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 Sloane'sA007405, the Dowling numbers.
Their number by dimension is row 4 of Sloane'sA039755, 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[edit]

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 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[edit]

Tesseract subspace 4.png Balanced ternary 4, positive grouped by weight.svg

The tesseract contains all of the 81 face centers.
So its set of positive face centers is the whole list from to .

Only the neutral permutation leaves the whole tesseract unchanged.
​0​0  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[edit]

3a 4 cubes with edge length (green) ​odd​0  00
projections
0Tesseract subspace 3a0.png   ​8​0 1Tesseract subspace 3a1.png   ​4​0 2Tesseract subspace 3a2.png   ​2​0 3Tesseract subspace 3a3.png   ​1​0
3b 12 cuboids with edge lengths (orange) and (green) ​even​green  00
1100, 0011
projections
0Tesseract subspace 3b00.png   ​12​6 5Tesseract subspace 3b05.png   ​3​1
11Tesseract subspace 3b11.png   ​0​6 6Tesseract subspace 3b06.png   ​0​1

2-dimensional[edit]

2a 6 squares with edge length (blue) ​even​0  00
1100, 0011
projections
0Tesseract subspace 2a0.png   ​12​0 5Tesseract subspace 2a5.png   ​3​0
2b 24 rectangles with edge lengths (green) and (blue) ​odd​green  00
1000
projections
0Tesseract subspace 2b00.png   ​14​2 6Tesseract subspace 2b06.png   ​14​14 12Tesseract subspace 2b12.png   ​14​6
5Tesseract subspace 2b05.png   ​8​2 11Tesseract subspace 2b11.png   ​4​14 17Tesseract subspace 2b17.png   ​2​6
2c 12 squares with edge length (green) ​even​bold  00
1100, 0011
projections
0Tesseract subspace 2c00.png   ​15​7 1Tesseract subspace 2c01.png   ​12​7 8Tesseract subspace 2c08.png   ​3​7 11Tesseract subspace 2c11.png   ​0​7
2d 16 rectangles with edge lengths (orange) and (blue)
1000
projections
0Tesseract subspace 2d00.png 3Tesseract subspace 2d03.png 8Tesseract subspace 2d08.png 15Tesseract subspace 2d15.png

1-dimensional[edit]

1a 4 line segments with edge length (between opposite blue points) ​odd​0  00
Tesseract subspace 1a0.png
0Vector +000.svg   ​14​0
Tesseract subspace 1a1.png
1Vector 0+00.svg   ​13​0
Tesseract subspace 1a2.png
2Vector 00+0.svg   ​11​0
Tesseract subspace 1a3.png
3Vector 000+.svg   ​7​0
1b 12 line segments with edge length (between opposite green points) ​even​green  00
1100, 0011
projections
Tesseract subspace 1b00.png
0Vector −+00.svg   ​15​1
Tesseract subspace 1b10.png
10Vector 00−+.svg   ​15​6
Tesseract subspace 1b01.png
1Vector ++00.svg   ​12​1
Tesseract subspace 1b11.png
11Vector 00++.svg   ​3​6
1c 16 line segments with edge length (between opposite yellow points)
1110
projections
Tesseract subspace 1c00.png
0Vector −−+0.svg
Tesseract subspace 1c01.png
1Vector +−+0.svg
Tesseract subspace 1c02.png
2Vector −++0.svg
Tesseract subspace 1c03.png
3Vector +++0.svg
1d 8 line segments with edge length (between opposite red points)
Tesseract subspace 1d0.png
0Vector −−−+.svg
Tesseract subspace 1d1.png
1Vector +−−+.svg
Tesseract subspace 1d2.png
2Vector −+−+.svg
Tesseract subspace 1d3.png
3Vector ++−+.svg
Tesseract subspace 1d4.png
4Vector −−++.svg
Tesseract subspace 1d5.png
5Vector +−++.svg
Tesseract subspace 1d6.png
6Vector −+++.svg
Tesseract subspace 1d7.png
7Vector ++++.svg

0-dimensional[edit]

permutations
Tesseract subspace 0.png

The origin has the coordinate Vector 0000.svg and number value . So its set of positive face centers is empty.

105 permutations in 5 conjugacy classes leave only the origin unchanged.
​15​0  00

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


Numbering[edit]

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

Code[edit]

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)"

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