Acronym	proh
Name	prismatorhombated hexadecachoron, prismatotruncated tesseract, runcitruncated tesseract
Cross sections	©
Circumradius	sqrt[4+2 sqrt(2)] = 2.613126
Vertex figure	©
Vertex layers	Layer Symmetry Subsymmetries   o3o3o4o o3o3o . o3o . o o . o4o . o3o4o 1 x3o3x4x 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 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°
Face vector	192, 480, 368, 80
Confer	Grünbaumian relatives: 2proh   segmentochora: ticagirco   related CRFs: pabdiproh   uniform relative: srico   decompositions: rico || proh   srit || proh   general polytopal classes: Wythoffian polychora   partial Stott expansions
External links

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