Acronym proh
Name prismatorhombated hexadecachoron,
prismatotruncated tesseract,
runcitruncated tesseract
 
Cross sections
 ©
Circumradius sqrt[4+2 sqrt(2)] = 2.613126
Vertex figure
 ©
Vertex layers
LayerSymmetrySubsymmetries
 o3o3o4o o3o3o . o3o . o o . o4o . o3o4o
1x3o3x4x x3o3x .
co first
x3o . x
trip first
x . x4x
op first
. o3x4x
tic first
2a x3o3w . x3x . w u . o4w . x3x4x
2b o . u4x
3 x3x3w . o3x . X x . x4w . x3o4w
4 x3w3x . x3w . w U . o4w . x3o4w
5a o3x3U . o3w . X w . x4w . x3x4x
5b x3U . x W . x4x
6a o3w3u . u3w . w U . u4x . o3x4x
opposite tic
6b u3w3o .
7a U3x3o . x3w . X w . x4w  
7b o3W . x W . x4x
8 x3w3x . U3x . w U . o4w
9 w3x3x . x3U . w x . x4w
10a w3o3x . w3x . X u . o4w
10b W3o . x o . u4x
11 x3o3x .
opposite co
w3u . w x . x4x
opposite op
12a   w3o . X  
12b U3x . x
13 w3x . w
14 x3o . X
15 x3x . w
16 o3x . x
opposite trip
(U=x+w, W=q+3x=w+u=x+U, X=x+2q=w+q)
Lace city
in approx. ASCII-art
 ©  
    x4x w4o   w4o x4x    
                         
x4x x4u w4x   w4x x4u x4x
                         
w4o w4x           w4x w4o
                         
                         
w4o w4x           w4x w4o
                         
x4x x4u w4x   w4x x4u x4x
                         
    x4x w4o   w4o x4x    
                o3x         o3x                
                                               
        x3x                         x3x        
                                               
x3o                                         x3o
        w3x                         w3x        
                                               
w3o             U3x         U3x             w3o
                                               
        w3u                         w3u        
                                               
w3x             W3o         W3o             w3x
        x3U                         x3U        
        U3x                         U3x        
x3w             o3W         o3W             x3w
                                               
        u3w                         u3w        
                                               
o3w             x3U         x3U             o3w
                                               
        x3w                         x3w        
o3x                                         o3x
                                               
        x3x                         x3x        
                                               
                x3o         x3o                
Coordinates ((1+2 sqrt(2))/2, (1+sqrt(2))/2, (1+sqrt(2))/2, 1/2)   & all permutations, all changes of sign
General of army (is itself convex)
Colonel of regiment (is itself locally convex – uniform polychoral members:
by cells: cho co girco oho op tic trip
sirpith 160802400
proh 0160024832
siphado 008160832
)
Dihedral angles
  • at {3} between co and trip:   150°
  • at {4} between op and trip:   arccos(-sqrt[2/3]) = 144.735610°
  • at {4} between co and op:   135°
  • at {8} between op and tic:   135°
  • at {3} between co and tic:   120°
Confer
Grünbaumian relatives:
2proh  
segmentochora:
ticagirco  
related CRFs:
pabdiproh  
uniform relative:
srico  
decompositions:
rico || proh   srit || proh  
general polytopal classes:
partial Stott expansions  
External
links
hedrondude   wikipedia   WikiChoron   quickfur

As abstract polytope proh is isomorphic to quiproh, thereby replacing octagons by octagrams, resp. replacing tic by quith and replacing op by stop.

Augmenting coatics onto the tic would lead to the srico (which would have an even larger symmetry)!

Note that proh can be thought of as the external blend of 1 rico + 16 copes + 32 triddips + 24 squacupes + 8 coatics. This decomposition is described as the degenerate segmentoteron xx3oo3xx4ox&#x. – Alternatively it can be decomposed into 1 srit + 16 octacoes + 32 opes + 24 squipufs + 8 sircoatics according to ox3xo3ox4xx&#x.


Incidence matrix according to Dynkin symbol

x3o3x4x

. . . . | 192 |   2   2  1 |  1  2  2  1  2 |  1  1  2 1
--------+-----+------------+----------------+-----------
x . . . |   2 | 192   *  * |  1  1  1  0  0 |  1  1  1 0
. . x . |   2 |   * 192  * |  0  1  0  1  1 |  1  0  1 1
. . . x |   2 |   *   * 96 |  0  0  2  0  2 |  0  1  2 1
--------+-----+------------+----------------+-----------
x3o . . |   3 |   3   0  0 | 64  *  *  *  * |  1  1  0 0
x . x . |   4 |   2   2  0 |  * 96  *  *  * |  1  0  1 0
x . . x |   4 |   2   0  2 |  *  * 96  *  * |  0  1  1 0
. o3x . |   3 |   0   3  0 |  *  *  * 64  * |  1  0  0 1
. . x4x |   8 |   0   4  4 |  *  *  *  * 48 |  0  0  1 1
--------+-----+------------+----------------+-----------
x3o3x .   12 |  12  12  0 |  4  6  0  4  0 | 16  *  * *
x3o . x    6 |   6   0  3 |  2  0  3  0  0 |  * 32  * *
x . x4x   16 |   8   8  8 |  0  4  4  0  2 |  *  * 24 *
. o3x4x   24 |   0  24 12 |  0  0  0  8  6 |  *  *  * 8

snubbed forms: x3o3x4s, x3o3β4x, β3o3x4x, β3o3β4x, β3o3x4β, x3o3β4β, β3o3β4β

x3/2o3/2x4x

.   .   . . | 192 |   2   2  1 |  1  2  2  1  2 |  1  1  2 1
------------+-----+------------+----------------+-----------
x   .   . . |   2 | 192   *  * |  1  1  1  0  0 |  1  1  1 0
.   .   x . |   2 |   * 192  * |  0  1  0  1  1 |  1  0  1 1
.   .   . x |   2 |   *   * 96 |  0  0  2  0  2 |  0  1  2 1
------------+-----+------------+----------------+-----------
x3/2o   . . |   3 |   3   0  0 | 64  *  *  *  * |  1  1  0 0
x   .   x . |   4 |   2   2  0 |  * 96  *  *  * |  1  0  1 0
x   .   . x |   4 |   2   0  2 |  *  * 96  *  * |  0  1  1 0
.   o3/2x . |   3 |   0   3  0 |  *  *  * 64  * |  1  0  0 1
.   .   x4x |   8 |   0   4  4 |  *  *  *  * 48 |  0  0  1 1
------------+-----+------------+----------------+-----------
x3/2o3/2x .   12 |  12  12  0 |  4  6  0  4  0 | 16  *  * *
x3/2o   . x    6 |   6   0  3 |  2  0  3  0  0 |  * 32  * *
x   .   x4x   16 |   8   8  8 |  0  4  4  0  2 |  *  * 24 *
.   o3/2x4x   24 |   0  24 12 |  0  0  0  8  6 |  *  *  * 8


oxxxxo3xxooxx4xxwwxx&#xt   → all non-central heights = 1/sqrt(2) = 0.707107
                             central height = 1
(tic || pseudo girco || pseudo (x,w)-sirco || pseudo (x,w)-sirco || pseudo girco || tic)

o.....3o.....4o.....      & | 48  *  * |  2  1  2  0  0  0  0  0  0 |  1  2  1  2  2  0  0  0  0  0  0  0 | 1  1  1  2  0 0 0
.o....3.o....4.o....      & |  * 96  * |  0  0  1  1  1  1  1  0  0 |  0  0  1  1  1  1  1  1  1  1  0  0 | 0  1  1  1  1 1 0
..o...3..o...4..o...      & |  *  * 48 |  0  0  0  0  0  0  2  2  1 |  0  0  0  0  0  0  0  2  1  2  1  2 | 0  1  0  0  2 1 1
----------------------------+----------+----------------------------+-------------------------------------+------------------
...... x..... ......      & |  2  0  0 | 48  *  *  *  *  *  *  *  * |  1  1  0  1  0  0  0  0  0  0  0  0 | 1  1  0  1  0 0 0
...... ...... x.....      & |  2  0  0 |  * 24  *  *  *  *  *  *  * |  0  2  0  0  2  0  0  0  0  0  0  0 | 1  0  1  2  0 0 0
oo....3oo....4oo....&#x   & |  1  1  0 |  *  * 96  *  *  *  *  *  * |  0  0  1  1  1  0  0  0  0  0  0  0 | 0  1  1  1  0 0 0
.x.... ...... ......      & |  0  2  0 |  *  *  * 48  *  *  *  *  * |  0  0  1  0  0  1  0  1  0  0  0  0 | 0  1  1  0  1 0 0
...... .x.... ......      & |  0  2  0 |  *  *  *  * 48  *  *  *  * |  0  0  0  1  0  0  1  0  1  0  0  0 | 0  1  0  1  0 1 0
...... ...... .x....      & |  0  2  0 |  *  *  *  *  * 48  *  *  * |  0  0  0  0  1  1  1  0  0  1  0  0 | 0  0  1  1  1 1 0
.oo...3.oo...4.oo...&#x   & |  0  1  1 |  *  *  *  *  *  * 96  *  * |  0  0  0  0  0  0  0  1  1  1  0  0 | 0  1  0  0  1 1 0
..x... ...... ......      & |  0  0  2 |  *  *  *  *  *  *  * 48  * |  0  0  0  0  0  0  0  1  0  0  1  1 | 0  1  0  0  1 0 1
..oo..3..oo..4..oo..&#x     |  0  0  2 |  *  *  *  *  *  *  *  * 24 |  0  0  0  0  0  0  0  0  0  2  0  2 | 0  0  0  0  2 1 1
----------------------------+----------+----------------------------+-------------------------------------+------------------
o.....3x..... ......      & |  3  0  0 |  3  0  0  0  0  0  0  0  0 | 16  *  *  *  *  *  *  *  *  *  *  * | 1  1  0  0  0 0 0
...... x.....4x.....      & |  8  0  0 |  4  4  0  0  0  0  0  0  0 |  * 12  *  *  *  *  *  *  *  *  *  * | 1  0  0  1  0 0 0
ox.... ...... ......&#x   & |  1  2  0 |  0  0  2  1  0  0  0  0  0 |  *  * 48  *  *  *  *  *  *  *  *  * | 0  1  1  0  0 0 0
...... xx.... ......&#x   & |  2  2  0 |  1  0  2  0  1  0  0  0  0 |  *  *  * 48  *  *  *  *  *  *  *  * | 0  1  0  1  0 0 0
...... ...... xx....&#x   & |  2  2  0 |  0  1  2  0  0  1  0  0  0 |  *  *  *  * 48  *  *  *  *  *  *  * | 0  0  1  1  0 0 0
.x.... ...... .x....      & |  0  4  0 |  0  0  0  2  0  2  0  0  0 |  *  *  *  *  * 24  *  *  *  *  *  * | 0  0  1  0  1 0 0
...... .x....4.x....      & |  0  8  0 |  0  0  0  0  4  4  0  0  0 |  *  *  *  *  *  * 12  *  *  *  *  * | 0  0  0  1  0 1 0
.xx... ...... ......&#x   & |  0  2  2 |  0  0  0  1  0  0  2  1  0 |  *  *  *  *  *  *  * 48  *  *  *  * | 0  1  0  0  1 0 0
...... .xo... ......&#x   & |  0  2  1 |  0  0  0  0  1  0  2  0  0 |  *  *  *  *  *  *  *  * 48  *  *  * | 0  1  0  0  0 1 0
...... ...... .xwwx.&#xt    |  0  4  4 |  0  0  0  0  0  2  4  0  2 |  *  *  *  *  *  *  *  *  * 24  *  * | 0  0  0  0  1 1 0
..x...3..o... ......      & |  0  0  3 |  0  0  0  0  0  0  0  3  0 |  *  *  *  *  *  *  *  *  *  * 16  * | 0  1  0  0  0 0 1
..xx.. ...... ......&#x     |  0  0  4 |  0  0  0  0  0  0  0  2  2 |  *  *  *  *  *  *  *  *  *  *  * 24 | 0  0  0  0  1 0 1
----------------------------+----------+----------------------------+-------------------------------------+------------------
o.....3x.....4x.....      &  24  0  0 | 24 12  0  0  0  0  0  0  0 |  8  6  0  0  0  0  0  0  0  0  0  0 | 2  *  *  *  * * *
oxx...3xxo... ......&#xt  &   3  6  3 |  3  0  6  3  3  0  6  3  0 |  1  0  3  3  0  0  0  3  3  0  1  0 | * 16  *  *  * * *
ox.... ...... xx....&#x   &   2  4  0 |  0  1  4  2  0  2  0  0  0 |  0  0  2  0  2  1  0  0  0  0  0  0 | *  * 24  *  * * *
...... xx....4xx....&#x   &   8  8  0 |  4  4  8  0  4  4  0  0  0 |  0  1  0  4  4  0  1  0  0  0  0  0 | *  *  * 12  * * *
.xxxx. ...... .xwwx.&#xt      0  8  8 |  0  0  0  4  0  4  8  4  4 |  0  0  0  0  0  2  0  4  0  2  0  2 | *  *  *  * 12 * *
...... .xoox.4.xwwx.&#xt      0 16  8 |  0  0  0  0  8  8 16  0  4 |  0  0  0  0  0  0  2  0  8  4  0  0 | *  *  *  *  * 6 *
..xx..3..oo.. ......&#x       0  0  6 |  0  0  0  0  0  0  0  6  3 |  0  0  0  0  0  0  0  0  0  0  2  3 | *  *  *  *  * * 8

wx3oo3xw *b3xx&#zx   → height = 0
(tegum sum of 2 mutually gyrated (w,x,x)-ricoes)

o.3o.3o. *b3o.     | 96  * |  2  2  1  0  0 |  1  1  2  2  2  0  0  0 | 1 1  2  1 0
.o3.o3.o *b3.o     |  * 96 |  0  0  1  2  2 |  0  0  0  2  2  1  2  1 | 0 1  2  1 1
-------------------+-------+----------------+-------------------------+------------
.. .. x.    ..     |  2  0 | 96  *  *  *  * |  1  0  1  1  0  0  0  0 | 1 1  1  0 0
.. .. ..    x.     |  2  0 |  * 96  *  *  * |  0  1  1  0  1  0  0  0 | 1 0  1  1 0
oo3oo3oo *b3oo&#x  |  1  1 |  *  * 96  *  * |  0  0  0  2  2  0  0  0 | 0 1  2  1 0
.x .. ..    ..     |  0  2 |  *  *  * 96  * |  0  0  0  1  0  1  1  0 | 0 1  1  0 1
.. .. ..    .x     |  0  2 |  *  *  *  * 96 |  0  0  0  0  1  0  1  1 | 0 0  1  1 1
-------------------+-------+----------------+-------------------------+------------
.. o.3x.    ..     |  3  0 |  3  0  0  0  0 | 32  *  *  *  *  *  *  * | 1 1  0  0 0
.. o. .. *b3x.     |  3  0 |  0  3  0  0  0 |  * 32  *  *  *  *  *  * | 1 0  0  1 0
.. .. x.    x.     |  4  0 |  2  2  0  0  0 |  *  * 48  *  *  *  *  * | 1 0  1  0 0
wx .. xw    ..&#zx |  4  4 |  2  0  4  2  0 |  *  *  * 48  *  *  *  * | 0 1  1  0 0
.. .. ..    xx&#x  |  2  2 |  0  1  2  0  1 |  *  *  *  * 96  *  *  * | 0 0  1  1 0
.x3.o ..    ..     |  0  3 |  0  0  0  3  0 |  *  *  *  *  * 32  *  * | 0 1  0  0 1
.x .. ..    .x     |  0  4 |  0  0  0  2  2 |  *  *  *  *  *  * 48  * | 0 0  1  0 1
.. .o .. *b3.x     |  0  3 |  0  0  0  0  3 |  *  *  *  *  *  *  * 32 | 0 0  0  1 1
-------------------+-------+----------------+-------------------------+------------
.. o.3x. *b3x.      12  0 | 12 12  0  0  0 |  4  4  6  0  0  0  0  0 | 8 *  *  * *
wx3oo3xw    ..&#zx  12 12 | 12  0 12 12  0 |  4  0  0  6  0  4  0  0 | * 8  *  * *
wx .. xw    xx&#zx   8  8 |  4  4  8  4  4 |  0  0  2  2  4  0  2  0 | * * 24  * *
.. oo .. *b3xx&#x    3  3 |  0  3  3  0  3 |  0  1  0  0  3  0  0  1 | * *  * 32 *
.x3.o .. *b3.x       0 12 |  0  0  0 12 12 |  0  0  0  0  0  4  6  4 | * *  *  * 8

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