Pentagrammic pyramid

Pentagrammic pyramid
Rank	3
Type	Segmentotope
Space	Spherical
Notation
Bowers style acronym	Stappy
Coxeter diagram	ox5/2oo&#x
Elements
Faces	5 triangles, 1 pentagram
Edges	5+5
Vertices	1+5
Vertex figures	1 pentagram, edge length 1
	5 isosceles triangles, edge lengths 1, 1, (√5-1)/2
Measures (edge length 1)
Circumradius	$\sqrt{\frac{5-\sqrt{5}}{8}} ≈ 0.58779$
Volume	$\frac{5-\sqrt{5}}{24} ≈ 0.11516$
Dihedral angles	3-5/2: $\arccos\left(\sqrt{\frac{5-2\sqrt{5}}{15}}\right) ≈ 79.18768°$
	3-3: $\arccos\left(\frac{\sqrt{5}}{3}\right) ≈ 41.81031°$
Height	$\sqrt{\frac{5+\sqrt{5}}{10}} ≈ 0.85065$
Related polytopes
Army	Non-RF peppy
Regiment	Stappy
Dual	Pentagrammic pyramid
Conjugate	Pentagonal pyramid
Convex core	Pentagonal pyramid
Abstract & topological properties
Orientable	Yes
Properties
Symmetry	H2×I, order 10
Convex	No
Nature	Tame

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:

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

Related polyhedra

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

External links

• Bowers, Jonathan. "Batch 2: Ike and Sissid Facetings" (#4 under sissid).

• Klitzing, Richard. "stappy".

• Wikipedia Contributors. "Pentagrammic pyramid".