A hemi-octahedron is an abstract regular polyhedron, containing half the faces of a regular octahedron.

Typeabstract regular polyhedron
globally projective polyhedron
Faces4 triangles
Vertex configuration3.3.3.3
Schläfli symbol{3,4}/2 or {3,4}3
Symmetry groupS4, order 24
Dual polyhedronhemicube
Euler characteristic 1

It has 4 triangular faces, 6 edges, and 3 vertices. Its dual polyhedron is the hemicube.

It can be realized as a projective polyhedron (a tessellation of the real projective plane by 4 triangles), which can be visualized by constructing the projective plane as a hemisphere where opposite points along the boundary are connected and dividing the hemisphere into four equal parts. It can be seen as a square pyramid without its base.

It can be represented symmetrically as a hexagonal or square Schlegel diagram:

It has an unexpected property that there are two distinct edges between every pair of vertices – any two vertices define a digon.

See also


  • McMullen, Peter; Schulte, Egon (December 2002), "6C. Projective Regular Polytopes", Abstract Regular Polytopes (1st ed.), Cambridge University Press, pp. 162–165, ISBN 0-521-81496-0
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