hex

Acronym hex, K-4.2 (alt: octit, trapt)

Name hexadecachoron,
tetracross (β₄),
orthoplex,
hemitesseract,
hemioctachoron,
16-cell,
aeroter(id),
tetrahedral antiprism,
vertex figure of tac,
(line-)octahedral tegum,
(line-)trigonal-antirprismatic tegum,
Gosset polytope 1_1,1,
8-3-stepprism,
lattice C₄ contact polytope (span of its small roots),
equatorial cross-section of vertex-first tac

`©`	`©`
	skelleton – vertex figure – net

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

Layer	Symmetry	Subsymmetries
	`o3o3o4o`	`o3o3o .`	`o3o . o`	`o . o4o`	`. o3o4o`
1	`x3o3o4o`	`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`
1	`x3o3o *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, 0, 0, 0)/sqrt(2) & all permutations, all changes of sign
as hemitesseract: (1, 1, 1, 1)/sqrt(8) & all permutations, all even changes of sign
as "the other" (mirrored) hemitesseract: (1, 1, 1, -1)/sqrt(8) & all permutations, all even changes of sign
as tegum product:
- (1, 1, 0, 0)/2 & all sign changes
- (0, 0, 1, 1)/2 & all sign changes
(the compound of the first 3 such oriented hexadecachora is sico, 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:	oct	tet
tho	4	8
hex	0	16

)

Dual tes

Dihedral angles

at {3} between tet and tet: 120°

Face vector 8, 24, 32, 16

Confer

more general:: xPo3o...o3o4o sns2sms s4oPo4s
variations:: xo3oo3ox&#q
facetings:: hatho
Grünbaumian relatives:: hex+8oct 2hex+8oct
general pyramid-antiprisms:: n-apt
compounds:: haddet sico
related segmentochora:: octpy squasc
related CRFs:: pex hex quawros pacsid pith
ambification:: ico
complex polytopes:: Shephard's generalized hex
general polytopal classes:: Wythoffian polychora Catalan polychora tetrahedrochora regular noble polytopes orthoplex partial Stott expansions segmentochora fundamental lace prisms bistratic lace towers lace simplices Coxeter-Elte-Gosset polytopes Hanner polytopes
analogs:: regular orthoplex O_n demihypercube D_n

External
links

Considering its cells, i.e. the tets, more as digonal antiprisms, then always 4 build a closed ring within edgewise connection each, 4 of witch thus are swirling around each other.

This polychoron is not only obtained as the vertex alternated hemiation of the tesseract, the later could be re-obtained from this one by the extension of either its even or its odd facets. In fact it happens to be the kernel of any 2 tesseracts of the great icositetrachoron.

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.)

When considered as tet antiprism, the analysis of the being used lacing facets shows that the half-height section results in a half-edge sized co.

Being the dual of tes and considering that one's coordinates, it is apparent that this solid is nothing but a hyperball wrt. the norm |x|+|y|+|z|+|w|.

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 0 1
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})

o. o. o.4o.    | 4 * ♦ 1 1  4 0 | 4 4  4 | 4 4
.o .o .o4.o    | * 4 ♦ 0 0  4 2 | 2 2  8 | 4 4
---------------+-----+----------+--------+----
x. .. .. ..    | 2 0 | 2 *  * * | 4 0  0 | 4 0
.. x. .. ..    | 2 0 | * 2  * * | 0 4  0 | 0 4
oo oo oo4oo&#x | 1 1 | * * 16 * | 1 1  2 | 2 2
.. .. .x ..    | 0 2 | * *  * 4 | 0 0  4 | 2 2
---------------+-----+----------+--------+----
xo .. .. ..&#x | 2 1 | 1 0  2 0 | 8 *  * | 2 0
.. xo .. ..&#x | 2 1 | 0 1  2 0 | * 8  * | 0 2
.. .. ox ..&#x | 1 2 | 0 0  2 1 | * * 16 | 1 1
---------------+-----+----------+--------+----
xo .. ox ..&#x ♦ 2 2 | 1 0  2 1 | 2 0  2 | 8 *
.. xo ox ..&#x ♦ 2 2 | 0 1  2 1 | 0 2  2 | * 8

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

or
o... o...    & | 8 ♦ 1  4 1 |  6  6 | 2 4 2
---------------+---+--------+-------+------
x... ....    & | 2 | 4  * * |  4  0 | 2 2 0
oo.. oo..&#x & | 2 | * 16 * |  2  2 | 1 2 1
o.o. o.o.&#x & | 2 | *  * 4 |  0  4 | 0 2 2
---------------+---+--------+-------+------
xo.. ....&#x & | 3 | 1  2 0 | 16  * | 1 1 0
ooo. ooo.&#x & | 3 | 0  2 1 |  * 16 | 0 1 1
---------------+---+--------+-------+------
xo.. ox..&#x & ♦ 4 | 2  4 0 |  4  0 | 4 * *
xo.o ....&#x & ♦ 4 | 1  4 1 |  2  2 | * 8 *
oooo oooo&#x   ♦ 4 | 0  4 2 |  0  4 | * * 4

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)

o. o.3o.3o.    | 2 * ♦  6  0 | 12 0 0 | 4 4
.o .o3.o3.o    | * 6 ♦  2  4 |  8 2 2 | 4 4
---------------+-----+-------+--------+----
oo oo3oo3oo&#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
.. .o3.x ..    | 0 3 |  0  3 |  * 4 * | 2 0
.. .. .x3.o    | 0 3 |  0  3 |  * * 4 | 0 2
---------------+-----+-------+--------+----
.. oo3ox ..&#x ♦ 1 3 |  3  3 |  3 1 0 | 8 *
.. .. ox3oo&#x ♦ 1 3 |  3  3 |  3 0 1 | * 8

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

according to the coloring subsymmetry for Möbius-Kantor polygon
©	4 * \| 2 2 2 0 \| 2 4 2 1 2 1 \| 4 4 r⁴b² vertices * 4 \| 0 2 2 2 \| 1 2 1 2 4 2 \| 4 4 r²b⁴ vertices ----+---------+-------------+---- 2 0 \| 4 * * * \| 1 2 1 0 0 0 \| 2 2 red square's sides 1 1 \| * 8 * * \| 1 1 0 1 1 0 \| 3 1 red octagon's sides 1 1 \| * * 8 * \| 0 1 1 0 1 1 \| 1 3 blue octagram's sides 0 2 \| * * * 4 \| 0 0 0 1 2 1 \| 2 2 blue square's sides ----+---------+-------------+---- 2 1 \| 1 2 0 0 \| 4 * * * * * \| 2 0 {(rrr)} 2 1 \| 1 1 1 0 \| * 8 * * * * \| 1 1 {(rrb)} connected to red square's side 2 1 \| 1 0 2 0 \| * * 4 * * * \| 0 2 {(rbb)} connected to red square's side 1 2 \| 0 2 0 1 \| * * * 4 * * \| 2 0 {(rrb)} connected to blue square's side 1 2 \| 0 1 1 1 \| * * * * 8 * \| 1 1 {(rbb)} connected to blue square's side 1 2 \| 0 0 2 1 \| * * * * * 4 \| 0 2 {(bbb)} ----+---------+-------------+---- 2 2 \| 1 3 1 1 \| 1 1 0 1 1 0 \| 8 * tet comprising 4 consec. octagon's vert. 2 2 \| 1 1 3 1 \| 0 1 1 0 1 1 \| * 8 tet comprising 4 consec. octagram's vert.

according to the coloring subsymmetry for Möbius-Kantor polygon

©

4 * | 2 2 2 0 | 2 4 2 1 2 1 | 4 4  r⁴b² vertices
* 4 | 0 2 2 2 | 1 2 1 2 4 2 | 4 4  r²b⁴ vertices
----+---------+-------------+----
2 0 | 4 * * * | 1 2 1 0 0 0 | 2 2  red square's sides
1 1 | * 8 * * | 1 1 0 1 1 0 | 3 1  red octagon's sides
1 1 | * * 8 * | 0 1 1 0 1 1 | 1 3  blue octagram's sides
0 2 | * * * 4 | 0 0 0 1 2 1 | 2 2  blue square's sides
----+---------+-------------+----
2 1 | 1 2 0 0 | 4 * * * * * | 2 0  {(rrr)}
2 1 | 1 1 1 0 | * 8 * * * * | 1 1  {(rrb)} connected to red square's side
2 1 | 1 0 2 0 | * * 4 * * * | 0 2  {(rbb)} connected to red square's side
1 2 | 0 2 0 1 | * * * 4 * * | 2 0  {(rrb)} connected to blue square's side
1 2 | 0 1 1 1 | * * * * 8 * | 1 1  {(rbb)} connected to blue square's side
1 2 | 0 0 2 1 | * * * * * 4 | 0 2  {(bbb)}
----+---------+-------------+----
2 2 | 1 3 1 1 | 1 1 0 1 1 0 | 8 *  tet comprising 4 consec. octagon's vert.
2 2 | 1 1 3 1 | 0 1 1 0 1 1 | * 8  tet comprising 4 consec. octagram's vert.

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