Acronym tes (alt.: squadip), K-4.20
Name tesseract,
octachoron,
"hypercube",
4D measure-polytope4),
8-cell,
geoter(id),
square duoprism,
vertex figure of icot
|,>,O device line prism prism prism = ||||
 
 
 ©   ©
 ©
Segmentochoron display
Cross sections
 ©
Circumradius 1
Edge radius sqrt(3)/2 = 0.866025
Face radius 1/sqrt(2) = 0.707107
Inradius 1/2
Vertex figure
 ©
Vertex layers
LayerSymmetrySubsymmetries
 o3o3o4o o3o3o . o3o . o o . o4o . o3o4o
1o3o3o4x o3o3o .
vertex first
o3o . x
edge first
o . o4x
{4} first
. o3o4x
cube first
2 o3o3q .
vertex figure
o3q . x q . o4x . o3o4x
opposite cube
3 o3q3o . q3o . x o . o4x
opposite {4}
 
4 q3o3o . o3o . x
opposite edge
 
5 o3o3o .
opposite vertex
 
Lace city
in approx. ASCII-art
 ©  
x4o   x4o
         
         
x4o   x4o
 ©  
x x   x x
         
         
x x   x x
o3o  q3o  o3q  o3o
                  
                  
o3o  q3o  o3q  o3o
Coordinates
  1. (1/2, 1/2, 1/2, 1/2)   & all changes of sign
    (axis parallel orientation)
  2. (1/sqrt(2), 1/sqrt(2), 0, 0)   & cyclic changes & all changes of sign
    (when inscribed into dual ico)
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:
by cells: cube
tes 8
)
Dual hex
Dihedral angles
Confer
general duoprisms:
4,n-dip   4,2n-dip   n,n-dip   n,m-dip   2n,m-dip   2n,2m-dip   n/d,m/b-dip  
compounds:
gico   socdip   trisqdip  
Grünbaumian relatives:
2tes   6tes  
derived facetings:
qo3oo3oo&#x   qo3oq3oo&#x   4D corner hypercubera   tidtes  
related CRFs:
cytau tes   ecubedpy   ecubpy  
decompositions:
tespy  
general polytopal classes:
regular   noble polytopes   hypercube   partial Stott expansions   segmentochora   bistratic lace towers   lace simplices  
External
links
hedrondude   wikipedia   polytopewiki   WikiChoron   mathworld   quickfur

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.)


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

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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