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Lithic polytopes

The Lithic polytopes are a class of finite abstractly regular polytopes built with Petrie polygon relations and, for ranks above 3, amalgamations of other Lithic polytopes. More specifically, Lithic polytopes are built using a Schläfli type as a base, and invoking Petrie polygon relations on its elements. As an example, the Klein icositetrahedral heptacosichoron (700-cell) {7,3,6:8,4} can be constructed by taking the base type {7,3,6} ($\rho$) and adding the relations $\rho =\rho /{⟨{\left({\rho }_0{\rho }_1{\rho }_2\right)}^8,{\left({\rho }_1{\rho }_2{\rho }_3\right)}^4⟩}^{\rho }$, which essentially means setting the Petrie polygon of its cell {7,3} to 8 and its vertex figure {3,6} to 4. It can also be constructed as the amalgamation of the Klein icositetrahedron {7,3}₈ and the dual Petrial cube {3,6}₄, which are both Lithic polyhedra.

Subclasses

There are various subclasses of Lithic polytopes, which are mainly used to exclude undesired infinite families.

Rank 3

There are two main rank 3 subclasses: the Greater Lithic polyhedra are all polyhedra of the form {p,q}_a. The Smaller Lithic polyhedra, with the Smaller usually omitted, are a subset of the Greater Lithic polyhedra excluding two infinite families, {4,4}_2n and {6,3}_2n. More information about them can be found here.

Rank 4

There are six main rank 4 subclasses: the Universal Lithic polychora, which are maximal for their facet and vertex figures (see more). For subclasses distinguishing infinite families, there are the Grander Lithic polychora, which includes all polychora of the form {p,q,r:a,b}_c, the Greater Lithic polychora, which excludes infinite families and the Smaller Lithic polychora, which excludes non-Smaller Lithic polyhedron facets and vertex figures. More information about them can be found here.