Some lace cities for Gosset-Polytopes

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

Some lace cities for Gosset-Polytopes

Postby wendy » Wed Nov 07, 2018 8:19 am

The dimensional coordinates are 4d by (n-4)d, the tetrahedra lie by
pairs in each of the 16-chora given as points in the second (n-4) figure.

N = (2,0,0,0) normal, or canonical 4-cross
E = (1,1,1,1) half-tesseract with even nr of (-) signs
O = (1,1,1,-1) half-tesseract with odd nr of (-) signs
x = (0,0,0,0) a point
X = (2,2,0,0) a 24-choron.

6D. The opposite space to 4d is 2d,

2_21. Triangle vertex x with edge centres N, E, O
1_22 Hexagon, vertices N,E,O,N,E,O, centre X

7D. The opposite space here is 3d.

3_21. Cube vertices x, face centres marked N,E,O so same letter on
opposite faces. 6*8 + 8 = 56
2_31 Octahedron x, centre point X gives 30 points
Edge centres of octahedron marked N,E,O, such that a diametric
square is marked in the same letter. 12*8+30 = 126

8D The opposite space here is 4d.

4_21 Xx, xX two orthogonal 24-cells makes 48
EE, OO, NN three bi-16choron prisms makes 192
The dream you dream alone is only a dream
the dream we dream together is reality.

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Re: Some lace cities for Gosset-Polytopes

Postby Klitzing » Wed Nov 07, 2018 9:11 am

Yep, Wendy,

naq = 3_21 indeed has
Code: Select all
     _p-------------p
   /      N     _/  |
p-----+-------p     |
|            O|     |
|  E  |       |  E  |
|      O      |     |
|     p - - - + - - p
|  /      N   | _/   
p-------------p     

where:
p = o3o3o *b3o (point)
N = x3o3o *b3o (hex)
E = o3o3x *b3o (gyro hex)
O = o3o3o *b3x (alt. gyro hex)

thus vertically: o3o3o *b3o3o3x (gee) || x3o3o *b3o3o3o (hax) || o3o3o *b3o3o3x (gee)
or diametrically: o3o3o3o3o *c3o (point) || x3o3o3o3o *c3o (jak) || o3o3o3o3x *c3o alt. jak || o3o3o3o3o *c3o (point)
or face-diagonally: x o3o3o *c3o3o (line) || o x3o3o *c3o3o (perp hin) || x o3o3o *c3o3x (taccup) || o o3o3x *c3o3o (perp alt hin) || x o3o3o *c3o3o (line)

laq = 2_31 indeed is
Code: Select all
                    p                   
                  /|\ \                 
                /    \  \               
              /       E   \             
            /     |    \    \           
          O             \     O         
        /            _  p _     \       
      /        _ E-      |  - _   \     
    /    _  N                   N _ \   
  /_  -                             - \ 
p _             |   X   |            _  p
  \ - _                        _  -   / 
    \   N _              _  N       /   
      \     - _|   _  -E          /     
        \       p               /       
          O     \             O         
            \    \    |     /           
              \   E       /             
                \  \    /               
                  \ \|/                 
                    p                   

where:
p = o3o3o *b3o (point)
N = x3o3o *b3o (hex)
E = o3o3x *b3o (gyro hex)
O = o3o3o *b3x (alt. gyro hex)
X = o3x3o *b3o (ico)

thus vertically: o3o3o *b3o3o3o (point) || x3o3o *b3o3o3o (hax) || o3o3o *b3o3x3o (rag) || x3o3o *b3o3o3o (hax) || o3o3o *b3o3o3o (point)
or face-2-face: x3o3o3o3o *c3o (jak) || o3o3o3o3o *c3x (mo) || o3o3o3o3x *c3o (alt. jak)
or line-2-line: o o3o3o *c3o3x (tac) || x x3o3o *c3o3o (hinnip) || compound of u o3o3o *c3o3o (perp u-line) and o o3o3o *c3x3o (rat) || x o3o3x *c3o3o (alt. hinnip) || o o3o3o *c3o3x (tac)


and additionally furthermore
lin = 1_32 similarily can be given as
Code: Select all
              O           E           
       N           X           N     
            E           O             
                                      
                                      
        N           X           N     
 O           S           T           E
      X           R           X       
           O           E             
                                      
              E           O           
       X           R           X     
E           T           S           O
     N           X           N       
                                      
                                      
             O           E           
      N           X           N       
           E           O             

where:
N = x3o3o *b3o (hex)
E = o3o3x *b3o (gyro hex)
O = o3o3o *b3x (alt. gyro hex)
X = o3x3o *b3o (ico)
R = o3o3x *b3x (rit)
S = x3o3o *b3x (gyro rit)
T = x3o3x *b3o (alt. gyro rit)

i.e. (N,E,O)-toe with hexagons X-centered plus inscribed (R,S,T)-oct
thus vertically: o3o3o3o3o *c3x (mo) || o3x3o3o3o *c3o (rojak) || o3o3o3x3o *c3o (alt. rojak) || o3o3o3o3o *c3x (mo)
or {4}-2-{4} diagonally: x3o3o *b3o3o3o (hax) || o3o3o *b3x3o3o (brag) || o3o3x *b3o3o3x (alt. sochax) || o3o3o *b3x3o3o (brag) || x3o3o *b3o3o3o (hax)

--- rk
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Re: Some lace cities for Gosset-Polytopes

Postby Klitzing » Mon Dec 16, 2019 6:25 pm

Not quite a lace city, but rather a very interesting representation of fy = 4_21 = o3o3o3o *c3o3o3o3x within a subsymmetry group of E8 occured to me today. Or better to say, I dreamed for that one already for quite long now, but now finally managed to fiddle it out. It is wrt. A2 x A2 x A2 x A2. There you can provide fy as the tegum sum (or convex hull) of

  • x3x o3o o3o o3o
  • o3o x3x o3o o3o
  • o3o o3o x3x o3o
  • o3o o3o o3o x3x
     
  • o3o x3o x3o x3o
  • o3o o3x o3x o3x
  • x3o o3o o3x x3o
  • o3x o3o x3o o3x
  • x3o x3o o3o o3x
  • o3x o3x o3o x3o
  • x3o o3x x3o o3o
  • o3x x3o o3x o3o
That is, 4 mutually perpendicular hexagons plus 8 (out of 32 combinatorically possible) triangle-triprisms (trittip).

Any 3 of the former block orient a mo = 1_22 = o3o3o3o3o *c3x within the thus spanned subspace, where the remaining vertices are provided by the 2 corresponding ones of the latter block each. The hull of that very pair of the latter block each, when taken alone, is just oddimo (= octadeca-diminished mo), i.e. nothing but the tegum sum of 2 tridual trittips.

Note that, while the choice of the orientation of a single trittip within that subspace (given the orientation of the hexagons) allows for 8=2^3 choices, the corresponding adjoin by the tridual counterpart always reduces that choice down to 4 possible oddimo instances each. Any 2 of those here being chosen oddimos are being selected such, that in the commonly being used 4D subspace the corresponding triangle-duoprisms (triddips) always are para-dual (or, equivalently, dual-para). - In fact, this characterisation already makes the choice of the 7 following items unique, when the first of that above 2nd block has been chosen.

So, within these 6D subspaces mo is not really fixed by the hexagons. And in fact this is just like for the cuboctahedron, which too is not fixed by the choice of an equatorial hexagon: the top triangle then still might flip between pointing forward or backward.

--- rk
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Re: Some lace cities for Gosset-Polytopes

Postby wendy » Tue Dec 17, 2019 1:26 pm

The usual way of finding all of this kind of stuff is to use spheres in lattices. Most of the lace cities i derive are from lattices.

The usual starting point is to calculate the density of the relevant lattice. I use Q units, which represent a sphere of diameter sqrt(2) in a unit cube. The units used by Coxeter and Sloane is U units, which represent a sphere of diameter 2 in a sphere of radius 1.

Q units correspond to 1/sqrt(S) for the trigonal lattices (A, D, E), where S = 1+n for An, 4 for Dn and 9-n for En. S is the number of 'stations' or standing points where the lattice might stand in a fixed symmetry. In other words, it represent the kind of 'deep holes' that a lattice has. A, D, E represent the trigonal lattices T1, T2, T3, which means that subsequent layers of A(n-1) are moved 1, 2, or 3 nodes around the symmetry (a polygon of branches, with S nodes). T0 and T4 are also used. T4 is 3_31, as a ring of 8 and zero height, has S = 16-2n, and T0 is simply prismatic layers of An, gives 2n. So if t is the trigonal number,

S = (2-t)^2 + (2-t)n.
The dream you dream alone is only a dream
the dream we dream together is reality.

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