# Pentagonal-hexagonal duoantiprism

(Redirected from 5-6 duoantiprism)
Pentagonal-hexagonal duoantiprism
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Rank4
TypeIsogonal
SpaceSpherical
Bowers style acronymPhiddap
Info
Coxeter diagrams10o2s12o
SymmetryI2(10)×I2(12)/2, order 240
ArmyPhiddap
RegimentPhiddap
Elements
Vertex figureGyrobifastigium
Cells60 digonal disphenoids, 12 pentagonal antiprisms, 10 hexagonal antiprisms
Faces120+120 isosceles triangles, 12 pentagons, 10 hexagons
Edges60+60+120
Vertices60
Measures (based on polygons of edge length 1)
Edge lengthsLacing (120): ${\displaystyle \sqrt{\frac{25-10\sqrt3-\sqrt5}{10}} ≈ 0.73780}$
Edges of pentagons (60): 1
Edges of hexagons (60): 1
Circumradius${\displaystyle \sqrt{\frac{15+\sqrt5}{10}} ≈ 1.31286}$
Central density1
Euler characteristic0
Related polytopes
DualPentagonal-hexagonal duoantitegum
Properties
ConvexYes
OrientableYes
NatureTame

The pentagonal-hexagonal duoantiprism or phiddap, also known as the 5-6 duoantiprism, is a convex isogonal polychoron that consists of 10 hexagonal antiprisms, 12 pentagonal antiprisms, and 60 digonal disphenoids. 2 hexagonal antiprisms, 2 pentagonal antiprisms, and 4 digonal disphenoids join at each vertex. It can be obtained through the process of alternating the decagonal-dodecagonal duoprism. However, it cannot be made uniform, as it generally has 3 edge lengths, which can be minimized to no fewer than 2 different sizes.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:${\displaystyle \sqrt{\frac{400+165\sqrt3+\sqrt{19955+11500\sqrt3}}{482}}}$ ≈ 1:1.35539.

## Vertex coordinates

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

• ${\displaystyle \left(±1,\,0,\,0,\,\sqrt{\frac{5+\sqrt5}{10}}\right),}$
• ${\displaystyle \left(±1,\,0,\,±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5-\sqrt5}{40}}\right),}$
• ${\displaystyle \left(±1,\,0,\,±\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}}\right),}$
• ${\displaystyle \left(±\frac12,\,±\frac{\sqrt3}{2},\,0,\,\sqrt{\frac{5+\sqrt5}{10}}\right),}$
• ${\displaystyle \left(±\frac12,\,±\frac{\sqrt3}{2},\,±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5-\sqrt5}{40}}\right),}$
• ${\displaystyle \left(±\frac12,\,±\frac{\sqrt3}{2},\,±\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}}\right),}$
• ${\displaystyle \left(0,\,±1,\,0,\,-\sqrt{\frac{5+\sqrt5}{10}}\right),}$
• ${\displaystyle \left(0,\,±1,\,±\frac{1+\sqrt5}{4},\,-\sqrt{\frac{5-\sqrt5}{40}}\right),}$
• ${\displaystyle \left(0,\,±1,\,±\frac12,\,\sqrt{\frac{5+2\sqrt5}{20}}\right),}$
• ${\displaystyle \left(±\frac{\sqrt3}{2},\,±\frac12,\,0,\,-\sqrt{\frac{5+\sqrt5}{10}}\right),}$
• ${\displaystyle \left(±\frac{\sqrt2}{2},\,±\frac12,\,±\frac{1+\sqrt5}{4},\,-\sqrt{\frac{5-\sqrt5}{40}}\right),}$
• ${\displaystyle \left(±\frac{\sqrt3}{2},\,±\frac12,\,±\frac12,\,\sqrt{\frac{5+2\sqrt5}{20}}\right),}$