Acronym spic
Name small prismatotetracontoctachoron,
runcinated icositetrachoron,
expanded icositetrachoron
Cross sections
 ©
Circumradius sqrt[2+sqrt(2)] = 1.847759
Vertex figure
 ©
Vertex layers
LayerSymmetrySubsymmetries
 o3o4o3o o3o4o . o3o . o o . o3o . o4o3o
1x3o4o3x x3o4o .
oct first
x3o . x
trip first
x . o3x
trip first
. o4o3x
oct first
2 x3o4x . x3q . o o . q3x . x4o3x
3 w3o4o . o3x . w w . x3o . o4o3w
4 o3x4x . w3o . x x . o3w . x4x3o
5 q3x4o . o3w . q q . w3o . o4x3q
6 o3x4x . q3x . w w . x3q . x4x3o
7 w3o4o . x3w . o o . w3x . o4o3w
8 x3o4x . o3x . X X . x3o . x4o3x
9 x3o4o .
opposite oct
w3x . x x . x3w . o4o3x
opposite oct
10   o3w . w w . w3o  
11 w3o . w w . o3w
12 x3w . x x . w3x
13 x3o . X X . o3x
14 w3x . o o . x3w
15 x3q . w w . q3x
16 w3o . q q . o3w
17 o3w . x x . w3o
18 x3o . w w . o3x
19 q3x . o o . x3q
20 o3x . x
opposite trip
x . x3o
opposite trip
(X=x+w=2x+q)
Lace city
in approx. ASCII-art
 ©  
                o4o         o4x         o4o                
                                                           
                                                           
                                                           
        x4o         x4x             x4x         x4o        
                                                           
                                                           
                                                           
o4o                         o4w                         o4o
                                                           
        x4x         w4o             w4o         x4x        
                                                           
                                                           
                                                           
o4x             o4w         q4q         o4w             o4x
                                                           
                                                           
                                                           
        x4x         w4o             w4o         x4x        
                                                           
o4o                         o4w                         o4o
                                                           
                                                           
                                                           
        x4o         x4x             x4x         x4o        
                                                           
                                                           
                                                           
                o4o         o4x         o4o                
 ©  
                    o3x             o3x                    
                                                           
                            q3x                            
        x3o                                     x3o        
                    o3w             o3w                    
                w3o                     w3o                
                                                           
        x3q                                     x3q        
                            w3x                            
x3o                                                     x3o
                    x3w             x3w                    
        w3o                                     w3o        
                                                           
        o3w                                     o3w        
                    w3x             w3x                    
o3x                                                     o3x
                            x3w                            
        q3x                                     q3x        
                                                           
                o3w                     o3w                
                    w3o             w3o                    
        o3x                                     o3x        
                            x3q                            
                                                           
                    x3o             x3o                    
Coordinates
  1. (0, 0, sqrt(2)/2, (2+sqrt(2))/2)                   & all permutations, all changes of sign
  2. (1/2, 1/2, (1+sqrt(2))/2, (1+sqrt(2))/2)       & all permutations, all changes of sign
    (vertex inscribed srit)
General of army (is itself convex)
Colonel of regiment (is itself locally convex – uniform polychoral members:
by cells: oct quith sirco sroh trip
siddic 4848000
inpac 48024096
spic 48000192
sirc 000480
& others)
Dihedral angles
  • at {4} between trip and trip:   arccos[-sqrt(8)/3] = 160.528779°
  • at {3} between oct and trip:   150°
Face vector 144, 576, 672, 240
Confer
related segmentochora:
oct || sirco   squippy || squacu  
uniform relative:
srit   odip   ticcup  
related CRFs:
owau prit   dapabdi spic  
variants:
q3o4o3x  
decompositions:
ico || spic  
ambification:
respic  
general polytopal classes:
Wythoffian polychora   partial Stott expansions  
External
links
hedrondude   wikipedia   polytopewiki   WikiChoron   quickfur

As abstract polytope spic is isomorphic to quippic, thereby replacing oct by inverted oct resp. an antipodium vertex figure by a frustrum one.

Diminishing spic by octasircoes at hexadecachoral positions results in the srit. Diminishing only at one cycle of 4 of those positions results in the cyted spic.

Applying an oct-first deep paradiminishing results in the bistratic central segment dapabdi spic. That one then could be further diminished at the medial layer, resulting in ticcup.

Stott expanding an hexadecachoral subset of spic's octs results in the owau prit.

Note that spic can be thought of as the external blend of 1 ico + 24 octpies + 96 tepes + 96 triddips + 24 opes. This decomposition is described as the degenerate segmentoteron ox3oo4oo3xx&#x.

By virtue of an outer symmetry this is a non-quasiregular monotoxal polychoron, that is all edges belong to the same equivalence class.


Incidence matrix according to Dynkin symbol

x3o4o3x

. . . . | 144    4   4 |   4   8   4 |  1  4  4  1
--------+-----+---------+-------------+------------
x . . . |   2 | 288   * |   2   2   0 |  1  2  1  0
. . . x |   2 |   * 288 |   0   2   2 |  0  1  2  1
--------+-----+---------+-------------+------------
x3o . . |   3 |   3   0 | 192   *   * |  1  1  0  0
x . . x |   4 |   2   2 |   * 288   * |  0  1  1  0
. . o3x |   3 |   0   3 |   *   * 192 |  0  0  1  1
--------+-----+---------+-------------+------------
x3o4o .    6 |  12   0 |   8   0   0 | 24  *  *  *
x3o . x    6 |   6   3 |   2   3   0 |  * 96  *  *
x . o3x    6 |   3   6 |   0   3   2 |  *  * 96  *
. o4o3x    6 |   0  12 |   0   0   8 |  *  *  * 24
or
. . . .    | 144    8 |   8   8 |  2   8
-----------+-----+-----+---------+-------
x . . .  & |   2 | 576 |   2   2 |  1   3
-----------+-----+-----+---------+-------
x3o . .  & |   3 |   3 | 384   * |  1   1
x . . x    |   4 |   4 |   * 288 |  0   2
-----------+-----+-----+---------+-------
x3o4o .  &    6 |  12 |   8   0 | 48   *
x3o . x  &    6 |   9 |   2   3 |  * 192

snubbed forms: β3o4o3x, β3o4o3β

x3o4/3o3/2x

. .   .   . | 144 |   4   4 |   4   8   4 |  1  4  4  1
------------+-----+---------+-------------+------------
x .   .   . |   2 | 288   * |   2   2   0 |  1  2  1  0
. .   .   x |   2 |   * 288 |   0   2   2 |  0  1  2  1
------------+-----+---------+-------------+------------
x3o   .   . |   3 |   3   0 | 192   *   * |  1  1  0  0
x .   .   x |   4 |   2   2 |   * 288   * |  0  1  1  0
. .   o3/2x |   3 |   0   3 |   *   * 192 |  0  0  1  1
------------+-----+---------+-------------+------------
x3o4/3o   .    6 |  12   0 |   8   0   0 | 24  *  *  *
x3o   .   x    6 |   6   3 |   2   3   0 |  * 96  *  *
x .   o3/2x    6 |   3   6 |   0   3   2 |  *  * 96  *
. o4/3o3/2x    6 |   0  12 |   0   0   8 |  *  *  * 24

x3/2o4o3/2x

.   . .   . | 144 |   4   4 |   4   8   4 |  1  4  4  1
------------+-----+---------+-------------+------------
x   . .   . |   2 | 288   * |   2   2   0 |  1  2  1  0
.   . .   x |   2 |   * 288 |   0   2   2 |  0  1  2  1
------------+-----+---------+-------------+------------
x3/2o .   . |   3 |   3   0 | 192   *   * |  1  1  0  0
x   . .   x |   4 |   2   2 |   * 288   * |  0  1  1  0
.   . o3/2x |   3 |   0   3 |   *   * 192 |  0  0  1  1
------------+-----+---------+-------------+------------
x3/2o4o   .    6 |  12   0 |   8   0   0 | 24  *  *  *
x3/2o .   x    6 |   6   3 |   2   3   0 |  * 96  *  *
x   . o3/2x    6 |   3   6 |   0   3   2 |  *  * 96  *
.   o4o3/2x    6 |   0  12 |   0   0   8 |  *  *  * 24


xxwoqowxx3oooxxxooo4oxoxoxoxo&#xt   → height(1,2) = height(2,3) = height(4,5) = height(5,6) = height(7,8) = height(8,9) = 1/2
                                      height(3,4) = height(6,7) = 1/sqrt(2)-1/2 = 0.207107
(oct || pseudo sirco || pseudo w-oct || pseudo tic || pseudo (q,x)-toe || pseudo tic || pseudo w-oct || pseudo sirco || oct)

o........3o........4o........      & | 12  *  *  *  * |  4  4  0  0  0  0  0  0  0  0  0  0 |  4  8  4  0  0  0  0  0  0  0  0  0  0  0  0  0 | 1  4  4  1  0  0  0  0 0 0  0  0
.o.......3.o.......4.o.......      & |  * 48  *  *  * |  0  1  2  2  1  2  0  0  0  0  0  0 |  0  2  2  1  2  2  2  1  2  2  0  0  0  0  0  0 | 0  1  2  1  1  2  1  2 0 0  0  0
..o......3..o......4..o......      & |  *  * 12  *  * |  0  0  0  0  4  0  4  0  0  0  0  0 |  0  0  0  0  0  4  0  0  0  8  4  0  0  0  0  0 | 0  0  0  1  0  0  4  4 1 0  0  0
...o.....3...o.....4...o.....      & |  *  *  * 48  * |  0  0  0  0  0  2  0  2  1  2  1  0 |  0  0  0  0  0  0  1  2  2  2  0  1  2  2  2  2 | 0  0  0  0  1  1  2  2 0 1  1  2
....o....3....o....4....o....        |  *  *  *  * 24 |  0  0  0  0  0  0  2  0  0  4  0  2 |  0  0  0  0  0  0  0  0  0  4  4  0  4  2  0  2 | 0  0  0  0  0  0  4  2 1 0  1  2
-------------------------------------+----------------+-------------------------------------+-------------------------------------------------+---------------------------------
x........ ......... .........      & |  2  0  0  0  0 | 24  *  *  *  *  *  *  *  *  *  *  * |  2  2  0  0  0  0  0  0  0  0  0  0  0  0  0  0 | 1  2  1  0  0  0  0  0 0 0  0  0
oo.......3oo.......4oo.......&#x   & |  1  1  0  0  0 |  * 48  *  *  *  *  *  *  *  *  *  * |  0  2  2  0  0  0  0  0  0  0  0  0  0  0  0  0 | 0  1  2  1  0  0  0  0 0 0  0  0
.x....... ......... .........      & |  0  2  0  0  0 |  *  * 48  *  *  *  *  *  *  *  *  * |  0  1  0  1  1  0  1  0  0  0  0  0  0  0  0  0 | 0  1  1  0  1  1  0  0 0 0  0  0
......... ......... .x.......      & |  0  2  0  0  0 |  *  *  * 48  *  *  *  *  *  *  *  * |  0  0  1  0  1  1  0  0  1  0  0  0  0  0  0  0 | 0  0  1  1  0  1  0  1 0 0  0  0
.oo......3.oo......4.oo......&#x   & |  0  1  1  0  0 |  *  *  *  * 48  *  *  *  *  *  *  * |  0  0  0  0  0  2  0  0  0  2  0  0  0  0  0  0 | 0  0  0  1  0  0  1  2 0 0  0  0
.o.o.....3.o.o.....4.o.o.....&#x   & |  0  1  0  1  0 |  *  *  *  *  * 96  *  *  *  *  *  * |  0  0  0  0  0  0  1  1  1  1  0  0  0  0  0  0 | 0  0  0  0  1  1  1  1 0 0  0  0
..o.o....3..o.o....4..o.o....&#x   & |  0  0  1  0  1 |  *  *  *  *  *  * 48  *  *  *  *  * |  0  0  0  0  0  0  0  0  0  2  2  0  0  0  0  0 | 0  0  0  0  0  0  2  1 1 0  0  0
......... ...x..... .........      & |  0  0  0  2  0 |  *  *  *  *  *  *  * 48  *  *  *  * |  0  0  0  0  0  0  0  1  0  0  0  1  1  0  1  0 | 0  0  0  0  1  0  1  0 0 1  0  1
......... ......... ...x.....      & |  0  0  0  2  0 |  *  *  *  *  *  *  *  * 24  *  *  * |  0  0  0  0  0  0  0  0  2  0  0  0  0  2  0  0 | 0  0  0  0  0  1  0  2 0 0  1  0
...oo....3...oo....4...oo....&#x   & |  0  0  0  1  1 |  *  *  *  *  *  *  *  *  * 96  *  * |  0  0  0  0  0  0  0  0  0  1  0  0  1  1  0  1 | 0  0  0  0  0  0  1  1 0 0  1  1
...o.o...3...o.o...4...o.o...&#x     |  0  0  0  2  0 |  *  *  *  *  *  *  *  *  *  * 24  * |  0  0  0  0  0  0  0  0  0  0  0  0  0  0  2  2 | 0  0  0  0  0  0  0  0 0 1  1  2
......... ....x.... .........        |  0  0  0  0  2 |  *  *  *  *  *  *  *  *  *  *  * 24 |  0  0  0  0  0  0  0  0  0  0  2  0  2  0  0  0 | 0  0  0  0  0  0  2  0 1 0  0  1
-------------------------------------+----------------+-------------------------------------+-------------------------------------------------+---------------------------------
x........3o........ .........      & |  3  0  0  0  0 |  3  0  0  0  0  0  0  0  0  0  0  0 | 16  *  *  *  *  *  *  *  *  *  *  *  *  *  *  * | 1  1  0  0  0  0  0  0 0 0  0  0
xx....... ......... .........&#x   & |  2  2  0  0  0 |  1  2  1  0  0  0  0  0  0  0  0  0 |  * 48  *  *  *  *  *  *  *  *  *  *  *  *  *  * | 0  1  1  0  0  0  0  0 0 0  0  0
......... ......... ox.......&#x   & |  1  2  0  0  0 |  0  2  0  1  0  0  0  0  0  0  0  0 |  *  * 48  *  *  *  *  *  *  *  *  *  *  *  *  * | 0  0  1  1  0  0  0  0 0 0  0  0
.x.......3.o....... .........      & |  0  3  0  0  0 |  0  0  3  0  0  0  0  0  0  0  0  0 |  *  *  * 16  *  *  *  *  *  *  *  *  *  *  *  * | 0  1  0  0  1  0  0  0 0 0  0  0
.x....... ......... .x.......      & |  0  4  0  0  0 |  0  0  2  2  0  0  0  0  0  0  0  0 |  *  *  *  * 24  *  *  *  *  *  *  *  *  *  *  * | 0  0  1  0  0  1  0  0 0 0  0  0
......... ......... .xo......&#x   & |  0  2  1  0  0 |  0  0  0  1  2  0  0  0  0  0  0  0 |  *  *  *  *  * 48  *  *  *  *  *  *  *  *  *  * | 0  0  0  1  0  0  0  1 0 0  0  0
.x.o..... ......... .........&#x   & |  0  2  0  1  0 |  0  0  1  0  0  2  0  0  0  0  0  0 |  *  *  *  *  *  * 48  *  *  *  *  *  *  *  *  * | 0  0  0  0  1  1  0  0 0 0  0  0
......... .o.x..... .........&#x   & |  0  1  0  2  0 |  0  0  0  0  0  2  0  1  0  0  0  0 |  *  *  *  *  *  *  * 48  *  *  *  *  *  *  *  * | 0  0  0  0  1  0  1  0 0 0  0  0
......... ......... .x.x.....&#x   & |  0  2  0  2  0 |  0  0  0  1  0  2  0  0  1  0  0  0 |  *  *  *  *  *  *  *  * 48  *  *  *  *  *  *  * | 0  0  0  0  0  1  0  1 0 0  0  0
.oooo....3.oooo....4.oooo....&#xr  & |  0  1  1  1  1 |  0  0  0  0  1  1  1  0  0  1  0  0 |  *  *  *  *  *  *  *  *  * 96  *  *  *  *  *  * | 0  0  0  0  0  0  1  1 0 0  0  0
......... ..o.x.... .........&#x   & |  0  0  1  0  2 |  0  0  0  0  0  0  2  0  0  0  0  1 |  *  *  *  *  *  *  *  *  *  * 48  *  *  *  *  * | 0  0  0  0  0  0  1  0 1 0  0  0
...o.....3...x..... .........      & |  0  0  0  3  0 |  0  0  0  0  0  0  0  3  0  0  0  0 |  *  *  *  *  *  *  *  *  *  *  * 16  *  *  *  * | 0  0  0  0  1  0  0  0 0 1  0  0
......... ...xx.... .........&#x   & |  0  0  0  2  2 |  0  0  0  0  0  0  0  1  0  2  0  1 |  *  *  *  *  *  *  *  *  *  *  *  * 48  *  *  * | 0  0  0  0  0  0  1  0 0 0  0  1
......... ......... ...xo....&#x   & |  0  0  0  2  1 |  0  0  0  0  0  0  0  0  1  2  0  0 |  *  *  *  *  *  *  *  *  *  *  *  *  * 48  *  * | 0  0  0  0  0  0  0  1 0 0  1  0
......... ...x.x... .........&#x     |  0  0  0  4  0 |  0  0  0  0  0  0  0  2  0  0  2  0 |  *  *  *  *  *  *  *  *  *  *  *  *  *  * 24  * | 0  0  0  0  0  0  0  0 0 1  0  1
...ooo...3...ooo...4...ooo...&#xr    |  0  0  0  2  1 |  0  0  0  0  0  0  0  0  0  2  1  0 |  *  *  *  *  *  *  *  *  *  *  *  *  *  *  * 48 | 0  0  0  0  0  0  0  0 0 0  1  1
-------------------------------------+----------------+-------------------------------------+-------------------------------------------------+---------------------------------
x........3o........4o........      &   6  0  0  0  0 | 12  0  0  0  0  0  0  0  0  0  0  0 |  8  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 | 2  *  *  *  *  *  *  * * *  *  *
xx.......3oo....... .........&#x   &   3  3  0  0  0 |  3  3  3  0  0  0  0  0  0  0  0  0 |  1  3  0  1  0  0  0  0  0  0  0  0  0  0  0  0 | * 16  *  *  *  *  *  * * *  *  *
xx....... ........  ox.......&#x   &   2  4  0  0  0 |  1  4  2  2  0  0  0  0  0  0  0  0 |  0  2  2  0  1  0  0  0  0  0  0  0  0  0  0  0 | *  * 24  *  *  *  *  * * *  *  *
......... ooo......4oxo......&#xt  &   1  4  1  0  0 |  0  4  0  4  4  0  0  0  0  0  0  0 |  0  0  4  0  0  4  0  0  0  0  0  0  0  0  0  0 | *  *  * 12  *  *  *  * * *  *  *
.x.o.....3.o.x..... .........&#x   &   0  3  0  3  0 |  0  0  3  0  0  6  0  3  0  0  0  0 |  0  0  0  1  0  0  3  3  0  0  0  1  0  0  0  0 | *  *  *  * 16  *  *  * * *  *  *
.x.o..... ......... .x.x.....&#x   &   0  4  0  2  0 |  0  0  2  2  0  4  0  0  1  0  0  0 |  0  0  0  0  1  0  2  0  2  0  0  0  0  0  0  0 | *  *  *  *  * 24  *  * * *  *  *
......... .ooxx.... .........&#xr  &   0  1  1  2  2 |  0  0  0  0  1  2  2  1  0  2  0  1 |  0  0  0  0  0  0  0  1  0  2  1  0  1  0  0  0 | *  *  *  *  *  * 48  * * *  *  *
......... ......... .xoxo....&#xr  &   0  2  1  2  1 |  0  0  0  1  2  2  1  0  1  2  0  0 |  0  0  0  0  0  1  0  0  1  2  0  0  0  1  0  0 | *  *  *  *  *  *  * 48 * *  *  *
......... ..o.x.o..4..o.o.o..&#xt      0  0  2  0  4 |  0  0  0  0  0  0  8  0  0  0  0  4 |  0  0  0  0  0  0  0  0  0  0  8  0  0  0  0  0 | *  *  *  *  *  *  *  * 6 *  *  *
...o.o...3...x.x... .........&#x       0  0  0  6  0 |  0  0  0  0  0  0  0  6  0  0  3  0 |  0  0  0  0  0  0  0  0  0  0  0  2  0  0  3  0 | *  *  *  *  *  *  *  * * 8  *  *
...oqo... ......... ...xox...&#xt      0  0  0  4  2 |  0  0  0  0  0  0  0  0  2  8  2  0 |  0  0  0  0  0  0  0  0  0  0  0  0  0  4  0  4 | *  *  *  *  *  *  *  * * * 12  *
......... ...xxx... .........&#xr      0  0  0  4  2 |  0  0  0  0  0  0  0  2  0  4  2  1 |  0  0  0  0  0  0  0  0  0  0  0  0  2  0  1  2 | *  *  *  *  *  *  *  * * *  * 24

qo3xx3oo4ox&#zx   → height = 0
(tegum sum of (q,x)-thex and srit)

o.3o.3o.4o.     | 48  * |  4   4   0  0 |  4   8   4  0  0  0 | 1  1  4  4  0  0
.o3.o3.o4.o     |  * 96 |  0   2   4  2 |  0   4   4  2  2  4 | 0  1  2  4  1  2
----------------+-------+---------------+---------------------+-----------------
.. x. .. ..     |  2  0 | 96   *   *  * |  2   2   0  0  0  0 | 1  0  2  1  0  0
oo3oo3oo4oo&#x  |  1  1 |  * 192   *  * |  0   2   2  0  0  0 | 0  1  1  2  0  0
.. .x .. ..     |  0  2 |  *   * 192  * |  0   1   0  1  1  1 | 0  0  1  1  1  1
.. .. .. .x     |  0  2 |  *   *   * 96 |  0   0   2  0  0  2 | 0  1  0  2  0  1
----------------+-------+---------------+---------------------+-----------------
.. x.3o. ..     |  3  0 |  3   0   0  0 | 64   *   *  *  *  * | 1  0  1  0  0  0
.. xx .. ..&#x  |  2  2 |  1   2   1  0 |  * 192   *  *  *  * | 0  0  1  1  0  0
.. .. .. ox&#x  |  1  2 |  0   2   0  1 |  *   * 192  *  *  * | 0  1  0  1  0  0
.o3.x .. ..     |  0  3 |  0   0   3  0 |  *   *   * 64  *  * | 0  0  0  0  1  1
.. .x3.o ..     |  0  3 |  0   0   3  0 |  *   *   *  * 64  * | 0  0  1  0  1  0
.. .x .. .x     |  0  4 |  0   0   2  2 |  *   *   *  *  * 96 | 0  0  0  1  0  1
----------------+-------+---------------+---------------------+-----------------
.. x.3o.4o.       6  0 | 12   0   0  0 |  8   0   0  0  0  0 | 8  *  *  *  *  *
qo .. oo4ox&#zx   2  4 |  0   8   0  4 |  0   0   8  0  0  0 | * 24  *  *  *  *
.. xx3oo ..&#x    3  3 |  3   3   3  0 |  1   3   0  0  1  0 | *  * 64  *  *  *
.. xx .. ox&#x    2  4 |  1   4   2  2 |  0   2   2  0  0  1 | *  *  * 96  *  *
.o3.x3.o ..       0  6 |  0   0  12  0 |  0   0   0  4  4  0 | *  *  *  * 16  *
.o3.x .. .x       0  6 |  0   0   6  3 |  0   0   0  2  0  3 | *  *  *  *  * 32

qoo3xxx3oqo *b3ooq&#zx   → height = 0
(tegum sum of 3 mutually gyrated (q,x)-thexes)

o..3o..3o.. *b3o..     | 48  *  * |  4  2  2  0  0  0 |  2  2  4  4   4  0  0  0  0  0 | 1  2  2  1  4 0  0 0
.o.3.o.3.o. *b3.o.     |  * 48  * |  0  2  0  4  2  0 |  0  0  4  0   4  2  2  4  0  0 | 0  2  0  1  4 1  2 0
..o3..o3..o *b3..o     |  *  * 48 |  0  0  2  0  2  4 |  0  0  0  4   4  0  0  4  2  2 | 0  0  2  1  4 0  2 1
-----------------------+----------+-------------------+--------------------------------+---------------------
... x.. ...    ...     |  2  0  0 | 96  *  *  *  *  * |  1  1  1  1   0  0  0  0  0  0 | 1  1  1  0  1 0  0 0
oo.3oo.3oo. *b3oo.&#x  |  1  1  0 |  * 96  *  *  *  * |  0  0  2  0   2  0  0  0  0  0 | 0  1  0  1  2 0  0 0
o.o3o.o3o.o *b3o.o&#x  |  1  0  1 |  *  * 96  *  *  * |  0  0  0  2   2  0  0  0  0  0 | 0  0  1  1  2 0  0 0
... .x. ...    ...     |  0  2  0 |  *  *  * 96  *  * |  0  0  1  0   0  1  1  1  0  0 | 0  1  0  0  1 1  1 0
.oo3.oo3.oo *b3.oo&#x  |  0  1  1 |  *  *  *  * 96  * |  0  0  0  0   2  0  0  2  0  0 | 0  0  0  1  2 0  1 0
... ..x ...    ...     |  0  0  2 |  *  *  *  *  * 96 |  0  0  0  1   0  0  0  1  1  1 | 0  0  1  0  1 0  1 1
-----------------------+----------+-------------------+--------------------------------+---------------------
... x..3o..    ...     |  3  0  0 |  3  0  0  0  0  0 | 32  *  *  *   *  *  *  *  *  * | 1  0  1  0  0 0  0 0
... x.. ... *b3o..     |  3  0  0 |  3  0  0  0  0  0 |  * 32  *  *   *  *  *  *  *  * | 1  1  0  0  0 0  0 0
... xx. ...    ...&#x  |  2  2  0 |  1  2  0  1  0  0 |  *  * 96  *   *  *  *  *  *  * | 0  1  0  0  1 0  0 0
... x.x ...    ...&#x  |  2  0  2 |  1  0  2  0  0  1 |  *  *  * 96   *  *  *  *  *  * | 0  0  1  0  1 0  0 0
ooo3ooo3ooo *b3ooo&#x  |  1  1  1 |  0  1  1  0  1  0 |  *  *  *  * 192  *  *  *  *  * | 0  0  0  1  1 0  0 0
.o.3.x. ...    ...     |  0  3  0 |  0  0  0  3  0  0 |  *  *  *  *   * 32  *  *  *  * | 0  0  0  0  0 1  1 0
... .x. ... *b3.o.     |  0  3  0 |  0  0  0  3  0  0 |  *  *  *  *   *  * 32  *  *  * | 0  1  0  0  0 1  0 0
... .xx ...    ...&#x  |  0  2  2 |  0  0  0  1  2  1 |  *  *  *  *   *  *  * 96  *  * | 0  0  0  0  1 0  1 0
..o3..x ...    ...     |  0  0  3 |  0  0  0  0  0  3 |  *  *  *  *   *  *  *  * 32  * | 0  0  0  0  0 0  1 1
... ..x3..o    ...     |  0  0  3 |  0  0  0  0  0  3 |  *  *  *  *   *  *  *  *  * 32 | 0  0  1  0  0 0  0 1
-----------------------+----------+-------------------+--------------------------------+---------------------
... x..3o.. *b3o..       6  0  0 | 12  0  0  0  0  0 |  4  4  0  0   0  0  0  0  0  0 | 8  *  *  *  * *  * *
... xx. ... *b3oo.&#x    3  3  0 |  3  3  0  3  0  0 |  0  1  3  0   0  0  1  0  0  0 | * 32  *  *  * *  * *
... x.x3o.o    ...&#x    3  0  3 |  3  0  3  0  0  3 |  1  0  0  3   0  0  0  0  0  1 | *  * 32  *  * *  * *
qoo ... oqo    ooq&#zx   2  2  2 |  0  4  4  0  4  0 |  0  0  0  0   8  0  0  0  0  0 | *  *  * 24  * *  * *
... xxx ...    ...&#x    2  2  2 |  1  2  2  1  2  1 |  0  0  1  1   2  0  0  1  0  0 | *  *  *  * 96 *  * *
.o.3.x. ... *b3.o.       0  6  0 |  0  0  0 12  0  0 |  0  0  0  0   0  4  4  0  0  0 | *  *  *  *  * 8  * *
.oo3.xx ...    ...&#x    0  3  3 |  0  0  0  3  3  3 |  0  0  0  0   0  1  0  3  1  0 | *  *  *  *  * * 32 *
..o3..x3..o    ...       0  0  6 |  0  0  0  0  0 12 |  0  0  0  0   0  0  0  0  4  4 | *  *  *  *  * *  * 8

oxqwU2qowxx3xxooo4oxoxo&#zx   → height = 0 (U=w+x=q+u=q+2x)
(tegum sum of equatorial (q,x)-toe with ticcup, (q,w)-ope, (w,x)-sircope, and (U,x)-ope)

o....2o....3o....4o....     | 24  *  *  *  * |  2  4  2  0  0  0  0  0  0  0  0  0 |  2  4  2  4  4  0  0  0  0  0  0  0  0  0  0  0 |  1  2  4  2 1 0  0  0  0  0  0 0
.o...2.o...3.o...4.o...     |  * 48  *  *  * |  0  2  0  1  2  1  2  0  0  0  0  0 |  2  2  2  0  2  2  1  1  2  2  0  0  0  0  0  0 |  1  2  2  2 0 1  1  1  0  0  0 0
..o..2..o..3..o..4..o..     |  *  * 12  *  * |  0  0  4  0  0  0  0  4  0  0  0  0 |  0  0  0  4  8  0  0  0  0  0  4  0  0  0  0  0 |  0  0  4  4 1 0  0  0  1  0  0 0
...o.2...o.3...o.4...o.     |  *  *  * 48  * |  0  0  0  0  0  0  2  1  2  2  1  0 |  0  0  0  0  2  0  0  2  1  2  2  1  2  2  2  0 |  0  0  1  2 0 0  1  2  1  1  2 0
....o2....o3....o4....o     |  *  *  *  * 12 |  0  0  0  0  0  0  0  0  0  0  4  4 |  0  0  0  0  0  0  0  0  0  0  0  0  0  8  4  4 |  0  0  0  0 0 0  0  0  1  4  4 1
----------------------------+----------------+-------------------------------------+-------------------------------------------------+---------------------------------
..... ..... x.... .....     |  2  0  0  0  0 | 24  *  *  *  *  *  *  *  *  *  *  * |  0  2  0  2  0  0  0  0  0  0  0  0  0  0  0  0 |  0  1  2  0 1 0  0  0  0  0  0 0
oo...2oo...3oo...4oo...&#x  |  1  1  0  0  0 |  * 96  *  *  *  *  *  *  *  *  *  * |  1  1  1  0  1  0  0  0  0  0  0  0  0  0  0  0 |  1  1  1  1 0 0  0  0  0  0  0 0
o.o..2o.o..3o.o..4o.o..&#x  |  1  0  1  0  0 |  *  * 48  *  *  *  *  *  *  *  *  * |  0  0  0  2  2  0  0  0  0  0  0  0  0  0  0  0 |  0  0  2  1 1 0  0  0  0  0  0 0
.x... ..... ..... .....     |  0  2  0  0  0 |  *  *  * 24  *  *  *  *  *  *  *  * |  2  0  0  0  0  2  0  0  0  0  0  0  0  0  0  0 |  1  2  0  0 0 1  0  0  0  0  0 0
..... ..... .x... .....     |  0  2  0  0  0 |  *  *  *  * 48  *  *  *  *  *  *  * |  0  1  0  0  0  1  1  0  1  0  0  0  0  0  0  0 |  0  1  1  0 0 1  1  0  0  0  0 0
..... ..... ..... .x...     |  0  2  0  0  0 |  *  *  *  *  * 24  *  *  *  *  *  * |  0  0  2  0  0  0  0  0  0  2  0  0  0  0  0  0 |  1  0  0  2 0 0  0  1  0  0  0 0
.o.o.2.o.o.3.o.o.4.o.o.&#x  |  0  1  0  1  0 |  *  *  *  *  *  * 96  *  *  *  *  * |  0  0  0  0  1  0  0  1  1  1  0  0  0  0  0  0 |  0  0  1  1 0 0  1  1  0  0  0 0
..oo.2..oo.3..oo.4..oo.&#x  |  0  0  1  1  0 |  *  *  *  *  *  *  * 48  *  *  *  * |  0  0  0  0  2  0  0  0  0  0  2  0  0  0  0  0 |  0  0  1  2 0 0  0  0  1  0  0 0
..... ...x. ..... .....     |  0  0  0  2  0 |  *  *  *  *  *  *  *  * 48  *  *  * |  0  0  0  0  0  0  0  1  0  0  0  1  1  1  0  0 |  0  0  0  0 0 0  1  1  0  1  1 0
..... ..... ..... ...x.     |  0  0  0  2  0 |  *  *  *  *  *  *  *  *  * 48  *  * |  0  0  0  0  0  0  0  0  0  1  1  0  1  0  1  0 |  0  0  0  1 0 0  0  1  1  0  1 0
...oo2...oo3...oo4...oo&#x  |  0  0  0  1  1 |  *  *  *  *  *  *  *  *  *  * 48  * |  0  0  0  0  0  0  0  0  0  0  0  0  0  2  2  0 |  0  0  0  0 0 0  0  0  1  1  2 0
..... ....x ..... .....     |  0  0  0  0  2 |  *  *  *  *  *  *  *  *  *  *  * 24 |  0  0  0  0  0  0  0  0  0  0  0  0  0  2  0  2 |  0  0  0  0 0 0  0  0  0  2  1 1
----------------------------+----------------+-------------------------------------+-------------------------------------------------+---------------------------------
ox... ..... ..... .....&#x  |  1  2  0  0  0 |  0  2  0  1  0  0  0  0  0  0  0  0 | 48  *  *  *  *  *  *  *  *  *  *  *  *  *  *  * |  1  1  0  0 0 0  0  0  0  0  0 0
..... ..... xx... .....&#x  |  2  2  0  0  0 |  1  2  0  0  1  0  0  0  0  0  0  0 |  * 48  *  *  *  *  *  *  *  *  *  *  *  *  *  * |  0  1  1  0 0 0  0  0  0  0  0 0
..... ..... ..... ox...&#x  |  1  2  0  0  0 |  0  2  0  0  0  1  0  0  0  0  0  0 |  *  * 48  *  *  *  *  *  *  *  *  *  *  *  *  * |  1  0  1  0 0 0  0  0  0  0  0 0
..... ..... x.o.. .....&#x  |  2  0  1  0  0 |  1  0  2  0  0  0  0  0  0  0  0  0 |  *  *  * 48  *  *  *  *  *  *  *  *  *  *  *  * |  0  0  1  0 1 0  0  0  0  0  0 0
oooo.2oooo.3oooo.4oooo.&#xr |  1  1  1  1  0 |  0  1  1  0  0  0  1  1  0  0  0  0 |  *  *  *  * 96  *  *  *  *  *  *  *  *  *  *  * |  0  0  1  1 0 0  0  0  0  0  0 0 cycle: (ABDC)
.x... ..... .x... .....     |  0  4  0  0  0 |  0  0  0  2  2  0  0  0  0  0  0  0 |  *  *  *  *  * 24  *  *  *  *  *  *  *  *  *  * |  0  1  0  0 0 1  0  0  0  0  0 0
..... .o...3.x... .....     |  0  3  0  0  0 |  0  0  0  0  3  0  0  0  0  0  0  0 |  *  *  *  *  *  * 16  *  *  *  *  *  *  *  *  * |  0  0  0  0 0 1  1  0  0  0  0 0
..... .o.x. ..... .....&#x  |  0  1  0  2  0 |  0  0  0  0  0  0  2  0  1  0  0  0 |  *  *  *  *  *  *  * 48  *  *  *  *  *  *  *  * |  0  0  0  0 0 0  1  1  0  0  0 0
..... ..... .x.o. .....&#x  |  0  2  0  1  0 |  0  0  0  0  1  0  2  0  0  0  0  0 |  *  *  *  *  *  *  *  * 48  *  *  *  *  *  *  * |  0  0  1  0 0 0  1  0  0  0  0 0
..... ..... ..... .x.x.&#x  |  0  2  0  2  0 |  0  0  0  0  0  1  2  0  0  1  0  0 |  *  *  *  *  *  *  *  *  * 48  *  *  *  *  *  * |  0  0  0  1 0 0  0  1  0  0  0 0
..... ..... ..... ..ox.&#x  |  0  0  1  2  0 |  0  0  0  0  0  0  0  2  0  1  0  0 |  *  *  *  *  *  *  *  *  *  * 48  *  *  *  *  * |  0  0  0  1 0 0  0  0  1  0  0 0
..... ...x.3...o. .....     |  0  0  0  3  0 |  0  0  0  0  0  0  0  0  3  0  0  0 |  *  *  *  *  *  *  *  *  *  *  * 16  *  *  *  * |  0  0  0  0 0 0  1  0  0  1  0 0
..... ...x. ..... ...x.     |  0  0  0  4  0 |  0  0  0  0  0  0  0  0  2  2  0  0 |  *  *  *  *  *  *  *  *  *  *  *  * 24  *  *  * |  0  0  0  0 0 0  1  0  0  0  1 0
..... ...xx ..... .....&#x  |  0  0  0  2  2 |  0  0  0  0  0  0  0  0  1  0  2  1 |  *  *  *  *  *  *  *  *  *  *  *  *  * 48  *  * |  0  0  0  0 0 0  0  0  0  1  1 0
..... ..... ..... ...xo&#x  |  0  0  0  2  1 |  0  0  0  0  0  0  0  0  0  1  2  0 |  *  *  *  *  *  *  *  *  *  *  *  *  *  * 48  * |  0  0  0  0 0 0  0  0  1  0  1 0
..... ....x3....o .....     |  0  0  0  0  3 |  0  0  0  0  0  0  0  0  0  0  0  3 |  *  *  *  *  *  *  *  *  *  *  *  *  *  *  * 16 |  0  0  0  0 0 0  0  0  0  1  0 1
----------------------------+----------------+-------------------------------------+-------------------------------------------------+---------------------------------
ox...2qo... ..... ox...&#zx   2  4  0  0  0 |  0  8  0  2  0  2  0  0  0  0  0  0 |  4  0  4  0  0  0  0  0  0  0  0  0  0  0  0  0 | 12  *  *  * * *  *  *  *  *  * *
ox... ..... xx... .....&#x    2  4  0  0  0 |  1  4  0  2  2  0  0  0  0  0  0  0 |  2  2  0  0  0  1  0  0  0  0  0  0  0  0  0  0 |  * 24  *  * * *  *  *  *  *  * *
..... ..... xxoo. .....&#xr   2  2  1  1  0 |  1  2  2  0  1  0  2  1  0  0  0  0 |  0  1  0  1  2  0  0  0  1  0  0  0  0  0  0  0 |  *  * 48  * * *  *  *  *  *  * * cycle: (ABDC)
..... ..... ..... oxox.&#xr   1  2  1  2  0 |  0  2  1  0  0  1  2  2  0  1  0  0 |  0  0  1  0  2  0  0  0  0  1  1  0  0  0  0  0 |  *  *  * 48 * *  *  *  *  *  * * cycle: (ABDC)
o.q.. ..... x.o..4o.o..&#zx   4  0  2  0  0 |  4  0  8  0  0  0  0  0  0  0  0  0 |  0  0  0  8  0  0  0  0  0  0  0  0  0  0  0  0 |  *  *  *  * 6 *  *  *  *  *  * *
.x...2.o...3.x... .....       0  6  0  0  0 |  0  0  0  3  6  0  0  0  0  0  0  0 |  0  0  0  0  0  3  2  0  0  0  0  0  0  0  0  0 |  *  *  *  * * 8  *  *  *  *  * *
..... .o.x.3.x.o. .....&#x    0  3  0  3  0 |  0  0  0  0  3  0  6  0  3  0  0  0 |  0  0  0  0  0  0  1  3  3  0  0  1  0  0  0  0 |  *  *  *  * * * 16  *  *  *  * *
..... .o.x. ..... .x.x.&#x    0  2  0  4  0 |  0  0  0  0  0  1  4  0  2  2  0  0 |  0  0  0  0  0  0  0  2  0  2  0  0  1  0  0  0 |  *  *  *  * * *  * 24  *  *  * *
..... ..... ..ooo4..oxo&#xt   0  0  1  4  1 |  0  0  0  0  0  0  0  4  0  4  4  0 |  0  0  0  0  0  0  0  0  0  0  4  0  0  0  4  0 |  *  *  *  * * *  *  * 12  *  * *
..... ...xx3...oo .....&#x    0  0  0  3  3 |  0  0  0  0  0  0  0  0  3  0  3  3 |  0  0  0  0  0  0  0  0  0  0  0  1  0  3  0  1 |  *  *  *  * * *  *  *  * 16  * *
..... ...xx ..... ...xo&#x    0  0  0  4  2 |  0  0  0  0  0  0  0  0  2  2  4  1 |  0  0  0  0  0  0  0  0  0  0  0  0  1  2  2  0 |  *  *  *  * * *  *  *  *  * 24 *
..... ....x3....o4....o       0  0  0  0  6 |  0  0  0  0  0  0  0  0  0  0  0 12 |  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  8 |  *  *  *  * * *  *  *  *  *  * 2

ooxxowq4oxoxwoq qowxoxo4qwoxxoo&#zx

o......4o...... o......4o......     | 8  *  *  *  *  * * |  4  4  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |  4  4  8  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 | 1 1  4  4  0  0  0  0  0  0  0  0 0 0
.o.....4.o..... .o.....4.o.....     | * 16  *  *  *  * * |  2  0  2  4  0  0  0  0  0  0  0  0  0  0  0  0 |  4  0  4  2  4  2  0  0  0  0  0  0  0  0  0  0  0  0  0  0 | 1 0  2  4  1  2  0  0  0  0  0  0 0 0
..o....4..o.... ..o....4..o....     | *  * 16  *  *  * * |  0  2  0  0  2  4  0  0  0  0  0  0  0  0  0  0 |  0  4  4  0  0  0  4  2  2  0  0  0  0  0  0  0  0  0  0  0 | 0 1  4  2  0  0  2  1  0  0  0  0 0 0
...o...4...o... ...o...4...o...     | *  *  * 64  *  * * |  0  0  0  1  0  1  1  1  1  1  1  1  0  0  0  0 |  0  0  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  0  0 | 0 0  1  1  1  1  1  1  1  1  1  1 0 0
....o..4....o.. ....o..4....o..     | *  *  *  * 16  * * |  0  0  0  0  0  0  0  0  0  0  4  0  2  2  0  0 |  0  0  0  0  0  0  0  0  0  0  0  2  2  4  0  0  0  4  4  0 | 0 0  0  0  1  0  0  0  2  0  2  4 1 0
.....o.4.....o. .....o.4.....o.     | *  *  *  *  * 16 * |  0  0  0  0  0  0  0  0  0  0  0  4  0  0  2  2 |  0  0  0  0  0  0  0  0  0  0  0  0  0  0  2  4  2  4  0  4 | 0 0  0  0  0  0  0  1  0  2  4  2 0 1
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oo.....4oo..... oo.....4oo.....&#x  | 1  1  0  0  0  0 0 | 32  *  *  *  *  *  *  *  *  *  *  *  *  *  *  * |  2  0  2  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 | 1 0  1  2  0  0  0  0  0  0  0  0 0 0
o.o....4o.o.... o.o....4o.o....&#x  | 1  0  1  0  0  0 0 |  * 32  *  *  *  *  *  *  *  *  *  *  *  *  *  * |  0  2  2  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 | 0 1  2  1  0  0  0  0  0  0  0  0 0 0
....... .x..... ....... .......     | 0  2  0  0  0  0 0 |  *  * 16  *  *  *  *  *  *  *  *  *  *  *  *  * |  2  0  0  0  2  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 | 1 0  0  2  0  1  0  0  0  0  0  0 0 0
.o.o...4.o.o... .o.o...4.o.o...&#x  | 0  1  0  1  0  0 0 |  *  *  * 64  *  *  *  *  *  *  *  *  *  *  *  * |  0  0  1  1  1  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0 | 0 0  1  1  1  1  0  0  0  0  0  0 0 0
..x.... ....... ....... .......     | 0  0  2  0  0  0 0 |  *  *  *  * 16  *  *  *  *  *  *  *  *  *  *  * |  0  2  0  0  0  0  2  0  0  0  0  0  0  0  0  0  0  0  0  0 | 0 1  2  0  0  0  1  0  0  0  0  0 0 0
..oo...4..oo... ..oo...4..oo...&#x  | 0  0  1  1  0  0 0 |  *  *  *  *  * 64  *  *  *  *  *  *  *  *  *  * |  0  0  1  0  0  0  1  1  1  0  0  0  0  0  0  0  0  0  0  0 | 0 0  1  1  0  0  1  1  0  0  0  0 0 0
...x... ....... ....... .......     | 0  0  0  2  0  0 0 |  *  *  *  *  *  * 32  *  *  *  *  *  *  *  *  * |  0  0  0  1  0  0  1  0  0  1  0  1  0  0  0  0  0  0  0  0 | 0 0  1  0  1  0  1  0  1  0  0  0 0 0
....... ...x... ....... .......     | 0  0  0  2  0  0 0 |  *  *  *  *  *  *  * 32  *  *  *  *  *  *  *  * |  0  0  0  0  1  0  0  1  0  0  1  0  0  0  1  0  0  0  0  0 | 0 0  0  1  0  1  0  1  0  1  0  0 0 0
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...o.o.4...o.o. ...o.o.4...o.o.&#x  | 0  0  0  1  0  1 0 |  *  *  *  *  *  *  *  *  *  *  * 64  *  *  *  * |  0  0  0  0  0  0  0  0  0  0  0  0  0  0  1  1  1  1  0  0 | 0 0  0  0  0  0  0  1  0  1  1  1 0 0
....... ....... ....... ....x..     | 0  0  0  0  2  0 0 |  *  *  *  *  *  *  *  *  *  *  *  * 16  *  *  * |  0  0  0  0  0  0  0  0  0  0  0  0  0  2  0  0  0  0  2  0 | 0 0  0  0  0  0  0  0  1  0  0  2 1 0
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