Acronym hex, K-4.2 (alt: octit, trapt)
Name hexadecachoron,
tetracross4),
hemitesseract,
hemioctachoron,
16-cell,
aeroter(id),
tetrahedral antiprism,
vertex figure of tac,
(line-)octahedral tegum,
(line-)trigonal-antirprismatic tegum,
Gosset polytope 11,1,
lattice C4 contact polytope (span of its small roots)
  ©   ©  
Segmentochoron display
Cross sections
 ©
Circumradius 1/sqrt(2) = 0.707107
Edge radius 1/2
Face radius 1/sqrt(6) = 0.408248
Inradius 1/sqrt(8) = 0.353553
Vertex figure
 ©
Vertex layers
LayerSymmetrySubsymmetries
 o3o3o4o o3o3o . o3o . o o . o4o . o3o4o
1x3o3o4o x3o3o .
tet first
x3o . o
{3} first
x . o4o
edge first
. o3o4o
vertex first
2 o3o3x .
opposite tet
o3o . q o . x4o . x3o4o
vertex figure
3   o3x . o
opposite {3}
x . o4o
opposite edge
. o3o4o
opposite vertex
 o3o3o *b3o o3o3o    . o3o . *b3o o . o    o . o3o *b3o
1x3o3o *b3o x3o3o    .
tet first
x3o . *b3o
tet first
x . o    o
edge first
. o3o *b3o
vertex first
2 o3o3x    .
opposite tet
o3o . *b3x
opposite tet
o . x    x . x3o *b3o
vertex figure
3     x . o    o
opposite edge
. o3o *b3o
opposite vertex
Lace city
in approx. ASCII-art
 ©  
    o4o    
           
o4o x4o o4o
           
    o4o    
    x3o    
o3o     o3o
    o3x    
 ©  
x o    o x
          
          
          
o x    x o
Coordinates
  • as orthoplex (tetracross):   (1/sqrt(2), 0, 0, 0)   & all permutations, all changes of sign
  • as hemitesseract:   (1/sqrt(8), 1/sqrt(8), 1/sqrt(8), 1/sqrt(8))   & all permutations, all even changes of sign
  • as "the other" (mirrored) hemitesseract:   (1/sqrt(8), 1/sqrt(8), 1/sqrt(8), -1/sqrt(8))   & all permutations, all even changes of sign
  • (the compound of those 3 such oriented hexadecachora is stico, vertex inscribed in the dual ico of the intersection kernel)
Volume 1/6 = 0.166667
Surface 4 sqrt(2)/3 = 1.885618
Rel. Roundness 3 π2/64 = 46.263771 %
General of army (is itself convex)
Colonel of regiment (is itself locally convex – uniform polychoral members:
by cells: octtet
tho 48
hex 016
)
Dual tes
Dihedral angles
  • at {3} between tet and tet:   120°
Confer
more general:
s4oPo4s  
variations:
xo3oo3ox&#q
Grünbaumian relatives:
hex+8oct   2hex+8oct  
general pyramid-antiprisms:
n-apt  
compounds:
haddet   stico  
related segmentochora:
octpy   squasc  
related CRFs:
pex hex   quawros   pacsid pith  
general polytopal classes:
tetrahedrochora   regular   noble polytopes   orthoplex   partial Stott expansions   segmentochora   fundamental lace prisms   bistratic lace towers   lace simplices   Coxeter-Elte-Gosset polytopes  
External
links
hedrondude   wikipedia   WikiChoron   mathworld   quickfur

Note that hex can be thought of as the external blend of 4 squascs. Further, the overlay of 2 such fully perpendicular decompositions would amount to the degenerate segmentoteron xo4oo ox4oo&#x.

The number of ways to color the hexadecachoron with different colors per cell is 16!/192 = 108 972 864 000. – This is because the color group is the permutation group of 16 elements and has size 16!, while the order of the pure rotational tesseractic group is 192. (The reflectional tesseractic group would have twice as many, i.e. 384 elements.)


Incidence matrix according to Dynkin symbol

x3o3o4o

. . . . | 8   6 | 12 |  8
--------+---+----+----+---
x . . . | 2 | 24 |  4 |  4
--------+---+----+----+---
x3o . . | 3 |  3 | 32 |  2
--------+---+----+----+---
x3o3o .  4 |  6 |  4 | 16

snubbed forms: β3o3o4o

x3o3o4/3o

. . .   . | 8   6 | 12 |  8
----------+---+----+----+---
x . .   . | 2 | 24 |  4 |  4
----------+---+----+----+---
x3o .   . | 3 |  3 | 32 |  2
----------+---+----+----+---
x3o3o   .  4 |  6 |  4 | 16

x3o3/2o4o

. .   . . | 8   6 | 12 |  8
----------+---+----+----+---
x .   . . | 2 | 24 |  4 |  4
----------+---+----+----+---
x3o   . . | 3 |  3 | 32 |  2
----------+---+----+----+---
x3o3/2o .  4 |  6 |  4 | 16

x3o3/2o4/3o

. .   .   . | 8   6 | 12 |  8
------------+---+----+----+---
x .   .   . | 2 | 24 |  4 |  4
------------+---+----+----+---
x3o   .   . | 3 |  3 | 32 |  2
------------+---+----+----+---
x3o3/2o   .  4 |  6 |  4 | 16

x3/2o3o4o

.   . . . | 8   6 | 12 |  8
----------+---+----+----+---
x   . . . | 2 | 24 |  4 |  4
----------+---+----+----+---
x3/2o . . | 3 |  3 | 32 |  2
----------+---+----+----+---
x3/2o3o .  4 |  6 |  4 | 16

x3/2o3o4/3o

.   . .   . | 8   6 | 12 |  8
------------+---+----+----+---
x   . .   . | 2 | 24 |  4 |  4
------------+---+----+----+---
x3/2o .   . | 3 |  3 | 32 |  2
------------+---+----+----+---
x3/2o3o   .  4 |  6 |  4 | 16

x3/2o3/2o4o

.   .   . . | 8   6 | 12 |  8
------------+---+----+----+---
x   .   . . | 2 | 24 |  4 |  4
------------+---+----+----+---
x3/2o   . . | 3 |  3 | 32 |  2
------------+---+----+----+---
x3/2o3/2o .  4 |  6 |  4 | 16

x3/2o3/2o4/3o

.   .   .   . | 8   6 | 12 |  8
--------------+---+----+----+---
x   .   .   . | 2 | 24 |  4 |  4
--------------+---+----+----+---
x3/2o   .   . | 3 |  3 | 32 |  2
--------------+---+----+----+---
x3/2o3/2o   .  4 |  6 |  4 | 16

x3o3o *b3o

. . .    . | 8   6 | 12 | 4 4
-----------+---+----+----+----
x . .    . | 2 | 24 |  4 | 2 2
-----------+---+----+----+----
x3o .    . | 3 |  3 | 32 | 1 1
-----------+---+----+----+----
x3o3o    .  4 |  6 |  4 | 8 *
x3o . *b3o  4 |  6 |  4 | * 8

snubbed forms: β3o3o *b3o

x3o3o *b3/2o

. . .      . | 8   6 | 12 | 4 4
-------------+---+----+----+----
x . .      . | 2 | 24 |  4 | 2 2
-------------+---+----+----+----
x3o .      . | 3 |  3 | 32 | 1 1
-------------+---+----+----+----
x3o3o      .  4 |  6 |  4 | 8 *
x3o . *b3/2o  4 |  6 |  4 | * 8

x3o3/2o *b3/2o

. .   .      . | 8   6 | 12 | 4 4
---------------+---+----+----+----
x .   .      . | 2 | 24 |  4 | 2 2
---------------+---+----+----+----
x3o   .      . | 3 |  3 | 32 | 1 1
---------------+---+----+----+----
x3o3/2o      .  4 |  6 |  4 | 8 *
x3o   . *b3/2o  4 |  6 |  4 | * 8

x3/2o3o *b3o

.   . .    . | 8   6 | 12 | 4 4
-------------+---+----+----+----
x   . .    . | 2 | 24 |  4 | 2 2
-------------+---+----+----+----
x3/2o .    . | 3 |  3 | 32 | 1 1
-------------+---+----+----+----
x3/2o3o    .  4 |  6 |  4 | 8 *
x3/2o . *b3o  4 |  6 |  4 | * 8

x3/2o3o *b3/2o

.   . .      . | 8   6 | 12 | 4 4
---------------+---+----+----+----
x   . .      . | 2 | 24 |  4 | 2 2
---------------+---+----+----+----
x3/2o .      . | 3 |  3 | 32 | 1 1
---------------+---+----+----+----
x3/2o3o      .  4 |  6 |  4 | 8 *
x3/2o . *b3/2o  4 |  6 |  4 | * 8

x3/2o3/2o *b3/2o

.   .   .      . | 8   6 | 12 | 4 4
-----------------+---+----+----+----
x   .   .      . | 2 | 24 |  4 | 2 2
-----------------+---+----+----+----
x3/2o   .      . | 3 |  3 | 32 | 1 1
-----------------+---+----+----+----
x3/2o3/2o      .  4 |  6 |  4 | 8 *
x3/2o   . *b3/2o  4 |  6 |  4 | * 8

s4o3o3o

demi( . . . . ) | 8   6 | 12 | 4 4
----------------+---+----+----+----
      s4o . .    2 | 24 |  4 | 2 2
----------------+---+----+----+----
sefa( s4o3o . ) | 3 |  3 | 32 | 1 1
----------------+---+----+----+----
      s4o3o .    4 |  6 |  4 | 8 *
sefa( s4o3o3o )  4 |  6 |  4 | * 8

starting figure: x4o3o3o

s2s4o3o

demi( . . . . ) | 8   3  3 |  9 3 | 3 1 4
----------------+---+-------+------+------
      s2s . .    2 | 12  * |  4 0 | 2 0 2
      . s4o .    2 |  * 12 |  2 2 | 1 1 2
----------------+---+-------+------+------
sefa( s2s4o . ) | 3 |  2  1 | 24 * | 1 0 1
sefa( . s4o3o ) | 3 |  0  3 |  * 8 | 0 1 1
----------------+---+-------+------+------
      s2s4o .    4 |  4  2 |  4 0 | 6 * *
      . s4o3o    4 |  0  6 |  0 4 | * 2 *
sefa( s2s4o3o )  4 |  3  3 |  3 1 | * * 8

starting figure: x x4o3o

s4o2s4o

demi( . . . . ) | 8  1  4 1 |  6  6 | 2 2 4
----------------+---+--------+-------+------
      s4o . .    2 | 4  * * |  4  0 | 2 0 2
      s 2 s .    2 | * 16 * |  2  2 | 1 1 2
      . . s4o    2 | *  * 4 |  0  4 | 0 2 2
----------------+---+--------+-------+------
sefa( s4o2s . ) | 3 | 1  2 0 | 16  * | 1 1 0
sefa( s 2 s4o ) | 3 | 0  2 1 |  * 16 | 0 1 1
----------------+---+--------+-------+------
      s4o2s .    4 | 2  4 0 |  4  0 | 4 * *
      s 2 s4o    4 | 0  4 2 |  0  4 | * 4 *
sefa( s4o2s4o )  4 | 1  4 1 |  2  2 | * * 8
or
demi( . . . . )    | 8  2  4 | 12 | 4 4
-------------------+---+------+----+----
      s4o . .    &  2 | 8  * |  4 | 2 2
      s 2 s .       2 | * 16 |  4 | 2 2
-------------------+---+------+----+----
sefa( s4o2s . )  & | 3 | 1  2 | 32 | 1 1
-------------------+---+------+----+----
      s4o2s .    &  4 | 2  4 |  4 | 8 *
sefa( s4o2s4o )     4 | 2  4 |  4 | * 8

starting figure: x4o x4o

s2s2s4o

demi( . . . . ) | 8  1 2 2 1 |  6 3 3 | 2 1 1 4
----------------+---+---------+--------+--------
      s2s . .    2 | 4 * * * |  4 0 0 | 2 0 0 2
      s 2 s .    2 | * 8 * * |  2 2 0 | 1 1 0 2
      . s2s .    2 | * * 8 * |  2 0 2 | 1 0 1 2
      . . s4o    2 | * * * 4 |  0 2 2 | 0 1 1 2
----------------+---+---------+--------+--------
sefa( s2s2s . ) | 3 | 1 1 1 0 | 16 * * | 1 0 0 1
sefa( s 2 s4o ) | 3 | 0 2 0 1 |  * 8 * | 0 1 0 1
sefa( . s2s4o ) | 3 | 0 0 2 1 |  * * 8 | 0 0 1 1
----------------+---+---------+--------+--------
      s2s2s .    4 | 2 2 2 0 |  4 0 0 | 4 * * *
      s 2 s4o    4 | 0 4 0 2 |  0 4 0 | * 2 * *
      . s2s4o    4 | 0 0 4 2 |  0 0 4 | * * 2 *
sefa( s2s2s4o )  4 | 1 2 2 1 |  2 1 1 | * * * 8

starting figure: x x x4o

s2s2s2s

demi( . . . .  ) | 8  1 1 1 1 1 1 | 3 3 3 3 | 1 1 1 1 4
-----------------+---+-------------+---------+----------
      s2s . .     2 | 4 * * * * * | 2 2 0 0 | 1 1 0 0 2
      s 2 s .     2 | * 4 * * * * | 2 0 2 0 | 1 0 1 0 2
      s . . s2*a  2 | * * 4 * * * | 0 2 2 0 | 0 1 1 0 2
      . s2s .     2 | * * * 4 * * | 2 0 0 2 | 1 0 0 1 2
      . s 2 s     2 | * * * * 4 * | 0 2 0 2 | 0 1 0 1 2
      . . s2s     2 | * * * * * 4 | 0 0 2 2 | 0 0 1 1 2
-----------------+---+-------------+---------+----------
sefa( s2s2s .  ) | 3 | 1 1 0 1 0 0 | 8 * * * | 1 0 0 0 1
sefa( s2s 2 s  ) | 3 | 1 0 1 0 1 0 | * 8 * * | 0 1 0 0 1
sefa( s 2 s2s  ) | 3 | 0 1 1 0 0 1 | * * 8 * | 0 0 1 0 1
sefa( . s2s2s  ) | 3 | 0 0 0 1 1 1 | * * * 8 | 0 0 0 1 1
-----------------+---+-------------+---------+----------
      s2s2s .     4 | 2 2 0 2 0 0 | 4 0 0 0 | 2 * * * *
      s2s 2 s     4 | 2 0 2 0 2 0 | 0 4 0 0 | * 2 * * *
      s 2 s2s     4 | 0 2 2 0 0 2 | 0 0 4 0 | * * 2 * *
      . s2s2s     4 | 0 0 0 2 2 2 | 0 0 0 4 | * * * 2 *
sefa( s2s2s2s  )  4 | 1 1 1 1 1 1 | 1 1 1 1 | * * * * 8

starting figure: x x x x

xo3oo3ox&#x   → height = 1/sqrt(2) = 0.707107
(tet || dual tet)

o.3o.3o.    | 4 *  3  3 0 | 3  6  3 0 | 1 3 3 1 0
.o3.o3.o    | * 4  0  3 3 | 0  3  6 3 | 0 1 3 3 1
------------+-----+--------+-----------+----------
x. .. ..    | 2 0 | 6  * * | 2  2  0 0 | 1 2 1 0 0
oo3oo3oo&#x | 1 1 | * 12 * | 0  2  2 0 | 0 1 2 1 0
.. .. .x    | 0 2 | *  * 6 | 0  0  2 2 | 0 0 1 2 1
------------+-----+--------+-----------+----------
x.3o. ..    | 3 0 | 3  0 0 | 4  *  * * | 1 1 0 0 0
xo .. ..&#x | 2 1 | 1  2 0 | * 12  * * | 0 1 1 0 0
.. .. ox&#x | 1 2 | 0  2 1 | *  * 12 * | 0 0 1 1 0
.. .o3.x    | 0 3 | 0  0 3 | *  *  * 4 | 0 0 0 1 1
------------+-----+--------+-----------+----------
x.3o.3o.     4 0 | 6  0 0 | 4  0  0 0 | 1 * * * *
xo3oo ..&#x  3 1 | 3  3 0 | 1  3  0 0 | * 4 * * *
xo .. ox&#x  2 2 | 1  4 1 | 0  2  2 0 | * * 6 * *
.. oo3ox&#x  1 3 | 0  3 3 | 0  0  3 1 | * * * 4 *
.o3.o3.x     0 4 | 0  0 6 | 0  0  0 4 | * * * * 1
or
o.3o.3o.    & | 8   3  3 | 3  9 | 1 4 3
--------------+---+-------+------+------
x. .. ..    & | 2 | 12  * | 2  2 | 1 2 1
oo3oo3oo&#x   | 2 |  * 12 | 0  4 | 0 2 2
--------------+---+-------+------+------
x.3o. ..    & | 3 |  3  0 | 8  * | 1 1 0
xo .. ..&#x & | 3 |  1  2 | * 24 | 0 1 1
--------------+---+-------+------+------
x.3o.3o.    &  4 |  6  0 | 4  0 | 2 * *
xo3oo ..&#x &  4 |  3  3 | 1  3 | * 8 *
xo .. ox&#x    4 |  2  4 | 0  4 | * * 6

oxo3ooo4ooo&#xt   → both heights = 1/sqrt(2) = 0.707107
(pt || pseudo oct || pt)

o..3o..4o..    | 1 * *  6  0 0 | 12 0  0 | 8 0
.o.3.o.4.o.    | * 6 *  1  4 1 |  4 4  4 | 4 4
..o3..o4..o    | * * 1  0  0 6 |  0 0 12 | 0 8
---------------+-------+--------+---------+----
oo.3oo.4oo.&#x | 1 1 0 | 6  * * |  4 0  0 | 4 0
.x. ... ...    | 0 2 0 | * 12 * |  1 2  1 | 2 2
.oo3.oo4.oo&#x | 0 1 1 | *  * 6 |  0 0  4 | 0 4
---------------+-------+--------+---------+----
ox. ... ...&#x | 1 2 0 | 2  1 0 | 12 *  * | 2 0
.x.3.o. ...    | 0 3 0 | 0  3 0 |  * 8  * | 1 1
.xo ... ...&#x | 0 2 1 | 0  1 2 |  * * 12 | 0 2
---------------+-------+--------+---------+----
ox.3oo. ...&#x  1 3 0 | 3  3 0 |  3 1  0 | 8 *
.xo3.oo ...&#x  0 3 1 | 0  3 3 |  0 1  3 | * 8
or
o..3o..4o..    & | 2 *   6  0 | 12 0 |  8
.o.3.o.4.o.      | * 6   2  4 |  8 4 |  8
-----------------+-----+-------+------+---
oo.3oo.4oo.&#x & | 1 1 | 12  * |  4 0 |  4
.x. ... ...      | 0 2 |  * 12 |  2 2 |  4
-----------------+-----+-------+------+---
ox. ... ...&#x & | 1 2 |  2  1 | 24 * |  2
.x.3.o. ...      | 0 3 |  0  3 |  * 8 |  2
-----------------+-----+-------+------+---
ox.3oo. ...&#x &  1 3 |  3  3 |  3 1 | 16

ooo3oxo3ooo&#xt   → both heights = 1/sqrt(2) = 0.707107
(pt || pseudo oct || pt)

o..3o..3o..    | 1 * *  6  0 0 | 12 0 0  0 | 4 4 0 0
.o.3.o.3.o.    | * 6 *  1  4 1 |  4 2 2  4 | 2 2 2 2
..o3..o3..o    | * * 1  0  0 6 |  0 0 0 12 | 0 0 4 4
---------------+-------+--------+-----------+--------
oo.3oo.3oo.&#x | 1 1 0 | 6  * * |  4 0 0  0 | 2 2 0 0
... .x. ...    | 0 2 0 | * 12 * |  1 1 1  1 | 1 1 1 1
.oo3.oo3.oo&#x | 0 1 1 | *  * 6 |  0 0 0  4 | 0 0 2 2
---------------+-------+--------+-----------+--------
... ox. ...&#x | 1 2 0 | 2  1 0 | 12 * *  * | 1 1 0 0
.o.3.x. ...    | 0 3 0 | 0  3 0 |  * 4 *  * | 1 0 1 0
... .x.3.o.    | 0 3 0 | 0  3 0 |  * * 4  * | 0 1 0 1
... .xo ...&#x | 0 2 1 | 0  1 2 |  * * * 12 | 0 0 1 1
---------------+-------+--------+-----------+--------
oo.3ox. ...&#x  1 3 0 | 3  3 0 |  3 1 0  0 | 4 * * *
... ox.3oo.&#x  1 3 0 | 3  3 0 |  3 0 1  0 | * 4 * *
.oo3.xo ...&#x  0 3 1 | 0  3 3 |  0 1 0  3 | * * 4 *
... .xo3.oo&#x  0 3 1 | 0  3 3 |  0 0 1  3 | * * * 4
or
o..3o..3o..    & | 2 *   6  0 | 12 0 0 | 4 4
.o.3.o.3.o.      | * 6   2  4 |  8 2 2 | 4 4
-----------------+-----+-------+--------+----
oo.3oo.3oo.&#x & | 1 1 | 12  * |  4 0 0 | 2 2
... .x. ...      | 0 2 |  * 12 |  2 1 1 | 2 2
-----------------+-----+-------+--------+----
... ox. ...&#x & | 1 2 |  2  1 | 24 * * | 1 1
.o.3.x. ...      | 0 3 |  0  3 |  * 4 * | 2 0
... .x.3.o.      | 0 3 |  0  3 |  * * 4 | 0 2
-----------------+-----+-------+--------+----
oo.3ox. ...&#x &  1 3 |  3  3 |  3 1 0 | 8 *
... ox.3oo.&#x &  1 3 |  3  3 |  3 0 1 | * 8

o(qo)o o(ox)o4o(oo)o&#xt   → both heights = 1/sqrt(2) = 0.707107
(pt || pseudo oct || pt)

o(..). o(..).4o(..).    & | 2 * *  2 4 0 0 |  8 4 0 |  8
.(o.). .(o.).4.(o.).      | * 2 *  2 0 4 0 |  8 0 4 |  8
.(.o). .(.o).4.(.o).      | * * 4  0 2 2 2 |  4 4 4 |  8
--------------------------+-------+---------+--------+---
o(o.). o(o.).4o(o.).&#x & | 1 1 0 | 4 * * * |  4 0 0 |  4
o(.o). o(.o).4o(.o).&#x & | 1 0 1 | * 8 * * |  2 2 0 |  4
.(oo). .(oo).4.(oo).&#x   | 0 1 1 | * * 8 * |  2 0 2 |  4
.(..). .(.x). .(..).      | 0 0 2 | * * * 4 |  0 2 2 |  4
--------------------------+-------+---------+--------+---
o(oo). o(oo).4o(oo).&#x & | 1 1 1 | 1 1 1 0 | 16 * * |  2
.(..). o(.x). .(..).&#x & | 1 0 2 | 0 2 0 1 |  * 8 * |  2
.(..). .(ox). .(..).&#x   | 0 1 2 | 0 0 2 1 |  * * 8 |  2
--------------------------+-------+---------+--------+---
.(..). o(ox). .(..).&#x &  1 1 2 | 1 2 2 1 |  2 1 1 | 16

o(qoo)o o(oqo)o o(ooq)o&#xt   → both heights = 1/sqrt(2) = 0.707107
(pt || pseudo oct || pt)

o(...). o(...). o(...).    | 1 * * * *  2 2 2 0 0 0 0 0 0 | 4 4 4 0 0 0 0 | 8 0
.(o..). .(o..). .(o..).    | * 2 * * *  1 0 0 2 2 1 0 0 0 | 2 2 0 4 2 2 0 | 4 4
.(.o.). .(.o.). .(.o.).    | * * 2 * *  0 1 0 2 0 0 2 1 0 | 2 0 2 4 2 0 2 | 4 4
.(..o). .(..o). .(..o).    | * * * 2 *  0 0 1 0 2 0 2 0 1 | 0 2 2 4 0 2 2 | 4 4
.(...)o .(...)o .(...)o    | * * * * 1  0 0 0 0 0 2 0 2 2 | 0 0 0 0 4 4 4 | 0 8
---------------------------+-----------+-------------------+---------------+----
o(o..). o(o..). o(o..).&#x | 1 1 0 0 0 | 2 * * * * * * * * | 2 2 0 0 0 0 0 | 4 0
o(.o.). o(.o.). o(.o.).&#x | 1 0 1 0 0 | * 2 * * * * * * * | 2 0 2 0 0 0 0 | 4 0
o(..o). o(..o). o(..o).&#x | 1 0 0 1 0 | * * 2 * * * * * * | 0 2 2 0 0 0 0 | 4 0
.(oo.). .(oo.). .(oo.).&#x | 0 1 1 0 0 | * * * 4 * * * * * | 1 0 0 2 1 0 0 | 2 2
.(o.o). .(o.o). .(o.o).&#x | 0 1 0 1 0 | * * * * 4 * * * * | 0 1 0 2 0 1 0 | 2 2
.(o..)o .(o..)o .(o..)o&#x | 0 1 0 0 1 | * * * * * 2 * * * | 0 0 0 0 2 2 0 | 0 4
.(.oo). .(.oo). .(.oo).&#x | 0 0 1 1 0 | * * * * * * 4 * * | 0 0 1 2 0 0 1 | 2 2
.(.o.)o .(.o.)o .(.o.)o&#x | 0 0 1 0 1 | * * * * * * * 2 * | 0 0 0 0 2 0 2 | 0 4
.(..o)o .(..o)o .(..o)o&#x | 0 0 0 1 1 | * * * * * * * * 2 | 0 0 0 0 0 2 2 | 0 4
---------------------------+-----------+-------------------+---------------+----
o(oo.). o(oo.). o(oo.).&#x | 1 1 1 0 0 | 1 1 0 1 0 0 0 0 0 | 4 * * * * * * | 2 0
o(o.o). o(o.o). o(o.o).&#x | 1 1 0 1 0 | 1 0 1 0 1 0 0 0 0 | * 4 * * * * * | 2 0
o(.oo). o(.oo). o(.oo).&#x | 1 0 1 1 0 | 0 1 1 0 0 0 1 0 0 | * * 4 * * * * | 2 0
.(ooo). .(ooo). .(ooo).&#x | 0 1 1 1 0 | 0 0 0 1 1 0 1 0 0 | * * * 8 * * * | 1 1
.(oo.)o .(oo.)o .(oo.)o&#x | 0 1 1 0 1 | 0 0 0 1 0 1 0 1 0 | * * * * 4 * * | 0 2
.(o.o)o .(o.o)o .(o.o)o&#x | 0 1 0 1 1 | 0 0 0 0 1 1 0 0 1 | * * * * * 4 * | 0 2
.(.oo)o .(.oo)o .(.oo)o&#x | 0 0 1 1 1 | 0 0 0 0 0 0 1 1 1 | * * * * * * 4 | 0 2
---------------------------+-----------+-------------------+---------------+----
o(ooo). o(ooo). o(ooo).&#x  1 1 1 1 0 | 1 1 1 1 1 0 1 0 0 | 1 1 1 1 0 0 0 | 8 *
.(ooo)o .(ooo)o .(ooo)o&#x  0 1 1 1 1 | 0 0 0 1 1 1 1 1 1 | 0 0 0 1 1 1 1 | * 8

xox oxo4ooo&#xt   → both heights = 1/2
(line || perp pseudo {4} || line)

o.. o..4o..    | 2 * *  1 4 1 0 0 0 | 4 4 4 0 0 | 4 4 0
.o. .o.4.o.    | * 4 *  0 2 0 2 2 0 | 1 4 2 1 4 | 2 4 2
..o ..o4..o    | * * 2  0 0 1 0 4 1 | 0 0 4 4 4 | 0 4 4
---------------+-------+-------------+-----------+------
x.. ... ...    | 2 0 0 | 1 * * * * * | 4 0 0 0 0 | 4 0 0
oo. oo.4oo.&#x | 1 1 0 | * 8 * * * * | 1 2 1 0 0 | 2 2 0
o.o o.o4o.o&#x | 1 0 1 | * * 2 * * * | 0 0 4 0 0 | 0 4 0
... .x. ...    | 0 2 0 | * * * 4 * * | 0 2 0 0 2 | 1 2 1
.oo .oo4.oo&#x | 0 1 1 | * * * * 8 * | 0 0 1 1 2 | 0 2 2
..x ... ...    | 0 0 2 | * * * * * 1 | 0 0 0 4 0 | 0 0 4
---------------+-------+-------------+-----------+------
xo. ... ...&#x | 2 1 0 | 1 2 0 0 0 0 | 4 * * * * | 2 0 0
... ox. ...&#x | 1 2 0 | 0 2 0 1 0 0 | * 8 * * * | 1 1 0
ooo ooo4ooo&#x | 1 1 1 | 0 1 1 0 1 0 | * * 8 * * | 0 2 0
.ox ... ...&#x | 0 1 2 | 0 0 0 0 2 1 | * * * 4 * | 0 0 2
... .xo ...&#x | 0 2 1 | 0 0 0 1 2 0 | * * * * 8 | 0 1 1
---------------+-------+-------------+-----------+------
xo. ox. ...&#x  2 2 0 | 1 4 0 1 0 0 | 2 2 0 0 0 | 4 * *
... oxo ...&#x  1 2 1 | 0 2 1 1 2 0 | 0 1 2 0 1 | * 8 *
.ox .xo ...&#x  0 2 2 | 0 0 0 1 4 1 | 0 0 0 2 2 | * * 4
or
o.. o..4o..    & | 4 *  1  4 1 0 | 4  4 4 | 4 4
.o. .o.4.o.      | * 4  0  4 0 2 | 2  8 2 | 4 4
-----------------+-----+----------+--------+----
x.. ... ...    & | 2 0 | 2  * * * | 4  0 0 | 4 0
oo. oo.4oo.&#x & | 1 1 | * 16 * * | 1  2 1 | 2 2
o.o o.o4o.o&#x   | 2 0 | *  * 2 * | 0  0 4 | 0 4
... .x. ...      | 0 2 | *  * * 4 | 0  4 0 | 2 2
-----------------+-----+----------+--------+----
xo. ... ...&#x & | 2 1 | 1  2 0 0 | 8  * * | 2 0
... ox. ...&#x & | 1 2 | 0  2 0 1 | * 16 * | 1 1
ooo ooo4ooo&#x   | 2 1 | 0  2 1 0 | *  * 8 | 0 2
-----------------+-----+----------+--------+----
xo. ox. ...&#x &  2 2 | 1  4 0 1 | 2  2 0 | 8 *
... oxo ...&#x    2 2 | 0  4 1 1 | 0  2 2 | * 8

xox oxo oxo&#xt   → both heights = 1/2
(line || perp pseudo {4} || line)

o.. o.. o..    | 2 * *  1 4 1 0 0 0 0 | 4 2 2 4 0 0 0 | 2 2 2 2 0 0
.o. .o. .o.    | * 4 *  0 2 0 1 1 2 0 | 1 2 2 2 1 2 2 | 1 1 2 2 1 1
..o ..o ..o    | * * 2  0 0 1 0 0 4 1 | 0 0 0 4 4 2 2 | 0 0 2 2 2 2
---------------+-------+---------------+---------------+------------
x.. ... ...    | 2 0 0 | 1 * * * * * * | 4 0 0 0 0 0 0 | 2 2 0 0 0 0
oo. oo. oo.&#x | 1 1 0 | * 8 * * * * * | 1 1 1 1 0 0 0 | 1 1 1 1 0 0
o.o o.o o.o&#x | 1 0 1 | * * 2 * * * * | 0 0 0 4 0 0 0 | 0 0 2 2 0 0
... .x. ...    | 0 2 0 | * * * 2 * * * | 0 2 0 0 0 2 0 | 1 0 2 0 1 0
... ... .x.    | 0 2 0 | * * * * 2 * * | 0 0 2 0 0 0 2 | 0 1 0 2 0 1
.oo .oo .oo&#x | 0 1 1 | * * * * * 8 * | 0 0 0 1 1 1 1 | 0 0 1 1 1 1
..x ... ...    | 0 0 2 | * * * * * * 1 | 0 0 0 0 4 0 0 | 0 0 0 0 2 2
---------------+-------+---------------+---------------+------------
xo. ... ...&#x | 2 1 0 | 1 2 0 0 0 0 0 | 4 * * * * * * | 1 1 0 0 0 0
... ox. ...&#x | 1 2 0 | 0 2 0 1 0 0 0 | * 4 * * * * * | 1 0 1 0 0 0
... ... ox.&#x | 1 2 0 | 0 2 0 0 1 0 0 | * * 4 * * * * | 0 1 0 1 0 0
ooo ooo ooo&#x | 1 1 1 | 0 1 1 0 0 1 0 | * * * 8 * * * | 0 0 1 1 0 0
.ox ... ...&#x | 0 1 2 | 0 0 0 0 0 2 1 | * * * * 4 * * | 0 0 0 0 1 1
... .xo ...&#x | 0 2 1 | 0 0 0 1 0 2 0 | * * * * * 4 * | 0 0 1 0 1 0
... ... .xo&#x | 0 2 1 | 0 0 0 0 1 2 0 | * * * * * * 4 | 0 0 0 1 0 1
---------------+-------+---------------+---------------+------------
xo. ox. ...&#x  2 2 0 | 1 4 0 1 0 0 0 | 2 2 0 0 0 0 0 | 2 * * * * *
xo. ... ox.&#x  2 2 0 | 1 4 0 0 1 0 0 | 2 0 2 0 0 0 0 | * 2 * * * *
... oxo ...&#x  1 2 1 | 0 2 1 1 0 2 0 | 0 1 0 2 0 1 0 | * * 4 * * *
... ... oxo&#x  1 2 1 | 0 2 1 0 1 2 0 | 0 0 1 2 0 0 1 | * * * 4 * *
.ox .xo ...&#x  0 2 2 | 0 0 0 1 0 4 1 | 0 0 0 0 2 2 0 | * * * * 2 *
.ox ... .xo&#x  0 2 2 | 0 0 0 0 1 4 1 | 0 0 0 0 2 0 2 | * * * * * 2
or
o.. o.. o..    & | 4 *  1  4 1 0 0 | 4 2 2 4 | 2 2 2 2
.o. .o. .o.      | * 4  0  4 0 1 1 | 2 4 4 2 | 2 2 2 2
-----------------+-----+------------+---------+--------
x.. ... ...    & | 2 0 | 2  * * * * | 4 0 0 0 | 2 2 0 0
oo. oo. oo.&#x & | 1 1 | * 16 * * * | 1 1 1 1 | 1 1 1 1
o.o o.o o.o&#x   | 2 0 | *  * 2 * * | 0 0 0 4 | 0 0 2 2
... .x. ...      | 0 2 | *  * * 2 * | 0 4 0 0 | 2 0 2 0
... ... .x.      | 0 2 | *  * * * 2 | 0 0 4 0 | 0 2 0 2
-----------------+-----+------------+---------+--------
xo. ... ...&#x & | 2 1 | 1  2 0 0 0 | 8 * * * | 1 1 0 0
... ox. ...&#x & | 1 2 | 0  2 0 1 0 | * 8 * * | 1 0 1 0
... ... ox.&#x & | 1 2 | 0  2 0 0 1 | * * 8 * | 0 1 0 1
ooo ooo ooo&#x   | 2 1 | 0  2 1 0 0 | * * * 8 | 0 0 1 1
-----------------+-----+------------+---------+--------
xo. ox. ...&#x &  2 2 | 1  4 0 1 0 | 2 2 0 0 | 4 * * *
xo. ... ox.&#x &  2 2 | 1  4 0 0 1 | 2 0 2 0 | * 4 * *
... oxo ...&#x    2 2 | 0  4 1 1 0 | 0 2 0 2 | * * 4 *
... ... oxo&#x    2 2 | 0  4 1 0 1 | 0 0 2 2 | * * * 4

xoo3oox oqo&#xt   → both heights = 1/sqrt(6) = 0.408248
({3} || perp q-line || dual {3})

o..3o.. o..    | 3 * *  2 2 2 0 0 | 1 4 2 1  4 0 0 | 2 4 2 0
.o.3.o. .o.    | * 2 *  0 3 0 3 0 | 0 3 0 0  6 3 0 | 1 3 3 1
..o3..o ..o    | * * 3  0 0 2 2 2 | 0 0 1 2  4 4 1 | 0 2 4 2
---------------+-------+-----------+----------------+--------
x.. ... ...    | 2 0 0 | 3 * * * * | 1 2 1 0  0 0 0 | 2 2 0 0
oo.3oo. oo.&#x | 1 1 0 | * 6 * * * | 0 2 0 0  2 0 0 | 1 2 1 0
o.o3o.o o.o&#x | 1 0 1 | * * 6 * * | 0 0 1 1  2 0 0 | 0 2 2 0
.oo3.oo .oo&#x | 0 1 1 | * * * 6 * | 0 0 0 0  2 2 0 | 0 1 2 1
... ..x ...    | 0 0 2 | * * * * 3 | 0 0 0 1  0 2 1 | 0 0 2 2
---------------+-------+-----------+----------------+--------
x..3o.. ...    | 3 0 0 | 3 0 0 0 0 | 1 * * *  * * * | 2 0 0 0
xo. ... ...&#x | 2 1 0 | 1 2 0 0 0 | * 6 * *  * * * | 1 1 0 0
x.o ... ...&#x | 2 0 1 | 1 0 2 0 0 | * * 3 *  * * * | 0 2 0 0
... o.x ...&#x | 1 0 2 | 0 0 2 0 1 | * * * 3  * * * | 0 0 2 0
ooo3ooo ooo&#x | 1 1 1 | 0 1 1 1 0 | * * * * 12 * * | 0 1 1 0
... .ox ...&#x | 0 1 2 | 0 0 0 2 1 | * * * *  * 6 * | 0 0 1 1
..o3..x ...    | 0 0 3 | 0 0 0 0 3 | * * * *  * * 1 | 0 0 0 2
---------------+-------+-----------+----------------+--------
xo.3oo. ...&#x  3 1 0 | 3 3 0 0 0 | 1 3 0 0  0 0 0 | 2 * * *
xoo ... ...&#x  2 1 1 | 1 2 2 1 0 | 0 1 1 0  2 0 0 | * 6 * *
... oox ...&#x  1 1 2 | 0 1 2 2 1 | 0 0 0 1  2 1 0 | * * 6 *
.oo3.ox ...&#x  0 1 3 | 0 0 0 3 3 | 0 0 0 0  0 3 1 | * * * 2
or
o..3o.. o..     & | 6 *  2  2 2 | 1  4 3  4 | 2  6
.o.3.o. .o.       | * 2  0  6 0 | 0  6 0  6 | 2  6
------------------+-----+--------+-----------+-----
x.. ... ...     & | 2 0 | 6  * * | 1  2 1  0 | 2  2
oo.3oo. oo.&#x  & | 1 1 | * 12 * | 0  2 0  2 | 1  3
o.o3o.o o.o&#x    | 2 0 | *  * 6 | 0  0 2  2 | 0  4
------------------+-----+--------+-----------+-----
x..3o.. ...     & | 3 0 | 3  0 0 | 2  * *  * | 2  0
xo. ... ...&#x  & | 2 1 | 1  2 0 | * 12 *  * | 1  1
x.o ... ...&#x  & | 3 0 | 1  0 2 | *  * 6  * | 0  2
ooo3ooo ooo&#x    | 2 1 | 0  2 1 | *  * * 12 | 0  2
------------------+-----+--------+-----------+-----
xo.3oo. ...&#x  &  3 1 | 3  3 0 | 1  3 0  0 | 4  *
xoo ... ...&#x  &  3 1 | 1  3 2 | 0  1 1  2 | * 12

oxoo3ooox&#xr   → all cyclical heights = sqrt(2/3) = 0.816497
                  in fact this lace simplex degenerates into a rhomb with diagonals:
                  height(1,3) = sqrt(2) = 1.414214
                  height(2,4) = sqrt(2/3) = 0.816497
(pt || ({3} || inv {3}) || pt)

o(..).3o(..).     & | 2 *  3 3 0 0 | 3 3  6 0 0 | 1 1 3 3
.(o.).3.(o.).     & | * 6  1 1 2 2 | 2 2  4 1 3 | 1 1 3 3
--------------------+-----+---------+------------+--------
o(o.).3o(o.).&#x  & | 1 1 | 6 * * * | 2 0  2 0 0 | 1 0 2 1
o(.o).3o(.o).&#x  & | 1 1 | * 6 * * | 0 2  2 0 0 | 0 1 1 2
.(x.). .(..).     & | 0 2 | * * 6 * | 1 1  0 1 1 | 1 1 1 1
.(oo).3.(oo).&#x  & | 0 2 | * * * 6 | 0 0  2 0 2 | 0 0 2 2
--------------------+-----+---------+------------+--------
o(x.). .(..).&#x  & | 1 2 | 2 0 1 0 | 6 *  * * * | 1 0 1 0
.(..). o(.x).&#x  & | 1 2 | 0 2 1 0 | * 6  * * * | 0 1 0 1
o(oo).3o(oo).&#x  & | 1 2 | 1 1 0 1 | * * 12 * * | 0 0 1 1
.(x.).3.(o.).     & | 0 3 | 0 0 3 0 | * *  * 2 * | 1 1 0 0
.(xo). .(..).     & | 0 3 | 0 0 1 2 | * *  * * 6 | 0 0 1 1
--------------------+-----+---------+------------+--------
o(x.).3o(o.).&#x  &  1 3 | 3 0 3 0 | 3 0  0 1 0 | 2 * * *
o(.o).3o(.x).&#x  &  1 3 | 0 3 3 0 | 0 3  0 1 0 | * 2 * *
o(xo). .(..).&#x  &  1 3 | 2 1 1 2 | 1 0  2 0 1 | * * 6 *
.(..). o(ox).&#x  &  1 3 | 1 2 1 2 | 0 1  2 0 1 | * * * 6

xo4oo ox4oo&#zx   → height = 0
(tegum sum of {4} and fully perp {4})
(tegum product of 2 {4})

o.4o. o.4o.    | 4 *  2  4 * |  8  4 |  8
.o4.o .o4.o    | * 4  0  4 2 |  4  8 |  8
---------------+-----+--------+-------+---
x. .. .. ..    | 2 0 | 4  * * |  4  0 |  4
oo4oo oo4oo&#x | 1 1 | * 16 * |  2  2 |  4
.. .. .x ..    | 0 2 | *  * 4 |  0  4 |  4
---------------+-----+--------+-------+---
xo .. .. ..&#x | 2 1 | 1  2 0 | 16  * |  2
.. .. ox ..&#x | 1 2 | 0  2 1 |  * 16 |  2
---------------+-----+--------+-------+---
xo .. ox ..&#x  2 2 | 1  4 1 |  2  2 | 16
or
o.4o. o.4o.    & | 8  2  4 | 12 |  8
-----------------+---+------+----+---
x. .. .. ..    & | 2 | 8  * |  4 |  4
oo4oo oo4oo&#x   | 2 | * 16 |  4 |  4
-----------------+---+------+----+---
xo .. .. ..&#x & | 3 | 1  2 | 32 |  2
-----------------+---+------+----+---
xo .. ox ..&#x    4 | 2  4 |  4 | 16

xo xo ox4oo&#zx   → height = 0
(tegum sum of {4} and fully perp {4})
(tegum product of 2 {4})

... 

xo xo ox ox&#zx   → height = 0
(tegum sum of {4} and fully perp {4})
(tegum product of 2 {4})

o. o. o. o.     | 4 *  1 1  4 0 0 | 4 4 2 2 | 2 2 2 2
.o .o .o .o     | * 4  0 0  4 1 1 | 2 2 4 4 | 2 2 2 2
----------------+-----+------------+---------+--------
x. .. .. ..     | 2 0 | 2 *  * * * | 4 0 0 0 | 2 2 0 0
.. x. .. ..     | 2 0 | * 2  * * * | 0 4 0 0 | 0 0 2 2
oo oo oo oo&#x  | 1 1 | * * 16 * * | 1 1 1 1 | 1 1 1 1
.. .. .x ..     | 0 2 | * *  * 2 * | 0 0 4 0 | 2 0 2 0
.. .. .. .x     | 0 2 | * *  * * 2 | 0 0 0 4 | 0 2 0 2
----------------+-----+------------+---------+--------
xo .. .. ..&#x  | 2 1 | 1 0  2 0 0 | 8 * * * | 1 1 0 0
.. xo .. ..&#x  | 2 1 | 0 1  2 0 0 | * 8 * * | 0 0 1 1
.. .. ox ..&#x  | 1 2 | 0 0  2 1 0 | * * 8 * | 1 0 1 0
.. .. .. ox&#x  | 1 2 | 0 0  2 0 1 | * * * 8 | 0 1 0 1
----------------+-----+------------+---------+--------
xo .. ox ..&#x   2 2 | 1 0  4 1 0 | 2 0 2 0 | 4 * * *
xo .. .. ox&#x   2 2 | 1 0  4 0 1 | 2 0 0 2 | * 4 * *
.. xo ox ..&#x   2 2 | 0 1  4 1 0 | 0 2 2 0 | * * 4 *
.. xo .. ox&#x   2 2 | 0 1  4 0 1 | 0 2 0 2 | * * * 4

xoxo oxox&#xr   → all cyclical heights = 1/sqrt(2) = 0.707107
                  (in fact this lace simplex degenerates into a square)

o... o...    | 2 * * *  1 2 1 2 0 0 0 0 0 0 | 2 1 2 1 2 2 2 0 0 0 0 0 | 1 1 1 2 1 2 0 0 0
.o.. .o..    | * 2 * *  0 2 0 0 1 2 1 0 0 0 | 1 2 0 0 2 2 0 1 2 2 0 0 | 1 0 2 1 0 2 1 1 0
..o. ..o.    | * * 2 *  0 0 1 0 0 2 0 1 2 0 | 0 0 0 0 2 0 2 2 1 2 2 1 | 0 0 1 0 1 2 1 2 1
...o ...o    | * * * 2  0 0 0 2 0 0 1 0 2 1 | 0 0 1 2 0 2 2 0 0 2 1 2 | 0 1 0 1 2 2 0 1 1
-------------+---------+---------------------+-------------------------+------------------
x... ....    | 2 0 0 0 | 1 * * * * * * * * * | 2 0 2 0 0 0 0 0 0 0 0 0 | 1 1 0 2 0 0 0 0 0
oo.. oo..&#x | 1 1 0 0 | * 4 * * * * * * * * | 1 1 0 0 1 1 0 0 0 0 0 0 | 1 0 1 1 0 1 0 0 0
o.o. o.o.&#x | 1 0 1 0 | * * 2 * * * * * * * | 0 0 0 0 2 0 2 0 0 0 0 0 | 0 0 1 0 1 2 0 0 0
o..o o..o&#x | 1 0 0 1 | * * * 4 * * * * * * | 0 0 1 1 0 1 1 0 0 0 0 0 | 0 1 0 1 1 1 0 0 0
.... .x..    | 0 2 0 0 | * * * * 1 * * * * * | 0 2 0 0 0 0 0 0 2 0 0 0 | 1 0 2 0 0 0 1 0 0
.oo. .oo.&#x | 0 1 1 0 | * * * * * 4 * * * * | 0 0 0 0 1 0 0 1 1 1 0 0 | 0 0 1 0 0 1 1 1 0
.o.o .o.o&#x | 0 1 0 1 | * * * * * * 2 * * * | 0 0 0 0 0 2 0 0 0 2 0 0 | 0 0 0 1 0 2 0 1 0
..x. ....    | 0 0 2 0 | * * * * * * * 1 * * | 0 0 0 0 0 0 0 2 0 0 2 0 | 0 0 0 0 0 0 1 2 1
..oo ..oo&#x | 0 0 1 1 | * * * * * * * * 4 * | 0 0 0 0 0 0 1 0 0 1 1 1 | 0 0 0 0 1 1 0 1 1
.... ...x    | 0 0 0 2 | * * * * * * * * * 1 | 0 0 0 2 0 0 0 0 0 0 0 2 | 0 1 0 0 2 0 0 0 1
-------------+---------+---------------------+-------------------------+------------------
xo.. ....&#x | 2 1 0 0 | 1 2 0 0 0 0 0 0 0 0 | 2 * * * * * * * * * * * | 1 0 0 1 0 0 0 0 0
.... ox..&#x | 1 2 0 0 | 0 2 0 0 1 0 0 0 0 0 | * 2 * * * * * * * * * * | 1 0 1 0 0 0 0 0 0
x..o ....&#x | 2 0 0 1 | 1 0 0 2 0 0 0 0 0 0 | * * 2 * * * * * * * * * | 0 1 0 1 0 0 0 0 0
.... o..x&#x | 1 0 0 2 | 0 0 0 2 0 0 0 0 0 1 | * * * 2 * * * * * * * * | 0 1 0 0 1 0 0 0 0
ooo. ooo.&#x | 1 1 1 0 | 0 1 1 0 0 1 0 0 0 0 | * * * * 4 * * * * * * * | 0 0 1 0 0 1 0 0 0
oo.o oo.o&#x | 1 1 0 1 | 0 1 0 1 0 0 1 0 0 0 | * * * * * 4 * * * * * * | 0 0 0 1 0 1 0 0 0
o.oo o.oo&#x | 1 0 1 1 | 0 0 1 1 0 0 0 0 1 0 | * * * * * * 4 * * * * * | 0 0 0 0 1 1 0 0 0
.ox. ....&#x | 0 1 2 0 | 0 0 0 0 0 2 0 1 0 0 | * * * * * * * 2 * * * * | 0 0 0 0 0 0 1 1 0
.... .xo.&#x | 0 2 1 0 | 0 0 0 0 1 2 0 0 0 0 | * * * * * * * * 2 * * * | 0 0 1 0 0 0 1 0 0
.ooo .ooo&#x | 0 1 1 1 | 0 0 0 0 0 1 1 0 1 0 | * * * * * * * * * 4 * * | 0 0 0 0 0 1 0 1 0
..xo ....&#x | 0 0 2 1 | 0 0 0 0 0 0 0 1 2 0 | * * * * * * * * * * 2 * | 0 0 0 0 0 0 0 1 1
.... ..ox&#x | 0 0 1 2 | 0 0 0 0 0 0 0 0 2 1 | * * * * * * * * * * * 2 | 0 0 0 0 1 0 0 0 1
-------------+---------+---------------------+-------------------------+------------------
xo.. ox..&#x  2 2 0 0 | 1 4 0 0 1 0 0 0 0 0 | 2 2 0 0 0 0 0 0 0 0 0 0 | 1 * * * * * * * *
x..o o..x&#x  2 0 0 2 | 1 0 0 4 0 0 0 0 0 1 | 0 0 2 2 0 0 0 0 0 0 0 0 | * 1 * * * * * * *
.... oxo.&#x  1 2 1 0 | 0 2 1 0 1 2 0 0 0 0 | 0 1 0 0 2 0 0 0 1 0 0 0 | * * 2 * * * * * *
xo.o ....&#x  2 1 0 1 | 1 2 0 2 0 0 1 0 0 0 | 1 0 1 0 0 2 0 0 0 0 0 0 | * * * 2 * * * * *
.... o.ox&#x  1 0 1 2 | 0 0 1 2 0 0 0 0 2 1 | 0 0 0 1 0 0 2 0 0 0 0 1 | * * * * 2 * * * *
oooo oooo&#x  1 1 1 1 | 0 1 1 1 0 1 1 0 1 0 | 0 0 0 0 1 1 1 0 0 1 0 0 | * * * * * 4 * * *
.ox. .xo.&#x  0 2 2 0 | 0 0 0 0 1 4 0 1 0 0 | 0 0 0 0 0 0 0 2 2 0 0 0 | * * * * * * 1 * *
.oxo ....&#x  0 1 2 1 | 0 0 0 0 0 2 1 1 2 0 | 0 0 0 0 0 0 0 1 0 2 1 0 | * * * * * * * 2 *
..xo ..ox&#x  0 0 2 2 | 0 0 0 0 0 0 0 1 4 1 | 0 0 0 0 0 0 0 0 0 0 2 2 | * * * * * * * * 1

qo ox3oo4oo&#zx   → height = 0
(tegum sum of q-line and perp oct)
(tegum product of q-line with oct)

o. o.3o.4o.    | 2 *   6  0 | 12 0 |  8
.o .o3.o4.o    | * 6   2  4 |  8 4 |  8
---------------+-----+-------+------+---
oo oo3oo4oo&#x | 1 1 | 12  * |  4 0 |  4
.. .x .. ..    | 0 2 |  * 12 |  2 2 |  4
---------------+-----+-------+------+---
.. ox .. ..&#x | 1 2 |  2  1 | 24 * |  2
.. .x3.o ..    | 0 3 |  0  3 |  * 8 |  2
---------------+-----+-------+------+---
.. ox3oo ..&#x  1 3 |  3  3 |  3 1 | 16

qo oo3ox3oo&#zx   → height = 0
(tegum sum of q-line and perp oct)
(tegum product of q-line with oct)

... 

qo os2os3os&#zx   → height = 0
(tegum sum of q-line and perp oct)
(tegum product of q-line with oct)

o. demi( o.2o.3o. )    | 2 *   6 0 0 |  6  6 0 0 | 2  6
.o demi( .o2.o3.o )    | * 6   2 2 2 |  4  4 1 3 | 2  6
-----------------------+-----+--------+-----------+-----
oo demi( oo2oo3oo )&#x | 1 1 | 12 * * |  2  2 0 0 | 1  3
..       .s2.s ..      | 0 2 |  * 6 * |  2  0 0 2 | 0  4
.. sefa( .. .s3.s )    | 0 2 |  * * 6 |  0  2 1 1 | 2  2
-----------------------+-----+--------+-----------+-----
oo       os2os ..  &#x | 1 2 |  2 1 0 | 12  * * * | 0  2
oo sefa( .. os3os )&#x | 1 2 |  2 0 1 |  * 12 * * | 1  1
..       .. .s3.s      | 0 3 |  0 0 3 |  *  * 2 * | 2  0
.. sefa( .s2.s3.s )    | 0 3 |  0 2 1 |  *  * * 6 | 0  2
-----------------------+-----+--------+-----------+-----
oo       .. os3os  &#x  1 3 |  3 0 3 |  0  3 1 0 | 4  *
oo sefa( os2os3os )&#x  1 3 |  3 2 1 |  2  1 0 1 | * 12

qooo oqoo ooqo oooq&#zx   → height = 0
(tegum sum of 4 pairwise perp q-lines)
(tegum product of 4 q-lines)

o... o... o... o...     | 2 * * *  2 2 2 0 0 0 | 4 4 4 0 |  8
.o.. .o.. .o.. .o..     | * 2 * *  2 0 0 2 2 0 | 4 4 0 4 |  8
..o. ..o. ..o. ..o.     | * * 2 *  0 2 0 2 0 2 | 4 0 4 4 |  8
...o ...o ...o ...o     | * * * 2  0 0 2 0 2 2 | 0 4 4 4 |  8
------------------------+---------+-------------+---------+---
oo.. oo.. oo.. oo..&#x  | 1 1 0 0 | 4 * * * * * | 2 2 0 0 |  4
o.o. o.o. o.o. o.o.&#x  | 1 0 1 0 | * 4 * * * * | 2 0 2 0 |  4
o..o o..o o..o o..o&#x  | 1 0 0 1 | * * 4 * * * | 0 2 2 0 |  4
.oo. .oo. .oo. .oo.&#x  | 0 1 1 0 | * * * 4 * * | 2 0 0 2 |  4
.o.o .o.o .o.o .o.o&#x  | 0 1 0 1 | * * * * 4 * | 0 2 0 2 |  4
..oo ..oo ..oo ..oo&#x  | 0 0 1 1 | * * * * * 4 | 0 0 2 2 |  4
------------------------+---------+-------------+---------+---
ooo. ooo. ooo. ooo.&#x  | 1 1 1 0 | 1 1 0 1 0 0 | 8 * * * |  2
oo.o oo.o oo.o oo.o&#x  | 1 1 0 1 | 1 0 1 0 1 0 | * 8 * * |  2
o.oo o.oo o.oo o.oo&#x  | 1 0 1 1 | 0 1 1 0 0 1 | * * 8 * |  2
.ooo .ooo .ooo .ooo&#x  | 0 1 1 1 | 0 0 0 1 1 1 | * * * 8 |  2
------------------------+---------+-------------+---------+---
oooo oooo oooo oooo&#x   1 1 1 1 | 1 1 1 1 1 1 | 1 1 1 1 | 16

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