Orbiform polytope

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A polytope is orbiform if all its edges are of equal length and it can be inscribed in a hypersphere.[1] All uniform polytopes are orbiform.

As many infinite families can be formed through repeated blending of other orbiforms, classification of all orbiforms is not known to be a particularly interesting problem. The primary motivation for studying orbiforms is in the discovery of scaliform polytopes, which have only orbiform elements.

Polygons[edit | edit source]

The planar orbiform polygons are simply the regular polygons.

Polyhedra[edit | edit source]

In 3 dimensions, there is a wide variety of orbiform polytopes that have yet to be fully studied. The orbiform polyhedra include 25 of the 92 Johnson solids as well as a variety of facetings of uniform polyhedra, plus various blends of these. Examples include the square and pentagonal pyramids, the cupolae, the cuploids, and cupolaic blends.

With the possibility of skew faces excluded, all faces of orbiform polyhedra are regular, and thus acrohedra. Indeed, some polyhedra specifically discovered in the search for acrohedra turned out to be orbiform, such as the polyhedra produced by Green's rules.

Properties[edit | edit source]

If an orbiform polytope has a circumradius of less than its edge length, its pyramid is also orbiform.