Prismatorhombated hexacosichoron Rank 4 Type Uniform Space Spherical Notation Bowers style acronym Prix Coxeter diagram x5x3o3x ( Elements Cells 1200 triangular prisms , 720 decagonal prisms , 600 cuboctahedra , 120 truncated dodecahedra Faces 2400+2400 triangles , 3600+3600 squares , 1440 decagons Edges 3600+7200+7200 Vertices 7200 Vertex figure Skewed rectangular pyramid , base edge lengths 1, √2 , 1, √2 ; lateral edge lengths √2 , √2 , √(5+√5 )/2 , √(5+√5 )/2 Measures (edge length 1) Circumradius
48
+
21
5
≈
9.74461
{\displaystyle \sqrt{48+21\sqrt5} ≈ 9.74461}
Hypervolume
25
3367
+
1475
5
4
≈
41657.5017
{\displaystyle 25\frac{3367+1475\sqrt5}{4} ≈ 41657.5017}
Dichoral angles Co–3–trip:
arccos
(
−
6
+
30
8
)
≈
172.23876
∘
{\displaystyle \arccos\left(-\frac{\sqrt6+\sqrt{30}}{8}\right) ≈ 172.23876^\circ}
Dip–4–trip:
arccos
(
−
10
+
2
5
15
)
≈
169.18768
∘
{\displaystyle \arccos\left(-\sqrt{\frac{10+2\sqrt5}{15}}\right) ≈ 169.18768^\circ}
Co–4–dip:
arccos
(
−
5
+
2
5
10
)
≈
166.71747
∘
{\displaystyle \arccos\left(-\sqrt{\frac{5+2\sqrt5}{10}}\right) ≈ 166.71747^\circ}
Tid–10–dip: 162° Tid–3–co:
arccos
(
−
7
+
3
5
4
)
≈
157.76124
∘
{\displaystyle \arccos\left(-\frac{\sqrt{7+3\sqrt5}}{4}\right) ≈ 157.76124^\circ}
Central density 1 Number of external pieces 2640 Level of complexity 16 Related polytopes Army Prix Regiment Prix Dual Rhombipyramidal heptachiliadiacosichoron Conjugate Quasiprismatorhombated grand hexacosichoron Abstract & topological properties Flag count230400 Euler characteristic 0 Orientable Yes Properties Symmetry H4 , order 14400Convex Yes Nature Tame
The prismatorhombated hexacosichoron , or prix , also commonly called the runcitruncated 120-cell , is a convex uniform polychoron that consists of 1200 triangular prisms , 720 decagonal prisms , 600 cuboctahedra , and 120 truncated dodecahedra . 1 triangular prism, 2 decagonal prisms, 1 cuboctahedron, and 1 truncated dodecahedron join at each vertex. As one of its names suggests, it can be obtained by runcitruncating the hecatonicosachoron .
The vertices of a prismatorhombated hexacosichoron of edge length 1 are given by all permutations of:
(
0
,
±
1
,
±
7
+
3
5
2
,
±
7
+
3
5
2
)
,
{\displaystyle \left(0,\,±1,\,±\frac{7+3\sqrt5}{2},\,±\frac{7+3\sqrt5}{2}\right),}
(
±
1
2
,
±
1
2
,
±
3
2
+
5
2
,
±
8
+
3
5
2
)
,
{\displaystyle \left(±\frac12,\,±\frac12,\,±3\frac{2+\sqrt5}{2},\,±\frac{8+3\sqrt5}{2}\right),}
(
±
1
2
,
±
1
2
,
±
3
+
2
5
2
,
±
9
+
4
5
2
)
,
{\displaystyle \left(±\frac12,\,±\frac12,\,±\frac{3+2\sqrt5}{2},\,±\frac{9+4\sqrt5}{2}\right),}
(
±
2
+
5
2
,
±
5
+
2
5
2
,
±
3
2
+
5
2
,
±
3
2
+
5
2
)
,
{\displaystyle \left(±\frac{2+\sqrt5}{2},\,±\frac{5+2\sqrt5}{2},\,±3\frac{2+\sqrt5}{2},\,±3\frac{2+\sqrt5}{2}\right),}
(
±
3
+
5
2
,
±
3
+
5
2
,
±
5
+
3
5
2
,
±
7
+
3
5
2
)
,
{\displaystyle \left(±\frac{3+\sqrt5}{2},\,±\frac{3+\sqrt5}{2},\,±\frac{5+3\sqrt5}{2},\,±\frac{7+3\sqrt5}{2}\right),}
Plus all even permutations of:
(
0
,
±
1
2
,
±
17
+
9
5
4
,
±
5
+
3
5
4
)
,
{\displaystyle \left(0,\,±\frac12,\,±\frac{17+9\sqrt5}{4},\,±\frac{5+3\sqrt5}{4}\right),}
(
0
,
±
5
+
5
4
,
±
13
+
7
5
4
,
±
3
2
+
5
2
)
,
{\displaystyle \left(0,\,±\frac{5+\sqrt5}{4},\,±\frac{13+7\sqrt5}{4},\,±3\frac{2+\sqrt5}{2}\right),}
(
0
,
±
2
+
5
2
,
±
19
+
7
5
4
,
±
3
3
+
5
4
)
,
{\displaystyle \left(0,\,±\frac{2+\sqrt5}{2},\,±\frac{19+7\sqrt5}{4},\,±3\frac{3+\sqrt5}{4}\right),}
(
0
,
±
3
1
+
5
4
,
±
17
+
7
5
4
,
±
5
+
2
5
2
)
,
{\displaystyle \left(0,\,±3\frac{1+\sqrt5}{4},\,±\frac{17+7\sqrt5}{4},\,±\frac{5+2\sqrt5}{2}\right),}
(
±
1
2
,
±
(
3
+
5
)
,
±
11
+
5
5
4
,
±
13
+
5
5
4
)
,
{\displaystyle \left(±\frac12,\,±(3+\sqrt5),\,±\frac{11+5\sqrt5}{4},\,±\frac{13+5\sqrt5}{4}\right),}
(
±
1
2
,
±
3
+
5
4
,
±
15
+
7
5
4
,
±
5
+
3
5
2
)
,
{\displaystyle \left(±\frac12,\,±\frac{3+\sqrt5}{4},\,±\frac{15+7\sqrt5}{4},\,±\frac{5+3\sqrt5}{2}\right),}
(
±
1
2
,
±
3
+
5
4
,
±
19
+
7
5
4
,
±
(
2
+
5
)
)
,
{\displaystyle \left(±\frac12,\,±\frac{3+\sqrt5}{4},\,±\frac{19+7\sqrt5}{4},\,±(2+\sqrt5)\right),}
(
±
1
2
,
±
3
+
5
4
,
±
17
+
9
5
4
,
±
3
+
5
2
)
,
{\displaystyle \left(±\frac12,\,±\frac{3+\sqrt5}{4},\,±\frac{17+9\sqrt5}{4},\,±\frac{3+\sqrt5}{2}\right),}
(
±
1
2
,
±
1
+
5
2
,
±
17
+
7
5
4
,
±
9
+
5
5
4
)
,
{\displaystyle \left(±\frac12,\,±\frac{1+\sqrt5}{2},\,±\frac{17+7\sqrt5}{4},\,±\frac{9+5\sqrt5}{4}\right),}
(
±
1
2
,
±
2
+
5
2
,
±
9
+
4
5
2
,
±
5
+
2
5
2
)
,
{\displaystyle \left(±\frac12,\,±\frac{2+\sqrt5}{2},\,±\frac{9+4\sqrt5}{2},\,±\frac{5+2\sqrt5}{2}\right),}
(
±
1
2
,
±
3
+
5
2
,
±
13
+
7
5
4
,
±
13
+
5
5
4
)
,
{\displaystyle \left(±\frac12,\,±\frac{3+\sqrt5}{2},\,±\frac{13+7\sqrt5}{4},\,±\frac{13+5\sqrt5}{4}\right),}
(
±
1
2
,
±
5
+
3
5
4
,
±
15
+
7
5
4
,
±
(
3
+
5
)
)
,
{\displaystyle \left(±\frac12,\,±\frac{5+3\sqrt5}{4},\,±\frac{15+7\sqrt5}{4},\,±(3+\sqrt5)\right),}
(
±
1
,
±
3
+
5
4
,
±
9
+
4
5
2
,
±
7
+
3
5
4
)
,
{\displaystyle \left(±1,\,±\frac{3+\sqrt5}{4},\,±\frac{9+4\sqrt5}{2},\,±\frac{7+3\sqrt5}{4}\right),}
(
±
1
,
±
2
+
5
2
,
±
15
+
7
5
4
,
±
11
+
5
5
4
)
,
{\displaystyle \left(±1,\,±\frac{2+\sqrt5}{2},\,±\frac{15+7\sqrt5}{4},\,±\frac{11+5\sqrt5}{4}\right),}
(
±
3
+
5
4
,
±
9
+
5
5
4
,
±
3
2
+
5
2
,
±
(
3
+
5
)
)
,
{\displaystyle \left(±\frac{3+\sqrt5}{4},\,±\frac{9+5\sqrt5}{4},\,±3\frac{2+\sqrt5}{2},\,±(3+\sqrt5)\right),}
(
±
3
+
5
4
,
±
1
+
5
2
,
±
17
+
9
5
4
,
±
2
+
5
2
)
,
{\displaystyle \left(±\frac{3+\sqrt5}{4},\,±\frac{1+\sqrt5}{2},\,±\frac{17+9\sqrt5}{4},\,±\frac{2+\sqrt5}{2}\right),}
(
±
3
+
5
4
,
±
3
1
+
5
4
,
±
9
+
4
5
2
,
±
3
+
5
2
)
,
{\displaystyle \left(±\frac{3+\sqrt5}{4},\,±3\frac{1+\sqrt5}{4},\,±\frac{9+4\sqrt5}{2},\,±\frac{3+\sqrt5}{2}\right),}
(
±
3
+
5
4
,
±
5
+
2
5
2
,
±
7
+
3
5
2
,
±
13
+
5
5
4
)
,
{\displaystyle \left(±\frac{3+\sqrt5}{4},\,±\frac{5+2\sqrt5}{2},\,±\frac{7+3\sqrt5}{2},\,±\frac{13+5\sqrt5}{4}\right),}
(
±
3
+
5
4
,
±
7
+
3
5
4
,
±
8
+
3
5
2
,
±
(
3
+
5
)
)
,
{\displaystyle \left(±\frac{3+\sqrt5}{4},\,±\frac{7+3\sqrt5}{4},\,±\frac{8+3\sqrt5}{2},\,±(3+\sqrt5)\right),}
(
±
3
+
5
4
,
±
5
+
2
5
2
,
±
5
+
3
5
2
,
±
13
+
5
5
4
)
,
{\displaystyle \left(±\frac{3+\sqrt5}{4},\,±\frac{5+2\sqrt5}{2},\,±\frac{5+3\sqrt5}{2},\,±\frac{13+5\sqrt5}{4}\right),}
(
±
1
+
5
2
,
±
5
+
5
4
,
±
5
+
3
5
4
,
±
9
+
4
5
2
)
,
{\displaystyle \left(±\frac{1+\sqrt5}{2},\,±\frac{5+\sqrt5}{4},\,±\frac{5+3\sqrt5}{4},\,±\frac{9+4\sqrt5}{2}\right),}
(
±
1
+
5
2
,
±
3
3
+
5
4
,
±
3
2
+
5
2
,
±
13
+
5
5
4
)
,
{\displaystyle \left(±\frac{1+\sqrt5}{2},\,±3\frac{3+\sqrt5}{4},\,±3\frac{2+\sqrt5}{2},\,±\frac{13+5\sqrt5}{4}\right),}
(
±
1
+
5
2
,
±
(
2
+
5
)
,
±
7
+
3
5
2
,
±
(
3
+
5
)
)
,
{\displaystyle \left(±\frac{1+\sqrt5}{2},\,±(2+\sqrt5),\,±\frac{7+3\sqrt5}{2},\,±(3+\sqrt5)\right),}
(
±
5
+
5
4
,
±
3
+
5
2
,
±
8
+
3
5
2
,
±
11
+
5
5
4
)
,
{\displaystyle \left(±\frac{5+\sqrt5}{4},\,±\frac{3+\sqrt5}{2},\,±\frac{8+3\sqrt5}{2},\,±\frac{11+5\sqrt5}{4}\right),}
(
±
2
+
5
2
,
±
9
+
5
5
4
,
±
5
+
3
5
2
,
±
11
+
5
5
4
)
,
{\displaystyle \left(±\frac{2+\sqrt5}{2},\,±\frac{9+5\sqrt5}{4},\,±\frac{5+3\sqrt5}{2},\,±\frac{11+5\sqrt5}{4}\right),}
(
±
2
+
5
2
,
±
3
+
5
2
,
±
5
+
3
5
4
,
±
19
+
7
5
4
)
,
{\displaystyle \left(±\frac{2+\sqrt5}{2},\,±\frac{3+\sqrt5}{2},\,±\frac{5+3\sqrt5}{4},\,±\frac{19+7\sqrt5}{4}\right),}
(
±
2
+
5
2
,
±
3
+
2
5
2
,
±
8
+
3
5
2
,
±
5
+
2
5
2
)
,
{\displaystyle \left(±\frac{2+\sqrt5}{2},\,±\frac{3+2\sqrt5}{2},\,±\frac{8+3\sqrt5}{2},\,±\frac{5+2\sqrt5}{2}\right),}
(
±
3
1
+
5
4
,
±
(
2
+
5
)
,
±
3
2
+
5
2
,
±
11
+
5
5
4
)
,
{\displaystyle \left(±3\frac{1+\sqrt5}{4},\,±(2+\sqrt5),\,±3\frac{2+\sqrt5}{2},\,±\frac{11+5\sqrt5}{4}\right),}
(
±
3
+
5
2
,
±
5
+
3
5
4
,
±
8
+
3
5
2
,
±
9
+
5
5
4
)
,
{\displaystyle \left(±\frac{3+\sqrt5}{2},\,±\frac{5+3\sqrt5}{4},\,±\frac{8+3\sqrt5}{2},\,±\frac{9+5\sqrt5}{4}\right),}
(
±
3
+
5
2
,
±
5
+
2
5
2
,
±
7
+
3
5
4
,
±
17
+
7
5
4
)
,
{\displaystyle \left(±\frac{3+\sqrt5}{2},\,±\frac{5+2\sqrt5}{2},\,±\frac{7+3\sqrt5}{4},\,±\frac{17+7\sqrt5}{4}\right),}
(
±
3
+
5
2
,
±
3
+
2
5
2
,
±
15
+
7
5
4
,
±
3
3
+
5
4
)
,
{\displaystyle \left(±\frac{3+\sqrt5}{2},\,±\frac{3+2\sqrt5}{2},\,±\frac{15+7\sqrt5}{4},\,±3\frac{3+\sqrt5}{4}\right),}
(
±
5
+
3
5
4
,
±
5
+
2
5
2
,
±
(
2
+
5
)
,
±
15
+
7
5
4
)
,
{\displaystyle \left(±\frac{5+3\sqrt5}{4},\,±\frac{5+2\sqrt5}{2},\,±(2+\sqrt5),\,±\frac{15+7\sqrt5}{4}\right),}
(
±
5
+
3
5
4
,
±
7
+
3
5
4
,
±
3
2
+
5
2
,
±
5
+
3
5
2
)
,
{\displaystyle \left(±\frac{5+3\sqrt5}{4},\,±\frac{7+3\sqrt5}{4},\,±3\frac{2+\sqrt5}{2},\,±\frac{5+3\sqrt5}{2}\right),}
(
±
5
+
3
5
4
,
±
3
+
2
5
2
,
±
7
+
3
5
2
,
±
9
+
5
5
4
)
,
{\displaystyle \left(±\frac{5+3\sqrt5}{4},\,±\frac{3+2\sqrt5}{2},\,±\frac{7+3\sqrt5}{2},\,±\frac{9+5\sqrt5}{4}\right),}
(
±
7
+
3
5
4
,
±
3
+
2
5
2
,
±
13
+
7
5
4
,
±
(
2
+
5
)
)
.
{\displaystyle \left(±\frac{7+3\sqrt5}{4},\,±\frac{3+2\sqrt5}{2},\,±\frac{13+7\sqrt5}{4},\,±(2+\sqrt5)\right).}
The prismatorhombated hexacosichoron has a semi-uniform variant of the form x5y3o3z that maintains its full symmetry. This variant uses 120 truncated dodecahedra of form x5y3o, 600 rhombitetratetrahedra of form y3o3z, 1200 triangular prisms of form x z3o, and 720 dipentagonal prisms of form z x5y as cells, with 3 edge lengths.
With edges of length a, b, and c (such that it forms a5b3o3c), its circumradius is given by
14
a
2
+
21
b
2
+
3
c
2
+
33
a
b
+
11
a
c
+
14
b
c
+
(
6
a
2
+
9
b
2
+
c
2
+
15
a
b
+
5
a
c
+
6
b
c
)
5
2
{\displaystyle \sqrt{\frac{14a^2+21b^2+3c^2+33ab+11ac+14bc+(6a^2+9b^2+c^2+15ab+5ac+6bc)\sqrt5}{2}}}
The prismatorhombated hexacosichoron is the colonel of a 3-member regiment that also includes the small prismatohexacosihecatonicosihecatonicosachoron and the small rhombiprismic hexacosihecatonicosachoron .
The segmentochoron truncated dodecahedron atop great rhombicosidodecahedron can be obtained as a cap of the prismatorhombated hexacosichoron.
