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
  ©   ©  
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 even permutations, all even changes of sign
  • as "the other" (mirrored) hemitesseract:   (1/sqrt(8), 1/sqrt(8), 1/sqrt(8), -1/sqrt(8))   & all even 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)
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  
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.


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})

... 

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

© 2004-2018
top of page