User:Tomruen/configuration



The configuration matrix shows the number of k-face elements along the diagonal, while the nondiagonal element show the incidence counts between all elements. The number of elements of its facets can be seen on the bottom row, left of the diagonal, and k-face elements above that. The top row, right of the diagonal represent the number of elements of the vertex figure. The second row contains the edge-figures, and so on. These figures are the duals of the k-faces of the dual polytope, which can be seen by rotating the matrix 180 degrees.

For regular n-polytopes, the there are only one type of element, so the matrix is n×n. For irregular polytopes, the matrix is expanded with one row per element type, which in the limit contains one row for every element. Like a general polyhedron with v vertices, e edges and f faces would have v+e+f total rows and columns.

8-honeycombs
o3o3o3o *c3o3o3o3o3x - goh o3o3o3o *c3o3o3o3o3x  (N → ∞)

. . . .   . . . . . |  N ♦  240 |  6720 |  60480 | 241920 | 483840 | 483840 | 138240 69120 | 17280 2160 -++--+---+++++--+--- . . . .   . . . . x |  2 | 120N ♦    56 |    756 |   4032 |  10080 |  12096 |   4032  2016 |   576  126 -++--+---+++++--+--- . . . .   . . . o3x |  3 |    3 | 2240N ♦     27 |    216 |    720 |   1080 |    432   216 |    72   27 -++--+---+++++--+--- . . . .   . . o3o3x ♦  4 |    6 |     4 | 15120N ♦     16 |     80 |    160 |     80    40 |    16   10 -++--+---+++++--+--- . . . .   . o3o3o3x ♦  5 |   10 |    10 |      5 | 48384N ♦     10 |     30 |     20    10 |     5    5 -++--+---+++++--+--- . . . .   o3o3o3o3x ♦  6 |   15 |    20 |     15 |      6 | 80640N ♦      6 |      6     3 |     2    3 -++--+---+++++--+--- . . o. *c3o3o3o3o3x ♦ 7 |   21 |    35 |     35 |     21 |      7 | 69120N |      2     1 |     1    2 -++--+---+++++--+--- . o3o. *c3o3o3o3o3x ♦ 8 |   28 |    56 |     70 |     56 |     28 |      8 | 17280N     * |     1    1 . . o3o *c3o3o3o3o3x ♦ 8 |   28 |    56 |     70 |     56 |     28 |      8 |      * 8640N |     0    2 -++--+---+++++--+--- o3o3o. *c3o3o3o3o3x ♦ 9 |   36 |    84 |    126 |    126 |     84 |     36 |      8     0 | 1920N    * . o3o3o *c3o3o3o3o3x ♦ 16 | 112 |   448 |   1120 |   1792 |   1792 |   1024 |    128   128 |     * 135N

Computation
The f-vector values, seen on the diagonal, are computed by systematically removing nodes (mirrors) from the Kaleidoscope. The element of a given set of removals is defined by the set of nodes connected to at least one ringed nodes. The number of elements of that type is computed from the full order of the Coxeter group divided by the order of the remaining mirrors. If groups of mirrors are not connected, the order is the product of all such connected groups remaining.

Truncated cuboctahedron
Example truncated cuboctahedron, with all mirrors active, all 1+3+3+1 fundamental domain simplex positions contain elements.

5-cell
x3o3o3o - pen

rectified 5-cell
o3x3o3o - rap

Truncated 5-cell
x3x3o3o - tip

Cantellated 5-cell
x3o3x3o - srip

runcinated 5-cell
x3o3o3x - spid

Bitruncated 5-cell
o3x3x3o - deca

Runcitruncated 5-cell
x3x3o3x - prip

Runcitruncated 5-cell
x3x3o3x - prip

Omnitruncated 5-cell
x3x3x3x - gippid

24-cell
x3o4o3o - ico

Rectified 24-cell
o3x4o3o - rico

Truncated 24-cell
x3x4o3o - tico

Cantellated 24-cell
x3o4x3o - sric

Runcinated 24-cell
x3o4o3x - spic

Bitruncated 24-cell
o3x4x3o - cont

Cantitruncated 24-cell
x3x4x3o - grico

Runcitruncated 24-cell
x3x4o3x - prico

Omnitruncated 24-cell
x3x4x3x - gippic

Snub 24-cell
Example: snub 24-cell

Omnitruncated tesseract
Example on omnitruncated tesseract. An omnitruncated 4-polytope will have 2^4-1 or 15 types of elements.

0_31
Example rectified 5-simplex

0_22
Example birectified 5-simplex

1_21
Example 5-demicube:

1_31
Example 6-demicube

2_21
Example on 2_21 polytope:

1_22
Example on 1_22 polytope:

0_221
Example Rectified 1_22 polytope

Omnitruncated 6-simplex
Example: Omnitruncated 6-simplex BIG TEST!

1_41
Example on 7-demicube:

3_21
Example on 3_21 polytope:

2_31
Example on 2_31 polytope:

1_32
Example on 1_32 polytope:

0_321
Example on rectified 1_32 polytope:

8-cube
Example on 8-cube. A regular n-polytope will have n types of elements, one for each dimension.

4_21
Example on 4_21 polytope:

2_41
Example on 2_41 polytope:

1_42
Example on 1_42 polytope:

0_421
Example on rectified 1_42 polytope: