Six Dimensional Regiment Counts

[Polyhedron Dude]

Sense my monitor has been out of commision, I decided to search out the members of the Spil regiment, spil being ooxoox. It is also called the runcinated heptapeton. There are 9 possible facet regiments: dot, spix, rappip, tratet, sarx, srippip, traco, hoct, and tratut. There are 16 ridge regiments: rap 1, rap 2, ope 1, ope 2, triddip 1, triddip 2, spid, tepe, tip, srip, shiddip, cope, thiddip 1, thiddip 2, tuttip, and tisdip. The spil regiment has 138 members plus 15 fissaries. One of them has two types of facets: topax and garpopip. I'm currently looking at the Sibpof regiment, sibpof being oxooxo - anyone wish to make a guess how many members it'll have.

[Klitzing]

Btw. topax = ?, just know that it is in spix regiment...

--- rk

[Polyhedron Dude]

Topax is the one with 6 garpops and 15 chopes.

[Klitzing]

Topax is the one with 6 garpops and 15 chopes.

Still don't know its long name, so.

None the less, calculation already becomes possible.

Here is the incidence matrix of its vertex figure. - Note that it is - at least combinatorically - selfdual!

Code: Select all

Vertex Pattern:
a     b                         
   c                         g   
                          h     i
d     e                         
   f                             

6 * | 2  2 1 | 2  4 2 | 3 2  a
* 3 | 0  4 2 | 2  4 4 | 4 2  g
----+--------+--------+----
2 0 | 6  * * | 1  2 0 | 2 1  ae q
1 1 | * 12 * | 1  1 1 | 2 1  ag q
1 1 | *  * 6 | 0  2 2 | 2 2  ai h
----+--------+--------+----
2 1 | 1  2 0 | 6  * * | 2 0  aeg qqq = verf(cube)
2 1 | 1  1 1 | * 12 * | 1 1  aeh qqh = verf(hip)
2 2 | 0  2 2 | *  * 6 | 1 1  ahbi qh(-q)h = verf(cho)
----+--------+--------+----
3 2 | 2  4 2 | 2  2 1 | 6 *  abfhi verf(chope)
4 2 | 2  4 4 | 0  4 2 | * 3  abdehi verf(garpop)


With that one then follows the according incidence matrix of topax itself:

Code: Select all

60 |   6  3 |  6  12  6 |  6 12  6 |  6 3
---+--------+-----------+----------+-----
 2 | 180  * |  2   2  1 |  2  4  2 |  3 2
 2 |   * 90 |  0   4  2 |  2  4  4 |  4 2
---+--------+-----------+----------+-----
 4 |   4  0 | 90   *  * |  1  2  0 |  2 1
 4 |   2  2 |  * 180  * |  1  1  1 |  2 1
 6 |   3  3 |  *   * 60 |  0  2  2 |  2 2
---+--------+-----------+----------+-----
 8 |   8  4 |  2   4  0 | 45  *  * |  2 0  cube
12 |  12  6 |  3   3  2 |  * 60  * |  1 1  hip
12 |  12 12 |  0   6  4 |  *  * 30 |  1 1  cho
---+--------+-----------+----------+-----
24 |  36 24 | 12  24  8 |  6  4  2 | 15 *  chope
30 |  60 30 | 15  30 20 |  0 10  5 |  * 6  garpop


--- rk

[wendy]

We're evidently talking about Ft9 and Ft18 here. Two different things, really.

The first letter is the dimension, here A=1, B=2, ..., F=6. The second letter is the symmetry as per 3d (t=tetra, o=octahedron, c=cube, q=343, i=335, d=533.

The number is to be rendered in binary, and the nodes are numbered 1,2,4,8,16,&c, the nodes simply add. So ft9 is 3,3,3,3,3 with xoox marking, ie x3o3o3x3o3o. It's been mentioned before as a kind of short name.

Prism products are written by running the elements together, so a pentagonal prism is AB5.

[Klitzing]

We're evidently talking about Ft9 and Ft18 here. Two different things, really.

The first letter is the dimension, here A=1, B=2, ..., F=6. The second letter is the symmetry as per 3d (t=tetra, o=octahedron, c=cube, q=343, i=335, d=533.

The number is to be rendered in binary, and the nodes are numbered 1,2,4,8,16,&c, the nodes simply add. So ft9 is 3,3,3,3,3 with xoox marking, ie x3o3o3x3o3o. It's been mentioned before as a kind of short name.

Prism products are written by running the elements together, so a pentagonal prism is AB5.

Yes, Wendy: spil = xooxoo = Ft9, sibpof = oxooxo = Ft18.

But Jonathan was looking for the count of members within either regiment, i.e. the number of possible uniform figures, which have the same set of vertices (= army) plus the same set of edges (= regiment).
(For sure, using his newer counting, i.e. with stricter sieve on what are considered allowed polytopes. E.g. rejecting fissaries etc.)

Cf. his page on the regiments, searching therein down for "Spil" resp. "Sibpof": those numbers there where still to be evaluated - and then individual names to be given and OBSAs to be coined ... (Likewise for cral = xoxoxo = Ft21, topal = xoxoox = Ft37, rag = oxooo'o = Fo2, etc.)

--- rk

[Polyhedron Dude]

I noticed a couple of miscounts on my regiment site - Ril should only have 7 members, this is due to 'hemi-clashing' - the same reason that there are no non-prismatic uniform polychora with dodecahedral hemis for cells. Staf has 19 members, scal has 138 members which has yet to be weeded for fissaries. I looked at the cral regiment a few months ago and it has 75 members, I also checked out rag and it lands at 138, but still needs to be cleared of fissaries. Sibpof has now been counted, but needs weeding of fissaries - it was a lot smaller than I expected - 96.

[wendy]

Every integer wythoff mirror-edge polytope on the simplex group (ie where the nodes are weighted by integers), can be constructed by gluing the various rectates together (ie they're part of the lattice A_n).

You can work out what's in the middle in a very easy way. The nodes are weighted 1, 2, 3, ..., n, done over modulo (n+1). Simply multiply the weightings of the nodes by the node-values, and add, mod n+1. The figure in the middle is the rectate equal to the modulus, if it's a 0 then there's a point there.

So, eg 3s0s0 (s=3 branch, nodes are numerical values). and 1s1s0 both have 0s0s1 (reversed tetrahedron)

1s0s0s1s0s0 adds to 5, ie 0s0s0s0s1s0. Also, 0s1s0s0s1s0 adds to 7 => 0, gives 0s0s0s0s0s0.

[Klitzing]

Hein?
Sorry Wendy, absolutely obscur, your magics...
--- rk

[wendy]

It is my most general form of the CD. One can hold stott-expansion as happening on a true coordinate system so the C-D branches are reduced to comma-roles, and the weightings are purely numeric. So instead of writing u for 2, f=1.618 &c, we write the branches as eg S, Q, F, etc, and write pure numbers between them, so 2 S 1 F 0 is the same as u3x5o. But you can use any number in there, like 15 S 12 F 96.

With the lattice prototypes, like SSS, one can write, eg 1 S 0 S 3 S 0,

[Klitzing]

Okay, the mere single CDs now are understood.
But what are you then doing with mod-wrap additions?
And what e.g. should I get from the line

So, eg 3s0s0 (s=3 branch, nodes are numerical values). and 1s1s0 both have 0s0s1 (reversed tetrahedron)

?
--- rk

[Polyhedron Dude]

Due to a 'typo' or better yet 'listo' there are more polypeta in the sibpof regiment. I weeded out the fissaries and assuming no 'typos' there are now 143 members plus 13 fissaries. 43 of these have double symmetry, 100 have single symmetry. The fissaries have single symmetry. It is so far the largest hoppic regiment, but topal has yet to be counted, it will be next.

[Polyhedron Dude]

I counted the Topal regiment - Topal = xoxoox. There are 139 members plus one fissary - so it looks like sibpof is the largest hoppic regiment.

[Polyhedron Dude]

The Rag regiment - oxooo'o now has 132 members plus 5 fissaries plus 2 pure compounds. Facet regiments include tac, rix 1, rix 2, scad 1, scad 2, rat, hexip, squatet, and troct. Ridge regiments include pen 1, pen 2, hex, rap, tepe 1, tepe 2, triddip 1, triddip 2, spid, ico, ope, and tisdip.

[Polyhedron Dude]

I now have all of the hoppic regiments counted . Hop is the 6-D simplex, the heptapeton. Here are the counts for the regiments:

ooooox - Hop - 1
ooooxo - Ril - 7
oooxoo - Bril - 18
ooooxx - Til - 1
oooxxo - Batal - 1
ooxxoo - Fe - 1
oooxox - Sril - 31
ooxoox - Spil - 138 + 15 F
ooxoxo - Sabril - 63
oxooox - Scal - 132 + 8 F (just recently cleared of fissaries)
oxooxo - Sibpof - 143 + 13 F
xoooox - Staf - 19
oooxxx - Gril - 1
ooxxxx - Gapil - 1
ooxxxo - Gabril - 1
oxxxxx - Gacal - 1
oxxxxo - Gibpof - 1
xxxxxx - Gotaf - 1
ooxoxx - Patal - 15
ooxxox - Pril - 3
oxoxxo - Bapril - 7
oxooxx - Catal - 37 + 1 F
oxoxox - Cral - 75
oxxoox - Copal - 7
xoooxx - Tocal - 15
xooxox - Topal - 139 + 1 F
oxoxxx - Cagral - 7
oxxoxx - Captal - 3
oxxxox - Copril - 3
xooxxx - Togral - 7
xxooxx - Tactaf - 5
xoxoxx - Tocral - 27
xoxxox - Taporf - 6
xoxxxx - Tagopal - 3
xxoxxx - Tacogral - 3

Grand total - 923 plus 38 fissaries!

[Klitzing]

I now have all of the hoppic regiments counted ...
Grand total - 923 plus 38 fissaries!

Congratulation! That's a new landmark!

And also named (incl. acronyms) them as well already?

--- rk

[Polyhedron Dude]

I now have all of the hoppic regiments counted ...
Grand total - 923 plus 38 fissaries!

Congratulation! That's a new landmark!

And also named (incl. acronyms) them as well already?

--- rk

I have yet to name the members of the larger regiments (30 members or larger), but got the smaller ones named. The next landmark will likely be the haxics - got two more regiments to count there - and they could have hundreds of members each. I'm also still working on the srog regiment and its looking like it will land in the 300-500 vicinity depending on fissary counts.

[Polyhedron Dude]

Lately I've been using POV-Ray to help get some regiment counts. So far I've coded it to search the trim regiment and to check the nit regiment (trim = xo8ox, nit = ooxo'o). I got it now to the point where it will only list dyadic members that are not compounds of other regiment members (It can't clear out pure compounds or fissaries yet, I have to do those by hand). The nit regiment listed the ones that I already knew, there was no new ones, but one of the fissaries that I had listed turned out to be a compound of other regiment members (I call these type of compounds IRCs or inter-regimental compounds).

The trim regiment has five nobles in it. The list has 250 IRCs, and 1243 others!!, each of these 1243 could either be true polytopes, fissaries, or pure compounds (possibly none). When coding, I left out the fissary members of the siphin regiment so they didn't get counted in as facets. I'm now coding in the sochax regiment, next in line will be sophax, mo, srog, brag, and brox.

[Polyhedron Dude]

Just finished sochax. Assuming no typos in my code, there are 1925 objects in the list!, one appears to be a pure compound, and I suspect many will be fissary. No IRCs should be on this list. Sochax is xoo8x and its verf is a -pen || rap, aka xooo || ooxo. Sochax is the 6-D version of siphin. This list excludes the 20 siphin fissaries and the one fissary in the spix regiment from its facet list, since these will lead only to fissaries. These regiments are starting to look scary .

[Polyhedron Dude]

I searched out both the mo regiment - oo6oo and the srog regiment - xoxoo'o. Mo (the 54-peton) is also called 1₂₂ and has 54 hins as its facets. The mo regiment has 84 members plus 58 fissaries and 2 pure compounds. The srog regiment has 290 members plus 63 fissaries and 2 pure compounds. Srog is the cantellated (small rhombated) hexacross (64-peton).

[Polyhedron Dude]

The Crag regiment is now counted and "defissed". Crag = xoxox'o = cellirhombated 64-peton (stericantellated hexacross). It's related regiment "cral" = oxoxox had 75 members and no fissaries, due to crag's ability to go sub-symmetric, or take on the symbol xox6x in demihexeract symmetry, it will have more members. I originally expected it would land around 250 or higher, but instead it wound up having a whopping - - - 123 , but then again it made defissing easier - there were no fissaries. This also lead to the counts of two skewvert regiments (those related to the 4-D skiviphado), both of which have 171 members - 171 = (123 - 75)*2 + 75. The two skewvert regiments are scrokix and gacrokix, which are xoxG and xGox respectively. Their verfs are similar to crag's, but are more skewed. Out of the 235 known non-prismatic polypeton regiments, only 17 has yet to be counted plus three more that needs defissing.

One of the ones needing defissing is trim = xo8ox (which is next) - I recently found two new classes of facets in there that I overlooked - two groups of tratuts, but they lead to face figures (at some square faces) shaped like the stella octangula - so all of the tratut containing ones are either fissary or a compound, whew - if it wasn't fissary, the count would of likely doubled. The current count of trim is 1244 which I suspect will be significantly lowered after defissing, my guess it'll land around 800. Out of the 1244, only 47 have full moic symmetry, the rest have jakic symmetry. The verf of trim looks like a stetched out demipenteract, Jak's verf is a demipenteract, I looked there and seen missing facets related to the tratuts, they were penps (pentachoron prisms). The jak regiment got four new figures, one was a compound and the others were fissaries (with stella octangula face figures at the squares). The compound has 216 penps and 72 hixes, I call it "pok" for prismato72peton, it is the compound of 36 hixips (hix prisms).

[Polyhedron Dude]

I finally got the Trim regiment defissed, took nearly two weeks! assuming no errors in my search, the count is now at 557, over half of them were fissary. The full symmetric ones dropped to 37. Regiments like xxo8ox and oxxo8ox now has 1077 members instead of 2443.

[Polyhedron Dude]

New regiment counts are in. I recently counted and defissed the scag regiment - xooox'o, it has 453 members! I also noticed that I made a few errors in the scal count, four of them didn't work and three more were fissary, so it now has 125 members instead of 132. The scag count also lead me to the cakix regiment count - 781. Cakix is the celliskewed hexeract, its symbol is xoo(o'x"x) = xooG.

Scag has thirteen possible facet regiments: spat, rin, scad1, scad2, rittip, cappix1, cappix2, traco, pirx1, pirx2, squatet, garx1, and garx2.
Scag's ridge regiments are rit, spid1, spid2, rico, cope, prip1, prip2, pen1, pen2, pen3, tip1, tip2, deca, tepe1, tepe2, tepe3, tepe4, triddip1, triddip2, tisdip, grip1, grip2, tuttip1, tuttip2, and thiddip - whew!

[Polyhedron Dude]

I now took a look at the brag regiment. Brag is the birectified hexacross = ooxoo'o and its verf is a troct. I originally thought it would land thousands of members, but it now looks to be 531 members plus 11 pure compounds. I have yet to defiss it, so the actual count may land around the 300 mark.

The face regiments of brag are rat, dot 1, dot 2, spix 1, spix 2, nit, icope, troct, hoct, squoct, squaco, and traco.
The ridge regiments are hex, rap 1, rap 2, rap 3, spid, ico, srip, ope 1, ope 2, ope 3, tisdip 1, tisdip 2, triddip 1, triddip 2, thiddip, tes, shiddip, rit, cope 1, cope 2, and cope 3. Rit is never activated.

[Polyhedron Dude]

Just finished sochax. Assuming no typos in my code, there are 1925 objects in the list!, one appears to be a pure compound, and I suspect many will be fissary. No IRCs should be on this list. Sochax is xoo8x and its verf is a -pen || rap, aka xooo || ooxo. Sochax is the 6-D version of siphin. This list excludes the 20 siphin fissaries and the one fissary in the spix regiment from its facet list, since these will lead only to fissaries. These regiments are starting to look scary .

The sochax code did have a couple typos, the end result landed it at 1860 members. I finally defissed it, nearly 2/3 were fissary and one was a pure compound. There are now 688 members in the sochax regiment, assuming no errors. The spreadsheet can be viewed below, just right click and save.

http://www.polytope.net/hedrondude/Sochax%20Regiment.xlr

In the spreadsheet, fissaries are blue, the compound is in green, the legitimate cases are in black.

Due to my hard drive crashing in May, I have lost the brag regiment info, so I'll need to code it back in to get the spreadsheet made again.

Below are some other regiment spreadsheets for crag, mo, scal, srog, and trim. Some of these may have the older names for the spix reigment instead of the newer ones, both of which are included in my polyteron spreadsheet. The mo spreadsheet also includes names.

http://www.polytope.net/hedrondude/Crag%20Regiment.xlr
http://www.polytope.net/hedrondude/Mo%20Regiment.xlr
http://www.polytope.net/hedrondude/Scal%20Regiment.xlr
http://www.polytope.net/hedrondude/Srog%20Regiment.xlr
http://www.polytope.net/hedrondude/Trim%20Regiment.xlr

The list of uniform polytera can be viewed below, Only the skatbacadint regiment has yet to be listed.

http://www.polytope.net/hedrondude/uniform%20polytera%20x.xlr

Jonathan B.

[polychoronlover]

The polyteron list is absolutely amazing!!! I was looking for Whytoffian members of regiments and I really wanted to know what the acronyms for some of them were, for example howoh, the cantellated hehad. Awesome!

I can even understand most of the LN abbreviations. For example, I gathered that t (as in topax) stands for ptero-, y stands for thyrido- (which seems to mean "windowed" wedges), i stands for inverti-, etc. I still haven't figured out what the "tT" in certain nit regiment members means, though, or whether the lowercase "q" means quasi- or not.

I also noticed symbols like [xooo,x]/2 for firx. I call these shapes hemi-wythoffian, because their double coverings have Whytoffian construction.

Again, great work on the list, and the 6-dimensional regiment counts!

EDIT: Only one computer in my house can open .xlr files. Are there any (free) programs that can open them besides LibreOffice Calc?

[Polyhedron Dude]

The tT is tritesseracti, this comes from the gicoes which are 3-tes compounds. For topax, I used toroidal for the t, but after looking at its verf, ptero makes more sense, the verf isn't a toroid, just some of the verf's cross sections are. The lower q is also quasi. You got the i and y right .

I've also been referring to the []/2 types as hemi-Wythoffian I'm not familiar with any free .xlr viewers, I use Microsoft Works. Concerning Wythoffians, I suspect that all Wythoffian regiments are now accounted for - in all dimensions, I just need to write down the proof, their symbols come in the following forms for dimensions 5 or greater:

ooo...oooo - simplexial, each o can be substituted for an o or an x.
ooo...ooo'x - cubical, the x must be an x, the o's can be either x or an o.
ooo...ooo'0 = ooo...oo6x - crossical - the 0 must be an o, the o's can be either x or o, the 6x represents the two branches in demicubic symmetry, the two branches must both be o or x. These are uniform under n-cubic and n-demicubic symmetries.
ooo...ooo8x - demicubical, the x must be an x, the top node of the 8 must be a o (or you could swap them), the o's and the bottom node of 8 can be either o or x.
oo8oo, ooo8oo, and oooo8oo - sporadic, all o's can be x's or o's. (only in 6-8 dimensions).
ooo...oox"x - stellicubical, the x's must be x nodes, the o's can be either x or o.
ooo...ooG, ooo...oGo, ... ooo...oGo...ooo,...,Goo...ooo - goccoics - the o's can be x nodes or o nodes, the G is gocco (o'x"x) or cotco (x'x"x).
Thats it!

Later, I'll reveal how I come up with the 6-D regiment lists, its a fun yet grueling task with 8 tricky phases that are sort of like solving Sudoku puzzles, 'find all the triangles' puzzles, 'match lock and key' puzzles, and many more all rolled up into one.

BTW Looks like rojak is going to be very bad - it contains nit, rat, spix, sarx, siphin, rix, hin, rappip, spiddip, srippip, and tratrip as its facet regiments. The bold ones have many members.

[Klitzing]

The polyteron list is absolutely amazing!!! I was looking for Whytoffian members of regiments and I really wanted to know what the acronyms for some of them were, for example howoh, the cantellated hehad. Awesome!

Thah = tetra-hemi-hexahedron is a hemi-figure and therefore not Wythoffian.
Tho = tesseracti-hemi-octachoron suffers the same problem one dimension up.
Hehad = hexadeca-hemi-decateron then is the relative of 5D.

It is indeed interesting that from 4D on a rectification of these figures is possible. But the property of being Wythoffian thereby surely is not retrieved.
The one derived from tho is ratho = rectified tesseracti-hemi-octachoron.
The one derived from hehad is rhohid = rectified hexadeca-hemi-decateron.

And, as you pointed out, an once more rectified rectification is possible for those figures from 5D on. That total operation alternatively is known as cantellation, even so Jonathan would rather like to speak of small rhombation here.
The one derived from hehad (resp. rhohid) then indeed is howoh. The only explanation I found for that acronym was "16W16", thus I'd assume it to be meant a "hexadeca-spheno-hexadecateron".

Yesterday I was working out the full incidence matrix of howoh. Here it is for your record:

Code: Select all

240 |   8   2 |   4   4   4   8  1 |  2  2  2   4   4  4 |  1 2  2  2
----+---------+--------------------+---------------------+-----------
  2 | 960   * |   1   1   1   1  0 |  1  1  1   1   1  1 |  1 1  1  1
  2 |   * 240 |   0   0   0   4  1 |  0  0  0   2   2  4 |  0 2  1  2
----+---------+--------------------+---------------------+-----------
  3 |   3   0 | 320   *   *   *  * |  1  1  0   1   0  0 |  1 0  1  1
  3 |   3   0 |   * 320   *   *  * |  0  1  1   0   0  1 |  1 1  0  1
  4 |   4   0 |   *   * 240   *  * |  1  0  1   0   1  0 |  1 1  1  0
  4 |   2   2 |   *   *   * 480  * |  0  0  0   1   1  1 |  0 1  1  1
  3 |   0   3 |   *   *   *   * 80 |  0  0  0   0   0  4 |  0 2  0  2
----+---------+--------------------+---------------------+-----------
  6 |  12   0 |   4   0   3   0  0 | 80  *  *   *   *  * |  1 0  1  0  thah
  6 |  12   0 |   4   4   0   0  0 |  * 80  *   *   *  * |  1 0  0  1  oct
 12 |  24   0 |   0   8   6   0  0 |  *  * 40   *   *  * |  1 1  0  0  co
  6 |   6   3 |   2   0   0   3  0 |  *  *  * 160   *  * |  0 0  1  1  trip
  8 |   8   4 |   0   0   2   4  0 |  *  *  *   * 120  * |  0 1  1  0  cube
 12 |  12  12 |   0   4   0   6  4 |  *  *  *   *   * 80 |  0 1  0  1  co
----+---------+--------------------+---------------------+-----------
 24 |  96   0 |  32  32  24   0  0 |  8  8  4   0   0  0 | 10 *  *  *  ratho
 96 | 192  96 |   0  64  48  96 32 |  0  0  8   0  24 16 |  * 5  *  *  rico
 12 |  24   6 |   8   0   6  12  0 |  2  0  0   4   3  0 |  * * 40  *  thahp
 30 |  60  30 |  20  20   0  30 10 |  0  5  0  10   0  5 |  * *  * 16  srip


--- rk

[polychoronlover]

I know Hehad isn't Wythoffian, but its double covering seems to be represented by the symbol x(ooo,o), and as such Howoh has the symbol x(oxo,x)/2.

I am sure this has all been discovered before, but I realized that operations that can be done to linear diagrams (like cantellation) can also be done to quasiregulars with non-linear diagrams, as long as they don't extend more than one node past the bifurcation.

Code: Select all

For example, the operation x---o---o---x---x can be applied to:
              o-...
             /
x---o---o---o
             \
              o-...
resulting in:
              x-...
             /
x---o---o---x
             \
              x-...


If the bifurcating branches have different labels, applying the "natural" operator will result in the figure having different edge lengths.

Later, I'll reveal how I come up with the 6-D regiment lists, its a fun yet grueling task with 8 tricky phases that are sort of like solving Sudoku puzzles, 'find all the triangles' puzzles, 'match lock and key' puzzles, and many more all rolled up into one.

Let me guess... I would make a list of possible facets and their regiments (which should have all their members enumerated already) that can appear from selecting coplanar/realmer/etc. vertices in the verf, then match them up with sets of ridges by making a ridge list, where each number represents a ridge, then filling it with zeroes and putting in 1 or 2 for all the ridges the facet touches, depending on whether the facet touches the ridge on one side or two.
So I have a database of facets, forming regiments, distinguished by what potential ridges they touch. I try each possible set of facets by adding their ridge lists until I get a list with only zeroes and twos, meaning each ridge touched is touched exactly twice, forming a closed polytope. I actually wrote a program to find honeycomb regiment members this way.
Then there is the problem of eliminating fissaries and compounds of other regiment members. Eliminating inter-regimental compounds is easy enough, but for fissaries I have had to look at the facet structures around each peak and lower-dimensional element and see if they close into more than one figure. I haven't written code to do this yet.

Anyway, I'd still be curious to see how you do it.

I have been thinking also of the locally nonconvex Wythoffian figures. For example, there is

Code: Select all

o-3-o-3-o-3-o
3  3/2
  o

o-3-o-3-o-3-o
    3  3/2
      o

o-3-o-3-o
3 3/23 3/2
  o   o


under hixic symmetry, which looks simple enough, but there is

Code: Select all

o-3-o-3-o
    3   3/2
    o-3-o

  o 
3 3/2
o-3-o
3   3
o3/2o

         o
        3
o--3--o--3--o
  3 3/2
    o


and more under hinnic symmetry, and under higher branched symmetries you have even more options.

[Polyhedron Dude]

The method you used are within some of my phases. Here are the 8 phases:

1. Regiment Map Phase - Using the symbol, I draw a projection of the verf with all possible edges (representing 2-faces). I use colored pens to color code the vertices and the edges in this complete verf. From this I start drawing the regiment map, finding all possible n-regiments inside the polytope into class types. I also calculate how many of each regiment shows up in the verf as well as the entire polytope - this step is one of the fun steps, although it can be a challenge.

2. Facet Ridge Regiments Phase - using the results of phase 1, I write a list of all facet regiments and their associated ridge regiments, I can also calculate how many times the facet meets at a ridge, for those that meet twice, I put () around them, nothing else needs to be connected here. If three or more facets meet at a ridge, I put [] around them, they will crash the polytope. I also write out the reverse version, each ridge regiment and their associated facet regiments. This step also helps me plan a strategy for step 4, to code it in the most efficient way possible.

3. Facet Ridge List Phase - Similar to phase 2, except in more detail, here I write down all possible facets, all members of the regiments with codes to represent the ridges each face has. I might use the numbers 1-7 in red to represent "srip1" and 1-7 in green for "srip2", and purple 1-7 for spid, etc. Some ridges and facets may have two orientations. I usually do phases 1-3 on paper.

4. Coding Phase - Using the info in the previous phases, I write code using POV-Ray. This code will check how many faces meet at a ridge. I use a simple function that gives 0, if there are 0 or 2 facets there and 1 otherwise. These numbers are added, if the total is 0 it passes the test. I also check "lock and key" values making sure the ridges match - we don't want a pirpop ridge matching with a sirdop ridge for instance. The polytopes that pass the tests will be stored in memory, then the code will check for inter-regimental compounds to kick them out. It then is coded to write the list into text form.

5. List Phase - I run the code, I may also need to debug it as well, if something seems off. It can take a few hours to run, depending on the straegy I used and the complexity of the facet regiments. When done, I copy the text and save it as an xlr file. I then search for any duplicates and remove them, this will happen on symmetric regiments where a polytope has two orientations like in the trim regiment xo8ox. I then search for potential pure compounds, they usually only have prismattic facets and no "major" symmetric facets - like siphin or rat regiment members. This list gives me my first count, which still includes fissaries.

6. Draw Facials Phase - This phase can be done anytime after the first phase. The facials include the edge figures, face figures, cell figures. Their edges are color coded also and drawn on paper. I then check each facial to see if fissary behavior can happen there.

7. Facial Code Phase - I then write POV-Ray code to draw the facials, but only the ones where fissariness can happen. It will take the input of several numbers, one for each facet regiment and this will draw the facials. The input numbers come from the list, for instant topax from the spix regiment will use the number 16, and fipyax - 37, and 0 if nothing is there. I also include pure compounds in each face regiment if there are any, but exclude fissaries.

8. Defissing Phase - Using the list from phase 5, I input the numbers and run the program. It draws the facials, I check them to make sure each facial is a single graph and doesn't split. If any of them split, it's fissary (or possibly a pure compound). I write an x or an F on the spreadsheet and color code the fissaries in blue. I also put 1's in a column so they can add up the fissaries. This phase can take a few days. It also helps me see if I have any errors in my original coding phase, for things will not connect right. If this happens I will need to do phase 4 over. Once finished I can give the final count on the regiment and update my regiment.xlr spreadsheet with this regiment count and any other regiments that behave similarly.