octs, ohoes, and hexats in compounds of three. Verf is a co w/ lightly inverted pseudo-triangles.

is this one new? (It wasn't on your last list, though its analogues in the ditetrahedronary polychora were not - I think this one corresponds to gadtacady in the gadtaxacy regiment.)

Not really, it had been on my unpublished updated list since 2015.

Ditetrahedronary distetrahedral hemihexahedral hemisquare tiling honeycomb. Cells are tets, thahs, and squats. Verf is a co w/ four of the tetrahedra replaced by cube shaped dents. This has cyclotetrahedral symmetry.

Is this one's verf the rectified thah, by any chance? It would make sense, given that the rectified thah should have triangles, squares, and bowties, which is what this one's verf has.

Yes, the verf is the rectified thah. Unfortunately it is not hemi-Wythoffian, as 2thah consists of two different types of edges. However, there is a tetracomb with ratho as its verf, which is hemi-Whythoffian (symbol [o3o3o3x3 *a *a3o3/2*c] / 2).

It might be a good idea to "group" these in a similar way to the ditetreahedronary polychora - ie: primary ((contains no tuts, co members, or squats), secondary (contains tuts, but no co members or squats), coic, ohoic, choic, tetracelli-choic (all these groups still lacking squats), and squatic).

Good idea, but I think it would be more important to group them by symmetry; first the o3o4o *b3o symmetry ones, then the o3o3o3o3 *a symmetry ones. In fact, I first thought of putting the members into two different categories based on their symmetry, but now I don't think this is a good idea. I could have one category for this regiment and divide it into subcategories based on symmetry and then sub-sub-categories based on your suggestion.

What regiment will you post next? The only larger regiments you haven't uploaded here yet are stut cadoca (trigonal cupolivert) and rich (which should be easy enough to derive from the earlier list of gacoca members).

Rich regiment. The number and compositions of stut cadoca regiment members are very easy to derive (from the stut phiddix regiment) so I'll only make it high priority to post them if I think that some of the verfs are different from the stut phiddix regiment verfs under the appropriate changes of edge length.

[username5243]

Interesting. Stut cadoca shouldn't have any new members - it's not like octet where the equal edge lengths allowed you to put squats in there.

I would think that, after rich, there aren't many regiments that are hard to derive...

Your mention of that tetracomb reminded me: I've been working on an Excel spreadsheet to list regiments of tetracombs, and might publish it on here when I'm done. (Of course, many have not been counted yet.)

[username5243]

Alright, I've finished the first draft of my list of tetracomb regiments. There's just one problem, the forum won't let me attach the list. (Whenever I try, it says "The extension xlsx is not allowed.") I don't know how to stop this from happening, and don't know how to publish it otherwise - any suggestions?

I'm fairly sure that there will be more uniform tetracombs than uniform polytera (both excluding prisms and the like), if only due to the fact that larger polychoron regiments can appear - spic, giddic, and afdec are all larger than any tessic regiment. Also, keep in mind that in all regiments that I know of, spic and giddic act like they have 32 members, and afdec acts like 99 - this happens because the icoic ones appear twice.

[Klitzing]

... There's just one problem, the forum won't let me attach the list. (Whenever I try, it says "The extension xlsx is not allowed.") I don't know how to stop this from happening, and don't know how to publish it otherwise - any suggestions? ...

How about attaching a zipped version of your spreadsheet?
--- rk

[username5243]

That seems to actually work. (That's a nice way to get around it that I didn't think of!) Thanks for the suggestion!

Anyway, some notes:

Several regiments remain uncounted, since they have no spherical 5D regiment with a similar verf. I'm not sure how one would go about counting these.

Many of the names - especially for star tetracombs - are of my own invention. They are not official names, so don't treat them as such.

Also, I'm not sure of the completeness of this list - I might well have missed some families - but it includes most of the obvious ones.

Let me know if you have any comments or counts of currently-unknown ones.

[polychoronlover]

Ricot [...] Number of members: 47

I don't think this is correct. This is the number of members for a duowedge-verfed shape with non-doublable symmetry. But ricot, taken as o4x3o3x4o, has only 23 members under this symmetry, the same as sibrid. These include 7 members under o3o3o4x3o symmetry. It also has 8 distinct members under x3o3x4o3o symmetry, and even more as x3o3x4o *b3x and x3o3x *b3x *b3x. Unless you actually enumerated all of these and found 16 new members, in which case I'd love to see how you did it.

[username5243]

Ricot [...] Number of members: 47

I don't think this is correct. This is the number of members for a duowedge-verfed shape with non-doublable symmetry. But ricot, taken as o4x3o3x4o, has only 23 members under this symmetry, the same as sibrid. These include 7 members under o3o3o4x3o symmetry. It also has 8 distinct members under x3o3x4o3o symmetry, and even more as x3o3x4o *b3x and x3o3x *b3x *b3x. Unless you actually enumerated all of these and found 16 new members, in which case I'd love to see how you did it.

Indeed, right you are. that's >31 members. I'd think the x3o3x4o *b3x would be somewhat similar to sibrant's hinnic members, but there's probably more - and I'm not sure how to begin counting quarter-tesseractic symmteric ones (which don't have an analoogous verfed polyteron regiment).

[polychoronlover]

There probably aren't any unique quarter-tesseractic symmetry ones, just as there are no unique rico regiment members with demitesseractic symmetry.

[polychoronlover]

There is a stut-cadoca-like tetracomb with a o3o3q |8/3| x3o3x verf and symbol x4o3o3o3x4/3 *b3e. It has at least 5 other Wythoffians in its regiment, with symbols x4o3o3o~x3/2*c, x4x3o3o3o4/3*b3e, x4/3o3o3o~x3/2*c, o3o3x4x4/3o3/2*b *c~*e, and o3o3o4x4/3x3/2*b *c~*e. It has tesses, quitits, wavitoths, gittiths, and octets. I propose that it be called small tritetrahedronary tesseractitristesseractiapeiratic tetracomb, or stat tatita. I've also been wondering if "tesseracti-" should be replaced by "tesserati-" to match "cubati-".

[username5243]

There is a stut-cadoca-like tetracomb with a o3o3q |8/3| x3o3x verf and symbol x4o3o3o3x4/3 *b3e. It has at least 5 other Wythoffians in its regiment, with symbols x4o3o3o~x3/2*c, x4x3o3o3o4/3*b3e, x4/3o3o3o~x3/2*c, o3o3x4x4/3o3/2*b *c~*e, and o3o3o4x4/3x3/2*b *c~*e. It has tesses, quitits, wavitoths, gittiths, and octets. I propose that it be called small tritetrahedronary tesseractitristesseractiapeiratic tetracomb, or stat tatita. I've also been wondering if "tesseracti-" should be replaced by "tesserati-" to match "cubati-".

Neat. This regiment should be pretty large because it contains the octet regiment as a facet. The octet regiment here should act like it has 53 members because it is appearing in cyclotetrahedral symmetry, with the cyclotetrahedral members of the octet regiment appearing twice each. (a similar thing happens in stut cadoca - ditatha appears in two different orientations, so if you were to make a list of its members, you'd have to distinguish between "a-ditathas" and "b-ditathas".) In fact, the stut cadoca regiment may well appear as a facet of this one, making it even larger. (I'm not sure what the largest regiment of tetracombs will actually be.)

I'm beginning to suspect that something similar exists in higher dimensions as well.

Anyway, I'd really like to fill in some of these counts, but I have no clue how Hedrondude does it for polytera. (And I suspect some of these regiments will be larger than any polyteron regiment.)

[Klitzing]

... and symbol x4o3o3o3x4/3 *b3e ...

There ought be some typo(s?) within that symbol, for it is bluddy nonsense!

First of all the last letter "e" probably is meant to re-reffer a priviously mentioned real node. Thus it ought be prefixed by an asterisk (*) as well. But then let's have a closer look on the so far encountered real nodes:

Code: Select all

x4o3o3o3x4/3 *b3*e
a b c d e


that is, the "4/3" should reconnect to the b-th node, i.e. the "o" between the "4" and the "3". And then from that (b-th) node you want to connect to the e-th node, i.e. the second "x". But then those 2 nodes where already joined by the "4/3", thus they cannot be connected by a "3" again!

Or, when the empty space between the "4/3" and the "*b" was placed by will (i.e. representing an independent graph, only re-connected by the astersks), then you aim for a second connection between the b-th and the e-th node by that "3" (besides the already mentioned one ...o3o3o3x...), but then there remains that other link "4/3" unconnected, having no other node symbol to which it has to be connected to.

That is, did you have that in mind:

Code: Select all

  +----------+
  v          |
             |
x4o3o3o3x4/3*b3*e
         ~~~  ~ |
        ^       |
        +-------+


, which has the problems highlighted by the "~"-underlinements?
Or did you have in mind:

Code: Select all

   *b3*e
   /   \
  v     v
 
x4o     x4/3(?)
   3   3
    o3o


, which lacks the node symbol marked "(?)"?

Please shed some further light onto that.
--- rk

PS: same holds most probably then for "x4x3o3o3o4/3*b3e" too, isn't it?

[polychoronlover]

Or, when the empty space between the "4/3" and the "*b" was placed by will (i.e. representing an independent graph, only re-connected by the astersks), then you aim for a second connection between the b-th and the e-th node by that "3" (besides the already mentioned one ...o3o3o3x...), but then there remains that other link "4/3" unconnected, having no other node symbol to which it has to be connected to.

That hypothesis is right. In fact, I meant x4o3o3o3x4/3*a *b3*e (and by analogy, x4x3o3o3o4/3*b3e is supposed to be x4x3o3o3o4/3*a *b3e).
Lesson learned: label the nodes before linearizing the symbol.

Neat. This regiment should be pretty large because it contains the octet regiment as a facet. The octet regiment here should act like it has 53 members because it is appearing in cyclotetrahedral symmetry, with the cyclotetrahedral members of the octet regiment appearing twice each. (a similar thing happens in stut cadoca - ditatha appears in two different orientations, so if you were to make a list of its members, you'd have to distinguish between "a-ditathas" and "b-ditathas".) In fact, the stut cadoca regiment may well appear as a facet of this one, making it even larger. (I'm not sure what the largest regiment of tetracombs will actually be.)

This could well be the largest. Other incredibly large regiments will probably include those of cypit (probably not quite as big), stadit (has a cube || co verf), skivtadia (x4o || x4o || o4x verf), scyropot, aftadia, and its conjugate (which I don't see on your spreadsheet), and icot (due to the large range of symmetries it can take on).

Anyway, I'd really like to fill in some of these counts, but I have no clue how Hedrondude does it for polytera. (And I suspect some of these regiments will be larger than any polyteron regiment.)

Don't worry, I have a plan for when it comes time to search these regiments

Anyway, on to the subject I was planning to talk about.

Prismatic honeycombs and regiments

Besides the one I listed in my last list of uniform honeycombs, there are 3 more copycats of the square hemiapeirogonal prismatic honeycomb. One has only half of the azips blended into squats, one has rows of azips going perpendicular to columns of azips (the columns being perpendicular to the prismatic layers), and one has half of the azips blended into squats and the other half in columns instead of rows.

There are also 5 more copycats of the ditrigonal triangular hemiapeirogonal (ditatha) prismatic honeycomb, formed by starting at the ditatha pseudoprismatic honeycomb (containing trips and squats) and selectively un-blending the squats into rows or columns of azips.

There is one new copycat each of the prismatic honeycombs of sossa, gossa, satsa, hatha, snassa, rassersa, rarsisresa, rosassa, and rorisassa, besides the one where the azips blend into squats. In each case, it has columns instead of rows of azips.

The alternate triangular prismatic honeycomb (which is like the ditatha prismatic honeycomb except each layer gets the opposite set of trips) has 13 copycats, if my count is correct, formed by replacing rows of azips by squats or columns of azips. In 7 cases, the trats decompose into sets of azaps as well.

The alternate trihexagonal prismatic honeycomb has one copycat, where the square tilings decompose into rows of azips.

Finally, there are an uncounted number of other chon regiment members, by letting it act like x~x * x~x * x~x (the least density of symmetry that keeps it uniform.) One of these has the same verf as the koho, the skew-octahemioctachoron.

[username5243]

Interesting. Also, I just realized that test should actually have more than 11 members due to all the prismatic and duoprismatic ones - but since no prismatics appeared on that list anywhere else, I didn't bother to count them.

I just realized something... Stut Tatita's verf his a q-tet || co with sides of length x(8/3), while stadit has a cube || co verf (also with sides of length x(8/3). As you probably know, that q-tet can be inscribed into that cube. I'm not sure if this means those two regiments are related, but they could be. Another similar case is with siphatit (q-tet ||oct verf) and scicot (cube || oct verf).

As I said, the amount of tetracombs is far greater than the number of uniform polytera - and that's mostly due to the appearance of icoic symmetric polychora. (Hey, be glad no hyics appear, or else there would be several with >1000 members...)

[polychoronlover]

There are some other cases where infinite cells of honeycombs can be decomposed into layers. In fact, I think there are so many that I should make a separate category for them, to keep their boringness from clogging up the other categories.

For example, the square tilings in members of the gacoca regiment can be decomposed into stacks of apeirogonal prisms, and they will remain uniform (at least the ones with full symmetry, but probably all of them). Parallel triangular tilings in certain octet regiment members can be decomposed into stacks of apeirogonal antiprisms, in fact multiple sets can as long as the boundaries between apeirogons from two stacks don't overlap. Stut cadoca might even have similar decompositions.

I just realized something... Stut Tatita's verf his a q-tet || co with sides of length x(8/3), while stadit has a cube || co verf (also with sides of length x(8/3). As you probably know, that q-tet can be inscribed into that cube. I'm not sure if this means those two regiments are related, but they could be. Another similar case is with siphatit (q-tet ||oct verf) and scicot (cube || oct verf).

Cool! One day I'll try to make a catalog of all relations like this. For example, I realized that o4o3x3o4x, x4o3x3o4x, and o3x3o4x3o can all be cut from o3x3o4o3x.

As I said, the amount of tetracombs is far greater than the number of uniform polytera - and that's mostly due to the appearance of icoic symmetric polychora. (Hey, be glad no hyics appear, or else there would be several with >1000 members...)

I know. It really makes me wonder how many uniform hyperbolic honeycombs there are. (Although, due to the problem of hyperbolic conjugates sometimes having infinite density, and also the problem of avoiding hypercycles as faces, the number might be smaller than it would be otherwise.)

[wendy]

A number of stary hyperbolic tilings of finite density are known. But there are not a lot of star-subgroups up there, and we mostly rely on Coxeter's theorm that starry symmetries are nade from regular ones.

The group o5o3o4o, for example, contains the subgroup o5/2o5o5/2o3*a of density 4, which leads to a number of uniform star-compounds. You can derive from this star-polytopes like x5x5/2o3o5/2*a.

There are nine 'convex' groups, but the only one that seems to contain usable subgroups is [5,3,4]. There are a number of non-wythoffian tilings, but these don't appear to lead to any useful star-groups either.

The conjucates of hyperbolic tilings are infinitely dense polytopes, and infinitely dense tilings. Thus the finite {5,3,6} produces an infinitely dense tiling {5/2,3,6}, and the tiling {5,3,4} produces an infinitely dense {5/2,3,4}. These polytopes are none the same useful, since they provide a filter to filter out numbers with too many places of phi-decimals. That is, the radii in {5,3,4} must be such that the isomorph fits into {5/2,3,4}.

[polychoronlover]

Finally, there are an uncounted number of other chon regiment members, by letting it act like x~x * x~x * x~x (the least density of symmetry that keeps it uniform.) One of these has the same verf as the koho, the skew-octahemioctachoron.

I did a search and found 4 more of these, excluding the ones with rows of azips instead of squats. In fact, this excludes any where two coplanar azips meet.

One of these 4 has the same verf as koho, as I said above. This verf is an oct faceting containing four equilateral triangles (cube verfs), two right triangles (azip verfs), and one crossed quadrilateral (sha verf). Another has a verf that can best be described as a square pyramid except where one of the triangles points down instead of up, it contains 4 cube verfs, 2 azip verfs, and 1 squat verf. These two honeycombs have 4 cubes around each verf. The other two are blends of the cubic honeycomb with already-discovered figures; one blends it with columns of square-apeirogonal duoprisms (one column per vertex) and the other with sha-prisms on half of all the honeycomb's layers. These two have 6 cubes around each verf.

It can be proven that all uniform facetings of a hypercubic tessellation have an even number of hypercube facets around each vertex. To see this, imagine the tessellations needed as facets to connect to the cubes that don't connect to cubes themselves; in n dimensions there are up to n sets of these, each in a perpendicular hyperplane. (In 2 dimensions, for example, sha has horizontal and vertical apeirogons. In 3 dimensions, cuhsquah has 3 sets of perpendicular squats.) Except that sometimes it's multiple facets in the same hyperplane, for example rows of apeirogons with blank rows in between. But for the purposes of the proof, those can be considered a single "facet" despite not being properly connected. Let's assume by induction that every possible hypercubic-tessellation-faceting in n - 1 dimensions has an even number of hypercubes around every vertex. Then, if one is used as an apeiratic facet in an n-cubic-tessellation-faceting, it must connect to an even number of n-cubes around each vertex, one at each (n - 1)-cube facet of itself. Any other (n - 1)-cubes in the same plane as the infinite facet - but not part of it - must be connected to two hypercubes above and below. Adding these pairs of n-cubes to the even number touching the infinite facet results in a total which is still even. Thus the total number of hypercubes whose facets lie in a particular hyperplane of the n-dimensional hypercubic-tessellation-faceting is even. However, as each n-cube has n sets of facets around each vertex, every n-cube around a vertex must have one facet lying in each of the n hyperplanes. So the even number of n-cubes derived earlier was the total number of n-cubes around each vertex, and thus each vertex has an even number of n-cubes around it.