# Pentagrammic pyramid

(Redirected from Stappy)
Pentagrammic pyramid
Rank3
TypeSegmentotope
Notation
Bowers style acronymStappy
Coxeter diagramox5/2oo&#x
Elements
Faces5 triangles, 1 pentagram
Edges5+5
Vertices1+5
Vertex figures1 pentagram, edge length 1
5 isosceles triangles, edge lengths 1, 1, (5-1)/2
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {5-{\sqrt {5}}}{8}}}\approx 0.58779}$
Volume${\displaystyle {\frac {5-{\sqrt {5}}}{24}}\approx 0.11516}$
Dihedral angles3-5/2: ${\displaystyle \arccos \left({\sqrt {\frac {5-2{\sqrt {5}}}{15}}}\right)\approx 79.18768^{\circ }}$
3-3: ${\displaystyle \arccos \left({\frac {\sqrt {5}}{3}}\right)\approx 41.81031^{\circ }}$
Height${\displaystyle {\sqrt {\frac {5+{\sqrt {5}}}{10}}}\approx 0.85065}$
Related polytopes
ArmyNon-RF peppy, edge lengths ${\displaystyle {\frac {{\sqrt {5}}-1}{2}}}$ (base), 1 (sides)
RegimentStappy
DualPentagrammic pyramid
ConjugatePentagonal pyramid
Convex corePentagonal pyramid
Abstract & topological properties
Flag count40
OrientableYes
Properties
SymmetryH2×I, order 10
ConvexNo
NatureTame

The pentagrammic pyramid, or stappy, is a pyramid with a pentagrammic base and 5 triangles as sides.

It is the vertex-first cap of the great icosahedron. A regular great icosahedron can be constructed by attaching two pentagrammic pyramids to the bases of a pentagrammic retroprism.

## Vertex coordinates

A pentagrammic pyramid of edge length 1 has the following vertices:

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

## Related polyhedra

Two pentagrammic pyramids can be attached at their bases to form a pentagrammic tegum.