# Pentagonal duoexpandoprism

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Pentagonal duoexpandoprism
Rank4
TypeIsogonal
Notation
Bowers style acronymPedep
Coxeter diagramxo5xx ox5xx&#zy
Elements
Cells25 tetragonal disphenoids, 50 wedges, 25 rectangular trapezoprisms, 10+10 pentagonal prisms
Faces100 isosceles triangles, 100 isosceles trapezoids, 50+50 rectangles, 20 pentagons
Edges50+50+100+100
Vertices100
Vertex figureMirror-symmetric triangular antiprism
Measures (based on two pentagonal-decagonal duoprisms of edge length 1)
Edge lengthsEdges of duoprisms (50+50+100): 1
Lacing edges (100): ${\displaystyle {\sqrt {\frac {5+{\sqrt {5}}}{5}}}\approx 1.20300}$
Circumradius${\displaystyle {\sqrt {\frac {10+3{\sqrt {5}}}{5}}}\approx 1.82802}$
Central density1
Related polytopes
ArmyPedep
RegimentPedep
DualPentagonal duoexpandotegum
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryH2≀S2, order 200
ConvexYes
NatureTame

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 ${\displaystyle c. It generally has circumradius ${\displaystyle {\sqrt {\frac {5a^{2}+5b^{2}+5ab+5c^{2}+(a^{2}+b^{2}+3ab+c^{2}){\sqrt {5}}}{10}}}}$.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:${\displaystyle {\frac {5+3{\sqrt {5}}-{\sqrt {50+18{\sqrt {5}}}}}{2}}}$ ≈ 1:1.10412.

## Vertex coordinates

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:

• ${\displaystyle \left(0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,0\right),}$
• ${\displaystyle \left(0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\sqrt {\frac {5+{\sqrt {5}}}{8}}}\right),}$
• ${\displaystyle \left(0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {5+2{\sqrt {5}}}}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,0\right),}$
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\sqrt {\frac {5+{\sqrt {5}}}{8}}}\right),}$
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {5+2{\sqrt {5}}}}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,0\right),}$
• ${\displaystyle \left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\sqrt {\frac {5+{\sqrt {5}}}{8}}}\right),}$
• ${\displaystyle \left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {5+2{\sqrt {5}}}}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{2}},\,0,\,0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right),}$
• ${\displaystyle \left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\sqrt {\frac {5+{\sqrt {5}}}{8}}},\,0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right),}$
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {5+2{\sqrt {5}}}}{2}},\,0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right),}$
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{2}},\,0,\,\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}}\right),}$
• ${\displaystyle \left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\sqrt {\frac {5+{\sqrt {5}}}{8}}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}}\right),}$
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {5+2{\sqrt {5}}}}{2}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}}\right),}$
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,0,\,\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}}\right),}$
• ${\displaystyle \left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\sqrt {\frac {5+{\sqrt {5}}}{8}}},\,\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}}\right),}$
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {5+2{\sqrt {5}}}}{2}},\,\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}}\right).}$