Acronym	duspid
Name	small-prismated-decachoron dual, triangular antitegmatic icosachoron, crystalic icosachoron
	©
Inradius	sqrt(5/8) = 0.790569
Dual	spid
Face vector	30, 70, 60, 20
Confer	general polytopal classes: Catalan polychora
External links

This polychoron can be obtained as the convex hull of 2 mutually dual pens and 2 mutually inverted raps, all of the same (pseudo) edge length a. The edges of those raps thereby become the long diagonals a = rr of the rhombs {(r,R)²}. Only the 3 consecutive lacings (all of size x) remain as true edges.

The corner angles of those rhombs are r = arccos(1/4) = 75.522488° resp. R = arccos(-1/4) = 104.477512°. The rhomb diagonals thus are rr = a = sqrt(5/2) = 1.581139 resp. RR = sqrt(3/2) = 1.224745.

Each cell is an according oblate rhombohedron. In fact, the vertices of those cells could be axially 4-colored, thereby resulting in the tegum sum description of the full polychoron.

Incidence matrix according to Dynkin symbol

m3o3o3m =
aooo3oaoo3ooao3oooa&#zxt   → height = 0
                             a = sqrt(5/2) = 1.581139

o...3o...3o...3o...     | 5  *  * * ♦  4  0  0 |  6  0 |  4
.o..3.o..3.o..3.o..     | * 10  * * ♦  2  3  0 |  6  3 |  6
..o.3..o.3..o.3..o.     | *  * 10 * ♦  0  3  2 |  3  6 |  6
...o3...o3...o3...o     | *  *  * 5 ♦  0  0  4 |  0  6 |  4
------------------------+-----------+----------+-------+---
oo..3oo..3oo..3oo..&#x  | 1  1  0 0 | 20  *  * |  3  0 |  3
.oo.3.oo.3.oo.3.oo.&#x  | 0  1  1 0 |  * 30  * |  2  2 |  4
..oo3..oo3..oo3..oo&#x  | 0  0  1 1 |  *  * 20 |  0  3 |  3
------------------------+-----------+----------+-------+---
.... oao. .... ....&#xt | 1  2  1 0 |  2  2  0 | 30  * |  2  {(r,R)2}
.... .... .oao ....&#xt | 0  1  2 1 |  0  2  2 |  * 30 |  2  {(r,R)2}
------------------------+-----------+----------+-------+---
.... oaoo3ooao ....&#xt | 1  3  3 1 |  3  6  3 |  3  3 | 20

or
o...3o...3o...3o...     & | 10  * ♦  4  0 |  6 |  4
.o..3.o..3.o..3.o..     & |  * 20 ♦  2  3 |  9 |  6
--------------------------+-------+-------+----+---
oo..3oo..3oo..3oo..&#x  & |  1  1 | 40  * |  3 |  3
.oo.3.oo.3.oo.3.oo.&#x    |  0  2 |  * 30 |  4 |  4
--------------------------+-------+-------+----+---
.... oao. .... ....&#xt & |  1  3 |  2  2 | 60 |  2  {(r,R)2}
--------------------------+-------+-------+----+---
.... oaoo3ooao ....&#xt   |  2  6 |  6  6 |  6 | 20