Six Dimensional Regiment Counts

[Polyhedron Dude]

Got a new regiment count, but this time it is not 6-D, it is 7-D. Behold the Naq regiment, containing 31 members plus two pure compounds. Naq is the E7 polytope, aka xoo8oo. There are several fissaries that I haven't listed here yet. This one I done by hand, I also coined names. The spreadsheet is located at:

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

The facet regiments in naq are hep, gee, ril, hax (never activated), jak, tacup, hixip, and trapen (found only in the compound zeb). The ridge regiments are hix 1, hix 2, tac, rix, hin, hexip, dot (never active), penp, and tratet. The peak regiments are pen, hex, rap, ope, tepe, and thiddip. The spire regiments are tet, oct, trip, and cube (never active). Finally the 0-2 dimensional regiments are point; dyad; trig and square. One interesting note, there are 7 nobles here, 6 contains a member in the jak regiment for facets, the 7th (called que) has bill (member of the ril regiment) for facets. I have yet to check for subsymmetrics.

[username5243]

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

--- rk

Figured I'd reply to this one.

I know Hedrondude featured a similar polyteron (Gatopin, great pteroprismated penteract, back when he was doing "Polyteron of the Day"). That one was in fawdint regiment (conjugate of span (x4o3o3x3o) regiment). So I would assume topax = pteroprismated hexateron.

[username5243]

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?

One other thing I wondered when I first opened it was, is there a rectified dah and han (members of hin regiment with firp and pinnip verf)? I figured they were in the nit regiment, but on looking on the regiments page, nothing I found could possibly expand as "rectified dah" and "rectified han". Then I checked it out. I'm fairly sure that the rectified dah is the one called hiquah (16quasi16teron) and the rectified han is the one called inoh (invertipenteracti16teron). Neat.

[Mecejide]

I think I found another error: staf is listed as having 19 regiment members, but it should have 23 (3*2^(6-3)-1), like rax.

[Mecejide]

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.

I'm pretty sure it also contains trahip.

[Klitzing]

Thoughts on hejak = x3o3o3o3o *c3x.

Here is a lace prism representation of it wrt. D4 object symmetry:

Code: Select all

            o       n             -- x3o3o *b3o3o (hin)
                              
                              
        e"      ON      e"        -- x3o3x *b3o3o (rin)
                              
                              
    o"      n'      o'      n"    -- x3o3o *b3x3o (sirhin)
                              
                              
e       NE      e'      OE    e   -- o3o3o *b3x3x (tot) + u3o3o *b3o3o (u-hin)
                              
                              
    n       o"      n"      o     -- x3o3o *b3o3x (siphin)


where:
n = x3o3o *b3o (hex)
o = o3o3x *b3o
e = o3o3o *b3x
N = u3o3o *b3o (u-hex)
O = o3o3u *b3o
E = o3o3o *b3u
n' = x3x3o *b3o (thex)
o' = o3x3x *b3o
e' = o3x3o *b3x
n" = o3o3x *b3x (rit)
o" = x3o3o *b3x
e" = x3o3x *b3o

--- rk

[Klitzing]

Similar thoughts on harjak = o3x3o3o3o *c3x provide

Code: Select all

               r1       H2H3       r1                  -- o3x3o *b3o3o (rin)
                                                    
                                                    
                                                    
          t3        A1        B1        t2             -- o3x3x *b3o3o (thin)
                                                    
                                                    
                                                    
     t1        I0        T1        I0        t1        -- o3x3o *b3x3o (bittit)
                                                    
                                                    
                                                    
r3        B3        T3        T2        A2        r2   -- x3o3o *b3x3x (gyro pirhin)
                                                    
                                                    
                                                    
    H1H2       A3        I0        B2       H1H3   
                                                    
                                                    
                                                    
          r3        t2        t3        r2             -- x3o3o *b3x3o (gyro sirhin)


Code: Select all

r1 = o3o3x *b3x,   r2 = x3o3o *b3x,   r3 = x3o3x *b3o   (rit)
H1 = u3o3o *b3o,   H2 = o3o3u *b3o,   H3 = o3o3o *b3u   (u-(hex)
t1 = x3x3o *b3o,   t2 = o3x3x *b3o,   t3 = o3x3o *b3u   (thex)
A1 = o3o3x *b3u,   A2 = u3o3o *b3x,   A3 = x3o3u *b3o   ((x,u)-rit)
B1 = o3o3u *b3x,   B2 = x3o3o *b3u,   B3 = u3o3x *b3o   ((u,x)-rit)
T1 = o3x3x *b3x,   T2 = x3x3o *b3x,   T3 = x3x3x *b3o   (tah)
I0 = o3u3o *b3o   (u-ico)

--- rk

[Mecejide]

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!

Does the 38 include the ones with hipquaxes?