Acronym	gidpith
Name	great disprismatotesseractihexadecachoron, great prismated tesseract, omnitruncated tesseract
Cross sections	©
Circumradius	sqrt[8+3 sqrt(2)] = 3.498949
Inradius wrt. hip	sqrt[27+12 sqrt(2)]/2 = 3.315515
Inradius wrt. op	(5+sqrt(2))/2 = 3.207107
Inradius wrt. toe	(2+3 sqrt(2))/2 = 3.121320
Inradius wrt. girco	sqrt[19+6 sqrt(2)]/2 = 2.621320
Vertex figure	©
Vertex layers	Layer Symmetry Subsymmetries   o3o3o4o o3o3o . o3o . o o . o4o . o3o4o 1 x3x3x4x x3x3x . toe first x3x . x hip first x . x4x op first . x3x4x girco first 2 x3x3w . x3u . w u . u4x . u3x4x 3a x3u3w . u3x . X x . U4x . x3u4x 3b D . x4w 4 x3U3x . x3U . w u . u4w . x3x4w 5 u3x3U . x3x . A W . x4w . x3x4w 6a u3w3u . x3W . x x . x4w . x3u4x 6b u3w . X 7a x3x3W . x3w . A U . u4w . u3x4x 7b Y . u4x 8a x3w3D . u3W . x w . x4w . x3x4x opposite girco 8b D3w3x . D3w . X W . U4x 8c B . x4x 9a W3x3x . D3U . w w . x4w   9b u3w . A W . U4x 9c B . x4x 10a u3w3u . W3x . X U . u4w 10b Y . u4x 11 U3x3u . x3Y . x x . x4w 12a x3U3x . W3u . w W . x4w 12b U3x . A 13a w3u3x . u3W . w u . u4w 13b x3U . A 14a w3x3x . Y3x . x x . U4x 14b D . x4w 15 x3x3x . opposite toe x3W . X u . u4x 16a   U3D . w x . x4x opposite op 16b w3u . A 17a W3u . x   17b w3D . X 18 w3x . A 19a W3x . x 19b w3u . X 20 x3x . A 21 U3x . w 22 x3u . X 23 u3x . w 24 x3x . x opposite hip (D=3x, U=2x+q=x+w, W=3x+q=u+w, X=x+2q, Y=4x+q, A=x+3q, B=5x+q)
Lace city in approx. ASCII-art	©   x4x u4x x4w x4w u4x x4x x4x D4x u4w u4w D4x x4x u4x D4x x4X x4X D4x u4x x4w u4w x4X x4X u4w x4w x4w u4w x4X x4X u4w x4w u4x D4x x4X x4X D4x u4x x4x D4x u4w u4w D4x x4x x4x u4x x4w x4w u4x x4x
	x3x x3x x3u x3u u3x u3x x3U x3U x3x x3x u3w x3W x3W u3w x3w x3w D3w u3W u3W D3w u3w D3U D3U u3w W3x W3x x3Y x3Y U3x W3u W3u U3x x3U u3W u3W x3U Y3x Y3x x3W x3W w3u U3D U3D w3u w3D W3u W3u w3D w3x w3x w3u W3x W3x w3u x3x x3x U3x U3x x3u x3u u3x u3x x3x x3x
Coordinates	((1+3 sqrt(2))/2, (1+2 sqrt(2))/2, (1+sqrt(2))/2, 1/2)   & all permutations, all changes of sign
Volume	2[131+92 sqrt(2)] = 522.215295
General of army	(is itself convex)
Colonel of regiment	(is itself locally convex – uniform polychoral members: by cells: girco hip op toe gidpith 8 32 24 16 )
Dihedral angles	at {6} between hip and toe:   150° at {4} between hip and op:   arccos[-sqrt(2/3)] = 144.735610° at {4} between op and toe:   135° at {8} between girco and op:   135° at {4} between girco and hip:   arccos[-1/sqrt(3)] = 125.264390° at {6} between girco and toe:   120°
Face vector	384, 768, 464, 80
Confer	segmentochora: toe || girco   related CRFs: prico   decompositions: proh || gidpith   tico || gidpith   general polytopal classes: Wythoffian polychora   partial Stott expansions   analogs: omnitruncated hypercube otCn
External links

As abstract polytope gidpith is isomorphic to gaquidpoth, thereby replacing the octagons by octagrams, resp. replacing the girco by quitco and the op by stop.

Augmenting toe || girco onto each girco of gidpith would lead to prico (which then would have an even larger symmetry)!

Note that gidpith can be thought of as the external blend of 1 tico + 16 topes + 32 thiddips + 24 squacupes + 8 toagircos. This decomposition is described as the degenerate segmentoteron xx3xx3xx4ox&#x. – Alternatively it also can be decomposed into 1 proh + 16 coatoes + 32 tricupes + 24 sodips + 8 ticagircos according to xx3ox3xx4xx&#x.

Incidence matrix according to Dynkin symbol

x3x3x4x

. . . . | 384 |   1   1   1   1 |  1  1  1  1  1  1 |  1  1  1 1
--------+-----+-----------------+-------------------+-----------
x . . . |   2 | 192   *   *   * |  1  1  1  0  0  0 |  1  1  1 0
. x . . |   2 |   * 192   *   * |  1  0  0  1  1  0 |  1  1  0 1
. . x . |   2 |   *   * 192   * |  0  1  0  1  0  1 |  1  0  1 1
. . . x |   2 |   *   *   * 192 |  0  0  1  0  1  1 |  0  1  1 1
--------+-----+-----------------+-------------------+-----------
x3x . . |   6 |   3   3   0   0 | 64  *  *  *  *  * |  1  1  0 0
x . x . |   4 |   2   0   2   0 |  * 96  *  *  *  * |  1  0  1 0
x . . x |   4 |   2   0   0   2 |  *  * 96  *  *  * |  0  1  1 0
. x3x . |   6 |   0   3   3   0 |  *  *  * 64  *  * |  1  0  0 1
. x . x |   4 |   0   2   0   2 |  *  *  *  * 96  * |  0  1  0 1
. . x4x |   8 |   0   0   4   4 |  *  *  *  *  * 48 |  0  0  1 1
--------+-----+-----------------+-------------------+-----------
x3x3x . ♦  24 |  12  12  12   0 |  4  6  0  4  0  0 | 16  *  * *
x3x . x ♦  12 |   6   6   0   6 |  2  0  3  0  3  0 |  * 32  * *
x . x4x ♦  16 |   8   0   8   8 |  0  4  4  0  0  2 |  *  * 24 *
. x3x4x ♦  48 |   0  24  24  24 |  0  0  0  8 12  6 |  *  *  * 8

snubbed forms: β3x3x4x, x3β3x4x, x3x3β4x, x3x3x4s, β3β3x4x, β3x3β4x, β3x3x4β, x3β3β4x, x3β3x4β, x3x3β4β, s3s3s4x, β3β3x4β, β3x3β4β, x3β3β4β, s3s3s4s

xuxxxxux3xxuxxuxx4xxxwwxxx&#xt   → all non-central heights = 1/sqrt(2) = 0.707107
                                   central height = 1
(girco || pseudo (u,x,x)-girco || pseudo (x,u,x)-girco || pseudo (x,x,w)-girco || pseudo (x,x,w)-girco || pseudo (x,u,x)-girco || pseudo (u,x,x)-girco || girco)

o.......3o.......4o.......      & | 96  *  *  * |  1  1  1  1  0  0  0  0  0  0  0  0  0 |  1  1  1  1  1  1  0  0  0  0  0  0  0  0  0 | 1  1  1  1 0  0 0
.o......3.o......4.o......      & |  * 96  *  * |  0  0  0  1  1  1  1  0  0  0  0  0  0 |  0  0  0  1  1  1  1  1  1  0  0  0  0  0  0 | 0  1  1  1 1  0 0
..o.....3..o.....4..o.....      & |  *  * 96  * |  0  0  0  0  0  0  1  1  1  1  0  0  0 |  0  0  0  1  0  0  0  1  1  1  1  1  0  0  0 | 0  1  1  0 1  1 0
...o....3...o....4...o....      & |  *  *  * 96 |  0  0  0  0  0  0  0  0  0  1  1  1  1 |  0  0  0  0  0  0  0  1  0  0  1  1  1  1  1 | 0  1  0  0 1  1 1
----------------------------------+-------------+----------------------------------------+----------------------------------------------+------------------
x....... ........ ........      & |  2  0  0  0 | 48  *  *  *  *  *  *  *  *  *  *  *  * |  1  1  0  1  0  0  0  0  0  0  0  0  0  0  0 | 1  1  1  0 0  0 0
........ x....... ........      & |  2  0  0  0 |  * 48  *  *  *  *  *  *  *  *  *  *  * |  1  0  1  0  1  0  0  0  0  0  0  0  0  0  0 | 1  1  0  1 0  0 0
........ ........ x.......      & |  2  0  0  0 |  *  * 48  *  *  *  *  *  *  *  *  *  * |  0  1  1  0  0  1  0  0  0  0  0  0  0  0  0 | 1  0  1  1 0  0 0
oo......3oo......4oo......&#x   & |  1  1  0  0 |  *  *  * 96  *  *  *  *  *  *  *  *  * |  0  0  0  1  1  1  0  0  0  0  0  0  0  0  0 | 0  1  1  1 0  0 0
........ .x...... ........      & |  0  2  0  0 |  *  *  *  * 48  *  *  *  *  *  *  *  * |  0  0  0  0  1  0  1  1  0  0  0  0  0  0  0 | 0  1  0  1 1  1 0
........ ........ .x......      & |  0  2  0  0 |  *  *  *  *  * 48  *  *  *  *  *  *  * |  0  0  0  0  0  1  1  0  1  0  0  0  0  0  0 | 0  0  1  1 1  1 0
.oo.....3.oo.....4.oo.....&#x   & |  0  1  1  0 |  *  *  *  *  *  * 96  *  *  *  *  *  * |  0  0  0  1  0  0  0  1  1  0  0  0  0  0  0 | 0  1  1  0 1  1 0
..x..... ........ ........      & |  0  0  2  0 |  *  *  *  *  *  *  * 48  *  *  *  *  * |  0  0  0  1  0  0  0  0  0  1  1  0  0  0  0 | 0  1  1  0 0  1 0
........ ........ ..x.....      & |  0  0  2  0 |  *  *  *  *  *  *  *  * 48  *  *  *  * |  0  0  0  0  0  0  0  0  1  1  0  1  0  0  0 | 0  0  1  0 1  1 0
..oo....3..oo....4..oo....&#x   & |  0  0  1  1 |  *  *  *  *  *  *  *  *  * 96  *  *  * |  0  0  0  0  0  0  0  1  0  0  1  1  0  0  0 | 0  1  0  0 1  1 0
...x.... ........ ........      & |  0  0  0  2 |  *  *  *  *  *  *  *  *  *  * 48  *  * |  0  0  0  0  0  0  0  0  0  0  1  0  1  1  0 | 0  1  0  0 0  1 1
........ ...x.... ........      & |  0  0  0  2 |  *  *  *  *  *  *  *  *  *  *  * 48  * |  0  0  0  0  0  0  0  1  0  0  0  0  1  0  1 | 0  1  0  0 1  0 1
...oo...3...oo...4...oo...&#x     |  0  0  0  2 |  *  *  *  *  *  *  *  *  *  *  *  * 48 |  0  0  0  0  0  0  0  0  0  0  0  1  0  1  1 | 0  0  0  0 1  1 1
----------------------------------+-------------+----------------------------------------+----------------------------------------------+------------------
x.......3x....... ........      & |  6  0  0  0 |  3  3  0  0  0  0  0  0  0  0  0  0  0 | 16  *  *  *  *  *  *  *  *  *  *  *  *  *  * | 1  1  0  0 0  0 0
x....... ........ x.......      & |  4  0  0  0 |  2  0  2  0  0  0  0  0  0  0  0  0  0 |  * 24  *  *  *  *  *  *  *  *  *  *  *  *  * | 1  0  1  0 0  0 0
........ x.......4x.......      & |  8  0  0  0 |  0  4  4  0  0  0  0  0  0  0  0  0  0 |  *  * 12  *  *  *  *  *  *  *  *  *  *  *  * | 1  0  0  1 0  0 0
xux..... ........ ........&#xt  & |  2  2  2  0 |  1  0  0  2  0  0  2  1  0  0  0  0  0 |  *  *  * 48  *  *  *  *  *  *  *  *  *  *  * | 0  1  1  0 0  0 0
........ xx...... ........&#x   & |  2  2  0  0 |  0  1  0  2  1  0  0  0  0  0  0  0  0 |  *  *  *  * 48  *  *  *  *  *  *  *  *  *  * | 0  1  0  1 0  0 0
........ ........ xx......&#x   & |  2  2  0  0 |  0  0  1  2  0  1  0  0  0  0  0  0  0 |  *  *  *  *  * 48  *  *  *  *  *  *  *  *  * | 0  0  1  1 0  0 0
........ .x......4.x......      & |  0  8  0  0 |  0  0  0  0  4  4  0  0  0  0  0  0  0 |  *  *  *  *  *  * 12  *  *  *  *  *  *  *  * | 0  0  0  1 1  0 0
........ .xux.... ........&#xt  & |  0  2  2  2 |  0  0  0  0  1  0  2  0  0  2  0  1  0 |  *  *  *  *  *  *  * 48  *  *  *  *  *  *  * | 0  1  0  0 1  0 0
........ ........ .xx.....&#x   & |  0  2  2  0 |  0  0  0  0  0  1  2  0  1  0  0  0  0 |  *  *  *  *  *  *  *  * 48  *  *  *  *  *  * | 0  0  1  0 1  0 0
..x..... ........ ..x.....      & |  0  0  4  0 |  0  0  0  0  0  0  0  2  2  0  0  0  0 |  *  *  *  *  *  *  *  *  * 24  *  *  *  *  * | 0  0  1  0 0  1 0
..xx.... ........ ........&#x   & |  0  0  2  2 |  0  0  0  0  0  0  0  1  0  2  1  0  0 |  *  *  *  *  *  *  *  *  *  * 48  *  *  *  * | 0  1  0  0 0  1 0
........ ........ ..xwwx..&#xt    |  0  0  4  4 |  0  0  0  0  0  0  0  0  2  4  0  0  2 |  *  *  *  *  *  *  *  *  *  *  * 24  *  *  * | 0  0  0  0 1  1 0
...x....3...x.... ........      & |  0  0  0  6 |  0  0  0  0  0  0  0  0  0  0  3  3  0 |  *  *  *  *  *  *  *  *  *  *  *  * 16  *  * | 0  1  0  0 0  0 1
...xx... ........ ........&#x     |  0  0  0  4 |  0  0  0  0  0  0  0  0  0  0  2  0  2 |  *  *  *  *  *  *  *  *  *  *  *  *  * 24  * | 0  0  0  0 0  1 1
........ ...xx... ........&#x     |  0  0  0  4 |  0  0  0  0  0  0  0  0  0  0  0  2  2 |  *  *  *  *  *  *  *  *  *  *  *  *  *  * 24 | 0  0  0  0 1  0 1
----------------------------------+-------------+----------------------------------------+----------------------------------------------+------------------
x.......3x.......4x.......      & ♦ 48  0  0  0 | 24 24 24  0  0  0  0  0  0  0  0  0  0 |  8 12  6  0  0  0  0  0  0  0  0  0  0  0  0 | 2  *  *  * *  * *
xuxx....3xxux.... ........&#xt  & ♦  6  6  6  6 |  3  3  0  6  3  0  6  3  0  6  3  3  0 |  1  0  0  3  3  0  0  3  0  0  3  0  1  0  0 | * 16  *  * *  * *
xux..... ........ xxx.....&#xt  & ♦  4  4  4  0 |  2  0  2  4  0  2  4  2  2  0  0  0  0 |  0  1  0  2  0  2  0  0  2  1  0  0  0  0  0 | *  * 24  * *  * *
........ xx......4xx......&#x   & ♦  8  8  0  0 |  0  4  4  8  4  4  0  0  0  0  0  0  0 |  0  0  1  0  4  4  1  0  0  0  0  0  0  0  0 | *  *  * 12 *  * *
........ .xuxxux.4.xxwwxx.&#xt    ♦  0 16 16 16 |  0  0  0  0  8  8 16  0  8 16  0  8  8 |  0  0  0  0  0  0  2  8  8  0  0  4  0  0  4 | *  *  *  * 6  * *
..xxxx.. ........ ..xwwx..&#xt    ♦  0  0  8  8 |  0  0  0  0  4  4  8  4  4  8  4  0  4 |  0  0  0  0  0  0  0  0  0  2  4  2  0  2  0 | *  *  *  * * 12 *
...xx...3...xx... ........&#x     ♦  0  0  0 12 |  0  0  0  0  0  0  0  0  0  0  6  6  6 |  0  0  0  0  0  0  0  0  0  0  0  0  2  3  3 | *  *  *  * *  * 8

wx3xx3xw *b3xx&#zx   → height = 0
(tegum sum of 2 mutually gyrated (w,x,x,x)-ticoes)

o.3o.3o. *b3o.     | 192   * |  1  1  1   1  0  0  0 |  1  1  1  1  1  1  0  0  0 | 1 1  1  1 0
.o3.o3.o *b3.o     |   * 192 |  0  0  0   1  1  1  1 |  0  0  0  1  1  1  1  1  1 | 0 1  1  1 1
-------------------+---------+-----------------------+----------------------------+------------
.. x. ..    ..     |   2   0 | 96  *  *   *  *  *  * |  1  1  0  0  1  0  0  0  0 | 1 1  0  1 0
.. .. x.    ..     |   2   0 |  * 96  *   *  *  *  * |  1  0  1  1  0  0  0  0  0 | 1 1  1  0 0
.. .. ..    x.     |   2   0 |  *  * 96   *  *  *  * |  0  1  1  0  0  1  0  0  0 | 1 0  1  1 0
oo3oo3oo *b3oo&#x  |   1   1 |  *  *  * 192  *  *  * |  0  0  0  1  1  1  0  0  0 | 0 1  1  1 0
.x .. ..    ..     |   0   2 |  *  *  *   * 96  *  * |  0  0  0  1  0  0  1  1  0 | 0 1  1  0 1
.. .x ..    ..     |   0   2 |  *  *  *   *  * 96  * |  0  0  0  0  1  0  1  0  1 | 0 1  0  1 1
.. .. ..    .x     |   0   2 |  *  *  *   *  *  * 96 |  0  0  0  0  0  1  0  1  1 | 0 0  1  1 1
-------------------+---------+-----------------------+----------------------------+------------
.. x.3x.    ..     |   6   0 |  3  3  0   0  0  0  0 | 32  *  *  *  *  *  *  *  * | 1 1  0  0 0
.. x. .. *b3x.     |   6   0 |  3  0  3   0  0  0  0 |  * 32  *  *  *  *  *  *  * | 1 0  0  1 0
.. .. x.    x.     |   4   0 |  0  2  2   0  0  0  0 |  *  * 48  *  *  *  *  *  * | 1 0  1  0 0
wx .. xw    ..&#zx |   4   4 |  0  2  0   4  2  0  0 |  *  *  * 48  *  *  *  *  * | 0 1  1  0 0
.. xx ..    ..&#x  |   2   2 |  1  0  0   2  0  1  0 |  *  *  *  * 96  *  *  *  * | 0 1  0  1 0
.. .. ..    xx&#x  |   2   2 |  0  0  1   2  0  0  1 |  *  *  *  *  * 96  *  *  * | 0 0  1  1 0
.x3.x ..    ..     |   0   6 |  0  0  0   0  3  3  0 |  *  *  *  *  *  * 32  *  * | 0 1  0  0 1
.x .. ..    .x     |   0   4 |  0  0  0   0  2  0  2 |  *  *  *  *  *  *  * 48  * | 0 0  1  0 1
.. .x .. *b3.x     |   0   6 |  0  0  0   0  0  3  3 |  *  *  *  *  *  *  *  * 32 | 0 0  0  1 1
-------------------+---------+-----------------------+----------------------------+------------
.. x.3x. *b3x.     ♦  24   0 | 12 12 12   0  0  0  0 |  4  4  6  0  0  0  0  0  0 | 8 *  *  * *
wx3xx3xw    ..&#zx ♦  24  24 | 12 12  0  24 12 12  0 |  4  0  0  6 12  0  4  0  0 | * 8  *  * *
wx .. xw    xx&#zx ♦   8   8 |  0  4  4   8  4  0  4 |  0  0  2  2  0  4  0  2  0 | * * 24  * *
.. xx .. *b3xx&#x  ♦   6   6 |  3  0  3   6  0  3  3 |  0  1  0  0  3  3  0  0  1 | * *  * 32 *
.x3.x .. *b3.x     ♦   0  24 |  0  0  0   0 12 12 12 |  0  0  0  0  0  0  4  6  4 | * *  *  * 8