Acronym	quarn
Name	quasirhombated penteract
Circumradius	[3-sqrt(2)]/2 = 0.792893
Inradius wrt. qrit	(sqrt(2)-1)/2 = 0.207107
Inradius wrt. rap	sqrt[(43-30 sqrt(2))/20] = 0.169351
Inradius wrt. tepe	sqrt[(17-12 sqrt(2))/8] = 0.0606602
Coordinates	(sqrt(2)-1, sqrt(2)-1, sqrt(2)-1, 1, 1)/2   & all permutations, all changes of sign
Volume	[1205-843 sqrt(2)]/30 = 0.427266
Colonel of regiment	wavinant
Dihedral angles (at margins)	at querco between qrit and qrit:   90° at oct between qrit and rap:   arccos(1/sqrt(5)) = 63.434949° at trip between qrit and tepe:   60° at tet between rap and tepe:   arccos[2/sqrt(5)] = 26.565051°
Face vector	320, 1280, 1520, 680, 122
Confer	general polytopal classes: Wythoffian polytera   analogs: quasirhombated hypercube qrbCn
External links

As abstract polytope quarn is isomorphic to sirn, thereby replacing querco by sirco and qrit by srit. – As such quarn is a lieutenant.

Incidence matrix according to Dynkin symbol

o3o3x3o4/3x

. . . .   . | 320 |   6   2 |   6   3   6  1 |   2   3   6  3 |  1  2  3
------------+-----+---------+----------------+----------------+---------
. . x .   . |   2 | 960   * |   2   1   1  0 |   1   2   2  1 |  1  1  2
. . . .   x |   2 |   * 320 |   0   0   3  1 |   0   0   3  3 |  0  1  3
------------+-----+---------+----------------+----------------+---------
. o3x .   . |   3 |   3   0 | 640   *   *  * |   1   1   1  0 |  1  1  1
. . x3o   . |   3 |   3   0 |   * 320   *  * |   0   2   0  1 |  1  0  2
. . x .   x |   4 |   2   2 |   *   * 480  * |   0   0   2  1 |  0  1  2
. . . o4/3x |   4 |   0   4 |   *   *   * 80 |   0   0   0  3 |  0  0  3
------------+-----+---------+----------------+----------------+---------
o3o3x .   . ♦   4 |   6   0 |   4   0   0  0 | 160   *   *  * |  1  1  0
. o3x3o   . ♦   6 |  12   0 |   4   4   0  0 |   * 160   *  * |  1  0  1
. o3x .   x ♦   6 |   6   3 |   2   0   3  0 |   *   * 320  * |  0  1  1
. . x3o4/3x ♦  24 |  24  24 |   0   8  12  6 |   *   *   * 40 |  0  0  2
------------+-----+---------+----------------+----------------+---------
o3o3x3o   . ♦  10 |  30   0 |  20  10   0  0 |   5   5   0  0 | 32  *  *
o3o3x .   x ♦   8 |  12   4 |   8   0   6  0 |   2   0   4  0 |  * 80  *
. o3x3o4/3x ♦  96 | 192  96 |  64  64  96 24 |   0  16  32  8 |  *  * 10

o3o3x3/2o4x

. . .   . . | 320 |   6   2 |   6   3   6  1 |   2   3   6  3 |  1  2  3
------------+-----+---------+----------------+----------------+---------
. . x   . . |   2 | 960   * |   2   1   1  0 |   1   2   2  1 |  1  1  2
. . .   . x |   2 |   * 320 |   0   0   3  1 |   0   0   3  3 |  0  1  3
------------+-----+---------+----------------+----------------+---------
. o3x   . . |   3 |   3   0 | 640   *   *  * |   1   1   1  0 |  1  1  1
. . x3/2o . |   3 |   3   0 |   * 320   *  * |   0   2   0  1 |  1  0  2
. . x   . x |   4 |   2   2 |   *   * 480  * |   0   0   2  1 |  0  1  2
. . .   o4x |   4 |   0   4 |   *   *   * 80 |   0   0   0  3 |  0  0  3
------------+-----+---------+----------------+----------------+---------
o3o3x   . . ♦   4 |   6   0 |   4   0   0  0 | 160   *   *  * |  1  1  0
. o3x3/2o . ♦   6 |  12   0 |   4   4   0  0 |   * 160   *  * |  1  0  1
. o3x   . x ♦   6 |   6   3 |   2   0   3  0 |   *   * 320  * |  0  1  1
. . x3/2o4x ♦  24 |  24  24 |   0   8  12  6 |   *   *   * 40 |  0  0  2
------------+-----+---------+----------------+----------------+---------
o3o3x3/2o . ♦  10 |  30   0 |  20  10   0  0 |   5   5   0  0 | 32  *  *
o3o3x   . x ♦   8 |  12   4 |   8   0   6  0 |   2   0   4  0 |  * 80  *
. o3x3/2o4x ♦  96 | 192  96 |  64  64  96 24 |   0  16  32  8 |  *  * 10

o3o3/2x3o4/3x

. .   . .   . | 320 |   6   2 |   6   3   6  1 |   2   3   6  3 |  1  2  3
--------------+-----+---------+----------------+----------------+---------
. .   x .   . |   2 | 960   * |   2   1   1  0 |   1   2   2  1 |  1  1  2
. .   . .   x |   2 |   * 320 |   0   0   3  1 |   0   0   3  3 |  0  1  3
--------------+-----+---------+----------------+----------------+---------
. o3/2x .   . |   3 |   3   0 | 640   *   *  * |   1   1   1  0 |  1  1  1
. .   x3o   . |   3 |   3   0 |   * 320   *  * |   0   2   0  1 |  1  0  2
. .   x .   x |   4 |   2   2 |   *   * 480  * |   0   0   2  1 |  0  1  2
. .   . o4/3x |   4 |   0   4 |   *   *   * 80 |   0   0   0  3 |  0  0  3
--------------+-----+---------+----------------+----------------+---------
o3o3/2x .   . ♦   4 |   6   0 |   4   0   0  0 | 160   *   *  * |  1  1  0
. o3/2x3o   . ♦   6 |  12   0 |   4   4   0  0 |   * 160   *  * |  1  0  1
. o3/2x .   x ♦   6 |   6   3 |   2   0   3  0 |   *   * 320  * |  0  1  1
. .   x3o4/3x ♦  24 |  24  24 |   0   8  12  6 |   *   *   * 40 |  0  0  2
--------------+-----+---------+----------------+----------------+---------
o3o3/2x3o   . ♦  10 |  30   0 |  20  10   0  0 |   5   5   0  0 | 32  *  *
o3o3/2x .   x ♦   8 |  12   4 |   8   0   6  0 |   2   0   4  0 |  * 80  *
. o3/2x3o4/3x ♦  96 | 192  96 |  64  64  96 24 |   0  16  32  8 |  *  * 10

o3o3/2x3/2o4x

. .   .   . . | 320 |   6   2 |   6   3   6  1 |   2   3   6  3 |  1  2  3
--------------+-----+---------+----------------+----------------+---------
. .   x   . . |   2 | 960   * |   2   1   1  0 |   1   2   2  1 |  1  1  2
. .   .   . x |   2 |   * 320 |   0   0   3  1 |   0   0   3  3 |  0  1  3
--------------+-----+---------+----------------+----------------+---------
. o3/2x   . . |   3 |   3   0 | 640   *   *  * |   1   1   1  0 |  1  1  1
. .   x3/2o . |   3 |   3   0 |   * 320   *  * |   0   2   0  1 |  1  0  2
. .   x   . x |   4 |   2   2 |   *   * 480  * |   0   0   2  1 |  0  1  2
. .   .   o4x |   4 |   0   4 |   *   *   * 80 |   0   0   0  3 |  0  0  3
--------------+-----+---------+----------------+----------------+---------
o3o3/2x   . . ♦   4 |   6   0 |   4   0   0  0 | 160   *   *  * |  1  1  0
. o3/2x3/2o . ♦   6 |  12   0 |   4   4   0  0 |   * 160   *  * |  1  0  1
. o3/2x   . x ♦   6 |   6   3 |   2   0   3  0 |   *   * 320  * |  0  1  1
. .   x3/2o4x ♦  24 |  24  24 |   0   8  12  6 |   *   *   * 40 |  0  0  2
--------------+-----+---------+----------------+----------------+---------
o3o3/2x3/2o . ♦  10 |  30   0 |  20  10   0  0 |   5   5   0  0 | 32  *  *
o3o3/2x   . x ♦   8 |  12   4 |   8   0   6  0 |   2   0   4  0 |  * 80  *
. o3/2x3/2o4x ♦  96 | 192  96 |  64  64  96 24 |   0  16  32  8 |  *  * 10

o3/2o3x3o4/3x

.   . . .   . | 320 |   6   2 |   6   3   6  1 |   2   3   6  3 |  1  2  3
--------------+-----+---------+----------------+----------------+---------
.   . x .   . |   2 | 960   * |   2   1   1  0 |   1   2   2  1 |  1  1  2
.   . . .   x |   2 |   * 320 |   0   0   3  1 |   0   0   3  3 |  0  1  3
--------------+-----+---------+----------------+----------------+---------
.   o3x .   . |   3 |   3   0 | 640   *   *  * |   1   1   1  0 |  1  1  1
.   . x3o   . |   3 |   3   0 |   * 320   *  * |   0   2   0  1 |  1  0  2
.   . x .   x |   4 |   2   2 |   *   * 480  * |   0   0   2  1 |  0  1  2
.   . . o4/3x |   4 |   0   4 |   *   *   * 80 |   0   0   0  3 |  0  0  3
--------------+-----+---------+----------------+----------------+---------
o3/2o3x .   . ♦   4 |   6   0 |   4   0   0  0 | 160   *   *  * |  1  1  0
.   o3x3o   . ♦   6 |  12   0 |   4   4   0  0 |   * 160   *  * |  1  0  1
.   o3x .   x ♦   6 |   6   3 |   2   0   3  0 |   *   * 320  * |  0  1  1
.   . x3o4/3x ♦  24 |  24  24 |   0   8  12  6 |   *   *   * 40 |  0  0  2
--------------+-----+---------+----------------+----------------+---------
o3/2o3x3o   . ♦  10 |  30   0 |  20  10   0  0 |   5   5   0  0 | 32  *  *
o3/2o3x .   x ♦   8 |  12   4 |   8   0   6  0 |   2   0   4  0 |  * 80  *
.   o3x3o4/3x ♦  96 | 192  96 |  64  64  96 24 |   0  16  32  8 |  *  * 10

o3/2o3x3/2o4x

.   . .   . . | 320 |   6   2 |   6   3   6  1 |   2   3   6  3 |  1  2  3
--------------+-----+---------+----------------+----------------+---------
.   . x   . . |   2 | 960   * |   2   1   1  0 |   1   2   2  1 |  1  1  2
.   . .   . x |   2 |   * 320 |   0   0   3  1 |   0   0   3  3 |  0  1  3
--------------+-----+---------+----------------+----------------+---------
.   o3x   . . |   3 |   3   0 | 640   *   *  * |   1   1   1  0 |  1  1  1
.   . x3/2o . |   3 |   3   0 |   * 320   *  * |   0   2   0  1 |  1  0  2
.   . x   . x |   4 |   2   2 |   *   * 480  * |   0   0   2  1 |  0  1  2
.   . .   o4x |   4 |   0   4 |   *   *   * 80 |   0   0   0  3 |  0  0  3
--------------+-----+---------+----------------+----------------+---------
o3/2o3x   . . ♦   4 |   6   0 |   4   0   0  0 | 160   *   *  * |  1  1  0
.   o3x3/2o . ♦   6 |  12   0 |   4   4   0  0 |   * 160   *  * |  1  0  1
.   o3x   . x ♦   6 |   6   3 |   2   0   3  0 |   *   * 320  * |  0  1  1
.   . x3/2o4x ♦  24 |  24  24 |   0   8  12  6 |   *   *   * 40 |  0  0  2
--------------+-----+---------+----------------+----------------+---------
o3/2o3x3/2o . ♦  10 |  30   0 |  20  10   0  0 |   5   5   0  0 | 32  *  *
o3/2o3x   . x ♦   8 |  12   4 |   8   0   6  0 |   2   0   4  0 |  * 80  *
.   o3x3/2o4x ♦  96 | 192  96 |  64  64  96 24 |   0  16  32  8 |  *  * 10

o3/2o3/2x3o4/3x

.   .   . .   . | 320 |   6   2 |   6   3   6  1 |   2   3   6  3 |  1  2  3
----------------+-----+---------+----------------+----------------+---------
.   .   x .   . |   2 | 960   * |   2   1   1  0 |   1   2   2  1 |  1  1  2
.   .   . .   x |   2 |   * 320 |   0   0   3  1 |   0   0   3  3 |  0  1  3
----------------+-----+---------+----------------+----------------+---------
.   o3/2x .   . |   3 |   3   0 | 640   *   *  * |   1   1   1  0 |  1  1  1
.   .   x3o   . |   3 |   3   0 |   * 320   *  * |   0   2   0  1 |  1  0  2
.   .   x .   x |   4 |   2   2 |   *   * 480  * |   0   0   2  1 |  0  1  2
.   .   . o4/3x |   4 |   0   4 |   *   *   * 80 |   0   0   0  3 |  0  0  3
----------------+-----+---------+----------------+----------------+---------
o3/2o3/2x .   . ♦   4 |   6   0 |   4   0   0  0 | 160   *   *  * |  1  1  0
.   o3/2x3o   . ♦   6 |  12   0 |   4   4   0  0 |   * 160   *  * |  1  0  1
.   o3/2x .   x ♦   6 |   6   3 |   2   0   3  0 |   *   * 320  * |  0  1  1
.   .   x3o4/3x ♦  24 |  24  24 |   0   8  12  6 |   *   *   * 40 |  0  0  2
----------------+-----+---------+----------------+----------------+---------
o3/2o3/2x3o   . ♦  10 |  30   0 |  20  10   0  0 |   5   5   0  0 | 32  *  *
o3/2o3/2x .   x ♦   8 |  12   4 |   8   0   6  0 |   2   0   4  0 |  * 80  *
.   o3/2x3o4/3x ♦  96 | 192  96 |  64  64  96 24 |   0  16  32  8 |  *  * 10

o3/2o3/2x3/2o4x

.   .   .   . . | 320 |   6   2 |   6   3   6  1 |   2   3   6  3 |  1  2  3
----------------+-----+---------+----------------+----------------+---------
.   .   x   . . |   2 | 960   * |   2   1   1  0 |   1   2   2  1 |  1  1  2
.   .   .   . x |   2 |   * 320 |   0   0   3  1 |   0   0   3  3 |  0  1  3
----------------+-----+---------+----------------+----------------+---------
.   o3/2x   . . |   3 |   3   0 | 640   *   *  * |   1   1   1  0 |  1  1  1
.   .   x3/2o . |   3 |   3   0 |   * 320   *  * |   0   2   0  1 |  1  0  2
.   .   x   . x |   4 |   2   2 |   *   * 480  * |   0   0   2  1 |  0  1  2
.   .   .   o4x |   4 |   0   4 |   *   *   * 80 |   0   0   0  3 |  0  0  3
----------------+-----+---------+----------------+----------------+---------
o3/2o3/2x   . . ♦   4 |   6   0 |   4   0   0  0 | 160   *   *  * |  1  1  0
.   o3/2x3/2o . ♦   6 |  12   0 |   4   4   0  0 |   * 160   *  * |  1  0  1
.   o3/2x   . x ♦   6 |   6   3 |   2   0   3  0 |   *   * 320  * |  0  1  1
.   .   x3/2o4x ♦  24 |  24  24 |   0   8  12  6 |   *   *   * 40 |  0  0  2
----------------+-----+---------+----------------+----------------+---------
o3/2o3/2x3/2o . ♦  10 |  30   0 |  20  10   0  0 |   5   5   0  0 | 32  *  *
o3/2o3/2x   . x ♦   8 |  12   4 |   8   0   6  0 |   2   0   4  0 |  * 80  *
.   o3/2x3/2o4x ♦  96 | 192  96 |  64  64  96 24 |   0  16  32  8 |  *  * 10