The pentagonal duoexpandoprism or pedep is a convex isogonal polychoron and the fourth member of the duoexpandoprism family. It consists of 20 pentagonal prisms of two kinds, 25 rectangular trapezoprisms , 50 wedges , and 25 tetragonal disphenoids . Each vertex joins 2 pentagonal prisms, 1 tetragonal disphenoid, 3 wedges, and 2 rectangular trapezoprisms. It can be obtained as the convex hull of two orthogonal pentagonal-decagonal duoprisms , or more generally pentagonal-dipentagonal duoprisms , and a subset of its variations can be constructed by expanding the cells of the pentagonal duoprism outward. However, it cannot be made uniform.
This is one of a total of five polychora that can be obtained as the convex hull of two orthogonal pentagonal-dipentagonal duoprisms. To produce variants of this polychoron, if the polychoron is written as ao5bc oa5cb&#zy, c must be in the range
c
<
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a
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4
{\displaystyle c<b+{\frac {a(1+{\sqrt {5}})}{4}}}
. It generally has circumradius
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{\displaystyle {\sqrt {\frac {5a^{2}+5b^{2}+5ab+5c^{2}+(a^{2}+b^{2}+3ab+c^{2}){\sqrt {5}}}{10}}}}
.
Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:
5
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3
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18
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{\displaystyle {\frac {5+3{\sqrt {5}}-{\sqrt {50+18{\sqrt {5}}}}}{2}}}
≈ 1:1.10412.
The vertices of a pentagonal duoexpandoprism, constructed as the convex hull of two orthogonal pentagonal-decagonal duoprisms of edge length 1, centered at the origin, are given by:
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{\displaystyle \left(0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,0\right),}
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{\displaystyle \left(0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\sqrt {\frac {5+{\sqrt {5}}}{8}}}\right),}
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{\displaystyle \left(0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {5+2{\sqrt {5}}}}{2}}\right),}
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40
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{\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,0\right),}
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{\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\sqrt {\frac {5+{\sqrt {5}}}{8}}}\right),}
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{\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {5+2{\sqrt {5}}}}{2}}\right),}
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{\displaystyle \left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,0\right),}
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{\displaystyle \left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\sqrt {\frac {5+{\sqrt {5}}}{8}}}\right),}
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{\displaystyle \left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {5+2{\sqrt {5}}}}{2}}\right),}
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{\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{2}},\,0,\,0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right),}
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{\displaystyle \left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\sqrt {\frac {5+{\sqrt {5}}}{8}}},\,0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right),}
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{\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {5+2{\sqrt {5}}}}{2}},\,0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right),}
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{\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{2}},\,0,\,\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}}\right),}
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{\displaystyle \left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\sqrt {\frac {5+{\sqrt {5}}}{8}}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}}\right),}
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{\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {5+2{\sqrt {5}}}}{2}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}}\right),}
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{\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,0,\,\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}}\right),}
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{\displaystyle \left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\sqrt {\frac {5+{\sqrt {5}}}{8}}},\,\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}}\right),}
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{\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {5+2{\sqrt {5}}}}{2}},\,\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}}\right).}