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:
(
±
1
2
,
±
1
2
,
±
4
+
3
5
2
,
±
8
+
3
5
2
)
,
{\displaystyle \left(\pm\frac12,\,\pm\frac12,\,\pm\frac{4+3\sqrt5}{2},\,\pm\frac{8+3\sqrt5}{2}\right),}
(
±
1
2
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±
3
2
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±
3
2
+
5
2
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±
3
2
+
5
2
)
,
{\displaystyle \left(\pm\frac12,\,\pm\frac32,\,\pm3\frac{2+\sqrt5}{2},\,\pm3\frac{2+\sqrt5}{2}\right),}
(
±
1
,
±
1
,
±
5
+
3
5
2
,
±
7
+
3
5
2
)
,
{\displaystyle \left(\pm1,\,\pm1,\,\pm\frac{5+3\sqrt5}{2},\,\pm\frac{7+3\sqrt5}{2}\right),}
plus all even permutations of:
(
0
,
±
1
2
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±
3
1
+
5
4
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±
3
5
+
3
5
4
)
,
{\displaystyle \left(0,\,\pm\frac12,\,\pm3\frac{1+\sqrt5}{4},\,\pm3\frac{5+3\sqrt5}{4}\right),}
(
0
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1
2
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±
5
1
+
5
4
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±
17
+
7
5
4
)
,
{\displaystyle \left(0,\,\pm\frac12,\,\pm5\frac{1+\sqrt5}{4},\,\pm\frac{17+7\sqrt5}{4}\right),}
(
0
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±
1
,
±
(
1
+
5
)
,
±
2
(
2
+
5
)
)
,
{\displaystyle \left(0,\,\pm1,\,\pm(1+\sqrt5),\,\pm2(2+\sqrt5)\right),}
(
0
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11
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3
5
4
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11
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5
5
4
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7
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2
5
2
)
,
{\displaystyle \left(0,\,\pm\frac{11+3\sqrt5}{4},\,\pm\frac{11+5\sqrt5}{4},\,\pm\frac{7+2\sqrt5}{2}\right),}
(
0
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5
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2
5
2
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13
+
5
5
4
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±
13
+
3
5
4
)
,
{\displaystyle \left(0,\,\pm\frac{5+2\sqrt5}{2},\,\pm\frac{13+5\sqrt5}{4},\,\pm\frac{13+3\sqrt5}{4}\right),}
(
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1
2
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1
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5
4
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3
3
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5
2
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±
7
+
5
5
4
)
,
{\displaystyle \left(\pm\frac12,\,\pm\frac{1+\sqrt5}{4},\,\pm3\frac{3+\sqrt5}{2},\,\pm\frac{7+5\sqrt5}{4}\right),}
(
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1
2
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±
1
+
5
2
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±
3
5
+
3
5
4
,
±
5
+
5
4
)
,
{\displaystyle \left(\pm\frac12,\,\pm\frac{1+\sqrt5}{2},\,\pm3\frac{5+3\sqrt5}{4},\,\pm\frac{5+\sqrt5}{4}\right),}
(
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1
2
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±
7
+
5
4
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3
1
+
5
4
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2
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2
+
5
)
)
,
{\displaystyle \left(\pm\frac12,\,\pm\frac{7+\sqrt5}{4},\,\pm3\frac{1+\sqrt5}{4},\,\pm2(2+\sqrt5)\right),}
(
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1
2
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3
1
+
5
4
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3
3
+
5
2
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±
3
3
+
5
4
)
,
{\displaystyle \left(\pm\frac12,\,\pm3\frac{1+\sqrt5}{4},\,\pm3\frac{3+\sqrt5}{2},\,\pm3\frac{3+\sqrt5}{4}\right),}
(
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1
2
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±
4
+
5
2
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3
2
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5
2
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±
7
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2
5
2
)
,
{\displaystyle \left(\pm\frac12,\,\pm\frac{4+\sqrt5}{2},\,\pm3\frac{2+\sqrt5}{2},\,\pm\frac{7+2\sqrt5}{2}\right),}
(
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1
2
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7
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3
5
4
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±
7
+
3
5
2
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±
13
+
3
5
4
)
,
{\displaystyle \left(\pm\frac12,\,\pm\frac{7+3\sqrt5}{4},\,\pm\frac{7+3\sqrt5}{2},\,\pm\frac{13+3\sqrt5}{4}\right),}
(
±
1
+
5
4
,
±
1
,
±
2
+
5
2
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3
5
+
3
5
4
)
,
{\displaystyle \left(\pm\frac{1+\sqrt5}{4},\,\pm1,\,\pm\frac{2+\sqrt5}{2},\,\pm3\frac{5+3\sqrt5}{4}\right),}
(
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1
+
5
4
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3
2
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5
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3
5
4
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2
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2
+
5
)
)
,
{\displaystyle \left(\pm\frac{1+\sqrt5}{4},\,\pm\frac32,\,\pm\frac{5+3\sqrt5}{4},\,\pm2(2+\sqrt5)\right),}
(
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1
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5
4
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3
3
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5
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,
{\displaystyle \left(\pm\frac{1+\sqrt5}{4},\,\pm3\frac{3+\sqrt5}{4},\,\pm\frac{5+3\sqrt5}{2},\,\pm\frac{7+2\sqrt5}{2}\right),}
(
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1
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)
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{\displaystyle \left(\pm\frac{1+\sqrt5}{4},\,\pm(2+\sqrt5),\,\pm3\frac{2+\sqrt5}{2},\,\pm\frac{13+3\sqrt5}{4}\right),}
(
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{\displaystyle \left(\pm1,\,\pm\frac{1+\sqrt5}{2},\,\pm3\frac{3+\sqrt5}{2},\,\pm(2+\sqrt5)\right),}
(
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1
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7
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)
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{\displaystyle \left(\pm1,\,\pm\frac{7+\sqrt5}{4},\,\pm3\frac{2+\sqrt5}{2},\,\pm\frac{13+5\sqrt5}{4}\right),}
(
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1
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3
1
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5
4
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17
+
7
5
4
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4
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2
)
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{\displaystyle \left(\pm1,\,\pm3\frac{1+\sqrt5}{4},\,\pm\frac{17+7\sqrt5}{4},\,\pm\frac{4+\sqrt5}{2}\right),}
(
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1
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5
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3
5
4
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8
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5
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11
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3
5
4
)
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{\displaystyle \left(\pm1,\,\pm\frac{5+3\sqrt5}{4},\,\pm\frac{8+3\sqrt5}{2},\,\pm\frac{11+3\sqrt5}{4}\right),}
(
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3
2
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1
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5
2
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17
+
7
5
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7
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3
5
4
)
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{\displaystyle \left(\pm\frac32,\,\pm\frac{1+\sqrt5}{2},\,\pm\frac{17+7\sqrt5}{4},\,\pm\frac{7+3\sqrt5}{4}\right),}
(
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3
2
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4
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11
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)
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{\displaystyle \left(\pm\frac32,\,\pm\frac{5+\sqrt5}{4},\,\pm\frac{7+3\sqrt5}{2},\,\pm\frac{11+5\sqrt5}{4}\right),}
(
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3
2
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2
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)
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{\displaystyle \left(\pm\frac32,\,\pm\frac{2+\sqrt5}{2},\,\pm\frac{8+3\sqrt5}{2},\,\pm\frac{5+2\sqrt5}{2}\right),}
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{\displaystyle \left(\pm\frac{1+\sqrt5}{2},\,\pm3\frac{3+\sqrt5}{4},\,\pm\frac{4+3\sqrt5}{2},\,\pm\frac{13+5\sqrt5}{4}\right),}
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5
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2
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2
)
,
{\displaystyle \left(\pm\frac{1+\sqrt5}{2},\,\pm\frac{11+3\sqrt5}{4},\,\pm\frac{7+5\sqrt5}{4},\,\pm3\frac{2+\sqrt5}{2}\right),}
(
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5
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4
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+
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2
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2
)
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{\displaystyle \left(\pm\frac{5+\sqrt5}{4},\,\pm\frac{7+\sqrt5}{4},\,\pm\frac{5+3\sqrt5}{2},\,\pm3\frac{2+\sqrt5}{2}\right),}
(
±
5
+
5
4
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±
5
+
3
5
4
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3
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4
+
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2
)
,
{\displaystyle \left(\pm\frac{5+\sqrt5}{4},\,\pm\frac{5+3\sqrt5}{4},\,\pm3\frac{3+\sqrt5}{2},\,\pm\frac{4+\sqrt5}{2}\right),}
(
±
5
+
5
4
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±
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1
+
5
)
,
±
8
+
3
5
2
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±
3
3
+
5
4
)
,
{\displaystyle \left(\pm\frac{5+\sqrt5}{4},\,\pm(1+\sqrt5),\,\pm\frac{8+3\sqrt5}{2},\,\pm3\frac{3+\sqrt5}{4}\right),}
(
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2
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5
2
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4
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3
3
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2
)
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{\displaystyle \left(\pm\frac{2+\sqrt5}{2},\,\pm\frac{7+\sqrt5}{4},\,\pm\frac{7+3\sqrt5}{4},\,\pm3\frac{3+\sqrt5}{2}\right),}
(
±
2
+
5
2
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4
+
5
2
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±
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+
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5
2
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±
3
2
+
5
2
)
,
{\displaystyle \left(\pm\frac{2+\sqrt5}{2},\,\pm\frac{4+\sqrt5}{2},\,\pm\frac{4+3\sqrt5}{2},\,\pm3\frac{2+\sqrt5}{2}\right),}
(
±
2
+
5
2
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±
3
3
+
5
4
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±
5
1
+
5
4
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±
7
+
3
5
2
)
,
{\displaystyle \left(\pm\frac{2+\sqrt5}{2},\,\pm3\frac{3+\sqrt5}{4},\,\pm5\frac{1+\sqrt5}{4},\,\pm\frac{7+3\sqrt5}{2}\right),}
(
±
7
+
5
4
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±
3
1
+
5
4
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±
8
+
3
5
2
,
±
(
2
+
5
)
)
,
{\displaystyle \left(\pm\frac{7+\sqrt5}{4},\,\pm3\frac{1+\sqrt5}{4},\,\pm\frac{8+3\sqrt5}{2},\,\pm(2+\sqrt5)\right),}
(
±
3
1
+
5
4
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±
4
+
5
2
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±
7
+
5
5
4
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±
7
+
3
5
2
)
,
{\displaystyle \left(\pm3\frac{1+\sqrt5}{4},\,\pm\frac{4+\sqrt5}{2},\,\pm\frac{7+5\sqrt5}{4},\,\pm\frac{7+3\sqrt5}{2}\right),}
(
±
3
1
+
5
4
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±
(
2
+
5
)
,
±
4
+
3
5
2
,
±
11
+
5
5
4
)
,
{\displaystyle \left(\pm3\frac{1+\sqrt5}{4},\,\pm(2+\sqrt5),\,\pm\frac{4+3\sqrt5}{2},\,\pm\frac{11+5\sqrt5}{4}\right),}
(
±
3
1
+
5
4
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±
7
+
5
5
4
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±
5
+
3
5
2
,
±
5
+
2
5
2
)
,
{\displaystyle \left(\pm3\frac{1+\sqrt5}{4},\,\pm\frac{7+5\sqrt5}{4},\,\pm\frac{5+3\sqrt5}{2},\,\pm\frac{5+2\sqrt5}{2}\right),}
(
±
5
+
3
5
4
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±
7
+
3
5
4
,
±
4
+
3
5
2
,
±
5
+
3
5
2
)
,
{\displaystyle \left(\pm\frac{5+3\sqrt5}{4},\,\pm\frac{7+3\sqrt5}{4},\,\pm\frac{4+3\sqrt5}{2},\,\pm\frac{5+3\sqrt5}{2}\right),}
(
±
5
+
3
5
4
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±
5
1
+
5
4
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±
3
2
+
5
2
,
±
(
2
+
5
)
)
,
{\displaystyle \left(\pm\frac{5+3\sqrt5}{4},\,\pm5\frac{1+\sqrt5}{4},\,\pm3\frac{2+\sqrt5}{2},\,\pm(2+\sqrt5)\right),}
(
±
(
1
+
5
)
,
±
7
+
3
5
4
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±
7
+
5
5
4
,
±
3
2
+
5
2
)
.
{\displaystyle \left(\pm(1+\sqrt5),\,\pm\frac{7+3\sqrt5}{4},\,\pm\frac{7+5\sqrt5}{4},\,\pm3\frac{2+\sqrt5}{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)\sqrt5}{2}}}
