The sceptre or S* blending four pentagonal cupolae , three pentagonal antiprisms , and a dodecahedron .
It is meant to be excavated from a great rhombicosidodecahedron , thus its important feature is the height between its two decagonal faces. Other polyhedra of the same height can be used in its place, for example, a pentagonal rotunda can replace a pentagonal cupola and a pentagonal antiprism.
The vertex coordinates of a sceptre, with the decagonal face nearest the dodecahedron centered at the origin and with edge length 1, are:
(
±
1
2
,
±
5
+
2
5
2
,
0
)
{\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {5+2{\sqrt {5}}}}{2}},\,0\right)}
(
±
3
+
5
4
,
±
5
+
5
8
,
0
)
{\displaystyle \left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\sqrt {\frac {5+{\sqrt {5}}}{8}}},\,0\right)}
(
±
1
+
5
2
,
0
,
0
)
{\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{2}},\,0,\,0\right)}
(
±
1
2
,
−
5
+
2
5
20
,
5
−
5
10
)
{\displaystyle \left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,{\sqrt {\frac {5-{\sqrt {5}}}{10}}}\right)}
(
±
1
+
5
4
,
5
−
5
40
,
5
−
5
10
)
{\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,{\sqrt {\frac {5-{\sqrt {5}}}{10}}}\right)}
(
0
,
5
+
5
10
,
5
−
5
10
)
{\displaystyle \left(0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,{\sqrt {\frac {5-{\sqrt {5}}}{10}}}\right)}
(
±
1
2
,
5
+
2
5
20
,
5
+
2
5
5
)
{\displaystyle \left(\pm {\frac {1}{2}},\,{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,{\sqrt {\frac {5+2{\sqrt {5}}}{5}}}\right)}
(
±
1
+
5
4
,
−
5
−
5
40
,
5
+
2
5
5
)
{\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,-{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,{\sqrt {\frac {5+2{\sqrt {5}}}{5}}}\right)}
(
0
,
−
5
+
5
10
,
5
+
2
5
5
)
{\displaystyle \left(0,\,-{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,{\sqrt {\frac {5+2{\sqrt {5}}}{5}}}\right)}
(
±
1
+
5
4
,
25
+
11
5
40
,
25
+
11
5
10
)
{\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {25+11{\sqrt {5}}}{40}}},\,{\sqrt {\frac {25+11{\sqrt {5}}}{10}}}\right)}
(
±
3
+
5
4
,
−
5
+
5
40
,
25
+
11
5
10
)
{\displaystyle \left(\pm {\frac {3+{\sqrt {5}}}{4}},\,-{\sqrt {\frac {5+{\sqrt {5}}}{40}}},\,{\sqrt {\frac {25+11{\sqrt {5}}}{10}}}\right)}
(
0
,
−
5
+
2
5
5
,
25
+
11
5
10
)
{\displaystyle \left(0,\,-{\sqrt {\frac {5+2{\sqrt {5}}}{5}}},\,{\sqrt {\frac {25+11{\sqrt {5}}}{10}}}\right)}
(
±
1
+
5
4
,
−
25
+
11
5
40
,
20
+
8
5
5
)
{\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,-{\sqrt {\frac {25+11{\sqrt {5}}}{40}}},\,{\sqrt {\frac {20+8{\sqrt {5}}}{5}}}\right)}
(
±
3
+
5
4
,
5
+
5
40
,
20
+
8
5
5
)
{\displaystyle \left(\pm {\frac {3+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5+{\sqrt {5}}}{40}}},\,{\sqrt {\frac {20+8{\sqrt {5}}}{5}}}\right)}
(
0
,
5
+
2
5
5
,
20
+
8
5
5
)
{\displaystyle \left(0,\,{\sqrt {\frac {5+2{\sqrt {5}}}{5}}},\,{\sqrt {\frac {20+8{\sqrt {5}}}{5}}}\right)}
(
±
1
2
,
−
5
+
2
5
20
,
65
+
29
5
10
)
{\displaystyle \left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,{\sqrt {\frac {65+29{\sqrt {5}}}{10}}}\right)}
(
±
1
+
5
4
,
5
−
5
40
,
65
+
29
5
10
)
{\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,{\sqrt {\frac {65+29{\sqrt {5}}}{10}}}\right)}
(
0
,
5
+
5
10
,
65
+
29
5
10
)
{\displaystyle \left(0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,{\sqrt {\frac {65+29{\sqrt {5}}}{10}}}\right)}
(
±
1
2
,
±
5
+
2
5
2
,
45
+
18
5
5
)
{\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {5+2{\sqrt {5}}}}{2}},\,{\sqrt {\frac {45+18{\sqrt {5}}}{5}}}\right)}
(
±
3
+
5
4
,
±
5
+
5
8
,
45
+
18
5
5
)
{\displaystyle \left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\sqrt {\frac {5+{\sqrt {5}}}{8}}},\,{\sqrt {\frac {45+18{\sqrt {5}}}{5}}}\right)}
(
±
1
+
5
2
,
0
,
45
+
18
5
5
)
{\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{2}},\,0,\,{\sqrt {\frac {45+18{\sqrt {5}}}{5}}}\right)}
(
±
1
2
,
5
+
2
5
20
,
125
+
41
5
10
)
{\displaystyle \left(\pm {\frac {1}{2}},\,{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,{\sqrt {\frac {125+41{\sqrt {5}}}{10}}}\right)}
(
±
1
+
5
4
,
−
5
−
5
40
,
125
+
41
5
10
)
{\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,-{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,{\sqrt {\frac {125+41{\sqrt {5}}}{10}}}\right)}
(
0
,
−
5
+
5
10
,
125
+
41
5
10
)
{\displaystyle \left(0,\,-{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,{\sqrt {\frac {125+41{\sqrt {5}}}{10}}}\right)}
(
±
1
2
,
−
5
+
2
5
20
,
80
+
32
5
5
)
{\displaystyle \left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,{\sqrt {\frac {80+32{\sqrt {5}}}{5}}}\right)}
(
±
1
+
5
4
,
5
−
5
40
,
80
+
32
5
5
)
{\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,{\sqrt {\frac {80+32{\sqrt {5}}}{5}}}\right)}
(
0
,
5
+
5
10
,
80
+
32
5
5
)
{\displaystyle \left(0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,{\sqrt {\frac {80+32{\sqrt {5}}}{5}}}\right)}
(
±
1
2
,
5
+
2
5
20
,
205
+
89
5
10
)
{\displaystyle \left(\pm {\frac {1}{2}},\,{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,{\sqrt {\frac {205+89{\sqrt {5}}}{10}}}\right)}
(
±
1
+
5
4
,
−
5
−
5
40
,
205
+
89
5
10
)
{\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,-{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,{\sqrt {\frac {205+89{\sqrt {5}}}{10}}}\right)}
(
0
,
−
5
+
5
10
,
205
+
89
5
10
)
{\displaystyle \left(0,\,-{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,{\sqrt {\frac {205+89{\sqrt {5}}}{10}}}\right)}
(
±
1
2
,
±
5
+
2
5
2
,
25
+
10
5
)
{\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {5+2{\sqrt {5}}}}{2}},\,{\sqrt {25+10{\sqrt {5}}}}\right)}
(
±
3
+
5
4
,
±
5
+
5
8
,
25
+
10
5
)
{\displaystyle \left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\sqrt {\frac {5+{\sqrt {5}}}{8}}},\,{\sqrt {25+10{\sqrt {5}}}}\right)}
(
±
1
+
5
2
,
0
,
25
+
10
5
)
{\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{2}},\,0,\,{\sqrt {25+10{\sqrt {5}}}}\right)}
