The great rhombated hecatonicosachoron , or grahi , also commonly called the cantitruncated 120-cell , is a convex uniform polychoron that consists of 1200 triangular prisms , 600 truncated tetrahedra , and 120 great rhombicosidodecahedra . 1 triangular prism, 1 truncated tetrahedron, and 2 great rhombicosidodecahedra join at each vertex. As one of its names suggests, it can be obtained by cantitruncating the hecatonicosachoron .
The vertices of a great rhombated hecatonicosachoron of edge length 1 are given by all permutations of:
(
±
1
2
,
±
1
2
,
±
3
+
2
5
2
,
±
5
2
+
5
2
)
,
{\displaystyle \left(±\frac12,\,±\frac12,\,±\frac{3+2\sqrt5}{2},\,±5\frac{2+\sqrt5}{2}\right),}
(
±
5
+
3
5
4
,
±
3
3
+
5
4
,
±
13
+
7
5
4
,
±
13
+
7
5
4
)
,
{\displaystyle \left(±\frac{5+3\sqrt5}{4},\,±3\frac{3+\sqrt5}{4},\,±\frac{13+7\sqrt5}{4},\,±\frac{13+7\sqrt5}{4}\right),}
(
±
7
+
3
5
4
,
±
7
+
3
5
4
,
±
11
+
7
5
4
,
±
15
+
7
5
4
)
,
{\displaystyle \left(±\frac{7+3\sqrt5}{4},\,±\frac{7+3\sqrt5}{4},\,±\frac{11+7\sqrt5}{4},\,±\frac{15+7\sqrt5}{4}\right),}
plus all even permutations of:
(
0
,
±
1
2
,
±
19
+
7
5
4
,
±
13
+
7
5
4
)
,
{\displaystyle \left(0,\,±\frac12,\,±\frac{19+7\sqrt5}{4},\,±\frac{13+7\sqrt5}{4}\right),}
(
0
,
±
1
2
,
±
3
7
+
3
5
4
,
±
7
+
5
5
4
)
,
{\displaystyle \left(0,\,±\frac12,\,±3\frac{7+3\sqrt5}{4},\,±\frac{7+5\sqrt5}{4}\right),}
(
0
,
±
3
+
5
4
,
±
9
+
4
5
2
,
±
11
+
7
5
4
)
,
{\displaystyle \left(0,\,±\frac{3+\sqrt5}{4},\,±\frac{9+4\sqrt5}{2},\,±\frac{11+7\sqrt5}{4}\right),}
(
0
,
±
3
+
5
4
,
±
11
+
4
5
2
,
±
9
+
5
5
4
)
,
{\displaystyle \left(0,\,±\frac{3+\sqrt5}{4},\,±\frac{11+4\sqrt5}{2},\,±\frac{9+5\sqrt5}{4}\right),}
(
0
,
±
1
+
5
2
,
±
(
5
+
2
5
)
,
±
5
+
3
5
2
)
,
{\displaystyle \left(0,\,±\frac{1+\sqrt5}{2},\,±(5+2\sqrt5),\,±\frac{5+3\sqrt5}{2}\right),}
(
±
1
2
,
±
13
+
5
5
4
,
±
7
+
3
5
2
,
±
5
3
+
5
4
)
,
{\displaystyle \left(±\frac12,\,±\frac{13+5\sqrt5}{4},\,±\frac{7+3\sqrt5}{2},\,±5\frac{3+\sqrt5}{4}\right),}
(
±
1
2
,
±
1
,
±
17
+
7
5
4
,
±
15
+
7
5
4
)
,
{\displaystyle \left(±\frac12,\,±1,\,±\frac{17+7\sqrt5}{4},\,±\frac{15+7\sqrt5}{4}\right),}
(
±
1
2
,
±
5
+
5
4
,
±
2
(
2
+
5
)
,
±
13
+
7
5
4
)
,
{\displaystyle \left(±\frac12,\,±\frac{5+\sqrt5}{4},\,±2(2+\sqrt5),\,±\frac{13+7\sqrt5}{4}\right),}
(
±
1
2
,
±
2
+
5
2
,
±
11
+
4
5
2
,
±
5
+
2
5
2
)
,
{\displaystyle \left(±\frac12,\,±\frac{2+\sqrt5}{2},\,±\frac{11+4\sqrt5}{2},\,±\frac{5+2\sqrt5}{2}\right),}
(
±
1
2
,
±
2
+
5
2
,
±
5
2
+
5
2
,
±
4
+
5
2
)
,
{\displaystyle \left(±\frac12,\,±\frac{2+\sqrt5}{2},\,±5\frac{2+\sqrt5}{2},\,±\frac{4+\sqrt5}{2}\right),}
(
±
1
2
,
±
3
1
+
5
4
,
±
(
5
+
2
5
)
,
±
11
+
5
5
4
)
,
{\displaystyle \left(±\frac12,\,±3\frac{1+\sqrt5}{4},\,±(5+2\sqrt5),\,±\frac{11+5\sqrt5}{4}\right),}
(
±
1
2
,
±
7
+
3
5
4
,
±
2
(
2
+
5
)
,
±
5
3
+
5
4
)
,
{\displaystyle \left(±\frac12,\,±\frac{7+3\sqrt5}{4},\,±2(2+\sqrt5),\,±5\frac{3+\sqrt5}{4}\right),}
(
±
1
,
±
3
+
5
4
,
±
5
2
+
5
2
,
±
7
+
3
5
4
)
,
{\displaystyle \left(±1,\,±\frac{3+\sqrt5}{4},\,±5\frac{2+\sqrt5}{2},\,±\frac{7+3\sqrt5}{4}\right),}
(
±
1
,
±
2
+
5
2
,
±
3
7
+
3
5
4
,
±
3
3
+
5
4
)
,
{\displaystyle \left(±1,\,±\frac{2+\sqrt5}{2},\,±3\frac{7+3\sqrt5}{4},\,±3\frac{3+\sqrt5}{4}\right),}
(
±
1
,
±
3
+
5
2
,
±
2
(
2
+
5
)
,
±
7
+
3
5
2
)
,
{\displaystyle \left(±1,\,±\frac{3+\sqrt5}{2},\,±2(2+\sqrt5),\,±\frac{7+3\sqrt5}{2}\right),}
(
±
1
,
±
5
+
3
5
4
,
±
9
+
4
5
2
,
±
13
+
5
5
4
)
,
{\displaystyle \left(±1,\,±\frac{5+3\sqrt5}{4},\,±\frac{9+4\sqrt5}{2},\,±\frac{13+5\sqrt5}{4}\right),}
(
±
3
+
5
4
,
±
11
+
5
5
4
,
±
13
+
7
5
4
,
±
5
3
+
5
4
)
,
{\displaystyle \left(±\frac{3+\sqrt5}{4},\,±\frac{11+5\sqrt5}{4},\,±\frac{13+7\sqrt5}{4},\,±5\frac{3+\sqrt5}{4}\right),}
(
±
3
+
5
4
,
±
3
1
+
5
4
,
±
5
2
+
5
2
,
±
3
+
5
2
)
,
{\displaystyle \left(±\frac{3+\sqrt5}{4},\,±3\frac{1+\sqrt5}{4},\,±5\frac{2+\sqrt5}{2},\,±\frac{3+\sqrt5}{2}\right),}
(
±
3
+
5
4
,
±
3
3
+
5
4
,
±
17
+
7
5
4
,
±
5
3
+
5
4
)
,
{\displaystyle \left(±\frac{3+\sqrt5}{4},\,±3\frac{3+\sqrt5}{4},\,±\frac{17+7\sqrt5}{4},\,±5\frac{3+\sqrt5}{4}\right),}
(
±
1
+
5
2
,
±
5
+
5
4
,
±
5
+
3
5
4
,
±
5
2
+
5
2
)
,
{\displaystyle \left(±\frac{1+\sqrt5}{2},\,±\frac{5+\sqrt5}{4},\,±\frac{5+3\sqrt5}{4},\,±5\frac{2+\sqrt5}{2}\right),}
(
±
1
+
5
2
,
±
5
+
2
5
2
,
±
15
+
7
5
4
,
±
5
3
+
5
4
)
,
{\displaystyle \left(±\frac{1+\sqrt5}{2},\,±\frac{5+2\sqrt5}{2},\,±\frac{15+7\sqrt5}{4},\,±5\frac{3+\sqrt5}{4}\right),}
(
±
5
+
5
4
,
±
3
1
+
5
4
,
±
3
7
+
3
5
4
,
±
7
+
3
5
4
)
,
{\displaystyle \left(±\frac{5+\sqrt5}{4},\,±3\frac{1+\sqrt5}{4},\,±3\frac{7+3\sqrt5}{4},\,±\frac{7+3\sqrt5}{4}\right),}
(
±
5
+
5
4
,
±
4
+
5
2
,
±
17
+
7
5
4
,
±
7
+
3
5
2
)
,
{\displaystyle \left(±\frac{5+\sqrt5}{4},\,±\frac{4+\sqrt5}{2},\,±\frac{17+7\sqrt5}{4},\,±\frac{7+3\sqrt5}{2}\right),}
(
±
5
+
5
4
,
±
7
+
3
5
4
,
±
19
+
7
5
4
,
±
13
+
5
5
4
)
,
{\displaystyle \left(±\frac{5+\sqrt5}{4},\,±\frac{7+3\sqrt5}{4},\,±\frac{19+7\sqrt5}{4},\,±\frac{13+5\sqrt5}{4}\right),}
(
±
2
+
5
2
,
±
11
+
5
5
4
,
±
11
+
7
5
4
,
±
7
+
3
5
2
)
,
{\displaystyle \left(±\frac{2+\sqrt5}{2},\,±\frac{11+5\sqrt5}{4},\,±\frac{11+7\sqrt5}{4},\,±\frac{7+3\sqrt5}{2}\right),}
(
±
2
+
5
2
,
±
5
+
3
5
2
,
±
13
+
7
5
4
,
±
13
+
5
5
4
)
,
{\displaystyle \left(±\frac{2+\sqrt5}{2},\,±\frac{5+3\sqrt5}{2},\,±\frac{13+7\sqrt5}{4},\,±\frac{13+5\sqrt5}{4}\right),}
(
±
3
1
+
5
4
,
±
9
+
5
5
4
,
±
15
+
7
5
4
,
±
13
+
5
5
4
)
,
{\displaystyle \left(±3\frac{1+\sqrt5}{4},\,±\frac{9+5\sqrt5}{4},\,±\frac{15+7\sqrt5}{4},\,±\frac{13+5\sqrt5}{4}\right),}
(
±
3
1
+
5
4
,
±
5
+
2
5
2
,
±
13
+
7
5
4
,
±
7
+
3
5
2
)
,
{\displaystyle \left(±3\frac{1+\sqrt5}{4},\,±\frac{5+2\sqrt5}{2},\,±\frac{13+7\sqrt5}{4},\,±\frac{7+3\sqrt5}{2}\right),}
(
±
3
+
5
2
,
±
5
+
3
5
4
,
±
11
+
4
5
2
,
±
7
+
3
5
4
)
,
{\displaystyle \left(±\frac{3+\sqrt5}{2},\,±\frac{5+3\sqrt5}{4},\,±\frac{11+4\sqrt5}{2},\,±\frac{7+3\sqrt5}{4}\right),}
(
±
3
+
5
2
,
±
4
+
5
2
,
±
15
+
7
5
4
,
±
13
+
7
5
4
)
,
{\displaystyle \left(±\frac{3+\sqrt5}{2},\,±\frac{4+\sqrt5}{2},\,±\frac{15+7\sqrt5}{4},\,±\frac{13+7\sqrt5}{4}\right),}
(
±
3
+
5
2
,
±
3
+
2
5
2
,
±
19
+
7
5
4
,
±
11
+
5
5
4
)
,
{\displaystyle \left(±\frac{3+\sqrt5}{2},\,±\frac{3+2\sqrt5}{2},\,±\frac{19+7\sqrt5}{4},\,±\frac{11+5\sqrt5}{4}\right),}
(
±
5
+
3
5
4
,
±
7
+
5
5
4
,
±
17
+
7
5
4
,
±
11
+
5
5
4
)
,
{\displaystyle \left(±\frac{5+3\sqrt5}{4},\,±\frac{7+5\sqrt5}{4},\,±\frac{17+7\sqrt5}{4},\,±\frac{11+5\sqrt5}{4}\right),}
(
±
4
+
5
2
,
±
7
+
3
5
4
,
±
(
5
+
2
5
)
,
±
3
3
+
5
4
)
,
{\displaystyle \left(±\frac{4+\sqrt5}{2},\,±\frac{7+3\sqrt5}{4},\,±(5+2\sqrt5),\,±3\frac{3+\sqrt5}{4}\right),}
(
±
4
+
5
2
,
±
3
+
2
5
2
,
±
9
+
4
5
2
,
±
5
+
2
5
2
)
,
{\displaystyle \left(±\frac{4+\sqrt5}{2},\,±\frac{3+2\sqrt5}{2},\,±\frac{9+4\sqrt5}{2},\,±\frac{5+2\sqrt5}{2}\right),}
(
±
7
+
3
5
4
,
±
3
+
2
5
2
,
±
17
+
7
5
4
,
±
5
+
3
5
2
)
,
{\displaystyle \left(±\frac{7+3\sqrt5}{4},\,±\frac{3+2\sqrt5}{2},\,±\frac{17+7\sqrt5}{4},\,±\frac{5+3\sqrt5}{2}\right),}
(
±
7
+
3
5
4
,
±
7
+
5
5
4
,
±
2
(
2
+
5
)
,
±
5
+
2
5
2
)
,
{\displaystyle \left(±\frac{7+3\sqrt5}{4},\,±\frac{7+5\sqrt5}{4},\,±2(2+\sqrt5),\,±\frac{5+2\sqrt5}{2}\right),}
(
±
3
+
2
5
2
,
±
3
3
+
5
4
,
±
9
+
5
5
4
,
±
2
(
2
+
5
)
)
.
{\displaystyle \left(±\frac{3+2\sqrt5}{2},\,±3\frac{3+\sqrt5}{4},\,±\frac{9+5\sqrt5}{4},\,±2(2+\sqrt5)\right).}
The great rhombated hecatonicosachoron has a semi-uniform variant of the form x5y3z3o that maintains its full symmetry. This variant uses 600 truncated tetrahedra of form y3z3o, 120 great rhombicosidodecahedra of form x5y3z, and 1200 triangular prisms of form x z3o as cells, with 3 edge lengths.
With edges of length a, b, and c (such that it forms a5b3c3o), its circumradius is given by
14
a
2
+
21
b
2
+
10
c
2
+
33
a
b
+
22
a
c
+
28
b
c
+
(
6
a
2
+
9
b
2
+
4
c
2
+
15
a
b
+
10
a
c
+
12
b
c
)
5
2
{\displaystyle \sqrt{\frac{14a^2+21b^2+10c^2+33ab+22ac+28bc+(6a^2+9b^2+4c^2+15ab+10ac+12bc)\sqrt5}{2}}}
)
