Tesseract and 16-cell faces
This list shows the k-faces of the tesseract and its dual 16-cell.
tesseract projection | |
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The convex hull of this projection is the Bilinski dodecahedron. A map from k-faces (−40 to 40) to lists of tesseract vertices (0 to 15) can be found here. |
16-cell projection | |||
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ternary weight | 0 | 1 | 2 | 3 | 4 |
---|---|---|---|---|---|
number of faces | 0 | 8 | 24 | 32 | 16 |
tesseract faces | t 4 tesseract |
t 3 cube |
t 2 square |
t 1 edge |
t 0 vertex |
16-cell faces | c 4 16-cell |
c 0 vertex |
c 1 edge |
c 2 triangle |
c 3 tetrahedron |
The sign vectors (with entries 1, 0, −1) in the following table are the face centers of the tesseract. (Compare example for the cube.)
The indices on the left (between −40 and 40) are their interpretation as little-endian balanced ternary numbers.
The columns to their right are their sums, patterns of non-zero entries, and Hamming weights (numbers of non-zero entries).
The default order of the table is first by Hamming weights, then by patterns, and then by sums.
index | b. t. vector |
s | p | w | tesseract face type |
tesseract vertices |
tesseract image |
16-cell face type |
16-cell image |
---|---|---|---|---|---|---|---|---|---|
0 | ![]() |
0 | 0 .... |
0 | t 4 tesseract |
!!!! !!!! !!!! !!!! | ![]() |
c 4 16-cell |
|
−1 | ![]() |
−1 | 1 !... |
1 | t 3 cube |
!.!. !.!. !.!. !.!. | ![]() |
c 0 vertex |
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1 | ![]() |
1 | 1 !... |
1 | t 3 cube |
.!.! .!.! .!.! .!.! | ![]() |
c 0 vertex |
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−3 | ![]() |
−1 | 2 .!.. |
1 | t 3 cube |
!!.. !!.. !!.. !!.. | ![]() |
c 0 vertex |
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3 | ![]() |
1 | 2 .!.. |
1 | t 3 cube |
..!! ..!! ..!! ..!! | ![]() |
c 0 vertex |
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−9 | ![]() |
−1 | 4 ..!. |
1 | t 3 cube |
!!!! .... !!!! .... | ![]() |
c 0 vertex |
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9 | ![]() |
1 | 4 ..!. |
1 | t 3 cube |
.... !!!! .... !!!! | ![]() |
c 0 vertex |
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−27 | ![]() |
−1 | 8 ...! |
1 | t 3 cube |
!!!! !!!! .... .... | ![]() |
c 0 vertex |
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27 | ![]() |
1 | 8 ...! |
1 | t 3 cube |
.... .... !!!! !!!! | ![]() |
c 0 vertex |
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−4 | ![]() |
−2 | 3 !!.. |
2 | t 2 square |
!... !... !... !... | ![]() |
c 1 edge |
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−2 | ![]() |
0 | 3 !!.. |
2 | t 2 square |
.!.. .!.. .!.. .!.. | ![]() |
c 1 edge |
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2 | ![]() |
0 | 3 !!.. |
2 | t 2 square |
..!. ..!. ..!. ..!. | ![]() |
c 1 edge |
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4 | ![]() |
2 | 3 !!.. |
2 | t 2 square |
...! ...! ...! ...! | ![]() |
c 1 edge |
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−10 | ![]() |
−2 | 5 !.!. |
2 | t 2 square |
!.!. .... !.!. .... | ![]() |
c 1 edge |
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−8 | ![]() |
0 | 5 !.!. |
2 | t 2 square |
.!.! .... .!.! .... | ![]() |
c 1 edge |
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8 | ![]() |
0 | 5 !.!. |
2 | t 2 square |
.... !.!. .... !.!. | ![]() |
c 1 edge |
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10 | ![]() |
2 | 5 !.!. |
2 | t 2 square |
.... .!.! .... .!.! | ![]() |
c 1 edge |
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−12 | ![]() |
−2 | 6 .!!. |
2 | t 2 square |
!!.. .... !!.. .... | ![]() |
c 1 edge |
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−6 | ![]() |
0 | 6 .!!. |
2 | t 2 square |
..!! .... ..!! .... | ![]() |
c 1 edge |
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6 | ![]() |
0 | 6 .!!. |
2 | t 2 square |
.... !!.. .... !!.. | ![]() |
c 1 edge |
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12 | ![]() |
2 | 6 .!!. |
2 | t 2 square |
.... ..!! .... ..!! | ![]() |
c 1 edge |
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−28 | ![]() |
−2 | 9 !..! |
2 | t 2 square |
!.!. !.!. .... .... | ![]() |
c 1 edge |
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−26 | ![]() |
0 | 9 !..! |
2 | t 2 square |
.!.! .!.! .... .... | ![]() |
c 1 edge |
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26 | ![]() |
0 | 9 !..! |
2 | t 2 square |
.... .... !.!. !.!. | ![]() |
c 1 edge |
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28 | ![]() |
2 | 9 !..! |
2 | t 2 square |
.... .... .!.! .!.! | ![]() |
c 1 edge |
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−30 | ![]() |
−2 | 10 .!.! |
2 | t 2 square |
!!.. !!.. .... .... | ![]() |
c 1 edge |
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−24 | ![]() |
0 | 10 .!.! |
2 | t 2 square |
..!! ..!! .... .... | ![]() |
c 1 edge |
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24 | ![]() |
0 | 10 .!.! |
2 | t 2 square |
.... .... !!.. !!.. | ![]() |
c 1 edge |
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30 | ![]() |
2 | 10 .!.! |
2 | t 2 square |
.... .... ..!! ..!! | ![]() |
c 1 edge |
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−36 | ![]() |
−2 | 12 ..!! |
2 | t 2 square |
!!!! .... .... .... | ![]() |
c 1 edge |
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−18 | ![]() |
0 | 12 ..!! |
2 | t 2 square |
.... !!!! .... .... | ![]() |
c 1 edge |
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18 | ![]() |
0 | 12 ..!! |
2 | t 2 square |
.... .... !!!! .... | ![]() |
c 1 edge |
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36 | ![]() |
2 | 12 ..!! |
2 | t 2 square |
.... .... .... !!!! | ![]() |
c 1 edge |
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−13 | ![]() |
−3 | 7 !!!. |
3 | t 1 edge |
0, 8 | ![]() |
c 2 triangle |
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−11 | ![]() |
−1 | 7 !!!. |
3 | t 1 edge |
1, 9 | ![]() |
c 2 triangle |
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−7 | ![]() |
−1 | 7 !!!. |
3 | t 1 edge |
2, 10 | ![]() |
c 2 triangle |
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5 | ![]() |
−1 | 7 !!!. |
3 | t 1 edge |
4, 12 | ![]() |
c 2 triangle |
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−5 | ![]() |
1 | 7 !!!. |
3 | t 1 edge |
3, 11 | ![]() |
c 2 triangle |
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7 | ![]() |
1 | 7 !!!. |
3 | t 1 edge |
5, 13 | ![]() |
c 2 triangle |
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11 | ![]() |
1 | 7 !!!. |
3 | t 1 edge |
6, 14 | ![]() |
c 2 triangle |
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13 | ![]() |
3 | 7 !!!. |
3 | t 1 edge |
7, 15 | ![]() |
c 2 triangle |
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−31 | ![]() |
−3 | 11 !!.! |
3 | t 1 edge |
0, 4 | ![]() |
c 2 triangle |
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−29 | ![]() |
−1 | 11 !!.! |
3 | t 1 edge |
1, 5 | ![]() |
c 2 triangle |
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−25 | ![]() |
−1 | 11 !!.! |
3 | t 1 edge |
2, 6 | ![]() |
c 2 triangle |
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23 | ![]() |
−1 | 11 !!.! |
3 | t 1 edge |
8, 12 | ![]() |
c 2 triangle |
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−23 | ![]() |
1 | 11 !!.! |
3 | t 1 edge |
3, 7 | ![]() |
c 2 triangle |
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25 | ![]() |
1 | 11 !!.! |
3 | t 1 edge |
9, 13 | ![]() |
c 2 triangle |
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29 | ![]() |
1 | 11 !!.! |
3 | t 1 edge |
10, 14 | ![]() |
c 2 triangle |
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31 | ![]() |
3 | 11 !!.! |
3 | t 1 edge |
11, 15 | ![]() |
c 2 triangle |
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−37 | ![]() |
−3 | 13 !.!! |
3 | t 1 edge |
0, 2 | ![]() |
c 2 triangle |
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−35 | ![]() |
−1 | 13 !.!! |
3 | t 1 edge |
1, 3 | ![]() |
c 2 triangle |
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−19 | ![]() |
−1 | 13 !.!! |
3 | t 1 edge |
4, 6 | ![]() |
c 2 triangle |
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17 | ![]() |
−1 | 13 !.!! |
3 | t 1 edge |
8, 10 | ![]() |
c 2 triangle |
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−17 | ![]() |
1 | 13 !.!! |
3 | t 1 edge |
5, 7 | ![]() |
c 2 triangle |
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19 | ![]() |
1 | 13 !.!! |
3 | t 1 edge |
9, 11 | ![]() |
c 2 triangle |
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35 | ![]() |
1 | 13 !.!! |
3 | t 1 edge |
12, 14 | ![]() |
c 2 triangle |
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37 | ![]() |
3 | 13 !.!! |
3 | t 1 edge |
13, 15 | ![]() |
c 2 triangle |
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−39 | ![]() |
−3 | 14 .!!! |
3 | t 1 edge |
0, 1 | ![]() |
c 2 triangle |
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−33 | ![]() |
−1 | 14 .!!! |
3 | t 1 edge |
2, 3 | ![]() |
c 2 triangle |
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−21 | ![]() |
−1 | 14 .!!! |
3 | t 1 edge |
4, 5 | ![]() |
c 2 triangle |
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15 | ![]() |
−1 | 14 .!!! |
3 | t 1 edge |
8, 9 | ![]() |
c 2 triangle |
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−15 | ![]() |
1 | 14 .!!! |
3 | t 1 edge |
6, 7 | ![]() |
c 2 triangle |
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21 | ![]() |
1 | 14 .!!! |
3 | t 1 edge |
10, 11 | ![]() |
c 2 triangle |
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33 | ![]() |
1 | 14 .!!! |
3 | t 1 edge |
12, 13 | ![]() |
c 2 triangle |
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39 | ![]() |
3 | 14 .!!! |
3 | t 1 edge |
14, 15 | ![]() |
c 2 triangle |
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−40 | ![]() |
−4 | 15 !!!! |
4 | t 0 vertex |
0 | ![]() |
c 3 tetrahedron |
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−38 | ![]() |
−2 | 15 !!!! |
4 | t 0 vertex |
1 | ![]() |
c 3 tetrahedron |
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−34 | ![]() |
−2 | 15 !!!! |
4 | t 0 vertex |
2 | ![]() |
c 3 tetrahedron |
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−22 | ![]() |
−2 | 15 !!!! |
4 | t 0 vertex |
4 | ![]() |
c 3 tetrahedron |
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14 | ![]() |
−2 | 15 !!!! |
4 | t 0 vertex |
8 | ![]() |
c 3 tetrahedron |
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−32 | ![]() |
0 | 15 !!!! |
4 | t 0 vertex |
3 | ![]() |
c 3 tetrahedron |
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−20 | ![]() |
0 | 15 !!!! |
4 | t 0 vertex |
5 | ![]() |
c 3 tetrahedron |
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−16 | ![]() |
0 | 15 !!!! |
4 | t 0 vertex |
6 | ![]() |
c 3 tetrahedron |
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16 | ![]() |
0 | 15 !!!! |
4 | t 0 vertex |
9 | ![]() |
c 3 tetrahedron |
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20 | ![]() |
0 | 15 !!!! |
4 | t 0 vertex |
10 | ![]() |
c 3 tetrahedron |
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32 | ![]() |
0 | 15 !!!! |
4 | t 0 vertex |
12 | ![]() |
c 3 tetrahedron |
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−14 | ![]() |
2 | 15 !!!! |
4 | t 0 vertex |
7 | ![]() |
c 3 tetrahedron |
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22 | ![]() |
2 | 15 !!!! |
4 | t 0 vertex |
11 | ![]() |
c 3 tetrahedron |
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34 | ![]() |
2 | 15 !!!! |
4 | t 0 vertex |
13 | ![]() |
c 3 tetrahedron |
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38 | ![]() |
2 | 15 !!!! |
4 | t 0 vertex |
14 | ![]() |
c 3 tetrahedron |
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40 | ![]() |
4 | 15 !!!! |
4 | t 0 vertex |
15 | ![]() |
c 3 tetrahedron |
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Python fragment |
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face_to_signs = {-40: '−−−−', -39: '0−−−', -38: '+−−−', -37: '−0−−', -36: '00−−', -35: '+0−−', -34: '−+−−', -33: '0+−−', -32: '++−−', -31: '−−0−', -30: '0−0−', -29: '+−0−', -28: '−00−', -27: '000−', -26: '+00−', -25: '−+0−', -24: '0+0−', -23: '++0−', -22: '−−+−', -21: '0−+−', -20: '+−+−', -19: '−0+−', -18: '00+−', -17: '+0+−', -16: '−++−', -15: '0++−', -14: '+++−', -13: '−−−0', -12: '0−−0', -11: '+−−0', -10: '−0−0', -9: '00−0', -8: '+0−0', -7: '−+−0', -6: '0+−0', -5: '++−0', -4: '−−00', -3: '0−00', -2: '+−00', -1: '−000', 0: '0000', 1: '+000', 2: '−+00', 3: '0+00', 4: '++00', 5: '−−+0', 6: '0−+0', 7: '+−+0', 8: '−0+0', 9: '00+0', 10: '+0+0', 11: '−++0', 12: '0++0', 13: '+++0', 14: '−−−+', 15: '0−−+', 16: '+−−+', 17: '−0−+', 18: '00−+', 19: '+0−+', 20: '−+−+', 21: '0+−+', 22: '++−+', 23: '−−0+', 24: '0−0+', 25: '+−0+', 26: '−00+', 27: '000+', 28: '+00+', 29: '−+0+', 30: '0+0+', 31: '++0+', 32: '−−++', 33: '0−++', 34: '+−++', 35: '−0++', 36: '00++', 37: '+0++', 38: '−+++', 39: '0+++', 40: '++++'}
for face_int, face_signs in face_to_signs.items():
indices = [i for i, x in enumerate(face_signs) if x != '0']
weight = len(indices)
tess_dim = 4 - weight
cross_dim = [4, 0, 1, 2, 3][weight]
tess_type = ['vertex', 'edge', 'square', 'cube', 'tesseract'][tess_dim]
cross_type = ['vertex', 'edge', 'triangle', 'tetrahedron', '16-cell'][cross_dim]
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