Acronym	quidpith (old: qippith)
Name	quasi(dis)prismatotesseractihexadecachoron
Cross sections	©
Circumradius	sqrt[(3-sqrt(2))/2] = 0.890446
Inradius wrt. tet	[2 sqrt(2)-1]/sqrt(8) = 0.646447
Inradius wrt. trip	-[3-sqrt(2)]/sqrt(12) = -0.457777
Inradius wrt. cube	(sqrt(2)-1)/2 = 0.207107
Coordinates	((sqrt(2)-1)/2, 1/2, 1/2, 1/2)   & all permutations, all changes of sign
Volume	[32 sqrt(2)-43]/6 = 0.375806
Surface	[96+4 sqrt(2)+24 sqrt(3)]/3 = 47.742025
General of army	tat
Colonel of regiment	gittith
Dihedral angles (at margins)	at {4} between cube and cube:   45° at {4} between cube and trip:   arccos(sqrt(2/3)) = 35.264390° at {3} between tet and trip:   30°
Face vector	64, 192, 208, 80
Confer	blends: gondip   gaquipadah   segmentochora: sistodip   general polytopal classes: Wythoffian polychora   analogs: quasiexpanded hypercube qeCn
External links

As abstract polytope quidpith is isomorphic to sidpith, thereby replacing inverted tet by prograde tet resp. a frustrum vertex figure by an antipodium one. – As such quidpith is a lieutenant.

Incidence matrix according to Dynkin symbol

x3o3o4/3x

. . .   . | 64 |  3  3 |  3  6  3 |  1  3  3 1
----------+----+-------+----------+-----------
x . .   . |  2 | 96  * |  2  2  0 |  1  2  1 0
. . .   x |  2 |  * 96 |  0  2  2 |  0  1  2 1
----------+----+-------+----------+-----------
x3o .   . |  3 |  3  0 | 64  *  * |  1  1  0 0
x . .   x |  4 |  2  2 |  * 96  * |  0  1  1 0
. . o4/3x |  4 |  0  4 |  *  * 48 |  0  0  1 1
----------+----+-------+----------+-----------
x3o3o   . ♦  4 |  6  0 |  4  0  0 | 16  *  * *
x3o .   x ♦  6 |  6  3 |  2  3  0 |  * 32  * *
x . o4/3x ♦  8 |  4  8 |  0  4  2 |  *  * 24 *
. o3o4/3x ♦  8 |  0 12 |  0  0  6 |  *  *  * 8

x3o3/2o4x

. .   . . | 64 |  3  3 |  3  6  3 |  1  3  3 1
----------+----+-------+----------+-----------
x .   . . |  2 | 96  * |  2  2  0 |  1  2  1 0
. .   . x |  2 |  * 96 |  0  2  2 |  0  1  2 1
----------+----+-------+----------+-----------
x3o   . . |  3 |  3  0 | 64  *  * |  1  1  0 0
x .   . x |  4 |  2  2 |  * 96  * |  0  1  1 0
. .   o4x |  4 |  0  4 |  *  * 48 |  0  0  1 1
----------+----+-------+----------+-----------
x3o3/2o . ♦  4 |  6  0 |  4  0  0 | 16  *  * *
x3o   . x ♦  6 |  6  3 |  2  3  0 |  * 32  * *
x .   o4x ♦  8 |  4  8 |  0  4  2 |  *  * 24 *
. o3/2o4x ♦  8 |  0 12 |  0  0  6 |  *  *  * 8

x3/2o3o4x

.   . . . | 64 |  3  3 |  3  6  3 |  1  3  3 1
----------+----+-------+----------+-----------
x   . . . |  2 | 96  * |  2  2  0 |  1  2  1 0
.   . . x |  2 |  * 96 |  0  2  2 |  0  1  2 1
----------+----+-------+----------+-----------
x3/2o . . |  3 |  3  0 | 64  *  * |  1  1  0 0
x   . . x |  4 |  2  2 |  * 96  * |  0  1  1 0
.   . o4x |  4 |  0  4 |  *  * 48 |  0  0  1 1
----------+----+-------+----------+-----------
x3/2o3o . ♦  4 |  6  0 |  4  0  0 | 16  *  * *
x3/2o . x ♦  6 |  6  3 |  2  3  0 |  * 32  * *
x   . o4x ♦  8 |  4  8 |  0  4  2 |  *  * 24 *
.   o3o4x ♦  8 |  0 12 |  0  0  6 |  *  *  * 8

x3/2o3/2o4/3x

.   .   .   . | 64 |  3  3 |  3  6  3 |  1  3  3 1
--------------+----+-------+----------+-----------
x   .   .   . |  2 | 96  * |  2  2  0 |  1  2  1 0
.   .   .   x |  2 |  * 96 |  0  2  2 |  0  1  2 1
--------------+----+-------+----------+-----------
x3/2o   .   . |  3 |  3  0 | 64  *  * |  1  1  0 0
x   .   .   x |  4 |  2  2 |  * 96  * |  0  1  1 0
.   .   o4/3x |  4 |  0  4 |  *  * 48 |  0  0  1 1
--------------+----+-------+----------+-----------
x3/2o3/2o   . ♦  4 |  6  0 |  4  0  0 | 16  *  * *
x3/2o   .   x ♦  6 |  6  3 |  2  3  0 |  * 32  * *
x   .   o4/3x ♦  8 |  4  8 |  0  4  2 |  *  * 24 *
.   o3/2o4/3x ♦  8 |  0 12 |  0  0  6 |  *  *  * 8

xo4/3xx ox4/3xx&#zx   → height = 0
(tegum sum of 2 interchanged sistodips.)

o.4/3o. o.4/3o.     & | 64 |  1  1  2  2 |  2  2  1  3  4 | 1 1  1  3  2
----------------------+----+-------------+----------------+-------------
x.   .. ..   ..     & |  2 | 32  *  *  * |  2  0  0  2  0 | 1 0  1  2  0
..   x. ..   ..     & |  2 |  * 32  *  * |  0  2  0  0  2 | 0 1  0  1  2
..   .. ..   x.     & |  2 |  *  * 64  * |  1  1  1  0  1 | 1 1  0  1  1
oo4/3oo oo4/3oo&#x    |  2 |  *  *  * 64 |  0  0  0  2  2 | 0 0  1  2  1
----------------------+----+-------------+----------------+-------------
x.   .. ..   x.     & |  4 |  2  0  2  0 | 32  *  *  *  * | 1 0  0  1  0
..   x. ..   x.     & |  4 |  0  2  2  0 |  * 32  *  *  * | 0 1  0  0  1
..   .. o.4/3x.     & |  4 |  0  0  4  0 |  *  * 16  *  * | 1 1  0  0  0
xo   .. ..   ..&#x  & |  3 |  1  0  0  2 |  *  *  * 64  * | 0 0  1  1  0
..   xx ..   ..&#x  & |  4 |  0  1  1  2 |  *  *  *  * 64 | 0 0  0  1  1
----------------------+----+-------------+----------------+-------------
x.   .. o.4/3x.     & ♦  8 |  4  0  8  0 |  4  0  2  0  0 | 8 *  *  *  *
..   x. o.4/3x.     & ♦  8 |  0  4  8  0 |  0  4  2  0  0 | * 8  *  *  *
xo   .. ox   ..&#x    ♦  4 |  2  0  0  4 |  0  0  0  4  0 | * * 16  *  *
xo   .. ..   xx&#x  & ♦  6 |  2  1  2  4 |  1  0  0  2  2 | * *  * 32  *
..   xx ..   xx&#x    ♦  8 |  0  4  4  4 |  0  2  0  0  4 | * *  *  * 16