 Rank3
TypeSegmentotope
SpaceSpherical
Notation
Bowers style acronymRastacu
Coxeter diagramox5/3xx&#x
Elements
Faces5 triangles, 5 squares, 1 pentagram, 1 decagram
Edges5+5+5+10
Vertices5+10
Vertex figures5 crossed isosceles trapezoids, edge lengths 1, 2, (1-5)/2, 2
10 scalene triangles, edge lengths 1, 2, (5-5)/2
Measures (edge length 1)
Circumradius$\frac{\sqrt{11-4\sqrt5}}{2} ≈ 0.71689$ Volume$\frac{4\sqrt5-5}{6} ≈ 0.65738$ Dihedral angles3–10/3: $\arccos\left(-\sqrt{\frac{5-2\sqrt5}{15}}\right) ≈ 100.81232°$ 3–4: $\arccos\left(\frac{\sqrt3-\sqrt{15}}{6}\right) ≈ 69.09484°$ 4–5/2: $\arccos\left(\sqrt{\frac{5-\sqrt5}{10}}\right) ≈ 58.28253°$ 4–10/3: $\arccos\left(\sqrt{\frac{5-\sqrt5}{10}}\right) ≈ 58.28253°$ Height$\sqrt{\frac{5+\sqrt5}{10}} ≈ 0.85065$ Related polytopes
ArmyNon-CRF Pecu
RegimentRastacu
DualSemibisected pentagrammic trapezohedron
ConjugatePentagonal cupola
Convex hullPentagonal cupola
Abstract & topological properties
OrientableYes
Properties
SymmetryH2×I, order 10
ConvexNo
NatureTame

The retrograde pentagrammic cupola or rastacu, also called the crossed pentagrammic cupola or sometimes just the pentagrammic cupola, is a cupola based on the pentagram, seen as a 5/3-gon rather than 5/2. It consists of 5 triangles, 5 squares, 1 pentagram, and 1 decagram.

It can be obtained as a segment of the quasirhombicosidodecahedron, just as its conjugate, the convex pentagonal cupola, can be obtained from the small rhombicosidodecahedron.

It also appears in some scaliform polychora, namely the retroprismatorhombisnub hecatonicosachoron and retroprismatorhombiretrosnub hecatonicosachoron.

## Vertex coordinates

A retrograde pentagrammic cupola of edge length 1 has vertices given by the following coordinates:

• $\left(±\frac12,\,-\sqrt{\frac{5-2\sqrt5}{20}},\,\sqrt{\frac{5+\sqrt5}{10}}\right),$ • $\left(±\frac{\sqrt5-1}{4},\,\sqrt{\frac{5-\sqrt5}{40}},\,\sqrt{\frac{5+\sqrt5}{10}}\right),$ • $\left(0,\,\sqrt{\frac{5-\sqrt5}{10}},\,\sqrt{\frac{5+\sqrt5}{10}}\right),$ • $\left(±\frac12,\,±\frac{\sqrt{5-2\sqrt5}}{2},\,0\right),$ • $\left(±\frac{3-\sqrt5}{4},\,±\sqrt{\frac{5-\sqrt5}{8}},\,0\right),$ • $\left(±\frac{\sqrt5-1}{2},\,0,\,0\right).$ ## Related polyhedra

The retrograde pentagrammic cupola is the pentagram-first cap of the quasirhombicosidodecahedron.