Castellated rhodoperihedral prism (EntityTopic, 15)
From Hi.gher. Space
The castellated rhodoperihedral prism is a bilbirothawroid discovered on February 4, 2014 by quickfur and confirmed by Klitzing.[1] Although it is not strictly a prism (as it's not the Cartesian product of a base shape with a digon) it is so named as it has similar structure to "real" prisms of uniform polyhedra.
Its cells are two rhodoperihedra, 24 pentagonal pyramids, 40 tetrahedra and 30 bilunabirotundae (bilbiro). The rhodoperihedra form the "ends" of the prism, and are orthogonal to the bilbiro, which form the "sides" of the prism.
Its faces are 40 "end" triangles (x3o), 60 "end" squares (x2x), 24 "end" pentagons (o5x), 120 "lunefacing side" triangles (xo&#x), 120 "rotundafacing side" triangles (xo&#x) and 60 "side" pentagons (ooooo3ooooo5ooooo&#xt).
It has 120 + 120 + 120 + 120 + 12 = 492 edges and 120 + 24 + 20 = 164 vertices.
Equations
 The hypervolumes of a castellated rhodoperihedral prism with side length l are given by:
total edge length = 492l
total surface area = (60 + 70√3 + 21√(25+10√5)) · l^{2}
surcell volume = ^{1}∕_{3} · (60φ^{2} + 197φ + 74 + 10√2) · l^{3}
bulk = Unknown
Projections
Centered on a rhodoperihedron:
Centered on a bilunabirotunda (J91):
Coordinates
The Cartesian coordinates of the castellated rhodoperihedral prism are:
Top and bottom x5o3x  Two φ^{2}scaled icosahedra  φscaled dodecahedron 




where φ=(1+√5)/2 is the Golden Ratio.
Incidence matrix
Dual: D4.3.1 dual
#  TXID  Va  Vb  Vc  Ea  Eb  Ec  Ed  Ee  3a  4a  5a  3b  3c  5b  C1a  C2a  C3a  C4a  Type  Name 

0  Va  = point  ;  
1  Vb  = point  ;  
2  Vc  = point  ;  
3  Ea  2  0  0  = digon  ;  
4  Eb  2  0  0  = digon  ;  
5  Ec  1  1  0  = digon  ;  
6  Ed  1  0  1  = digon  ;  
7  Ee  0  2  0  = digon  ;  
8  3a  3  0  0  3  0  0  0  0  = triangle  ;  
9  4a  4  0  0  2  2  0  0  0  = square  ;  
10  5a  5  0  0  0  5  0  0  0  = pentagon  ;  
11  3b  2  1  0  0  1  2  0  0  = triangle  ;  
12  3c  2  0  1  1  0  0  2  0  = triangle  ;  
13  5b  2  2  1  0  0  2  2  1  = pentagon  ;  
14  C1a  60  0  0  60  60  0  0  0  20  30  12  0  0  0  = rhodoperihedron  ;  
15  C2a  5  1  0  0  5  5  0  0  0  0  1  5  0  0  = pentagonal pyramid  ;  
16  C3a  3  0  1  3  0  0  3  0  1  0  0  0  3  0  = tetrahedron  ;  
17  C4a  8  4  2  4  4  8  8  2  0  2  0  4  4  4  = bilunabirotunda  ;  
18  H4.1a  120  24  20  120  120  120  120  12  40  60  24  120  120  60  2  24  40  30  = castellated rhodoperihedral prism  ; 
Usage as facets
This polytope does not currently appear as facets in any higherdimensional polytopes in the database.
Software models
Notable Tetrashapes  
Regular:  pyrochoron • aerochoron • geochoron • xylochoron • hydrochoron • cosmochoron 
Powertopes:  triangular octagoltriate • square octagoltriate • hexagonal octagoltriate • octagonal octagoltriate 
Circular:  glome • cubinder • duocylinder • spherinder • sphone • cylindrone • dicone • coninder 
Torii:  tiger • torisphere • spheritorus • torinder • ditorus 