Template:Tesseract subspaces; overview

Nested in the following collapsible tables are projections of all 116 subspaces together with a list of the positive face centers they contain.
The balanced ternary coordinates suggest a tesseract with ±1 vertex coordinates, i.e. with edge length 2. Anyway, the lengths mentioned below refer to a tesseract with edge length 1.

Each subspace is the set of fixed points of at least one permutation. If there is more than one, they are shown in a 16×24 matrix.

76 subspaces of 9 types have a unique self-inverse permutation. The pair $(m,n)$ of this permutation is shown next to the projection of the subspace.
The self-inverse permutation is unique (e.g. a 180° rotation), but there can be other permutations with the same set of fixed points (e.g. two 90° rotations).

4-dimensional

The tesseract contains all of the 81 face centers. So its set of positive face centers is the whole list from $1$ to $40$. Only the neutral permutation leaves the whole tesseract unchanged.    00

(4, 12, 16, 8): [ 
    (1, 3, 9, 27, 2, 4, 8, 10, 6, 12, 26, 28, 24, 30, 18, 36, 5, 7, 11, 13, 23, 25, 29, 31, 17, 19, 35, 37, 15, 21, 33, 39, 14, 16, 20, 22, 32, 34, 38, 40)
]

3-dimensional

3a 4 cubes with edge length $1$ (green)    00
list (3, 6, 4, 0): [ (1, 3, 9, 2, 4, 8, 10, 6, 12, 5, 7, 11, 13), (1, 3, 27, 2, 4, 26, 28, 24, 30, 23, 25, 29, 31), # 3a0, 3a1 (1, 9, 27, 8, 10, 26, 28, 18, 36, 17, 19, 35, 37), (3, 9, 27, 6, 12, 24, 30, 18, 36, 15, 21, 33, 39) # 3a2, 3a3 ] projections 0     1     2     3     coordinates 0 1 2 3

3b 12 cuboids with edge lengths $1$ (orange) and ${\sqrt {2}}$ (green)    00

list (2, 3, 4, 4): [ (1, 3, 2, 4, 18, 17, 19, 15, 21, 14, 16, 20, 22), (1, 9, 8, 10, 24, 23, 25, 15, 33, 14, 16, 32, 34), # 3b00, 3b01 (1, 27, 6, 26, 28, 5, 7, 21, 33, 20, 22, 32, 34), (3, 9, 6, 12, 26, 23, 29, 17, 35, 14, 20, 32, 38), # 3b02, 3b03 (3, 27, 8, 24, 30, 5, 11, 19, 35, 16, 22, 32, 38), (9, 27, 2, 18, 36, 7, 11, 25, 29, 16, 20, 34, 38), # 3b04, 3b05 (9, 27, 4, 18, 36, 5, 13, 23, 31, 14, 22, 32, 40), (3, 27, 10, 24, 30, 7, 13, 17, 37, 14, 20, 34, 40), # 3b06, 3b07 (3, 9, 6, 12, 28, 25, 31, 19, 37, 16, 22, 34, 40), (1, 27, 12, 26, 28, 11, 13, 15, 39, 14, 16, 38, 40), # 3b08, 3b09 (1, 9, 8, 10, 30, 29, 31, 21, 39, 20, 22, 38, 40), (1, 3, 2, 4, 36, 35, 37, 33, 39, 32, 34, 38, 40) # 3b10, 3b11 ] 1100, 0011 projections 0     5     11     6     coordinates 0 5 11 6 1010, 0101 projections 1     4     10     7     coordinates 1 4 10 7 0110, 1001 projections 3     2     8     9     coordinates 3 2 8 9

2-dimensional

2a 6 squares with edge length $1$ (blue)    00
list (2, 2, 0, 0): [ (1, 3, 2, 4), (1, 9, 8, 10), (3, 9, 6, 12), (1, 27, 26, 28), (3, 27, 24, 30), (9, 27, 18, 36) # 2a0..2a5 ] 1100, 0011 projections 0     5     coordinates 0 5 permutations 0 5 1010, 0101 projections 1     4     coordinates 1 4 permutations 1 4 0110, 1001 projections 2     3     coordinates 2 3 permutations 2 3

2b 24 rectangles with edge lengths $1$ (green) and ${\sqrt {2}}$ (blue)    00

list (1, 1, 2, 0): [ (1, 6, 5, 7), (3, 8, 5, 11), (9, 2, 7, 11), (9, 4, 5, 13), (3, 10, 7, 13), (1, 12, 11, 13), # 2b00..2b05 (1, 24, 23, 25), (3, 26, 23, 29), (27, 2, 25, 29), (27, 4, 23, 31), (3, 28, 25, 31), (1, 30, 29, 31), # 2b06..2b11 (1, 18, 17, 19), (9, 26, 17, 35), (27, 8, 19, 35), (27, 10, 17, 37), (9, 28, 19, 37), (1, 36, 35, 37), # 2b12..2b17 (3, 18, 15, 21), (9, 24, 15, 33), (27, 6, 21, 33), (27, 12, 15, 39), (9, 30, 21, 39), (3, 36, 33, 39) # 2b18..2b23 ] 1000 projections 0     6     12     5     11     17     coordinates 0 6 12 5 11 17 0100 projections 1     7     18     4     10     23     coordinates 1 7 18 4 10 23 0010 projections 2     13     19     3     16     22     coordinates 2 13 19 3 16 22 0001 projections 8     14     20     9     15     21     coordinates 8 14 20 9 15 21

2c 12 squares with edge length ${\sqrt {2}}$ (green)    00

list (0, 2, 0, 2): [ (2, 18, 16, 20), (4, 18, 14, 22), (8, 24, 16, 32), (6, 26, 20, 32), (10, 24, 14, 34), (6, 28, 22, 34), # 2c00..2c05 (12, 26, 14, 38), (8, 30, 22, 38), (2, 36, 34, 38), (12, 28, 16, 40), (10, 30, 20, 40), (4, 36, 32, 40) # 2c06..2c11 ] 1100, 0011 projections 0     1     8     11     coordinates 0 1 8 11 1010, 0101 projections 2     4     7     10     coordinates 2 4 7 10 0110, 1001 projections 3     6     5     9     coordinates 3 6 5 9

2d 16 rectangles with edge lengths $1$ (orange) and ${\sqrt {3}}$ (blue)

list (1, 0, 1, 2): [ (1, 15, 14, 16), (3, 17, 14, 20), (3, 19, 16, 22), (1, 21, 20, 22), # 2d00..2d03 (9, 23, 14, 32), (27, 5, 22, 32), (9, 25, 16, 34), (27, 7, 20, 34), # 2d04..2d07 (1, 33, 32, 34), (27, 11, 16, 38), (9, 29, 20, 38), (3, 35, 32, 38), # 2d08..2d11 (27, 13, 14, 40), (9, 31, 22, 40), (3, 37, 34, 40), (1, 39, 38, 40) # 2d12..2d15 ] 1000 projections 0 3 8 15 coordinates 0 3 8 15 permutations 0 3 8 15 0100 projections 1 2 11 14 coordinates 1 2 11 14 permutations 1 2 11 14 0010 projections 4 6 10 13 coordinates 4 6 10 13 permutations 4 6 10 13 0001 projections 5 7 9 12 coordinates 5 7 9 12 permutations 5 7 9 12

1-dimensional

1a 4 line segments with edge length $1$ (between opposite blue points)    00
list (1, 0, 0, 0): [ (1,), (3,), (9,), (27,) # 1a0..1a3 ]
0	1	2	3
permutations

1b 12 line segments with edge length ${\sqrt {2}}$ (between opposite green points)    00
list (0, 1, 0, 0): [ (2,), (4,), (8,), (10,), (6,), (12,), (26,), (28,), (24,), (30,), (18,), (36,) # 1b00..1b11 ] 1100, 0011 projections 0     10     1     11     permutations 0 10 1 11 1010, 0101 projections 2     8     3     9     permutations 2 8 3 9 0110, 1001 projections 4     6     5     7     permutations 4 6 5 7

1c 16 line segments with edge length ${\sqrt {3}}$ (between opposite yellow points)
list (0, 0, 1, 0): [ (5,), (7,), (11,), (13,), (23,), (25,), (29,), (31,), (17,), (19,), (35,), (37,), (15,), (21,), (33,), (39,) # 1c00..1c15 ] 1110 projections 0 1 2 3 permutations 0 1 2 3 1101 projections 4 5 6 7 permutations 4 5 6 7 1011 projections 8 9 10 11 permutations 8 9 10 11 0111 projections 12 13 14 15 permutations 12 13 14 15

1d 8 line segments with edge length ${\sqrt {4}}=2$ (between opposite red points)
list (0, 0, 0, 1): [ (14,), (16,), (20,), (22,), (32,), (34,), (38,), (40,) # 1d0..1d7 ]
0	1	2	3	4	5	6	7
permutations

0-dimensional

permutations

The origin has the coordinate and number value $0$. So its set of positive face centers is empty. 105 permutations in 5 conjugacy classes leave only the origin unchanged.    00

(0, 0, 0, 0): [ 
    ()
]