The great rhombated hexacosichoron , or grix , also commonly called the cantitruncated 600-cell , is a convex uniform polychoron that consists of 720 pentagonal prisms , 600 truncated octahedra , and 120 truncated icosahedra . 1 pentagonal prism, 1 truncated icosahedron, and 2 truncated octahedra join at each vertex. As one of its names suggests, it can be obtained by cantitruncating the hexacosichoron .
The vertices of a great rhombated hexacosichoron of edge length 1 are given by all permutations of:
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8
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3
5
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{\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {4+3{\sqrt {5}}}{2}},\,\pm {\frac {8+3{\sqrt {5}}}{2}}\right)}
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{\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {3}{2}},\,\pm 3{\frac {2+{\sqrt {5}}}{2}},\,\pm 3{\frac {2+{\sqrt {5}}}{2}}\right)}
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1
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1
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5
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7
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3
5
2
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{\displaystyle \left(\pm 1,\,\pm 1,\,\pm {\frac {5+3{\sqrt {5}}}{2}},\,\pm {\frac {7+3{\sqrt {5}}}{2}}\right)}
plus all even permutations of:
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1
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1
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3
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3
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{\displaystyle \left(0,\,\pm {\frac {1}{2}},\,\pm 3{\frac {1+{\sqrt {5}}}{4}},\,\pm 3{\frac {5+3{\sqrt {5}}}{4}}\right)}
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1
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5
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17
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7
5
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{\displaystyle \left(0,\,\pm {\frac {1}{2}},\,\pm 5{\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {17+7{\sqrt {5}}}{4}}\right)}
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1
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2
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{\displaystyle \left(0,\,\pm 1,\,\pm (1+{\sqrt {5}}),\,\pm 2(2+{\sqrt {5}})\right)}
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{\displaystyle \left(0,\,\pm {\frac {11+3{\sqrt {5}}}{4}},\,\pm {\frac {11+5{\sqrt {5}}}{4}},\,\pm {\frac {7+2{\sqrt {5}}}{2}}\right)}
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13
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{\displaystyle \left(0,\,\pm {\frac {5+2{\sqrt {5}}}{2}},\,\pm {\frac {13+5{\sqrt {5}}}{4}},\,\pm {\frac {13+3{\sqrt {5}}}{4}}\right)}
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{\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm 3{\frac {3+{\sqrt {5}}}{2}},\,\pm {\frac {7+5{\sqrt {5}}}{4}}\right)}
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1
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5
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3
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3
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4
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{\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm 3{\frac {5+3{\sqrt {5}}}{4}},\,\pm {\frac {5+{\sqrt {5}}}{4}}\right)}
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2
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{\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {7+{\sqrt {5}}}{4}},\,\pm 3{\frac {1+{\sqrt {5}}}{4}},\,\pm 2(2+{\sqrt {5}})\right)}
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3
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3
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{\displaystyle \left(\pm {\frac {1}{2}},\,\pm 3{\frac {1+{\sqrt {5}}}{4}},\,\pm 3{\frac {3+{\sqrt {5}}}{2}},\,\pm 3{\frac {3+{\sqrt {5}}}{4}}\right)}
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{\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {4+{\sqrt {5}}}{2}},\,\pm 3{\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {7+2{\sqrt {5}}}{2}}\right)}
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{\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {7+3{\sqrt {5}}}{4}},\,\pm {\frac {7+3{\sqrt {5}}}{2}},\,\pm {\frac {13+3{\sqrt {5}}}{4}}\right)}
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5
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{\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm 1,\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm 3{\frac {5+3{\sqrt {5}}}{4}}\right)}
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{\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {3}{2}},\,\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm 2(2+{\sqrt {5}})\right)}
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{\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm 3{\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {5+3{\sqrt {5}}}{2}},\,\pm {\frac {7+2{\sqrt {5}}}{2}}\right)}
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{\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm (2+{\sqrt {5}}),\,\pm 3{\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {13+3{\sqrt {5}}}{4}}\right)}
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{\displaystyle \left(\pm 1,\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm 3{\frac {3+{\sqrt {5}}}{2}},\,\pm (2+{\sqrt {5}})\right)}
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{\displaystyle \left(\pm 1,\,\pm {\frac {7+{\sqrt {5}}}{4}},\,\pm 3{\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {13+5{\sqrt {5}}}{4}}\right)}
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{\displaystyle \left(\pm 1,\,\pm 3{\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {17+7{\sqrt {5}}}{4}},\,\pm {\frac {4+{\sqrt {5}}}{2}}\right)}
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{\displaystyle \left(\pm 1,\,\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm {\frac {8+3{\sqrt {5}}}{2}},\,\pm {\frac {11+3{\sqrt {5}}}{4}}\right)}
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{\displaystyle \left(\pm {\frac {3}{2}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {17+7{\sqrt {5}}}{4}},\,\pm {\frac {7+3{\sqrt {5}}}{4}}\right)}
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{\displaystyle \left(\pm {\frac {3}{2}},\,\pm {\frac {5+{\sqrt {5}}}{4}},\,\pm {\frac {7+3{\sqrt {5}}}{2}},\,\pm {\frac {11+5{\sqrt {5}}}{4}}\right)}
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{\displaystyle \left(\pm {\frac {3}{2}},\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {8+3{\sqrt {5}}}{2}},\,\pm {\frac {5+2{\sqrt {5}}}{2}}\right)}
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{\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm 3{\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {4+3{\sqrt {5}}}{2}},\,\pm {\frac {13+5{\sqrt {5}}}{4}}\right)}
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{\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {11+3{\sqrt {5}}}{4}},\,\pm {\frac {7+5{\sqrt {5}}}{4}},\,\pm 3{\frac {2+{\sqrt {5}}}{2}}\right)}
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{\displaystyle \left(\pm {\frac {5+{\sqrt {5}}}{4}},\,\pm {\frac {7+{\sqrt {5}}}{4}},\,\pm {\frac {5+3{\sqrt {5}}}{2}},\,\pm 3{\frac {2+{\sqrt {5}}}{2}}\right)}
(
±
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{\displaystyle \left(\pm {\frac {5+{\sqrt {5}}}{4}},\,\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm 3{\frac {3+{\sqrt {5}}}{2}},\,\pm {\frac {4+{\sqrt {5}}}{2}}\right)}
(
±
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{\displaystyle \left(\pm {\frac {5+{\sqrt {5}}}{4}},\,\pm (1+{\sqrt {5}}),\,\pm {\frac {8+3{\sqrt {5}}}{2}},\,\pm 3{\frac {3+{\sqrt {5}}}{4}}\right)}
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{\displaystyle \left(\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {7+{\sqrt {5}}}{4}},\,\pm {\frac {7+3{\sqrt {5}}}{4}},\,\pm 3{\frac {3+{\sqrt {5}}}{2}}\right)}
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2
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{\displaystyle \left(\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {4+{\sqrt {5}}}{2}},\,\pm {\frac {4+3{\sqrt {5}}}{2}},\,\pm 3{\frac {2+{\sqrt {5}}}{2}}\right)}
(
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2
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{\displaystyle \left(\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm 3{\frac {3+{\sqrt {5}}}{4}},\,\pm 5{\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {7+3{\sqrt {5}}}{2}}\right)}
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±
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4
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{\displaystyle \left(\pm {\frac {7+{\sqrt {5}}}{4}},\,\pm 3{\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {8+3{\sqrt {5}}}{2}},\,\pm (2+{\sqrt {5}})\right)}
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{\displaystyle \left(\pm 3{\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {4+{\sqrt {5}}}{2}},\,\pm {\frac {7+5{\sqrt {5}}}{4}},\,\pm {\frac {7+3{\sqrt {5}}}{2}}\right)}
(
±
3
1
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4
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±
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2
+
5
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±
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5
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±
11
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5
5
4
)
{\displaystyle \left(\pm 3{\frac {1+{\sqrt {5}}}{4}},\,\pm (2+{\sqrt {5}}),\,\pm {\frac {4+3{\sqrt {5}}}{2}},\,\pm {\frac {11+5{\sqrt {5}}}{4}}\right)}
(
±
3
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5
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5
2
)
{\displaystyle \left(\pm 3{\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {7+5{\sqrt {5}}}{4}},\,\pm {\frac {5+3{\sqrt {5}}}{2}},\,\pm {\frac {5+2{\sqrt {5}}}{2}}\right)}
(
±
5
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3
5
4
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±
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5
4
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2
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{\displaystyle \left(\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm {\frac {7+3{\sqrt {5}}}{4}},\,\pm {\frac {4+3{\sqrt {5}}}{2}},\,\pm {\frac {5+3{\sqrt {5}}}{2}}\right)}
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5
)
)
{\displaystyle \left(\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm 5{\frac {1+{\sqrt {5}}}{4}},\,\pm 3{\frac {2+{\sqrt {5}}}{2}},\,\pm (2+{\sqrt {5}})\right)}
(
±
(
1
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5
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3
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2
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{\displaystyle \left(\pm (1+{\sqrt {5}}),\,\pm {\frac {7+3{\sqrt {5}}}{4}},\,\pm {\frac {7+5{\sqrt {5}}}{4}},\,\pm 3{\frac {2+{\sqrt {5}}}{2}}\right)}
The great rhombated hexacosichoron has a semi-uniform variant of the form o5z3y3x that maintains its full symmetry. This variant uses 120 truncated icosahedra of form o5y3x, 600 great rhombitetratetrahedra of form x3y3z, and 720 pentagonal prisms of form x z5o as cells, with 3 edge lengths.
With edges of length a, b, and c so that it forms o5c3b3a, its circumradius is given by
3
a
2
+
10
b
2
+
21
c
2
+
10
a
b
+
14
a
c
+
28
b
c
+
(
a
2
+
4
b
2
+
9
c
2
+
4
a
b
+
6
a
c
+
12
b
c
)
5
2
{\displaystyle {\sqrt {\frac {3a^{2}+10b^{2}+21c^{2}+10ab+14ac+28bc+(a^{2}+4b^{2}+9c^{2}+4ab+6ac+12bc){\sqrt {5}}}{2}}}}
Name OBSA Schläfli symbol CD diagram Image Hecatonicosachoron hi {5,3,3} Rectified hecatonicosachoron rahi r{5,3,3} Rectified hexacosichoron rox r{3,3,5} Hexacosichoron ex {3,3,5} Truncated hecatonicosachoron thi t{5,3,3} Small rhombated hecatonicosachoron srahi rr{5,3,3} Small disprismatohexacosihecatonicosachoron sidpixhi t0,3 {5,3,3} Hexacosihecatonicosachoron xhi 2t{5,3,3} Small rhombated hexacosichoron srix rr{3,3,5} Truncated hexacosichoron tex t{3,3,5} Great rhombated hecatonicosachoron grahi tr{5,3,3} Prismatorhombated hexacosichoron prix t0,1,3 {5,3,3} Prismatorhombated hecatonicosachoron prahi t0,2,3 {5,3,3} Great rhombated hexacosichoron grix tr{3,3,5} Great disprismatohexacosihecatonicosachoron gidpixhi t0,1,2,3 {5,3,3}
