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 (1/2, 1/2, 1/2, 1/2)   & all changes of sign
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  
Grünbaumian relatives:
2tes   6tes  
related CRFs:
cytau tes   ecubedpy   ecubpy  
decompositions:
tespy  
general polytopal classes:
regular   noble polytopes   hypercube   partial Stott expansions   segmentochora   bistratic lace towers  
External
links
hedrondude   wikipedia   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.


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: 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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