Hemiobelisk

A hemiobelisk is one half of an obelisk, or elongated square pyramid. It is obtained by cutting an equilateral obelisk with unit edge lengths along a plane passing through opposite corners of its square base and the apex of the pyramid.

It is implemented in the Wolfram Language as PolyhedronData["Hemiobelisk"].

This solid is one of two having 7 vertices, 11 edges, and 6 faces (the other being the hemicube) and is an example of one of the 7 topological classes of convex hexahedron.

When constructed from a unit elongated square pyramid, the solid consists of two unit squares, an isosceles right triangle with unit base lengths, two unit equilateral triangles, and an isosceles right pentagon with base length and all other sides of length 1.

A configuration of congruent copies gives the hemiobelisk Heesch number at least 1 in three dimensions. Pegg (2026) gave such a construction with 24 surrounding copies. For the geometric realization used in that construction, the vertices of a congruent copy can be taken as

This realization is one of three congruent pieces in a polyhedron dissection of a heptahedron. The heptahedron is obtained by attaching a regular tetrahedron to a triangular face of the space-filling polyhedron formed by truncating a regular tetrahedron at each vertex, one quarter of the way along each edge (Pegg 2026). In the coordinates above, each hemiobelisk has volume .

This realization has nonzero Dehn invariant, so it is not a space-filling polyhedron. The edges having dihedral angles that are rational multiples of contribute zero, while the remaining edges give

where denotes the tensor product over the rationals, denotes the equivalence class of the dihedral angle modulo rational multiples of , and is a basis angle used in the Dehn invariant entry. Since is not an allowed rational cosine at a rational multiple of by Niven's theorem, is irrational, so and therefore are nonzero.

The skeleton of the hemiobelisk is the hemiobelisk graph.