# Pentagonal duoantiprism

Pentagonal duoantiprism
Rank4
TypeIsogonal
Notation
Bowers style acronymPedap
Coxeter diagrams10o2s10o ()
Elements
Cells50 tetragonal disphenoids, 20 pentagonal antiprisms
Faces200 isosceles triangles, 20 pentagons
Edges100+100
Vertices50
Vertex figureGyrobifastigium
Measures (based on pentagons of edge length 1)
Edge lengthsLacing (100): ${\displaystyle {\sqrt {\frac {5-{\sqrt {5}}}{5}}}\approx 0.74350}$
Edges of pentagons (100): 1
Circumradius${\displaystyle {\sqrt {\frac {5+{\sqrt {5}}}{5}}}\approx 1.20300}$
Central density1
Related polytopes
ArmyPedap
RegimentPedap
DualPentagonal duoantitegum
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
Symmetry(I2(10)≀S2)/2, order 400
ConvexYes
NatureTame

The pentagonal duoantiprism or pedap, also known as the pentagonal-pentagonal duoantiprism, the 5 duoantiprism or the 5-5 duoantiprism, is a convex isogonal polychoron that consists of 20 pentagonal antiprisms and 50 tetragonal disphenoids. 4 pentagonal antiprisms and 4 tetragonal disphenoids join at each vertex. It can be obtained through the process of alternating the decagonal duoprism. However, it cannot be made uniform, and has two edge lengths. It is the second in an infinite family of isogonal pentagonal dihedral swirlchora.

The ratio between the longest and shortest edges is 1:${\displaystyle {\frac {\sqrt {5+{\sqrt {5}}}}{2}}}$ ≈ 1:1.34500.

## Vertex coordinates

The vertices of a pentagonal duoantiprism based on pentagons of edge length 1, centered at the origin, are given by:

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