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
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4
{\displaystyle c < b+\frac{a(1+\sqrt5)}{4}}
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{\displaystyle \sqrt{\frac{5a^2+5b^2+5ab+5c^2+(a^2+b^2+3ab+c^2)\sqrt5}{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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2
{\displaystyle \frac{5+3\sqrt5-\sqrt{50+18\sqrt5}}{2}}
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+\sqrt5}{10}},\,±\frac{1+\sqrt5}{2},\,0\right),}
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{\displaystyle \left(0,\,\sqrt{\frac{5+\sqrt5}{10}},\,±\frac{3+\sqrt5}{4},\,±\sqrt{\frac{5+\sqrt5}{8}}\right),}
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{\displaystyle \left(0,\,\sqrt{\frac{5+\sqrt5}{10}},\,±\frac12,\,±\frac{\sqrt{5+2\sqrt5}}{2}\right),}
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40
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{\displaystyle \left(±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5-\sqrt5}{40}},\,±\frac{1+\sqrt5}{2},\,0\right),}
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{\displaystyle \left(±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5-\sqrt5}{40}},\,±\frac{3+\sqrt5}{4},\,±\sqrt{\frac{5+\sqrt5}{8}}\right),}
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{\displaystyle \left(±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5-\sqrt5}{40}},\,±\frac12,\,±\frac{\sqrt{5+2\sqrt5}}{2}\right),}
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{\displaystyle \left(±\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}},\,±\frac{1+\sqrt5}{2},\,0\right),}
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{\displaystyle \left(±\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}},\,±\frac{3+\sqrt5}{4},\,±\sqrt{\frac{5+\sqrt5}{8}}\right),}
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{\displaystyle \left(±\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}},\,±\frac12,\,±\frac{\sqrt{5+2\sqrt5}}{2}\right),}
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{\displaystyle \left(±\frac{1+\sqrt5}{2},\,0,\,0,\,\sqrt{\frac{5+\sqrt5}{10}}\right),}
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{\displaystyle \left(±\frac{3+\sqrt5}{4},\,±\sqrt{\frac{5+\sqrt5}{8}},\,0,\,\sqrt{\frac{5+\sqrt5}{10}}\right),}
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2
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{\displaystyle \left(±\frac12,\,±\frac{\sqrt{5+2\sqrt5}}{2},\,0,\,\sqrt{\frac{5+\sqrt5}{10}}\right),}
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{\displaystyle \left(±\frac{1+\sqrt5}{2},\,0,\,±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5-\sqrt5}{40}}\right),}
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{\displaystyle \left(±\frac{3+\sqrt5}{4},\,±\sqrt{\frac{5+\sqrt5}{8}},\,±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5-\sqrt5}{40}}\right),}
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{\displaystyle \left(±\frac12,\,±\frac{\sqrt{5+2\sqrt5}}{2},\,±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5-\sqrt5}{40}}\right),}
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{\displaystyle \left(±\frac{1+\sqrt5}{4},\,0,\,±\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}}\right),}
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{\displaystyle \left(±\frac{3+\sqrt5}{4},\,±\sqrt{\frac{5+\sqrt5}{8}},\,±\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}}\right),}
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{\displaystyle \left(±\frac12,\,±\frac{\sqrt{5+2\sqrt5}}{2},\,±\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}}\right).}
