# Pentagonal tetraswirlprism

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Pentagonal tetraswirlprism
Rank4
TypeIsogonal
Elements
Cells200 phyllic disphenoids, 100 rhombic disphenoids, 40 pentagonal gyroprisms
Faces400+400 scalene triangles, 40 pentagons
Edges100+100+200+200
Vertices100
Vertex figure12-vertex polyhedron with 4 tetragons and 12 triangles
Measures (based on pentagonal duoprisms of edge length 1)
Edge lengthsShort side edges (100): ${\displaystyle {\sqrt {\frac {5+2{\sqrt {5}}}{5}}}-1\approx 0.37638}$
Medium side edges (100): ${\displaystyle {\sqrt {\frac {5-{\sqrt {5}}}{5}}}\approx 0.74350}$
Long side edges (200): ${\displaystyle {\sqrt {\frac {20+4{\sqrt {5}}-{\sqrt {250+110{\sqrt {5}}}}}{10}}}\approx 0.81694}$
Edges of pentagons (200): 1
Circumradius${\displaystyle {\sqrt {\frac {5+{\sqrt {5}}}{5}}}\approx 1.20300}$
Central density1
Related polytopes
DualPentagonal tetraswirltegum
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
Symmetry(I2(20)≀S2)+/4, order 400
ConvexYes
NatureTame

The pentagonal tetraswirlprism is a convex isogonal polychoron and member of the duoprismatic swirlprism family that consists of 40 pentagonal gyroprisms, 100 rhombic disphenoids, and 200 phyllic disphenoids. 4 pentagonal gyroprisms, 4 rhombic disphenoids, and 8 phyllic disphenoids join at each vertex. It can be obtained as a subsymmetrical faceting of the icosagonal duoprism. It is the fourth in an infinite family of isogonal pentagonal dihedral swirlchora.

The ratio between the longest and shortest edges is 1:${\displaystyle {\frac {{\sqrt {5}}+{\sqrt {5+2{\sqrt {5}}}}}{2}}}$ ≈ 1:2.65688.