Acronym proh
prismatotruncated tesseract,
runcitruncated tesseract

Cross sections
` ©`
Vertex figure
` ©`
Vertex layers
 Layer Symmetry Subsymmetries o3o3o4o o3o3o . o3o . o o . o4o . o3o4o 1 x3o3x4x x3o3x .co first x3o . xtrip first x . x4xop first . o3x4xtic 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 . o3x4xopposite 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 . x4xopposite op 12a w3o . X 12b U3x . x 13 w3x . w 14 x3o . X 15 x3x . w 16 o3x . xopposite 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 16 0 8 0 24 0 0 proh 0 16 0 0 24 8 32 siphado 0 0 8 16 0 8 32
)
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

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
```