Pentagrammic pyramid

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.

The smallest known holyhedron (polyhedron where every face has a hole) by face count is based on two pentagrammic pyramids with a ring of trapezoidal holes symmetrically cut out of their triangular faces.