ico

Acronym ico (alt.: jot)

Name icositetrachoron,
24-cell,
xylochoron,
octaplex,
rectified hexadecachoron,
birectified tesseract,
hyperdiamond,
joined tesseract,
trisoctachoron,
vertex figure of hext,
Voronoi cell of lattice F₄,
Voronoi cell of lattice D₄*,
surtegmated tesseract,
Gosset polytope 0_1,1,1,
lattice B₄ contact polytope (span of its small roots),
lattice D₄ contact polytope (span of its roots),
lattice F₄ contact polytope (span of its small roots),
equatorial cross-section of hex-first rat

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(edge-skeleton, readily
visualizing oct || pseudo co || oct)

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displaying the vertex sets
of 3 vertex-inscribed hexes

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showing its relation
to swirl symmetry

Cross sections

©

Circumradius 1

Edge radius sqrt(3)/2 = 0.866025

Face radius sqrt(2/3) = 0.816497

Inradius 1/sqrt(2) = 0.707107

Vertex figure

©

Vertex layers

Layer	Symmetry	Subsymmetries
	`o3o4o3o`	`o3o4o .`	`o3o . o`	`o . o3o`	`. o4o3o`
1	`x3o4o3o`	`x3o4o .` oct first	`x3o . o` {3} first	`x . o3o` edge first	`. o4o3o` vertex first
2		`o3x4o .`	`o3x . q`	`o . q3o`	`. x4o3o` vertex figure
3		`x3o4o .` opposite oct	`x3x . o`	`x . o3q`	`. o4o3q`
4			`x3o . q`	`u . o3o`	`. x4o3o`
5			`o3x . o` opposite {3}	`x . q3o`	`. o4o3o` opposite vertex
6				`o . o3q`
7				`x . o3o` opposite edge
	`o3o3o4o`	`o3o3o .`	`o3o . o`	`o . o4o`	`. o3o4o`
1	`o3x3o4o`	`o3x3o .` oct first	`o3x . o` {3} first	`o . o4o` vertex first	`. x3o4o` oct first
2		`x3o3x .`	`x3o . q`	`x . x4o` vertex figure	`. o3x4o`
3a		`o3x3o .` opposite oct	`x3x . o`	`u . o4o`	`. x3o4o` opposite oct
3b		`o3x3o .` opposite oct	`x3x . o`	`o . o4q`	`. x3o4o` opposite oct
4			`o3x . q`	`x . x4o`
5			`x3o . o` opposite {3}	`o . o4o` opposite vertex
	`o3o3o *b3o`	`o3o3o .`	`o3o . *b3o`	`o . o o`	`. o3o *b3o`
1	`o3x3o *b3o`	`o3x3o .` oct first	`o3x . *b3o` oct first	`o . o o` vertex first	`. x3o *b3o` oct first
2		`x3o3x .`	`x3o . *b3x`	`x . x x` vertex figure	`. o3x *b3x`
3a		`o3x3o .` opposite oct	`o3x . *b3o` opposite oct	`u . o o`	`. x3o *b3o` opposite oct
3b				`o . u o`
3c				`o . o u`
4				`x . x x`
5				`o . o o` opposite vertex

Lace city
in approx. ASCII-art

©	o3o -- o3o4o (point) o3o q3o o3q o3o -- o3o4x (cube) o3q q3o -- q3o4o (q-oct) o3o q3o o3q o3o -- o3o4x (cube) o3o -- o3o4o (point)

©	o3x x3o -- x3o4o (oct) x3o x3x o3x -- o3x4o (co) o3x x3o -- x3o4o (oct)

©

o4o  x4o  o4o		-- x3o4o (oct)
             
x4o  o4q  x4o		-- o3x4o (co)
             
o4o  x4o  o4o		-- x3o4o (oct)

    \    \    \    \    \
      \    \    \    \    +-- o o4o (point)
        \    \    \    +----- x x4o (cube)
          \    \    +-------- compound (q-oct) of:
            \    \            o o4q (dual q-{4})
              \    \          u o4o (u-line)
                \    +------- x x4o (cube)
                  +---------- o o4o (point)

this representation also displays the tegum sum of 2 perpendicular q-{4}'s and a tes

Lace hyper city
in approx. ASCII-art

Coordinates

1. (1, 0, 0, 0) & all permutations, all changes of sign
  (vertex inscribed q-hex)
2. (1/2, 1/2, 1/2, 1/2) & all permutations, all changes of sign
  (vertex inscribed tes)
or just (1/sqrt(2), 1/sqrt(2), 0, 0) & all permutations, all changes of sign
(in dual positioning)
(the compound of those 2 such oriented icositetrachora is stoc)

Volume 2

Surface 8 sqrt(2) = 11.313708

General of army (is itself convex)

Colonel of regiment

(is itself locally convex

by cells:	cho	co	cube	oct	oho	thah
hoh honho	8	4	0	0	0	24
hohoh	8	0	0	8	0	16
huhoh	8	0	0	0	0	16
dod honho	4	4	0	0	4	16
hodho	4	0	0	16	4	8
odho	4	0	0	0	4	8
ihi	0	12	0	0	0	24
ohuhoh	0	8	0	8	0	16
ghahoh	0	8	0	0	0	16
doh honho	0	4	0	8	8	8
ratho	0	4	0	8	0	8
gico (compound)	0	0	24	0	0	0
ico	0	0	0	24	0	0
shahoh	0	0	0	16	8	0
oh	0	0	0	8	8	0

)

Dual (selfdual, in different orientation)

Dihedral angles

at {3} between oct and oct: 120°

Face vector 24, 96, 96, 24

Confer

more general:: n-tet-swirl
Grünbaumian relatives:: 2ico 2ico+48{4}+128{3} 2ico+2gico ico+gico+72{4} ico+gico+24co
compounds:: stoc chi dox bitapna kitapna bitefa kitefa
segmentochora:: oct || co {6} || oct cubpy
further diminishings:: xo3ox qo3oq&zx
related CRFs:: pabdico pexic bicyte ausodip pacsrit poxic pocsric cytau tes ecubedpy ecubpy tridico
uniform relative:: hex tes
decompositions:: icopy
ambification:: rico
ambification pre-image:: hex
general polytopal classes:: Wythoffian polychora Catalan polychora regular noble polytopes bistratic lace towers lace simplices partial Stott expansions Coxeter-Elte-Gosset polytopes
analogs:: rectified orthoplex rO_n birectified hypercube brC_n

External
links

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Considering its cells, i.e. the octs, more as trigonal antiprisms, then always 6 build a closed ring each, 4 of witch thus are swirling around each other. This ring of 6 is best seen in the edge-first projection picture at the right.

As can be seen from what is mentioned above (at coordinates or the last of the pictures): the hex-diminished ico is nothing but the tes. Conversely, the 16-diminished ico (corresponding to the vertex directions of tes) is nothing but the hex. In fact, ico is the hull of the 3-hex compound {3,4,3}[3{3,3,4}]2{3,4,3} (sico).

Note that ico can be thought of as the external blend of 24 octpies. This decomposition is described as the degenerate segmentoteron ox3oo4oo3oo&#x.

The desmic configuration of the vertices of the projective compound of 3 tetrahedra (mentioned here) can, by doubling each point into antipodal pairs, transfered into elliptical space of the next dimension, then becoming the points of this icositetrachoron. In fact, the former vertex sets of the 3 tetrahedra here become the vertices of the 3 inscribed hexes. Projective lines of the former setup here would become great circles for sure.

The number of ways to color the icositetrachoron with different colors per cell is 24!/576 = 357 378 279 398 345 917 071 360 000. – This is because the color group is the permutation group of 24 elements and has size 24!, while the order of the pure rotational icositetrachoral group is 576. (The reflectional icositetrachoral group would have twice as many, i.e. 1152 elements.)

"join", as being used in its name "joined tesseract", here simply is meant as the according Conway operator. As such it represents nothing else but the dual of the rectified dual. And indeed, within 4D the rectified dual is the birectified form. And the birectified tes is just ico in its o3x3o4o form. Finally, this ico is also selfdual. I.e. this term just refers explicitely to o3m3o4o.

Its name "trisoctachoron" obviously more applies to its D₄ subsymmetry, i.e. o3x3o *b3o.

Being selfdual and considering the second set of coordinates, it is apparent that this solid is nothing but a hyperball wrt. the norm max(|x|+|y|, |x|+|z|, |x|+|w|, |y|+|z|, |y|+|w|, |z|+|w|).

Incidence matrix according to Dynkin symbol

x3o4o3o

. . . . | 24 ♦  8 | 12 |  6
--------+----+----+----+---
x . . . |  2 | 96 |  3 |  3
--------+----+----+----+---
x3o . . |  3 |  3 | 96 |  2
--------+----+----+----+---
x3o4o . ♦  6 | 12 |  8 | 24

snubbed forms: β3o4o3o

x3o4o3/2o

. . .   . | 24 ♦  8 | 12 |  6
----------+----+----+----+---
x . .   . |  2 | 96 |  3 |  3
----------+----+----+----+---
x3o .   . |  3 |  3 | 96 |  2
----------+----+----+----+---
x3o4o   . ♦  6 | 12 |  8 | 24

x3o4/3o3o

. .   . . | 24 ♦  8 | 12 |  6
----------+----+----+----+---
x .   . . |  2 | 96 |  3 |  3
----------+----+----+----+---
x3o   . . |  3 |  3 | 96 |  2
----------+----+----+----+---
x3o4/3o . ♦  6 | 12 |  8 | 24

x3o4/3o3/2o

. .   .   . | 24 ♦  8 | 12 |  6
------------+----+----+----+---
x .   .   . |  2 | 96 |  3 |  3
------------+----+----+----+---
x3o   .   . |  3 |  3 | 96 |  2
------------+----+----+----+---
x3o4/3o   . ♦  6 | 12 |  8 | 24

x3/2o4o3o

.   . . . | 24 ♦  8 | 12 |  6
----------+----+----+----+---
x   . . . |  2 | 96 |  3 |  3
----------+----+----+----+---
x3/2o . . |  3 |  3 | 96 |  2
----------+----+----+----+---
x3/2o4o . ♦  6 | 12 |  8 | 24

x3/2o4o3/2o

.   . .   . | 24 ♦  8 | 12 |  6
------------+----+----+----+---
x   . .   . |  2 | 96 |  3 |  3
------------+----+----+----+---
x3/2o .   . |  3 |  3 | 96 |  2
------------+----+----+----+---
x3/2o4o   . ♦  6 | 12 |  8 | 24

x3/2o4/3o3o

.   .   . . | 24 ♦  8 | 12 |  6
------------+----+----+----+---
x   .   . . |  2 | 96 |  3 |  3
------------+----+----+----+---
x3/2o   . . |  3 |  3 | 96 |  2
------------+----+----+----+---
x3/2o4/3o . ♦  6 | 12 |  8 | 24

x3/2o4/3o3/2o

.   .   .   . | 24 ♦  8 | 12 |  6
--------------+----+----+----+---
x   .   .   . |  2 | 96 |  3 |  3
--------------+----+----+----+---
x3/2o   .   . |  3 |  3 | 96 |  2
--------------+----+----+----+---
x3/2o4/3o   . ♦  6 | 12 |  8 | 24

o3x3o4o

. . . . | 24 ♦  8 |  4  8 |  4 2
--------+----+----+-------+-----
. x . . |  2 | 96 |  1  2 |  2 1
--------+----+----+-------+-----
o3x . . |  3 |  3 | 32  * |  2 0
. x3o . |  3 |  3 |  * 64 |  1 1
--------+----+----+-------+-----
o3x3o . ♦  6 | 12 |  4  4 | 16 *
. x3o4o ♦  6 | 12 |  0  8 |  * 8

snubbed forms: o3β3o4o

o3x3o4/3o

. . .   . | 24 ♦  8 |  4  8 |  4 2
----------+----+----+-------+-----
. x .   . |  2 | 96 |  1  2 |  2 1
----------+----+----+-------+-----
o3x .   . |  3 |  3 | 32  * |  2 0
. x3o   . |  3 |  3 |  * 64 |  1 1
----------+----+----+-------+-----
o3x3o   . ♦  6 | 12 |  4  4 | 16 *
. x3o4/3o ♦  6 | 12 |  0  8 |  * 8

o3x3/2o4o

. .   . . | 24 ♦  8 |  4  8 |  4 2
----------+----+----+-------+-----
. x   . . |  2 | 96 |  1  2 |  2 1
----------+----+----+-------+-----
o3x   . . |  3 |  3 | 32  * |  2 0
. x3/2o . |  3 |  3 |  * 64 |  1 1
----------+----+----+-------+-----
o3x3/2o . ♦  6 | 12 |  4  4 | 16 *
. x3/2o4o ♦  6 | 12 |  0  8 |  * 8

o3x3/2o4/3o

. .   .   . | 24 ♦  8 |  4  8 |  4 2
------------+----+----+-------+-----
. x   .   . |  2 | 96 |  1  2 |  2 1
------------+----+----+-------+-----
o3x   .   . |  3 |  3 | 32  * |  2 0
. x3/2o   . |  3 |  3 |  * 64 |  1 1
------------+----+----+-------+-----
o3x3/2o   . ♦  6 | 12 |  4  4 | 16 *
. x3/2o4/3o ♦  6 | 12 |  0  8 |  * 8

o3/2x3o4o

.   . . . | 24 ♦  8 |  4  8 |  4 2
----------+----+----+-------+-----
.   x . . |  2 | 96 |  1  2 |  2 1
----------+----+----+-------+-----
o3/2x . . |  3 |  3 | 32  * |  2 0
.   x3o . |  3 |  3 |  * 64 |  1 1
----------+----+----+-------+-----
o3/2x3o . ♦  6 | 12 |  4  4 | 16 *
.   x3o4o ♦  6 | 12 |  0  8 |  * 8

o3/2x3o4/3o

.   . .   . | 24 ♦  8 |  4  8 |  4 2
------------+----+----+-------+-----
.   x .   . |  2 | 96 |  1  2 |  2 1
------------+----+----+-------+-----
o3/2x .   . |  3 |  3 | 32  * |  2 0
.   x3o   . |  3 |  3 |  * 64 |  1 1
------------+----+----+-------+-----
o3/2x3o   . ♦  6 | 12 |  4  4 | 16 *
.   x3o4/3o ♦  6 | 12 |  0  8 |  * 8

o3/2x3/2o4o

.   .   . . | 24 ♦  8 |  4  8 |  4 2
------------+----+----+-------+-----
.   x   . . |  2 | 96 |  1  2 |  2 1
------------+----+----+-------+-----
o3/2x   . . |  3 |  3 | 32  * |  2 0
.   x3/2o . |  3 |  3 |  * 64 |  1 1
------------+----+----+-------+-----
o3/2x3/2o . ♦  6 | 12 |  4  4 | 16 *
.   x3/2o4o ♦  6 | 12 |  0  8 |  * 8

o3/2x3/2o4/3o

.   .   .   . | 24 ♦  8 |  4  8 |  4 2
--------------+----+----+-------+-----
.   x   .   . |  2 | 96 |  1  2 |  2 1
--------------+----+----+-------+-----
o3/2x   .   . |  3 |  3 | 32  * |  2 0
.   x3/2o   . |  3 |  3 |  * 64 |  1 1
--------------+----+----+-------+-----
o3/2x3/2o   . ♦  6 | 12 |  4  4 | 16 *
.   x3/2o4/3o ♦  6 | 12 |  0  8 |  * 8

o3x3o *b3o

. . .    . | 24 ♦  8 |  4  4  4 | 2 2 2
-----------+----+----+----------+------
. x .    . |  2 | 96 |  1  1  1 | 1 1 1
-----------+----+----+----------+------
o3x .    . |  3 |  3 | 32  *  * | 1 1 0
. x3o    . |  3 |  3 |  * 32  * | 1 0 1
. x . *b3o |  3 |  3 |  *  * 32 | 0 1 1
-----------+----+----+----------+------
o3x3o    . ♦  6 | 12 |  4  4  0 | 8 * *
o3x . *b3o ♦  6 | 12 |  4  0  4 | * 8 *
. x3o *b3o ♦  6 | 12 |  0  4  4 | * * 8

snubbed forms: o3β3o *b3o

o3x3o *b3/2o

. . .      . | 24 ♦  8 |  4  4  4 | 2 2 2
-------------+----+----+----------+------
. x .      . |  2 | 96 |  1  1  1 | 1 1 1
-------------+----+----+----------+------
o3x .      . |  3 |  3 | 32  *  * | 1 1 0
. x3o      . |  3 |  3 |  * 32  * | 1 0 1
. x . *b3/2o |  3 |  3 |  *  * 32 | 0 1 1
-------------+----+----+----------+------
o3x3o      . ♦  6 | 12 |  4  4  0 | 8 * *
o3x . *b3/2o ♦  6 | 12 |  4  0  4 | * 8 *
. x3o *b3/2o ♦  6 | 12 |  0  4  4 | * * 8

o3x3/2o *b3/2o

. .   .      . | 24 ♦  8 |  4  4  4 | 2 2 2
---------------+----+----+----------+------
. x   .      . |  2 | 96 |  1  1  1 | 1 1 1
---------------+----+----+----------+------
o3x   .      . |  3 |  3 | 32  *  * | 1 1 0
. x3/2o      . |  3 |  3 |  * 32  * | 1 0 1
. x   . *b3/2o |  3 |  3 |  *  * 32 | 0 1 1
---------------+----+----+----------+------
o3x3/2o      . ♦  6 | 12 |  4  4  0 | 8 * *
o3x   . *b3/2o ♦  6 | 12 |  4  0  4 | * 8 *
. x3/2o *b3/2o ♦  6 | 12 |  0  4  4 | * * 8

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

.   .   .      . | 24 ♦  8 |  4  4  4 | 2 2 2
-----------------+----+----+----------+------
.   x   .      . |  2 | 96 |  1  1  1 | 1 1 1
-----------------+----+----+----------+------
o3/2x   .      . |  3 |  3 | 32  *  * | 1 1 0
.   x3/2o      . |  3 |  3 |  * 32  * | 1 0 1
.   x   . *b3/2o |  3 |  3 |  *  * 32 | 0 1 1
-----------------+----+----+----------+------
o3/2x3/2o      . ♦  6 | 12 |  4  4  0 | 8 * *
o3/2x   . *b3/2o ♦  6 | 12 |  4  0  4 | * 8 *
.   x3/2o *b3/2o ♦  6 | 12 |  0  4  4 | * * 8

xox3oxo4ooo&#xt   → both heights = 1/sqrt(2) = 0.707107
(oct || pseudo co || oct)

o..3o..4o..     | 6  * * ♦  4  4  0  0  0 | 4  4  4 0  0  0 0 | 1 4 1 0 0
.o.3.o.4.o.     | * 12 * ♦  0  2  4  2  0 | 0  1  4 2  4  1 0 | 0 2 2 2 0
..o3..o4..o     | *  * 6 ♦  0  0  0  4  4 | 0  0  0 0  4  4 4 | 0 0 1 4 1
----------------+--------+----------------+-------------------+----------
x.. ... ...     | 2  0 0 | 12  *  *  *  * | 2  1  0 0  0  0 0 | 1 2 0 0 0
oo.3oo.4oo.&#x  | 1  1 0 |  * 24  *  *  * | 0  1  2 0  0  0 0 | 0 2 1 0 0
... .x. ...     | 0  2 0 |  *  * 24  *  * | 0  0  1 1  1  0 0 | 0 1 1 1 0
.oo3.oo4.oo&#x  | 0  1 1 |  *  *  * 24  * | 0  0  0 0  2  1 0 | 0 0 1 2 0
..x ... ...     | 0  0 2 |  *  *  *  * 12 | 0  0  0 0  0  1 2 | 0 0 0 2 1
----------------+--------+----------------+-------------------+----------
x..3o.. ...     | 3  0 0 |  3  0  0  0  0 | 8  *  * *  *  * * | 1 1 0 0 0
xo. ... ...&#x  | 2  1 0 |  1  2  0  0  0 | * 12  * *  *  * * | 0 2 0 0 0
... ox. ...&#x  | 1  2 0 |  0  2  1  0  0 | *  * 24 *  *  * * | 0 1 1 0 0
.o.3.x. ...     | 0  3 0 |  0  0  3  0  0 | *  *  * 8  *  * * | 0 1 0 1 0
... .xo ...&#x  | 0  2 1 |  0  0  1  2  0 | *  *  * * 24  * * | 0 0 1 1 0
.ox ... ...&#x  | 0  1 2 |  0  0  0  2  1 | *  *  * *  * 12 * | 0 0 0 2 0
..x3..o ...     | 0  0 3 |  0  0  0  0  3 | *  *  * *  *  * 8 | 0 0 0 1 1
----------------+--------+----------------+-------------------+----------
x..3o..4o..     ♦ 6  0 0 | 12  0  0  0  0 | 8  0  0 0  0  0 0 | 1 * * * *
xo.3ox. ...&#x  ♦ 3  3 0 |  3  6  3  0  0 | 1  3  3 1  0  0 0 | * 8 * * *
... oxo4ooo&#xt ♦ 1  4 1 |  0  4  4  4  0 | 0  0  4 0  4  0 0 | * * 6 * *
.ox3.xo ...&#x  ♦ 0  3 3 |  0  0  3  6  3 | 0  0  0 1  3  3 1 | * * * 8 *
..x3..o4..o     ♦ 0  0 6 |  0  0  0  0 12 | 0  0  0 0  0  0 8 | * * * * 1

or
o..3o..4o..     & | 12  * ♦  4  4  0 |  4  4  4 0 | 1  4 1
.o.3.o.4.o.       |  * 12 ♦  0  4  4 |  0  2  8 2 | 0  4 2
------------------+-------+----------+------------+-------
x.. ... ...     & |  2  0 | 24  *  * |  2  1  0 0 | 1  2 0
oo.3oo.4oo.&#x  & |  1  1 |  * 48  * |  0  1  2 0 | 0  2 1
... .x. ...       |  0  2 |  *  * 24 |  0  0  2 1 | 0  2 1
------------------+-------+----------+------------+-------
x..3o.. ...     & |  3  0 |  3  0  0 | 16  *  * * | 1  1 0
xo. ... ...&#x  & |  2  1 |  1  2  0 |  * 24  * * | 0  2 0
... ox. ...&#x  & |  1  2 |  0  2  1 |  *  * 48 * | 0  1 1
.o.3.x. ...       |  0  3 |  0  0  3 |  *  *  * 8 | 0  2 0
------------------+-------+----------+------------+-------
x..3o..4o..     & ♦  6  0 | 12  0  0 |  8  0  0 0 | 2  * *
xo.3ox. ...&#x  & ♦  3  3 |  3  6  3 |  1  3  3 1 | * 16 *
... oxo4ooo&#xt   ♦  2  4 |  0  8  4 |  0  0  8 0 | *  * 6

oxo3xox3oxo&#xt   → both heights = 1/sqrt(2) = 0.707107
(oct || pseudo co || oct)

o..3o..3o..     | 6  * * ♦  4  4  0  0  0  0 | 2 2  2  4  2 0 0  0  0  0 0 0 | 1 2 2 1 0 0 0
.o.3.o.3.o.     | * 12 * ♦  0  2  2  2  2  0 | 0 0  2  1  2 1 1  2  1  2 0 0 | 0 1 1 2 1 1 0
..o3..o3..o     | *  * 6 ♦  0  0  0  0  4  4 | 0 0  0  0  0 0 0  2  4  2 2 2 | 0 0 0 1 2 2 1
----------------+--------+-------------------+-------------------------------+--------------
... x.. ...     | 2  0 0 | 12  *  *  *  *  * | 1 1  0  1  0 0 0  0  0  0 0 0 | 1 1 1 0 0 0 0
oo. oo. oo.&#x  | 1  1 0 |  * 24  *  *  *  * | 0 0  1  1  1 0 0  0  0  0 0 0 | 0 1 1 1 0 0 0
.x. ... ...     | 0  2 0 |  *  * 12  *  *  * | 0 0  1  0  0 1 0  1  0  0 0 0 | 0 1 0 1 1 0 0
... ... .x.     | 0  2 0 |  *  *  * 12  *  * | 0 0  0  0  1 0 1  1  0  0 0 0 | 0 0 1 1 0 1 0
.oo .oo .oo&#x  | 0  1 1 |  *  *  *  * 24  * | 0 0  0  0  0 0 0  1  1  1 0 0 | 0 0 0 1 1 1 0
... ..x ...     | 0  0 2 |  *  *  *  *  * 12 | 0 0  0  0  0 0 0  0  1  0 1 1 | 0 0 0 0 1 1 1
----------------+--------+-------------------+-------------------------------+--------------
o..3x.. ...     | 3  0 0 |  3  0  0  0  0  0 | 4 *  *  *  * * *  *  *  * * * | 1 1 0 0 0 0 0
... x..3o..     | 3  0 0 |  3  0  0  0  0  0 | * 4  *  *  * * *  *  *  * * * | 1 0 1 0 0 0 0
ox. ... ...&#x  | 1  2 0 |  0  2  1  0  0  0 | * * 12  *  * * *  *  *  * * * | 0 1 0 1 0 0 0
... xo. ...&#x  | 2  1 0 |  1  2  0  0  0  0 | * *  * 12  * * *  *  *  * * * | 0 1 1 0 0 0 0
... ... ox.&#x  | 1  2 0 |  0  2  0  1  0  0 | * *  *  * 12 * *  *  *  * * * | 0 0 1 1 0 0 0
.x.3.o. ...     | 0  3 0 |  0  0  3  0  0  0 | * *  *  *  * 4 *  *  *  * * * | 0 1 0 0 1 0 0
... .o.3.x.     | 0  3 0 |  0  0  0  3  0  0 | * *  *  *  * * 4  *  *  * * * | 0 0 1 0 0 1 0
.xo ... ...&#x  | 0  2 1 |  0  0  1  0  2  0 | * *  *  *  * * * 12  *  * * * | 0 0 0 1 1 0 0
... .ox ...&#x  | 0  1 2 |  0  0  0  0  2  1 | * *  *  *  * * *  * 12  * * * | 0 0 0 0 1 1 0
... ... .xo&#x  | 0  2 1 |  0  0  0  1  2  0 | * *  *  *  * * *  *  * 12 * * | 0 0 0 1 0 1 0
..o3..x ...     | 0  0 3 |  0  0  0  0  0  3 | * *  *  *  * * *  *  *  * 4 * | 0 0 0 0 1 0 1
... ..x3..o     | 0  0 3 |  0  0  0  0  0  3 | * *  *  *  * * *  *  *  * * 4 | 0 0 0 0 0 1 1
----------------+--------+-------------------+-------------------------------+--------------
o..3x..3o..     ♦ 6  0 0 | 12  0  0  0  0  0 | 4 4  0  0  0 0 0  0  0  0 0 0 | 1 * * * * * *
ox.3xo. ...&#x  ♦ 3  3 0 |  3  6  3  0  0  0 | 1 0  3  3  0 1 0  0  0  0 0 0 | * 4 * * * * *
... xo.3ox.&#x  ♦ 3  3 0 |  3  6  0  3  0  0 | 0 1  0  3  3 0 1  0  0  0 0 0 | * * 4 * * * *
oxo ... oxo&#xt ♦ 1  4 1 |  0  4  2  2  4  0 | 0 0  2  0  2 0 0  2  0  2 0 0 | * * * 6 * * *
.xo3.ox ...&#x  ♦ 0  3 3 |  0  0  3  0  6  3 | 0 0  0  0  0 1 0  3  3  0 1 0 | * * * * 4 * *
... .ox3.xo&#x  ♦ 0  3 3 |  0  0  0  3  6  3 | 0 0  0  0  0 0 1  0  3  3 0 1 | * * * * * 4 *
..o3..x3..o     ♦ 0  0 6 |  0  0  0  0  0 12 | 0 0  0  0  0 0 0  0  0  0 4 4 | * * * * * * 1

or
o..3o..3o..     & | 12  * ♦  4  4  0  0 | 2 2  2  4  2 0 0 | 1 2 2 1
.o.3.o.3.o.       |  * 12 ♦  0  4  2  2 | 0 0  4  2  4 1 1 | 0 2 2 2
------------------+-------+-------------+------------------+--------
... x.. ...     & |  2  0 | 24  *  *  * | 1 1  0  1  0 0 0 | 1 1 1 0
oo. oo. oo.&#x  & |  1  1 |  * 48  *  * | 0 0  1  1  1 0 0 | 0 1 1 1
.x. ... ...       |  0  2 |  *  * 12  * | 0 0  2  0  0 1 0 | 0 2 0 1
... ... .x.       |  0  2 |  *  *  * 12 | 0 0  0  0  2 0 1 | 0 0 2 1
------------------+-------+-------------+------------------+--------
o..3x.. ...     & |  3  0 |  3  0  0  0 | 8 *  *  *  * * * | 1 1 0 0
... x..3o..     & |  3  0 |  3  0  0  0 | * 8  *  *  * * * | 1 0 1 0
ox. ... ...&#x  & |  1  2 |  0  2  1  0 | * * 24  *  * * * | 0 1 0 1
... xo. ...&#x  & |  2  1 |  1  2  0  0 | * *  * 24  * * * | 0 1 1 0
... ... ox.&#x  & |  1  2 |  0  2  0  1 | * *  *  * 24 * * | 0 0 1 1
.x.3.o. ...       |  0  3 |  0  0  3  0 | * *  *  *  * 4 * | 0 2 0 0
... .o.3.x.       |  0  3 |  0  0  0  3 | * *  *  *  * * 4 | 0 0 2 0
------------------+-------+-------------+------------------+--------
o..3x..3o..     & ♦  6  0 | 12  0  0  0 | 4 4  0  0  0 0 0 | 2 * * *
ox.3xo. ...&#x  & ♦  3  3 |  3  6  3  0 | 1 0  3  3  0 1 0 | * 8 * *
... xo.3ox.&#x  & ♦  3  3 |  3  6  0  3 | 0 1  0  3  3 0 1 | * * 8 *
oxo ... oxo&#xt   ♦  2  4 |  0  8  2  2 | 0 0  4  0  4 0 0 | * * * 6

ooqoo3ooooo4oxoxo&#xt   → all heights = 1/2
(pt  || pseudo cube || pseudo q-oct || pseudo cube || pt)

o....3o....4o....     | 1 * * * * ♦ 8  0  0 0  0  0 0 | 12  0  0  0  0 | 6  0 0
.o...3.o...4.o...     | * 8 * * * ♦ 1  3  3 1  0  0 0 |  3  6  3  0  0 | 3  3 0
..o..3..o..4..o..     | * * 6 * * ♦ 0  0  4 0  4  0 0 |  0  4  4  4  0 | 1  4 1
...o.3...o.4...o.     | * * * 8 * ♦ 0  0  0 1  3  3 1 |  0  0  3  6  3 | 0  3 3
....o3....o4....o     | * * * * 1 ♦ 0  0  0 0  0  0 8 |  0  0  0  0 12 | 0  0 6
----------------------+-----------+-------------------+----------------+-------
oo...3oo...4oo...&#x  | 1 1 0 0 0 | 8  *  * *  *  * * |  3  0  0  0  0 | 3  0 0
..... ..... .x...     | 0 2 0 0 0 | * 12  * *  *  * * |  1  2  0  0  0 | 2  1 0
.oo..3.oo..4.oo..&#x  | 0 1 1 0 0 | *  * 24 *  *  * * |  0  2  1  0  0 | 1  2 0
.o.o.3.o.o.4.o.o.&#x  | 0 1 0 1 0 | *  *  * 8  *  * * |  0  0  3  0  0 | 0  3 0
..oo.3..oo.4..oo.&#x  | 0 0 1 1 0 | *  *  * * 24  * * |  0  0  1  2  0 | 0  2 1
..... ..... ...x.     | 0 0 0 2 0 | *  *  * *  * 12 * |  0  0  0  2  1 | 0  1 2
...oo3...oo4...oo&#x  | 0 0 0 1 1 | *  *  * *  *  * 8 |  0  0  0  0  3 | 0  0 3
----------------------+-----------+-------------------+----------------+-------
..... ..... ox...&#x  | 1 2 0 0 0 | 2  1  0 0  0  0 0 | 12  *  *  *  * | 2  0 0
..... ..... .xo..&#x  | 0 2 1 0 0 | 0  1  2 0  0  0 0 |  * 24  *  *  * | 1  1 0
.ooo.3.ooo.4.ooo.&#xt | 0 1 1 1 0 | 0  0  1 1  1  0 0 |  *  * 24  *  * | 0  2 0
..... ..... ..ox.&#x  | 0 0 1 2 0 | 0  0  0 0  2  1 0 |  *  *  * 24  * | 0  1 1
..... ..... ...xo&#x  | 0 0 0 2 1 | 0  0  0 0  0  1 2 |  *  *  *  * 12 | 0  0 2
----------------------+-----------+-------------------+----------------+-------
..... ooo..4oxo..&#xt ♦ 1 4 1 0 0 | 4  4  4 0  0  0 0 |  4  4  0  0  0 | 6  * *
.oqo. ..... .xox.&#xt ♦ 0 2 2 2 0 | 0  1  4 2  4  1 0 |  0  2  4  2  0 | * 12 *
..... ..ooo4..oxo&#xt ♦ 0 0 1 4 1 | 0  0  0 0  4  4 4 |  0  0  0  4  4 | *  * 6

or
o....3o....4o....      & | 2  * * ♦  8  0  0 0 | 12  0  0 |  6  0
.o...3.o...4.o...      & | * 16 * ♦  1  3  3 1 |  3  6  3 |  3  3
..o..3..o..4..o..        | *  * 6 ♦  0  0  8 0 |  0  8  4 |  2  4
-------------------------+--------+------------+----------+------
oo...3oo...4oo...&#x   & | 1  1 0 | 16  *  * * |  3  0  0 |  3  0
..... ..... .x...      & | 0  2 0 |  * 24  * * |  1  2  0 |  2  1
.oo..3.oo..4.oo..&#x   & | 0  1 1 |  *  * 48 * |  0  2  1 |  1  2
.o.o.3.o.o.4.o.o.&#x     | 0  2 0 |  *  *  * 8 |  0  0  3 |  0  3
-------------------------+--------+------------+----------+------
..... ..... ox...&#x   & | 1  2 0 |  2  1  0 0 | 24  *  * |  2  0
..... ..... .xo..&#x   & | 0  2 1 |  0  1  2 0 |  * 48  * |  1  1
.ooo.3.ooo.4.ooo.&#xt    | 0  2 1 |  0  0  2 1 |  *  * 24 |  0  2
-------------------------+--------+------------+----------+------
..... ooo..4oxo..&#xt  & ♦ 1  4 1 |  4  4  4 0 |  4  4  0 | 12  *
.oqo. ..... .xox.&#xt    ♦ 0  4 2 |  0  2  8 2 |  0  4  4 |  * 12

ox(uoo)xo ox(ouo)xo ox(oou)xo&#xt   → all non-zero heights = 1/2
(pt  || pseudo cube || pseudo q-oct || pseudo cube || pt)

o.(...).. o.(...).. o.(...)..     & | 2  * * ♦  8  0  0 0 | 12  0  0 |  6  0
.o(...).. .o(...).. .o(...)..     & | * 16 * ♦  1  3  3 1 |  3  6  3 |  3  3
..(o..).. ..(o..).. ..(o..)..     & | *  * 6 ♦  0  0  8 0 |  0  8  4 |  2  4
------------------------------------+--------+------------+----------+------
oo(...).. oo(...).. oo(...)..&#x  & | 1  1 0 | 16  *  * * |  3  0  0 |  3  0
.x(...).. ..(...).. ..(...)..     & | 0  2 0 |  * 24  * * |  1  2  0 |  2  1
.o(o..).. .o(o..).. .o(o..)..&#x  & | 0  1 1 |  *  * 48 * |  0  2  1 |  1  2
.o(...)o. .o(...)o. .o(...)o.&#x    | 0  2 0 |  *  *  * 8 |  0  0  3 |  0  3
------------------------------------+--------+------------+----------+------
ox(...).. ..(...).. ..(...)..&#x  & | 1  2 0 |  2  1  0 0 | 24  *  * |  2  0
.x(.o.).. ..(...).. ..(...)..&#x  & | 0  2 1 |  0  1  2 0 |  * 48  * |  1  1
.o(o..)o. .o(o..)o. .o(o..)o.&#x  & | 0  2 1 |  0  0  2 1 |  *  * 24 |  0  2
------------------------------------+--------+------------+----------+------
ox(.o.).. ..(...).. ox(.o.)..&#xt & ♦ 1  4 1 |  4  4  4 0 |  4  4  0 | 12  *
.x(.oo)x. ..(...).. ..(...)..&#xr & ♦ 0  4 2 |  0  2  8 2 |  0  4  4 |  * 12

o3m3o4o =
qo3oo3oo4ox&#zx   → height = 0
(tegum sum of q-hex and tes)

o.3o.3o.4o.     | 8  * ♦  8  0 | 12 |  6
.o3.o3.o4.o     | * 16 ♦  4  4 | 12 |  6
----------------+------+-------+----+---
oo3oo3oo4oo&#x  | 1  1 | 64  * |  3 |  3
.. .. .. .x     | 0  2 |  * 32 |  3 |  3
----------------+------+-------+----+---
.. .. .. ox&#x  | 1  2 |  2  1 | 96 |  2
----------------+------+-------+----+---
qo .. oo4ox&#zx ♦ 2  4 |  8  4 |  8 | 24

qoo3ooo3oqo *b3ooq&#zx   → height = 0
(tegum sum of 3 mutually gyrated q-hexs)

o..3o..3o.. *b3o..     | 8 * * ♦  4  4  0 | 12 |  6
.o.3.o.3.o. *b3.o.     | * 8 * ♦  4  0  4 | 12 |  6
..o3..o3..o *b3..o     | * * 8 ♦  0  4  4 | 12 |  6
-----------------------+-------+----------+----+---
oo.3oo.3oo. *b3oo.&#x  | 1 1 0 | 32  *  * |  3 |  3
o.o3o.o3o.o *b3o.o&#x  | 1 0 1 |  * 32  * |  3 |  3
.oo3.oo3.oo *b3.oo&#x  | 0 1 1 |  *  * 32 |  3 |  3
-----------------------+-------+----------+----+---
ooo3ooo3ooo *b3ooo&#x  | 1 1 1 |  1  1  1 | 96 |  2
-----------------------+-------+----------+----+---
qoo ... oqo    ooq&#zx ♦ 2 2 2 |  4  4  4 |  8 | 24

qo xo3ox4oo&#zx   → height = 0
(tegum sum of (q,x)-ope and para co)

o. o.3o.4o.     & | 12  * ♦  4  4  0 |  4  4  4 0 | 1 1  4
.o .o3.o4.o       |  * 12 ♦  0  4  4 |  0  2  8 2 | 2 0  4
------------------+-------+----------+------------+-------
.. x. .. ..     & |  2  0 | 24  *  * |  2  1  0 0 | 0 1  2
oo oo3oo4oo&#x  & |  1  1 |  * 48  * |  0  1  2 0 | 1 0  2
.. .. .x ..       |  0  2 |  *  * 24 |  0  0  2 1 | 1 0  2
------------------+-------+----------+------------+-------
.. x.3o. ..     & |  3  0 |  3  0  0 | 16  *  * * | 0 1  1
.. xo .. ..&#x  & |  2  1 |  1  2  0 |  * 24  * * | 0 0  2
.. .. ox ..&#x  & |  1  2 |  0  2  1 |  *  * 48 * | 1 0  1
.. .o3.x ..       |  0  3 |  0  0  3 |  *  *  * 8 | 0 0  2
------------------+-------+----------+------------+-------
qo .. ox4oo&#zx   ♦  2  4 |  0  8  4 |  0  0  8 0 | 6 *  *
.. x.3o.4o.     & ♦  6  0 | 12  0  0 |  8  0  0 0 | * 2  *
.. xo3ox ..&#x  & ♦  3  3 |  3  6  3 |  1  3  3 1 | * * 16

oxo4ooq oxo4qoo&#zx   → all heights = 0 - except not existing lacing(1,3)
(tegum sum of 2 perpendicular q-{4}'s and a tes)

o..4o.. o..4o..     & | 8  * ♦  8  0 |  8  4 |  4 2
.o.4.o. .o.4.o.       | * 16 ♦  4  4 |  8  4 |  4 2
----------------------+------+-------+-------+-----
oo.4oo. oo.4oo.&#x  & | 1  1 | 64  * |  2  1 |  2 1
.x. ... ... ...     & | 0  2 |  * 32 |  2  1 |  2 1
----------------------+------+-------+-------+-----
ox. ... ... ...&#x  & | 1  2 |  2  1 | 64  * |  1 1
.xo ... ... ...&#x  & | 1  2 |  2  1 |  * 32 |  2 0
----------------------+------+-------+-------+-----
oxo ... oxo ...&#xt   ♦ 2  4 |  8  4 |  4  4 | 16 *
ox.4oo. ... qo.&#zx & ♦ 2  4 |  8  4 |  8  0 |  * 8

xxo3xox oqo3ooq&#zx   → all heights = 0
(tegum sum of unit-{6} and 2 bidual (x,q)-triddips)

o..3o.. o..3o..     | 6 * * ♦ 1 1  3  3 0  0 0 | 3 3  6 0  0  0 0 | 3 3 0
.o.3.o. .o.3.o.     | * 9 * ♦ 0 0  2  0 2  4 0 | 1 0  4 1  4  2 0 | 2 2 2
..o3..o ..o3..o     | * * 9 ♦ 0 0  0  2 0  4 2 | 0 1  4 0  2  4 1 | 2 2 2
--------------------+-------+------------------+------------------+------
x.. ... ... ...     | 2 0 0 | 3 *  *  * *  * * | 0 3  0 0  0  0 0 | 3 0 0
... x.. ... ...     | 2 0 0 | * 3  *  * *  * * | 3 0  0 0  0  0 0 | 0 3 0
oo.3oo. oo.3oo.&#x  | 1 1 0 | * * 18  * *  * * | 1 0  2 0  0  0 0 | 1 2 0
o.o3o.o o.o3o.o&#x  | 1 0 1 | * *  * 18 *  * * | 0 1  2 0  0  0 0 | 2 1 0
.x. ... ... ...     | 0 2 0 | * *  *  * 9  * * | 0 0  0 1  2  0 0 | 1 0 2
.oo3.oo .oo3.oo&#x  | 0 1 1 | * *  *  * * 36 * | 0 0  1 0  1  1 0 | 1 1 1
... ..x ... ...     | 0 0 2 | * *  *  * *  * 9 | 0 0  0 0  0  2 1 | 0 1 2
--------------------+-------+------------------+------------------+------
... xo. ... ...&#x  | 2 1 0 | 0 1  2  0 0  0 0 | 9 *  * *  *  * * | 0 2 0
x.o ... ... ...&#x  | 2 0 1 | 1 0  0  2 0  0 0 | * 9  * *  *  * * | 2 0 0
ooo3ooo ooo3ooo&#x  | 1 1 1 | 0 0  1  1 0  1 0 | * * 36 *  *  * * | 1 1 0
.x.3.o. ... ...     | 0 3 0 | 0 0  0  0 3  0 0 | * *  * 3  *  * * | 0 0 2
.xo ... ... ...&#x  | 0 2 1 | 0 0  0  0 1  2 0 | * *  * * 18  * * | 1 0 1
... .ox ... ...&#x  | 0 1 2 | 0 0  0  0 0  2 1 | * *  * *  * 18 * | 0 1 1
..o3..x ... ...     | 0 0 3 | 0 0  0  0 0  0 3 | * *  * *  *  * 3 | 0 0 2
--------------------+-------+------------------+------------------+------
xxo ... ... ooq&#zx ♦ 2 2 2 | 1 0  2  4 1  4 0 | 0 2  4 0  2  0 0 | 9 * *
... xox oqo ...&#zx ♦ 2 2 2 | 0 1  4  2 0  4 1 | 2 0  4 0  0  2 0 | * 9 *
.xo3.ox ... ...&#x  ♦ 0 3 3 | 0 0  0  0 3  6 3 | 0 0  0 1  3  3 1 | * * 6

or
o..3o.. o..3o..       | 6  * ♦ 2  6  0  0 |  6  6 0  0 |  6 0
.o.3.o. .o.3.o.     & | * 18 ♦ 0  2  2  4 |  1  4 1  6 |  4 2
----------------------+------+------------+------------+-----
x.. ... ... ...     & | 2  0 | 6  *  *  * |  3  0 0  0 |  3 0
oo.3oo. oo.3oo.&#x  & | 1  1 | * 36  *  * |  1  2 0  0 |  3 0
.x. ... ... ...     & | 0  2 | *  * 18  * |  0  0 1  2 |  1 2
.oo3.oo .oo3.oo&#x    | 0  2 | *  *  * 36 |  0  1 0  2 |  2 1
----------------------+------+------------+------------+-----
... xo. ... ...&#x  & | 2  1 | 1  2  0  0 | 18  * *  * |  2 0
ooo3ooo ooo3ooo&#x    | 1  2 | 0  2  0  1 |  * 36 *  * |  2 0
.x.3.o. ... ...     & | 0  3 | 0  0  3  0 |  *  * 6  * |  0 2
.xo ... ... ...&#x  & | 0  3 | 0  0  1  2 |  *  * * 36 |  1 1
----------------------+------+------------+------------+-----
xxo ... ... ooq&#zx & ♦ 2  4 | 1  6  1  4 |  2  4 0  2 | 18 *
.xo3.ox ... ...&#x    ♦ 0  6 | 0  0  6  6 |  0  0 2  6 |  * 6

uooox ouoox oouox oooux&#zx   → all existing heights = 0
(tegum sum of 4 pairwise perpendicular u-lines and a tes)

o.... o.... o.... o....     | 2 * * *  * ♦  8  0  0  0 0 0 0 0 | 4 4 4 0 0 0 0 0 0 0 0 0 | 2 2 2 0 0 0
.o... .o... .o... .o...     | * 2 * *  * ♦  0  8  0  0 0 0 0 0 | 0 0 0 4 4 4 0 0 0 0 0 0 | 2 0 0 2 2 0
..o.. ..o.. ..o.. ..o..     | * * 2 *  * ♦  0  0  8  0 0 0 0 0 | 0 0 0 0 0 0 4 4 4 0 0 0 | 0 2 0 2 0 2
...o. ...o. ...o. ...o.     | * * * 2  * ♦  0  0  0  8 0 0 0 0 | 0 0 0 0 0 0 0 0 0 4 4 4 | 0 0 2 0 2 2
....o ....o ....o ....o     | * * * * 16 ♦  1  1  1  1 1 1 1 1 | 1 1 1 1 1 1 1 1 1 1 1 1 | 1 1 1 1 1 1
----------------------------+------------+---------------------+-------------------------+------------
o...o o...o o...o o...o&#x  | 1 0 0 0  1 | 16  *  *  * * * * * | 1 1 1 0 0 0 0 0 0 0 0 0 | 1 1 1 0 0 0
.o..o .o..o .o..o .o..o&#x  | 0 1 0 0  1 |  * 16  *  * * * * * | 0 0 0 1 1 1 0 0 0 0 0 0 | 1 0 0 1 1 0
..o.o ..o.o ..o.o ..o.o&#x  | 0 0 1 0  1 |  *  * 16  * * * * * | 0 0 0 0 0 0 1 1 1 0 0 0 | 0 1 0 1 0 1
...oo ...oo ...oo ...oo&#x  | 0 0 0 1  1 |  *  *  * 16 * * * * | 0 0 0 0 0 0 0 0 0 1 1 1 | 0 0 1 0 1 1
....x ..... ..... .....     | 0 0 0 0  2 |  *  *  *  * 8 * * * | 0 0 0 1 0 0 1 0 0 1 0 0 | 0 0 0 1 1 1
..... ....x ..... .....     | 0 0 0 0  2 |  *  *  *  * * 8 * * | 1 0 0 0 0 0 0 1 0 0 1 0 | 0 1 1 0 0 1
..... ..... ....x .....     | 0 0 0 0  2 |  *  *  *  * * * 8 * | 0 1 0 0 1 0 0 0 0 0 0 1 | 1 0 1 0 1 0
..... ..... ..... ....x     | 0 0 0 0  2 |  *  *  *  * * * * 8 | 0 0 1 0 0 1 0 0 1 0 0 0 | 1 1 0 1 0 0
----------------------------+------------+---------------------+-------------------------+------------
..... o...x ..... .....&#x  | 1 0 0 0  2 |  2  0  0  0 0 1 0 0 | 8 * * * * * * * * * * * | 0 1 1 0 0 0
..... ..... o...x .....&#x  | 1 0 0 0  2 |  2  0  0  0 0 0 1 0 | * 8 * * * * * * * * * * | 1 0 1 0 0 0
..... ..... ..... o...x&#x  | 1 0 0 0  2 |  2  0  0  0 0 0 0 1 | * * 8 * * * * * * * * * | 1 1 0 0 0 0
.o..x ..... ..... .....&#x  | 0 1 0 0  2 |  0  2  0  0 1 0 0 0 | * * * 8 * * * * * * * * | 0 0 0 1 1 0
..... ..... .o..x .....&#x  | 0 1 0 0  2 |  0  2  0  0 0 0 1 0 | * * * * 8 * * * * * * * | 1 0 0 0 1 0
..... ..... ..... .o..x&#x  | 0 1 0 0  2 |  0  2  0  0 0 0 0 1 | * * * * * 8 * * * * * * | 1 0 0 1 0 0
..o.x ..... ..... .....&#x  | 0 0 1 0  2 |  0  0  2  0 1 0 0 0 | * * * * * * 8 * * * * * | 0 0 0 1 0 1
..... ..o.x ..... .....&#x  | 0 0 1 0  2 |  0  0  2  0 0 1 0 0 | * * * * * * * 8 * * * * | 0 1 0 0 0 1
..... ..... ..... ..o.x&#x  | 0 0 1 0  2 |  0  0  2  0 0 0 0 1 | * * * * * * * * 8 * * * | 0 1 0 1 0 0
...ox ..... ..... .....&#x  | 0 0 0 1  2 |  0  0  0  2 1 0 0 0 | * * * * * * * * * 8 * * | 0 0 0 0 1 1
..... ...ox ..... .....&#x  | 0 0 0 1  2 |  0  0  0  2 0 1 0 0 | * * * * * * * * * * 8 * | 0 0 1 0 0 1
..... ..... ...ox .....&#x  | 0 0 0 1  2 |  0  0  0  2 0 0 1 0 | * * * * * * * * * * * 8 | 0 0 1 0 1 0
----------------------------+------------+---------------------+-------------------------+------------
..... ..... oo..x oo..x&#xt ♦ 1 1 0 0  4 |  4  4  0  0 0 0 2 2 | 0 2 2 0 2 2 0 0 0 0 0 0 | 4 * * * * * (tower: a-e-b)
..... o.o.x ..... o.o.x&#xt ♦ 1 0 1 0  4 |  4  0  4  0 0 2 0 2 | 2 0 2 0 0 0 0 2 2 0 0 0 | * 4 * * * * (tower: a-e-c)
..... o..ox o..ox .....&#xt ♦ 1 0 0 1  4 |  4  0  0  4 0 2 2 0 | 2 2 0 0 0 0 0 0 0 0 2 2 | * * 4 * * * (tower: a-e-d)
.oo.x ..... ..... .oo.x&#xt ♦ 0 1 1 0  4 |  0  4  4  0 2 0 0 2 | 0 0 0 2 0 2 2 0 2 0 0 0 | * * * 4 * * (tower: b-e-c)
.o.ox ..... .o.ox .....&#xt ♦ 0 1 0 1  4 |  0  4  0  4 2 0 2 0 | 0 0 0 2 2 0 0 0 0 2 0 2 | * * * * 4 * (tower: b-e-d)
..oox ..oox ..... .....&#xt ♦ 0 0 1 1  4 |  0  0  4  4 2 2 0 0 | 0 0 0 0 0 0 2 2 0 2 2 0 | * * * * * 4 (tower: c-e-d)

(qo)(qo)(qo) (ox)(xo)(ox)4(oo)(oq)(oo)&#xt   → both heights = 1/sqrt(2) = 0.707107
(oct || pseudo co || oct)

(o.)(..)(..) (o.)(..)(..)4(o.)(..)(..)     & | 4 * * * ♦  4  4 0  0  0 0  0 |  4  4  4 0  0 0 | 1 1  4 0
(.o)(..)(..) (.o)(..)(..)4(.o)(..)(..)     & | * 8 * * ♦  2  0 2  2  2 0  0 |  4  0  2 2  4 0 | 1 0  4 1
(..)(o.)(..) (..)(o.)(..)4(..)(o.)(..)       | * * 8 * ♦  0  2 0  2  0 2  2 |  0  4  2 0  4 2 | 0 1  4 1
(..)(.o)(..) (..)(.o)(..)4(..)(.o)(..)       | * * * 4 ♦  0  0 0  0  4 0  4 |  0  0  0 2  8 2 | 0 0  4 2
---------------------------------------------+---------+--------------------+-----------------+---------
(oo)(..)(..) (oo)(..)(..)4(oo)(..)(..)&#x  & | 1 1 0 0 | 16  * *  *  * *  * |  2  0  1 0  0 0 | 1 0  2 0
(o.)(o.)(..) (o.)(o.)(..)4(o.)(o.)(..)&#x  & | 1 0 1 0 |  * 16 *  *  * *  * |  0  2  1 0  0 0 | 0 1  2 0
(..)(..)(..) (.x)(..)(..) (..)(..)(..)       | 0 2 0 0 |  *  * 8  *  * *  * |  2  0  0 1  0 0 | 1 0  2 0
(.o)(o.)(..) (.o)(o.)(..)4(.o)(o.)(..)&#x  & | 0 1 1 0 |  *  * * 16  * *  * |  0  0  1 0  2 0 | 0 0  2 1
(.o)(.o)(..) (.o)(.o)(..)4(.o)(.o)(..)&#x  & | 0 1 0 1 |  *  * *  * 16 *  * |  0  0  0 1  2 0 | 0 0  2 1
(..)(..)(..) (..)(x.)(..) (..)(..)(..)       | 0 0 2 0 |  *  * *  *  * 8  * |  0  2  0 0  0 1 | 0 1  2 0
(..)(oo)(..) (..)(oo)(..)4(..)(oo)(..)&#x    | 0 0 1 1 |  *  * *  *  * * 16 |  0  0  0 0  2 1 | 0 0  2 1
---------------------------------------------+---------+--------------------+-----------------+---------
(..)(..)(..) (ox)(..)(..) (..)(..)(..)&#x  & | 1 2 0 0 |  2  0 1  0  0 0  0 | 16  *  * *  * * | 1 0  1 0
(..)(..)(..) (o.)(x.)(..) (..)(..)(..)&#x  & | 1 0 2 0 |  0  2 0  0  0 1  0 |  * 16  * *  * * | 0 1  1 0
(oo)(o.)(..) (oo)(o.)(..)4(oo)(o.)(..)&#x  & | 1 1 1 0 |  1  1 0  1  0 0  0 |  *  * 16 *  * * | 0 0  2 0
(..)(..)(..) (.x)(.o)(..) (..)(..)(..)&#x  & | 0 2 0 1 |  0  0 1  0  2 0  0 |  *  *  * 8  * * | 0 0  2 0
(.o)(oo)(..) (.o)(oo)(..)4(.o)(oo)(..)&#x  & | 0 1 1 1 |  0  0 0  1  1 0  1 |  *  *  * * 32 * | 0 0  1 1
(..)(..)(..) (..)(xo)(..) (..)(..)(..)&#x    | 0 0 2 1 |  0  0 0  0  0 1  2 |  *  *  * *  * 8 | 0 0  2 0
---------------------------------------------+---------+--------------------+-----------------+---------
(qo)(..)(..) (ox)(..)(..) (..)(..)(..)&#zx & ♦ 2 4 0 0 |  8  0 4  0  0 0  0 |  8  0  0 0  0 0 | 2 *  * *
(..)(..)(..) (o.)(x.)(o.)4(o.)(o.)(o.)&#xt   ♦ 2 0 4 0 |  0  8 0  0  0 4  0 |  0  8  0 0  0 0 | * 2  * *
(..)(..)(..) (ox)(xo)(..) (..)(..)(..)&#xr & ♦ 1 2 2 1 |  2  2 1  2  2 1  2 |  1  1  2 1  2 1 | * * 16 *
(.o)(qo)(.o) (..)(..)(..) (.o)(oq)(.o)&#zx   ♦ 0 2 2 2 |  0  0 0  4  4 0  4 |  0  0  0 0  8 0 | * *  * 4

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