Category:Tesseract subspaces (image set)

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The shapes of subspaces with dimension 1 to 3 ordered by inclusion
The 76 self-inverse permutations in 9 conjugacy classes

These are the sets of fixed points of permutations of the tesseract, i.e. mainly its equivalents of symmetry axes and mirror planes,
which have 1, 2, and 3 dimensions (plus the origin and the whole space as least and greatest elements).

Their total number is 116. That is entry 4 of , the Dowling numbers.
Their number by dimension is row 4 of , the B-analogs of Stirling numbers of the second kind.

 4-dim. 1 1 3-dim. 4, 12 16 2-dim. 6, 24, 12, 16 58 1-dim. 4, 12, 16, 8 40 0-dim. 1 1 116

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 ${\displaystyle (m,n)}$ 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 ${\displaystyle 1}$ to ${\displaystyle 40}$. 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 ${\displaystyle 1}$ (green)  00
projections
0    1    2    3
3b 12 cuboids with edge lengths ${\displaystyle 1}$ (orange) and ${\displaystyle {\sqrt {2}}}$ (green)  00
1100, 0011
projections
0    5
11    6

2-dimensional

2a 6 squares with edge length ${\displaystyle 1}$ (blue)  00
1100, 0011
projections
0    5
2b 24 rectangles with edge lengths ${\displaystyle 1}$ (green) and ${\displaystyle {\sqrt {2}}}$ (blue)  00
1000
projections
0    6    12
5    11    17
2c 12 squares with edge length ${\displaystyle {\sqrt {2}}}$ (green)  00
1100, 0011
projections
0    1    8    11
2d 16 rectangles with edge lengths ${\displaystyle 1}$ (orange) and ${\displaystyle {\sqrt {3}}}$ (blue)
1000
projections
0 3 8 15

1-dimensional

1a 4 line segments with edge length ${\displaystyle 1}$ (between opposite blue points)  00

0

1

2

3
1b 12 line segments with edge length ${\displaystyle {\sqrt {2}}}$ (between opposite green points)  00
1100, 0011
projections

0

10

1

11
1c 16 line segments with edge length ${\displaystyle {\sqrt {3}}}$ (between opposite yellow points)
1110
projections

0

1

2

3
1d 8 line segments with edge length ${\displaystyle {\sqrt {4}}=2}$ (between opposite red points)

0

1

2

3

4

5

6

7

0-dimensional

permutations
 The origin has the coordinate and number value ${\displaystyle 0}$. 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): [
()
]


Numbering

The numbers used in the filename refer to the the colexicographic ordering of the positive vertices.
This allows to use the same identifiers for all dimensions. (E.g. cube subspace 2b5 and tesseract subspace 2b05 are the same rectangle with vertices 11 and 13.)
But it is more intuitive to use lexicographic order that takes all face centers into account. (It allows sorting by patterns like 1000.)
The following table shows the lexicographic order of the four kinds of subspaces where the orders differ. (compare code)

3b 0, 11,     1, 10,     3, 8,     2, 9,     4, 7,     5, 6 0, 5, 6, 11, 12, 17,     1, 4, 7, 10, 18, 23,     2, 3, 13, 16, 19, 22,     8, 9, 14, 15, 20, 21 3, 6, 5, 9,     2, 4, 7, 10,     0, 1, 8, 11 0, 3, 8, 15,     1, 2, 11, 14,     4, 6, 10, 13,     5, 7, 9, 12

Code

These images have been rendered with POV-Ray, and the calculations have been done with Python. The code can be found on GitHub.
The main POV-Ray file is subspaces.pov.
The colored code sections shown above are from the dictionary in e1_store_subspaces.py.

Subcategories

This category has the following 4 subcategories, out of 4 total.

Pages in category "Tesseract subspaces (image set)"

The following 4 pages are in this category, out of 4 total.