Acronym cuba octu
Name cube atop (oct,co)-ursachoron,
(oct,co)-ursachoron based wedge
Circumradius sqrt[354+171 sqrt(2)+100 sqrt(5)+63 sqrt(10)]/16 = 1.994779
Lace city
in approx. ASCII-art 
       o3o4x         
x3o4o    f3o4o  o3x4o
Dihedral angles
  • at cube between cubaco and octacube:   arccos([-4+3 sqrt(2)-sqrt(5)]/sqrt(19-10 sqrt(2))) = 154.747391°
  • at oct between cubaco and octu:   arccos([3-sqrt(2)+sqrt(5)]/sqrt[-4+8 sqrt(2)-2 sqrt(5)+4 sqrt(10)]) = 13.821698°
  • at co between cubaco and octu:   arccos(sqrt[(4+sqrt(2)+sqrt(5)+sqrt(10))/(-6+8 sqrt(2)-3 sqrt(5)+4 sqrt(10))]) = 11.430911°
  • ...
Face vector 32, 150, 244, 159, 35
Confer
general polytopal classes:
segmentotera  

This ursachoron based wedge might look to exist more generally for any ofx3xooNooo&#xt || o3oNx. None the less, their heights calculate to be zero both for N=3 and N=5. (E.g., for N=5 one would get .xofo3.ooox5.oxoo&#xt, the mono-diminished vertex first rotunda of ex.)


Incidence matrix

ofx3xoo4ooo&#xt || o3o4x   height = sqrt[sqrt(2)-2 sqrt(5)+sqrt(10)]/2 = 0.161520
                           shear = sqrt[-12+8 sqrt(2)+6 sqrt(5)-4 sqrt(10)]/4 = 0.071154

12 * * * |  4  2  2 0  0  0  0  0 | 2 2  4  4  1  4  1  0  0 0  0  0 0 | 1 2 2 2 2  4  0  2  0 0 0 0  0 0 0 | 1 1 2 2  1 0 0
 * 6 * * |  0  4  0 1  4  0  0  0 | 0 0  4  0  4  8  0  4  4 0  0  0 0 | 0 1 4 0 0  4  8  4  4 1 0 0  0 0 0 | 1 0 1 4  4 1 0
 * * 6 * |  0  0  0 1  0  4  4  0 | 0 0  0  0  4  0  0  4  0 4  8  4 0 | 0 0 4 0 0  0  8  0  4 0 1 4  4 1 0 | 1 0 0 4  4 1 1
 * * * 8 |  0  0  3 0  3  0  3  3 | 0 0  0  3  0  6  3  3  6 0  3  6 3 | 0 0 0 3 1  3  3  6  6 3 0 1  3 3 1 | 0 1 3 1  3 3 1
---------+------------------------+------------------------------------+------------------------------------+---------------
 2 0 0 0 | 24  *  * *  *  *  *  * | 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
 1 1 0 0 |  * 24  * *  *  *  *  * | 0 0  2  0  1  2  0  0  0 0  0  0 0 | 0 1 2 0 0  2  2  1  0 0 0 0  0 0 0 | 1 0 1 2  1 0 0
 1 0 0 1 |  *  * 24 *  *  *  *  * | 0 0  0  2  0  2  1  0  0 0  0  0 0 | 0 0 0 2 1  2  1  2  0 0 0 0  0 0 0 | 0 1 2 1  1 0 0
 0 1 1 0 |  *  *  * 6  *  *  *  * | 0 0  0  0  4  0  0  4  0 0  0  0 0 | 0 0 4 0 0  0  8  0  4 0 0 0  0 0 0 | 1 0 0 4  4 1 0
 0 1 0 1 |  *  *  * * 24  *  *  * | 0 0  0  0  0  2  0  1  2 0  0  0 0 | 0 0 0 0 0  1  2  2  2 1 0 0  0 0 0 | 0 0 1 1  2 1 0
 0 0 2 0 |  *  *  * *  * 12  *  * | 0 0  0  0  1  0  0  0  0 2  2  0 0 | 0 0 2 0 0  0  2  0  0 0 1 2  1 0 0 | 1 0 0 2  1 0 1
 0 0 1 1 |  *  *  * *  *  * 24  * | 0 0  0  0  0  0  0  1  0 0  2  2 0 | 0 0 0 0 0  0  2  0  2 0 0 1  2 1 0 | 0 0 0 1  2 1 1
 0 0 0 2 |  *  *  * *  *  *  * 12 | 0 0  0  0  0  0  1  0  2 0  0  2 2 | 0 0 0 2 0  0  0  2  2 2 0 0  1 2 1 | 0 1 2 0  1 2 1
---------+------------------------+------------------------------------+------------------------------------+---------------
 4 0 0 0 |  4  0  0 0  0  0  0  0 | 6 *  *  *  *  *  *  *  * *  *  * * | 1 1 0 1 0  0  0  0  0 0 0 0  0 0 0 | 1 1 1 0  0 0 0
 3 0 0 0 |  3  0  0 0  0  0  0  0 | * 8  *  *  *  *  *  *  * *  *  * * | 1 0 1 0 1  0  0  0  0 0 0 0  0 0 0 | 1 1 0 1  0 0 0
 2 1 0 0 |  1  2  0 0  0  0  0  0 | * * 24  *  *  *  *  *  * *  *  * * | 0 1 1 0 0  1  0  0  0 0 0 0  0 0 0 | 1 0 1 1  0 0 0
 2 0 0 1 |  1  0  2 0  0  0  0  0 | * *  * 24  *  *  *  *  * *  *  * * | 0 0 0 1 1  1  0  0  0 0 0 0  0 0 0 | 0 1 1 1  0 0 0
 1 2 2 0 |  0  2  0 2  0  1  0  0 | * *  *  * 12  *  *  *  * *  *  * * | 0 0 2 0 0  0  2  0  0 0 0 0  0 0 0 | 1 0 0 2  1 0 0
 1 1 0 1 |  0  1  1 0  1  0  0  0 | * *  *  *  * 48  *  *  * *  *  * * | 0 0 0 0 0  1  1  1  0 0 0 0  0 0 0 | 0 0 1 1  1 0 0
 1 0 0 2 |  0  0  2 0  0  0  0  1 | * *  *  *  *  * 12  *  * *  *  * * | 0 0 0 2 0  0  0  2  0 0 0 0  0 0 0 | 0 1 2 0  1 0 0
 0 1 1 1 |  0  0  0 1  1  0  1  0 | * *  *  *  *  *  * 24  * *  *  * * | 0 0 0 0 0  0  2  0  2 0 0 0  0 0 0 | 0 0 0 1  2 1 0
 0 1 0 2 |  0  0  0 0  2  0  0  1 | * *  *  *  *  *  *  * 24 *  *  * * | 0 0 0 0 0  0  0  1  1 1 0 0  0 0 0 | 0 0 1 0  1 1 0
 0 0 3 0 |  0  0  0 0  0  3  0  0 | * *  *  *  *  *  *  *  * 8  *  * * | 0 0 1 0 0  0  0  0  0 0 1 1  0 0 0 | 1 0 0 1  0 0 1
 0 0 2 1 |  0  0  0 0  0  1  2  0 | * *  *  *  *  *  *  *  * * 24  * * | 0 0 0 0 0  0  1  0  0 0 0 1  1 0 0 | 0 0 0 1  1 0 1
 0 0 1 2 |  0  0  0 0  0  0  2  1 | * *  *  *  *  *  *  *  * *  * 24 * | 0 0 0 0 0  0  0  0  1 0 0 0  1 1 0 | 0 0 0 0  1 1 1
 0 0 0 4 |  0  0  0 0  0  0  0  4 | * *  *  *  *  *  *  *  * *  *  * 6 | 0 0 0 0 0  0  0  0  0 1 0 0  0 1 1 | 0 1 1 0  0 1 1
---------+------------------------+------------------------------------+------------------------------------+---------------
12 0 0 0 | 24  0  0 0  0  0  0  0 | 6 8  0  0  0  0  0  0  0 0  0  0 0 | 1 * * * *  *  *  *  * * * *  * * * | 1 1 0 0  0 0 0  co
 4 1 0 0 |  4  4  0 0  0  0  0  0 | 1 0  4  0  0  0  0  0  0 0  0  0 0 | * 6 * * *  *  *  *  * * * *  * * * | 1 0 1 0  0 0 0  squippy
 3 3 3 0 |  3  6  0 3  0  3  0  0 | 0 1  3  0  3  0  0  0  0 1  0  0 0 | * * 8 * *  *  *  *  * * * *  * * * | 1 0 0 1  0 0 0  teddi
 4 0 0 4 |  4  0  8 0  0  0  0  4 | 1 0  0  4  0  0  4  0  0 0  0  0 0 | * * * 6 *  *  *  *  * * * *  * * * | 0 1 1 0  0 0 0  squap
 3 0 0 1 |  3  0  3 0  0  0  0  0 | 0 1  0  3  0  0  0  0  0 0  0  0 0 | * * * * 8  *  *  *  * * * *  * * * | 0 1 0 1  0 0 0  tet
 2 1 0 1 |  1  2  2 0  1  0  0  0 | 0 0  1  1  0  2  0  0  0 0  0  0 0 | * * * * * 24  *  *  * * * *  * * * | 0 0 1 1  0 0 0  tet
 1 2 2 1 |  0  2  1 2  2  1  2  0 | 0 0  0  0  1  2  0  2  0 0  1  0 0 | * * * * *  * 24  *  * * * *  * * * | 0 0 0 1  1 0 0  peppy
 1 1 0 2 |  0  1  2 0  2  0  0  1 | 0 0  0  0  0  2  1  0  1 0  0  0 0 | * * * * *  *  * 24  * * * *  * * * | 0 0 1 0  1 0 0  tet
 0 1 1 2 |  0  0  0 1  2  0  2  1 | 0 0  0  0  0  0  0  2  1 0  0  1 0 | * * * * *  *  *  * 24 * * *  * * * | 0 0 0 0  1 1 0  tet
 0 1 0 4 |  0  0  0 0  4  0  0  4 | 0 0  0  0  0  0  0  0  4 0  0  0 1 | * * * * *  *  *  *  * 6 * *  * * * | 0 0 1 0  0 1 0  squippy
 0 0 6 0 |  0  0  0 0  0 12  0  0 | 0 0  0  0  0  0  0  0  0 8  0  0 0 | * * * * *  *  *  *  * * 1 *  * * * | 1 0 0 0  0 0 1  oct
 0 0 3 1 |  0  0  0 0  0  3  3  0 | 0 0  0  0  0  0  0  0  0 1  3  0 0 | * * * * *  *  *  *  * * * 8  * * * | 0 0 0 1  0 0 1  tet
 0 0 2 2 |  0  0  0 0  0  1  4  1 | 0 0  0  0  0  0  0  0  0 0  2  2 0 | * * * * *  *  *  *  * * * * 12 * * | 0 0 0 0  1 0 1  tet
 0 0 1 4 |  0  0  0 0  0  0  4  4 | 0 0  0  0  0  0  0  0  0 0  0  4 1 | * * * * *  *  *  *  * * * *  * 6 * | 0 0 0 0  0 1 1  squippy
 0 0 0 8 |  0  0  0 0  0  0  0 12 | 0 0  0  0  0  0  0  0  0 0  0  0 6 | * * * * *  *  *  *  * * * *  * * 1 | 0 1 0 0  0 0 1  cube
---------+------------------------+------------------------------------+------------------------------------+---------------
12 6 6 0 | 24 24  0 6  0 12  0  0 | 6 8 24  0 12  0  0  0  0 8  0  0 0 | 1 6 8 0 0  0  0  0  0 0 1 0  0 0 0 | 1 * * *  * * *  octu
12 0 0 8 | 24  0 24 0  0  0  0 12 | 6 8  0 24  0  0 12  0  0 0  0  0 6 | 1 0 0 6 8  0  0  0  0 0 0 0  0 0 1 | * 1 * *  * * *  cubaco
 4 1 0 4 |  4  4  8 0  4  0  0  4 | 1 0  4  4  0  8  4  0  4 0  0  0 1 | 0 1 0 1 0  4  0  4  0 1 0 0  0 0 0 | * * 6 *  * * *  squappy
 3 3 3 1 |  3  6  3 3  3  3  3  0 | 0 1  3  3  3  6  0  3  0 1  3  0 0 | 0 0 1 0 1  3  3  0  0 0 0 1  0 0 0 | * * * 8  * * *  teddipy
 1 2 2 2 |  0  2  2 2  4  1  4  1 | 0 0  0  0  1  4  1  4  2 0  2  2 0 | 0 0 0 0 0  0  2  2  2 0 0 0  1 0 0 | * * * * 12 * *  peppypy
 0 1 1 4 |  0  0  0 1  4  0  4  4 | 0 0  0  0  0  0  0  4  4 0  0  4 1 | 0 0 0 0 0  0  0  0  4 1 0 0  0 1 0 | * * * *  * 6 *  squasc
 0 0 6 8 |  0  0  0 0  0 12 24 12 | 0 0  0  0  0  0  0  0  0 8 24 24 6 | 0 0 0 0 0  0  0  0  0 0 1 8 12 6 1 | * * * *  * * 1  octacube

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