Acronym | tes (alt.: squadip), K-4.20 | |||||||||||||||||||||||||||||||||||
Name |
tesseract, octachoron, "hypercube", 4D measure-polytope (γ4), 8-cell, geoter(id), square duoprism, vertex figure of icot | |||||||||||||||||||||||||||||||||||
|,>,O device | line prism prism prism = |||| | |||||||||||||||||||||||||||||||||||
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Segmentochoron display |
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Cross sections |
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Circumradius | 1 | |||||||||||||||||||||||||||||||||||
Edge radius | sqrt(3)/2 = 0.866025 | |||||||||||||||||||||||||||||||||||
Face radius | 1/sqrt(2) = 0.707107 | |||||||||||||||||||||||||||||||||||
Inradius | 1/2 | |||||||||||||||||||||||||||||||||||
Vertex figure |
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Vertex layers |
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Lace city in approx. ASCII-art |
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o3o q3o o3q o3o o3o q3o o3q o3o | ||||||||||||||||||||||||||||||||||||
Coordinates |
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Volume | 1 | |||||||||||||||||||||||||||||||||||
Surface | 8 | |||||||||||||||||||||||||||||||||||
Rel. Roundness | π2/32 = 30.842514 % | |||||||||||||||||||||||||||||||||||
Pattern |
A---B---A---B--- | | | | C---D---C---D--- | | | | A---B---A---B--- | | | | C---D---C---D--- | | | | | |||||||||||||||||||||||||||||||||||
General of army | (is itself convex) | |||||||||||||||||||||||||||||||||||
Colonel of regiment |
(is itself locally convex
– uniform polychoral members:
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Dual | hex | |||||||||||||||||||||||||||||||||||
Dihedral angles | ||||||||||||||||||||||||||||||||||||
Face vector | 16, 32, 24, 8 | |||||||||||||||||||||||||||||||||||
Confer |
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External links |
Augmenting cubpies onto the cubes would lead to the ico (which would have an even larger symmetry)!
The tesseract displays the easiest non-trivial swirl symmetry: Its cells can be divided into 2 rings of 4 cubes each.
Note that tes can be thought of as the external blend of 8 cubpies. This decomposition is described as the degenerate segmentoteron oo3oo3oo4ox&#x.
The number of ways to color the tesseract with different colors per cell is 8!/192 = 210. – This is because the color group is the permutation group of 8 elements and has size 8!, while the order of the pure rotational tesseractic group is 192. (The reflectional tesseractic group would have twice as many, i.e. 384 elements.)
The second set of coordinates provided above shows that there exists a 4-coloring of the vertices according to the 4 cyclic changes, i.e. there are 4 vertex inscribable differently colored q-squares. Each cube thereby will become a square prism with 2 diagonally colored bases each. The pattern above simply shows this 4-coloring, when applied to the 4,4-duoprism representation. (A similar 3-coloring of the vertices by means of vertex inscribed q-squares within 3D would just span the co.)
Being the dual of hex and considering that one's coordinates, it is apparent that this solid is nothing but a hyperball wrt. the norm max(|x|, |y|, |z|, |w|).
Incidence matrix according to Dynkin symbol
o3o3o4x . . . . | 16 ♦ 4 | 6 | 4 --------+----+----+----+-- . . . x | 2 | 32 | 3 | 3 --------+----+----+----+-- . . o4x | 4 | 4 | 24 | 2 --------+----+----+----+-- . o3o4x ♦ 8 | 12 | 6 | 8 snubbed forms: o3o3o4s, o3o3o4β
o3o3o4/3x . . . . | 16 ♦ 4 | 6 | 4 ----------+----+----+----+-- . . . x | 2 | 32 | 3 | 3 ----------+----+----+----+-- . . o4/3x | 4 | 4 | 24 | 2 ----------+----+----+----+-- . o3o4/3x ♦ 8 | 12 | 6 | 8
o3o3/2o4x . . . . | 16 ♦ 4 | 6 | 4 ----------+----+----+----+-- . . . x | 2 | 32 | 3 | 3 ----------+----+----+----+-- . . o4x | 4 | 4 | 24 | 2 ----------+----+----+----+-- . o3/2o4x ♦ 8 | 12 | 6 | 8
o3o3/2o4/3x . . . . | 16 ♦ 4 | 6 | 4 ------------+----+----+----+-- . . . x | 2 | 32 | 3 | 3 ------------+----+----+----+-- . . o4/3x | 4 | 4 | 24 | 2 ------------+----+----+----+-- . o3/2o4/3x ♦ 8 | 12 | 6 | 8
o3/2o3o4x . . . . | 16 ♦ 4 | 6 | 4 ----------+----+----+----+-- . . . x | 2 | 32 | 3 | 3 ----------+----+----+----+-- . . o4x | 4 | 4 | 24 | 2 ----------+----+----+----+-- . o3o4x ♦ 8 | 12 | 6 | 8
o3/2o3o4/3x . . . . | 16 ♦ 4 | 6 | 4 ------------+----+----+----+-- . . . x | 2 | 32 | 3 | 3 ------------+----+----+----+-- . . o4/3x | 4 | 4 | 24 | 2 ------------+----+----+----+-- . o3o4/3x ♦ 8 | 12 | 6 | 8
o3/2o3/2o4x . . . . | 16 ♦ 4 | 6 | 4 ------------+----+----+----+-- . . . x | 2 | 32 | 3 | 3 ------------+----+----+----+-- . . o4x | 4 | 4 | 24 | 2 ------------+----+----+----+-- . o3/2o4x ♦ 8 | 12 | 6 | 8
o3/2o3/2o4/3x . . . . | 16 ♦ 4 | 6 | 4 --------------+----+----+----+-- . . . x | 2 | 32 | 3 | 3 --------------+----+----+----+-- . . o4/3x | 4 | 4 | 24 | 2 --------------+----+----+----+-- . o3/2o4/3x ♦ 8 | 12 | 6 | 8
x4o x4o . . . . | 16 ♦ 2 2 | 1 4 1 | 2 2 --------+----+-------+--------+---- x . . . | 2 | 16 * | 1 2 0 | 2 1 . . x . | 2 | * 16 | 0 2 1 | 1 2 --------+----+-------+--------+---- x4o . . | 4 | 4 0 | 4 * * | 2 0 x . x . | 4 | 2 2 | * 16 * | 1 1 . . x4o | 4 | 0 4 | * * 4 | 0 2 --------+----+-------+--------+---- x4o x . ♦ 8 | 8 4 | 2 4 0 | 4 * x . x4o ♦ 8 | 4 8 | 0 4 2 | * 4 snubbed forms: s4o2s4o
x4o x4/3o . . . . | 16 ♦ 2 2 | 1 4 1 | 2 2 ----------+----+-------+--------+---- x . . . | 2 | 16 * | 1 2 0 | 2 1 . . x . | 2 | * 16 | 0 2 1 | 1 2 ----------+----+-------+--------+---- x4o . . | 4 | 4 0 | 4 * * | 2 0 x . x . | 4 | 2 2 | * 16 * | 1 1 . . x4/3o | 4 | 0 4 | * * 4 | 0 2 ----------+----+-------+--------+---- x4o x . ♦ 8 | 8 4 | 2 4 0 | 4 * x . x4/3o ♦ 8 | 4 8 | 0 4 2 | * 4
x4/3o x4/3o . . . . | 16 ♦ 2 2 | 1 4 1 | 2 2 ------------+----+-------+--------+---- x . . . | 2 | 16 * | 1 2 0 | 2 1 . . x . | 2 | * 16 | 0 2 1 | 1 2 ------------+----+-------+--------+---- x4/3o . . | 4 | 4 0 | 4 * * | 2 0 x . x . | 4 | 2 2 | * 16 * | 1 1 . . x4/3o | 4 | 0 4 | * * 4 | 0 2 ------------+----+-------+--------+---- x4/3o x . ♦ 8 | 8 4 | 2 4 0 | 4 * x . x4/3o ♦ 8 | 4 8 | 0 4 2 | * 4
x o3o4x . . . . | 16 ♦ 1 3 | 3 3 | 3 1 --------+----+------+-------+---- x . . . | 2 | 8 * | 3 0 | 3 0 . . . x | 2 | * 24 | 1 2 | 2 1 --------+----+------+-------+---- x . . x | 4 | 2 2 | 12 * | 2 0 . . o4x | 4 | 0 4 | * 12 | 1 1 --------+----+------+-------+---- x . o4x ♦ 8 | 4 8 | 4 2 | 6 * . o3o4x ♦ 8 | 0 12 | 0 6 | * 2 snubbed forms: x2o3o4s, s2o3o4s
x o3o4/3x . . . . | 16 ♦ 1 3 | 3 3 | 3 1 ----------+----+------+-------+---- x . . . | 2 | 8 * | 3 0 | 3 0 . . . x | 2 | * 24 | 1 2 | 2 1 ----------+----+------+-------+---- x . . x | 4 | 2 2 | 12 * | 2 0 . . o4/3x | 4 | 0 4 | * 12 | 1 1 ----------+----+------+-------+---- x . o4/3x ♦ 8 | 4 8 | 4 2 | 6 * . o3o4/3x ♦ 8 | 0 12 | 0 6 | * 2
x o3/2o4x . . . . | 16 ♦ 1 3 | 3 3 | 3 1 ----------+----+------+-------+---- x . . . | 2 | 8 * | 3 0 | 3 0 . . . x | 2 | * 24 | 1 2 | 2 1 ----------+----+------+-------+---- x . . x | 4 | 2 2 | 12 * | 2 0 . . o4x | 4 | 0 4 | * 12 | 1 1 ----------+----+------+-------+---- x . o4x ♦ 8 | 4 8 | 4 2 | 6 * . o3/2o4x ♦ 8 | 0 12 | 0 6 | * 2
x o3/2o4/3x . . . . | 16 ♦ 1 3 | 3 3 | 3 1 ------------+----+------+-------+---- x . . . | 2 | 8 * | 3 0 | 3 0 . . . x | 2 | * 24 | 1 2 | 2 1 ------------+----+------+-------+---- x . . x | 4 | 2 2 | 12 * | 2 0 . . o4/3x | 4 | 0 4 | * 12 | 1 1 ------------+----+------+-------+---- x . o4/3x ♦ 8 | 4 8 | 4 2 | 6 * . o3/2o4/3x ♦ 8 | 0 12 | 0 6 | * 2
x x x4o . . . . | 16 ♦ 1 1 2 | 1 2 2 1 | 2 1 1 --------+----+--------+---------+------ x . . . | 2 | 8 * * | 1 2 0 0 | 2 1 0 . x . . | 2 | * 8 * | 1 0 2 0 | 2 0 1 . . x . | 2 | * * 16 | 0 1 1 1 | 1 1 1 --------+----+--------+---------+------ x x . . | 4 | 2 2 0 | 4 * * * | 2 0 0 x . x . | 4 | 2 0 2 | * 8 * * | 1 1 0 . x x . | 4 | 0 2 2 | * * 8 * | 1 0 1 . . x4o | 4 | 0 0 4 | * * * 4 | 0 1 1 --------+----+--------+---------+------ x x x . ♦ 8 | 4 4 4 | 2 2 2 0 | 4 * * x . x4o ♦ 8 | 4 0 8 | 0 4 0 2 | * 2 * . x x4o ♦ 8 | 0 4 8 | 0 0 4 2 | * * 2 snubbed forms: x2s2s4o, s2s2s4o
x x x4/3o . . . . | 16 ♦ 1 1 2 | 1 2 2 1 | 2 1 1 ----------+----+--------+---------+------ x . . . | 2 | 8 * * | 1 2 0 0 | 2 1 0 . x . . | 2 | * 8 * | 1 0 2 0 | 2 0 1 . . x . | 2 | * * 16 | 0 1 1 1 | 1 1 1 ----------+----+--------+---------+------ x x . . | 4 | 2 2 0 | 4 * * * | 2 0 0 x . x . | 4 | 2 0 2 | * 8 * * | 1 1 0 . x x . | 4 | 0 2 2 | * * 8 * | 1 0 1 . . x4/3o | 4 | 0 0 4 | * * * 4 | 0 1 1 ----------+----+--------+---------+------ x x x . ♦ 8 | 4 4 4 | 2 2 2 0 | 4 * * x . x4/3o ♦ 8 | 4 0 8 | 0 4 0 2 | * 2 * . x x4/3o ♦ 8 | 0 4 8 | 0 0 4 2 | * * 2
x x x x . . . . | 16 ♦ 1 1 1 1 | 1 1 1 1 1 1 | 1 1 1 1 --------+----+---------+-------------+-------- x . . . | 2 | 8 * * * | 1 1 1 0 0 0 | 1 1 1 0 . x . . | 2 | * 8 * * | 1 0 0 1 1 0 | 1 1 0 1 . . x . | 2 | * * 8 * | 0 1 0 1 0 1 | 1 0 1 1 . . . x | 2 | * * * 8 | 0 0 1 0 1 1 | 0 1 1 1 --------+----+---------+-------------+-------- x x . . | 4 | 2 2 0 0 | 4 * * * * * | 1 1 0 0 x . x . | 4 | 2 0 2 0 | * 4 * * * * | 1 0 1 0 x . . x | 4 | 2 0 0 2 | * * 4 * * * | 0 1 1 0 . x x . | 4 | 0 2 2 0 | * * * 4 * * | 1 0 0 1 . x . x | 4 | 0 2 0 2 | * * * * 4 * | 0 1 0 1 . . x x | 4 | 0 0 2 2 | * * * * * 4 | 0 0 1 1 --------+----+---------+-------------+-------- x x x . ♦ 8 | 4 4 4 0 | 2 2 0 2 0 0 | 2 * * * x x . x ♦ 8 | 4 4 0 4 | 2 0 2 0 2 0 | * 2 * * x . x x ♦ 8 | 4 0 4 4 | 0 2 2 0 0 2 | * * 2 * . x x x ♦ 8 | 0 4 4 4 | 0 0 0 2 2 2 | * * * 2 snubbed forms: x2s2s2s, s2s2s2s
s2x2s4x demi( . . . . ) | 16 ♦ 1 1 1 1 | 1 1 1 2 1 | 1 1 2 ----------------+----+---------+-----------+------ demi( . x . . ) | 2 | 8 * * * | 1 1 0 0 1 | 0 1 2 demi( . . . x ) | 2 | * 8 * * | 1 0 1 1 0 | 1 1 1 s 2 s . | 2 | * * 8 * | 0 1 0 2 0 | 1 0 2 sefa( . . s4x ) | 2 | * * * 8 | 0 0 1 1 1 | 1 1 1 ----------------+----+---------+-----------+------ demi( . x . x ) | 4 | 2 2 0 0 | 4 * * * * | 0 1 1 s2x2s . | 4 | 2 0 2 0 | * 4 * * * | 0 0 2 . . s4x | 4 | 0 2 0 2 | * * 4 * * | 1 1 0 sefa( s 2 s4x ) | 4 | 0 1 2 1 | * * * 8 * | 1 0 1 sefa( . x2s4x ) | 4 | 2 0 0 2 | * * * * 4 | 0 1 1 ----------------+----+---------+-----------+------ s 2 s4x ♦ 8 | 0 4 4 4 | 0 0 2 4 0 | 2 * * . x2s4x ♦ 8 | 4 4 0 4 | 2 0 2 0 2 | * 2 * sefa( s2x2s4x ) ♦ 8 | 4 2 4 2 | 1 2 0 2 1 | * * 4 starting figure: x x x4x
s4x s4x demi( . . ) demi( . . ) | 16 ♦ 1 1 1 1 | 1 1 1 1 1 1 | 1 1 1 1 ------------------------+----+---------+-------------+-------- demi( . . ) demi( . x ) | 2 | 8 * * * | 1 1 0 1 0 0 | 1 1 1 0 demi( . . ) sefa( s4x ) | 2 | * 8 * * | 1 0 1 0 1 0 | 1 1 0 1 demi( . x ) demi( . . ) | 2 | * * 8 * | 0 1 1 0 0 1 | 1 0 1 1 sefa( s4x ) demi( . . ) | 2 | * * * 8 | 0 0 0 1 1 1 | 0 1 1 1 ------------------------+----+---------+-------------+-------- demi( . . ) s4x | 4 | 2 2 0 0 | 4 * * * * * | 1 1 0 0 demi( . x ) demi( . x ) | 4 | 2 0 2 0 | * 4 * * * * | 1 0 1 0 demi( . x ) sefa( s4x ) | 4 | 0 2 2 0 | * * 4 * * * | 1 0 0 1 sefa( s4x ) demi( . x ) | 4 | 2 0 0 2 | * * * 4 * * | 0 1 1 0 sefa( s4x ) sefa( s4x ) | 4 | 0 2 0 2 | * * * * 4 * | 0 1 0 1 s4x demi( . . ) | 4 | 0 0 2 2 | * * * * * 4 | 0 0 1 1 ------------------------+----+---------+-------------+-------- demi( . x ) s4x ♦ 8 | 4 4 4 0 | 2 2 2 0 0 0 | 2 * * * sefa( s4x ) s4x ♦ 8 | 4 4 0 4 | 2 0 0 2 2 0 | * 2 * * s4x demi( . x ) ♦ 8 | 4 0 4 4 | 0 2 0 2 0 2 | * * 2 * s4x sefa( s4x ) ♦ 8 | 0 4 4 4 | 0 0 2 0 2 2 | * * * 2 starting figure: x4x x4x
oo3oo4xx&#x → height = 1
(cube || cube)
o.3o.4o. | 8 * ♦ 3 1 0 | 3 3 0 | 1 3 0
.o3.o4.o | * 8 ♦ 0 1 3 | 0 3 3 | 0 3 1
------------+-----+---------+--------+------
.. .. x. | 2 0 | 12 * * | 2 1 0 | 1 2 0
oo3oo4oo&#x | 1 1 | * 8 * | 0 3 0 | 0 3 0
.. .. .x | 0 2 | * * 12 | 0 1 2 | 0 2 1
------------+-----+---------+--------+------
.. o.4x. | 4 0 | 4 0 0 | 6 * * | 1 1 0
.. .. xx&#x | 2 2 | 1 2 1 | * 12 * | 0 2 0
.. .o4.x | 0 4 | 0 0 4 | * * 6 | 0 1 1
------------+-----+---------+--------+------
o.3o.4x. ♦ 8 0 | 12 0 0 | 6 0 0 | 1 * *
.. oo4xx&#x ♦ 4 4 | 4 4 4 | 1 4 1 | * 6 *
.o3.o4.x ♦ 0 8 | 0 0 12 | 0 0 6 | * * 1
xx xx4oo&#x → height = 1
(cube || cube)
o. o.4o. | 8 * ♦ 1 2 1 0 0 | 2 1 1 2 0 0 | 1 2 1 0
.o .o4.o | * 8 ♦ 0 0 1 1 2 | 0 0 1 2 2 1 | 0 2 1 1
------------+-----+-----------+-------------+--------
x. .. .. | 2 0 | 4 * * * * | 2 0 1 0 0 0 | 1 2 0 0
.. x. .. | 2 0 | * 8 * * * | 1 1 0 1 0 0 | 1 1 1 0
oo oo4oo&#x | 1 1 | * * 8 * * | 0 0 1 2 0 0 | 0 2 1 0
.x .. .. | 0 2 | * * * 4 * | 0 0 1 0 2 0 | 0 2 0 1
.. .x .. | 0 2 | * * * * 8 | 0 0 0 1 1 1 | 0 1 1 1
------------+-----+-----------+-------------+--------
x. x. .. | 4 0 | 2 2 0 0 0 | 4 * * * * * | 1 1 0 0
.. x.4o. | 4 0 | 0 4 0 0 0 | * 2 * * * * | 1 0 1 0
xx .. ..&#x | 2 2 | 1 0 2 1 0 | * * 4 * * * | 0 2 0 0
.. xx ..&#x | 2 2 | 0 1 2 0 1 | * * * 8 * * | 0 0 2 0
.x .x .. | 0 4 | 0 0 0 2 2 | * * * * 4 * | 0 1 0 1
.. .x4.o | 0 4 | 0 0 0 0 4 | * * * * * 2 | 0 0 1 1
------------+-----+-----------+-------------+--------
x. x.4o. ♦ 8 0 | 4 8 0 0 0 | 4 2 0 0 0 0 | 1 * * *
xx xx ..&#x ♦ 4 4 | 2 2 4 2 2 | 1 0 4 0 1 0 | * 4 * *
.. xx4oo&#x ♦ 4 4 | 0 4 4 0 4 | 0 1 0 4 0 1 | * * 2 *
.x .x4.o ♦ 0 8 | 0 0 0 4 8 | 0 0 0 0 4 2 | * * * 1
xx xx xx&#x → height = 1
(cube || cube)
o. o. o. | 8 * ♦ 1 1 1 1 0 0 0 | 1 1 1 1 1 1 0 0 0 | 1 1 1 1 0
.o .o .o | * 8 ♦ 0 0 0 1 1 1 1 | 0 0 0 1 1 1 1 1 1 | 0 1 1 1 1
------------+-----+---------------+-------------------+----------
x. .. .. | 2 0 | 4 * * * * * * | 1 1 0 1 0 0 0 0 0 | 1 1 1 0 0
.. x. .. | 2 0 | * 4 * * * * * | 1 0 1 0 1 0 0 0 0 | 1 1 0 1 0
.. .. x. | 2 0 | * * 4 * * * * | 0 1 1 0 0 1 0 0 0 | 1 0 1 1 0
oo oo oo&#x | 1 1 | * * * 8 * * * | 0 0 0 1 1 1 0 0 0 | 0 1 1 1 0
.x .. .. | 0 2 | * * * * 4 * * | 0 0 0 1 0 0 1 1 0 | 0 1 1 0 1
.. .x .. | 0 2 | * * * * * 4 * | 0 0 0 0 1 0 1 0 1 | 0 1 0 1 1
.. .. .x | 0 2 | * * * * * * 4 | 0 0 0 0 0 1 0 1 1 | 0 0 1 1 1
------------+-----+---------------+-------------------+----------
x. x. .. | 4 0 | 2 2 0 0 0 0 0 | 2 * * * * * * * * | 1 1 0 0 0
x. .. x. | 4 0 | 2 0 2 0 0 0 0 | * 2 * * * * * * * | 1 0 1 0 0
.. x. x. | 4 0 | 0 2 2 0 0 0 0 | * * 2 * * * * * * | 1 0 0 1 0
xx .. ..&#x | 2 2 | 1 0 0 2 1 0 0 | * * * 4 * * * * * | 0 1 1 0 0
.. xx ..&#x | 2 2 | 0 1 0 2 0 1 0 | * * * * 4 * * * * | 0 1 0 1 0
.. .. xx&#x | 2 2 | 0 0 1 2 0 0 1 | * * * * * 4 * * * | 0 0 1 1 0
.x .x .. | 0 4 | 0 0 0 0 2 2 0 | * * * * * * 2 * * | 0 1 0 0 1
.x .. .x | 0 4 | 0 0 0 0 2 0 2 | * * * * * * * 2 * | 0 0 1 0 1
.. .x .x | 0 4 | 0 0 0 0 0 2 2 | * * * * * * * * 2 | 0 0 0 1 1
------------+-----+---------------+-------------------+----------
x. x. x. ♦ 8 0 | 4 4 4 0 0 0 0 | 2 2 2 0 0 0 0 0 0 | 1 * * * *
xx xx ..&#x ♦ 4 4 | 2 2 0 4 2 2 0 | 1 0 0 2 2 0 1 0 0 | * 2 * * *
xx .. xx&#x ♦ 4 4 | 2 0 2 4 2 0 2 | 0 1 0 2 0 2 0 1 0 | * * 2 * *
.. xx xx&#x ♦ 4 4 | 0 2 2 4 0 2 2 | 0 0 1 0 2 2 0 0 1 | * * * 2 *
.x .x .x ♦ 0 8 | 0 0 0 0 4 4 4 | 0 0 0 0 0 0 2 2 2 | * * * * 1
oqo xxx4ooo&#xt → both heights = 1/sqrt(2) = 0.707107 ({4} || pseudo (q,x,x)-cube || {4}) o.. o..4o.. | 4 * * ♦ 2 2 0 0 0 | 1 2 1 0 0 0 | 2 2 0 .o. .o.4.o. | * 8 * ♦ 0 1 2 1 0 | 0 2 1 1 2 0 | 1 2 1 ..o ..o4..o | * * 4 ♦ 0 0 0 2 2 | 0 0 1 0 4 1 | 0 2 2 ----------------+-------+-----------+-------------+------ ... x.. ... | 2 0 0 | 4 * * * * | 1 2 0 0 0 0 | 2 1 0 oo. oo.4oo.&#x | 1 1 0 | * 8 * * * | 0 2 1 0 0 0 | 1 2 0 ... .x. ... | 0 2 0 | * * 8 * * | 0 1 0 1 1 0 | 1 1 1 .oo .oo4.oo&#x | 0 1 1 | * * * 8 * | 0 0 1 0 2 0 | 0 2 1 ... ..x ... | 0 0 2 | * * * * 4 | 0 0 0 0 2 1 | 0 1 2 ----------------+-------+-----------+-------------+------ ... x..4o.. | 4 0 0 | 4 0 0 0 0 | 1 * * * * * | 2 0 0 ... xx. ...&#x | 2 2 0 | 1 2 1 0 0 | * 8 * * * * | 1 1 0 oqo ... ...&#xt | 1 2 1 | 0 2 0 2 0 | * * 4 * * * | 0 2 0 ... .x.4.o. | 0 4 0 | 0 0 4 0 0 | * * * 2 * * | 1 0 1 ... .xx ...&#x | 0 2 2 | 0 0 1 2 1 | * * * * 8 * | 0 1 1 ... ..x4..o | 0 0 4 | 0 0 0 0 4 | * * * * * 1 | 0 0 2 ----------------+-------+-----------+-------------+------ ... xx.4oo.&#x ♦ 4 4 0 | 4 4 4 0 0 | 1 4 0 1 0 0 | 2 * * oqo xxx ...&#xt ♦ 2 4 2 | 1 4 2 4 1 | 0 2 2 0 2 0 | * 4 * ... .xx4.oo&#x ♦ 0 4 4 | 0 0 4 4 4 | 0 0 0 1 4 1 | * * 2
or o.. o..4o.. & | 8 * ♦ 2 2 0 | 1 4 1 0 | 2 2 .o. .o.4.o. | * 8 ♦ 0 2 2 | 0 4 1 1 | 2 2 -------------------+-----+--------+----------+---- ... x.. ... & | 2 0 | 8 * * | 1 2 0 0 | 2 1 oo. oo.4oo.&#x & | 1 1 | * 16 * | 0 2 1 0 | 1 2 ... .x. ... | 0 2 | * * 8 | 0 2 0 1 | 2 1 -------------------+-----+--------+----------+---- ... x..4o.. & | 4 0 | 4 0 0 | 2 * * * | 2 0 ... xx. ...&#x & | 2 2 | 1 2 1 | * 16 * * | 1 1 oqo ... ...&#xt | 2 2 | 0 4 0 | * * 4 * | 0 2 ... .x.4.o. | 0 4 | 0 0 4 | * * * 2 | 2 0 -------------------+-----+--------+----------+---- ... xx.4oo.&#x & ♦ 4 4 | 4 4 4 | 1 4 0 1 | 4 * oqo xxx ...&#xt ♦ 4 4 | 2 8 2 | 0 4 2 0 | * 4
oqo xxx xxx&#xt → both heights = 1/sqrt(2) = 0.707107 ({4} || pseudo (q,x,x)-cube || {4}) o.. o.. o.. | 4 * * ♦ 1 1 2 0 0 0 0 0 | 1 2 2 1 0 0 0 0 | 2 1 1 0 .o. .o. .o. | * 8 * ♦ 0 0 1 1 1 1 0 0 | 0 1 1 1 1 1 1 0 | 1 1 1 1 ..o ..o ..o | * * 4 ♦ 0 0 0 0 0 2 1 1 | 0 0 0 1 0 2 2 1 | 0 1 1 2 ----------------+-------+-----------------+-----------------+-------- ... x.. ... | 2 0 0 | 2 * * * * * * * | 1 2 0 0 0 0 0 0 | 2 1 0 0 ... ... x.. | 2 0 0 | * 2 * * * * * * | 1 0 2 0 0 0 0 0 | 2 0 1 0 oo. oo. oo.&#x | 1 1 0 | * * 8 * * * * * | 0 1 1 1 0 0 0 0 | 1 1 1 0 ... .x. ... | 0 2 0 | * * * 4 * * * * | 0 1 0 0 1 1 0 0 | 1 1 0 1 ... ... .x. | 0 2 0 | * * * * 4 * * * | 0 0 1 0 1 0 1 0 | 1 0 1 1 .oo .oo .oo&#x | 0 1 1 | * * * * * 8 * * | 0 0 0 1 0 1 1 0 | 0 1 1 1 ... ..x ... | 0 0 2 | * * * * * * 2 * | 0 0 0 0 0 2 0 1 | 0 1 0 2 ... ... ..x | 0 0 2 | * * * * * * * 2 | 0 0 0 0 0 0 2 1 | 0 0 1 2 ----------------+-------+-----------------+-----------------+-------- ... x.. x.. | 4 0 0 | 2 2 0 0 0 0 0 0 | 1 * * * * * * * | 2 0 0 0 ... xx. ...&#x | 2 2 0 | 1 0 2 1 0 0 0 0 | * 4 * * * * * * | 1 1 0 0 ... ... xx.&#x | 2 2 0 | 0 1 2 0 1 0 0 0 | * * 4 * * * * * | 1 0 1 0 oqo ... ...&#xt | 1 2 1 | 0 0 2 0 0 2 0 0 | * * * 4 * * * * | 0 1 1 0 ... .x. .x. | 0 4 0 | 0 0 0 2 2 0 0 0 | * * * * 2 * * * | 1 0 0 1 ... .xx ...&#x | 0 2 2 | 0 0 0 1 0 2 1 0 | * * * * * 4 * * | 0 1 0 1 ... ... .xx&#x | 0 2 2 | 0 0 0 0 1 2 0 1 | * * * * * * 4 * | 0 0 1 1 ... ..x ..x | 0 0 4 | 0 0 0 0 0 0 2 2 | * * * * * * * 1 | 0 0 0 2 ----------------+-------+-----------------+-----------------+-------- ... xx. xx.&#x ♦ 4 4 0 | 2 2 4 2 2 0 0 0 | 1 2 2 0 1 0 0 0 | 2 * * * oqo xxx ...&#xt ♦ 2 4 2 | 1 0 4 2 0 4 1 0 | 0 2 0 2 0 2 0 0 | * 2 * * oqo ... xxx&#xt ♦ 2 4 2 | 0 1 4 0 2 4 0 1 | 0 0 2 2 0 0 2 0 | * * 2 * ... .xx .xx&#x ♦ 0 4 4 | 0 0 0 2 2 4 2 2 | 0 0 0 0 1 2 2 1 | * * * 2
or o.. o.. o.. & | 8 * ♦ 1 1 2 0 0 | 1 2 2 1 0 | 2 1 1 .o. .o. .o. | * 8 ♦ 0 0 2 1 1 | 0 2 2 1 1 | 2 1 1 -------------------+-----+------------+-----------+------ ... x.. ... & | 2 0 | 4 * * * * | 1 2 0 0 0 | 2 1 0 ... ... x.. & | 2 0 | * 4 * * * | 1 0 2 0 0 | 2 0 1 oo. oo. oo.&#x & | 1 1 | * * 16 * * | 0 1 1 1 0 | 1 1 1 ... .x. ... | 0 2 | * * * 4 * | 0 2 0 0 1 | 2 1 0 ... ... .x. | 0 2 | * * * * 4 | 0 0 2 0 1 | 2 0 1 -------------------+-----+------------+-----------+------ ... x.. x.. & | 4 0 | 2 2 0 0 0 | 2 * * * * | 2 0 0 ... xx. ...&#x & | 2 2 | 1 0 2 1 0 | * 8 * * * | 1 1 0 ... ... xx.&#x & | 2 2 | 0 1 2 0 1 | * * 8 * * | 1 0 1 oqo ... ...&#xt | 2 2 | 0 0 4 0 0 | * * * 4 * | 0 1 1 ... .x. .x. | 0 4 | 0 0 0 2 2 | * * * * 2 | 2 0 0 -------------------+-----+------------+-----------+------ ... xx. xx.&#x & ♦ 4 4 | 2 2 4 2 2 | 1 2 2 0 1 | 4 * * oqo xxx ...&#xt ♦ 4 4 | 2 0 8 2 0 | 0 4 0 2 0 | * 2 * oqo ... xxx&#xt ♦ 4 4 | 0 2 8 0 2 | 0 0 4 2 0 | * * 2
xxxx oqoo3ooqo&#xt → all heights = 1/sqrt(3) = 0.577350 (line || pseudo (q,x)-trip || pseudo inv (q,x)-trip || line) o... o...3o... | 2 * * * ♦ 1 3 0 0 0 0 0 | 3 3 0 0 0 | 3 1 0 .o.. .o..3.o.. | * 6 * * ♦ 0 1 1 2 0 0 0 | 1 2 2 1 0 | 2 1 1 ..o. ..o.3..o. | * * 6 * ♦ 0 0 0 2 1 1 0 | 0 1 2 2 1 | 1 1 2 ...o ...o3...o | * * * 2 ♦ 0 0 0 0 0 3 1 | 0 0 0 3 3 | 0 1 3 -------------------+---------+----------------+-----------+------ x... .... .... | 2 0 0 0 | 1 * * * * * * | 3 0 0 0 0 | 3 0 0 oo.. oo..3oo..&#x | 1 1 0 0 | * 6 * * * * * | 1 2 0 0 0 | 2 1 0 .x.. .... .... | 0 2 0 0 | * * 3 * * * * | 1 0 2 0 0 | 2 0 1 .oo. .oo.3.oo.&#x | 0 1 1 0 | * * * 12 * * * | 0 1 1 1 0 | 1 1 1 ..x. .... .... | 0 0 2 0 | * * * * 3 * * | 0 0 2 0 1 | 1 0 2 ..oo ..oo3..oo&#x | 0 0 1 1 | * * * * * 6 * | 0 0 0 2 1 | 0 1 2 ...x .... .... | 0 0 0 2 | * * * * * * 1 | 0 0 0 0 3 | 0 0 3 -------------------+---------+----------------+-----------+------ xx.. .... ....&#x | 2 2 0 0 | 1 2 1 0 0 0 0 | 3 * * * * | 2 0 0 .... oqo. ....&#xt | 1 2 1 0 | 0 2 0 2 0 0 0 | * 6 * * * | 1 1 0 .xx. .... ....&#x | 0 2 2 0 | 0 0 1 2 1 0 0 | * * 6 * * | 1 0 1 .... .... .oqo&#xt | 0 1 2 1 | 0 0 0 2 0 2 0 | * * * 6 * | 0 1 1 ..xx .... ....&#x | 0 0 2 2 | 0 0 0 0 1 2 1 | * * * * 3 | 0 0 2 -------------------+---------+----------------+-----------+------ xxx. oqo. ....&#xt ♦ 2 4 2 0 | 1 4 2 4 1 0 0 | 2 2 2 0 0 | 3 * * .... oqoo3ooqo&#xt ♦ 1 3 3 1 | 0 3 0 6 0 3 0 | 0 3 0 3 0 | * 2 * .xxx .... .oqo&#xt ♦ 0 2 4 2 | 0 0 1 4 2 4 1 | 0 0 2 2 2 | * * 3
or o... o...3o... & | 4 * ♦ 1 3 0 0 | 3 3 0 | 3 1 .o.. .o..3.o.. & | * 12 ♦ 0 1 1 2 | 1 3 2 | 3 1 ----------------------+------+-----------+--------+---- x... .... .... & | 2 0 | 2 * * * | 3 0 0 | 3 0 oo.. oo..3oo..&#x & | 1 1 | * 12 * * | 1 2 0 | 2 1 .x.. .... .... & | 0 2 | * * 6 * | 1 0 2 | 3 0 .oo. .oo.3.oo.&#x | 0 2 | * * * 12 | 0 2 1 | 2 1 ----------------------+------+-----------+--------+---- xx.. .... ....&#x & | 2 2 | 1 2 1 0 | 6 * * | 2 0 .... oqo. ....&#xt & | 1 3 | 0 2 0 2 | * 12 * | 1 1 .xx. .... ....&#x | 0 4 | 0 0 2 2 | * * 6 | 2 0 ----------------------+------+-----------+--------+---- xxx. oqo. ....&#xt & ♦ 2 6 | 1 4 3 4 | 2 2 2 | 6 * .... oqoo3ooqo&#xt ♦ 2 6 | 0 6 0 6 | 0 6 0 | * 2
oqooo3ooqoo3oooqo&#xt → all heights = 1/2 (pt || pseudo q-tet || pseudo q-oct || pseudo dual q-tet || pt) o....3o....3o.... | 1 * * * * ♦ 4 0 0 0 | 6 0 0 | 4 0 .o...3.o...3.o... | * 4 * * * ♦ 1 3 0 0 | 3 3 0 | 3 1 ..o..3..o..3..o.. | * * 6 * * ♦ 0 2 2 0 | 1 4 1 | 2 2 ...o.3...o.3...o. | * * * 4 * ♦ 0 0 3 1 | 0 3 3 | 1 3 ....o3....o3....o | * * * * 1 ♦ 0 0 0 4 | 0 0 6 | 0 4 ----------------------+-----------+-----------+--------+---- oo...3oo...3oo...&#x | 1 1 0 0 0 | 4 * * * | 3 0 0 | 3 0 .oo..3.oo..3.oo..&#x | 0 1 1 0 0 | * 12 * * | 1 2 0 | 2 1 ..oo.3..oo.3..oo.&#x | 0 0 1 1 0 | * * 12 * | 0 2 1 | 1 2 ...oo3...oo3...oo&#x | 0 0 0 1 1 | * * * 4 | 0 0 3 | 0 3 ----------------------+-----------+-----------+--------+---- oqo.. ..... .....&#xt | 1 2 1 0 0 | 2 2 0 0 | 6 * * | 2 0 ..... .oqo. .....&#xt | 0 1 2 1 0 | 0 2 2 0 | * 12 * | 1 1 ..... ..... ..oqo&#xt | 0 0 1 2 1 | 0 0 2 2 | * * 6 | 0 2 ----------------------+-----------+-----------+--------+---- oqoo.3ooqo. .....&#xt ♦ 1 3 3 1 0 | 3 6 3 0 | 3 3 0 | 4 * ..... .oqoo3.ooqo&#xt ♦ 0 1 3 3 1 | 0 3 6 3 | 0 3 3 | * 4
or o....3o....3o.... & | 2 * * ♦ 4 0 | 6 0 | 4 .o...3.o...3.o... & | * 8 * ♦ 1 3 | 3 3 | 4 ..o..3..o..3..o.. | * * 6 ♦ 0 4 | 2 4 | 4 -------------------------+-------+------+-------+-- oo...3oo...3oo...&#x & | 1 1 0 | 8 * | 3 0 | 3 .oo..3.oo..3.oo..&#x & | 0 1 1 | * 24 | 1 2 | 3 -------------------------+-------+------+-------+-- oqo.. ..... .....&#xt & | 1 2 1 | 2 2 | 12 * | 2 ..... .oqo. .....&#xt | 0 2 2 | 0 4 | * 12 | 2 -------------------------+-------+------+-------+-- oqoo.3ooqo. .....&#xt & ♦ 1 4 3 | 3 9 | 3 3 | 8
qo3oo3oq *b3oo&#zx → height = 0 (tegum sum of 2 mutually gyrated q-hexes) o.3o.3o. *b3o. | 8 * ♦ 4 | 6 | 4 .o3.o3.o *b3.o | * 8 ♦ 4 | 6 | 4 -------------------+-----+----+----+-- oo3oo3oo *b3oo&#x | 1 1 | 32 | 3 | 3 -------------------+-----+----+----+-- qo .. oq ..&#zx | 2 2 | 4 | 24 | 2 -------------------+-----+----+----+-- qo3oo3oq ..&#zx ♦ 4 4 | 12 | 6 | 8
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