Orbiform (no ontology)

From Hi.gher. Space

An n-dimensional polytope is said to be orbiform if it can be inscribed in an n-dimensional sphere. Equivalently, its vertices lie on an n-dimensional sphere.


Richard Klitzing's segmentochora are all orbiform.

Pyramids of orbiform polytopes

An n-dimensional orbiform polytope can be made into an (n+1)-dimensional pyramid by adding a point, the apex of the pyramid, at the center of the circumscribing sphere, displaced in the (n+1)'th direction. The resulting pyramid will have equal-length edges from the apex to the vertices of the original polytope (the base of the pyramid).

If the orbiform polytope is CRF with unit edge length, then the corresponding (n+1)-dimensional pyramid can also be made CRF, provided the radius of the circumscribing sphere is < 1. If the radius is > 1, then the edges from apex to base will be longer than unit edge length, and therefore cannot be CRF. If the radius is exactly 1, then a CRF pyramid would be degenerate, having height 0 (though non-CRF pyramids are still possible, but would require edge length > 1).

Non-orbiform CRF n-dimensional polytopes cannot form CRF (n+1)-dimensional pyramids, because some apex-to-base edges will be non-unit length. However, certain CRF pseudopyramids exist, if the apex is allowed to be larger than a point (such as an edge or a polygon, or an (n-3)-dimensional polytope).