orthoplex

Coined in 1991 by John Horton Conway and Neil Sloane. Blend of orthant +‎ complex, since it has one facet for each orthant.[1]

orthoplex (plural orthoplexes)

1. (geometry) A convex polytope analogous to an octahedron (3 dimensions) or 16-cell (4 dimensions).
   • 1990, Mathematica Journal, volumes 1-2, page 84:

     We made use of another way of considering the 120 cells, starting with the eight at the vertices of an orthoplex, that is, in the cells of a hypercube.

   • 1991, J. H. Conway, N. J. A. Sloane, “The Cell Structures of Certain Lattices”, in Peter Hilton, Friedrich Hirzebruch, Reinhold Remmert, editors, Miscellanea Mathematica, page 90:

     It is remarkable that the four-dimensional orthoplex is the same polytope as the four-dimensional hemicube.

   • 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things, page 412:

     The combinatorics of this case apply to all members of the Gosset series; in every case, their cells are simplexes and orthoplexes, the latter appearing with only half symmetry.

• (polytope analogous to an octahedron): cocube, cross-polytope, hyperoctahedron

• (polytope analogous to an octahedron): 16-cell, 4-orthoplex, hexadecachoron, octahedron

• 4-orthoplex

• simplex

• hypercube

• Turán graph