Retrograde pentagrammic cupola

Retrograde pentagrammic cupola
Rank	3
Type	Segmentotope
Space	Spherical
Notation
Bowers style acronym	Rastacu
Coxeter diagram	ox5/3xx&#x
Elements
Faces	5 triangles, 5 squares, 1 pentagram, 1 decagram
Edges	5+5+5+10
Vertices	5+10
Vertex figures	5 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 angles	3–10/3: $\arccos\left(-\sqrt{\frac{5-2\sqrt5}{15}}\right) ≈ 100.81232°$
	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
Army	Non-CRF Pecu
Regiment	Rastacu
Dual	Semibisected pentagrammic trapezohedron
Conjugate	Pentagonal cupola
Convex hull	Pentagonal cupola
Abstract & topological properties
Orientable	Yes
Properties
Symmetry	H2×I, order 10
Convex	No
Nature	Tame

Not to be confused with Pentagrammic cuploid.

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.

External links

• Klitzing, Richard. "rastacu".