Section

Not to be confused with Cross-section.

A section is a certain subordering of a polytope that preserves adjacency between elements. Vertex figures, edge figures, facets, ridges, faces, and cells are all examples of sections. Sections are important in the definition of abstract polytopes, where they are used to define strong connectivity and the diamond condition.

Definitions

Abstract polytopes

A section of a polytope 𝓟 is the set of elements between two others. A section between elements x  and y , written $x/y$, is

$x/y=\{z\in {\mathcal {P}}:x\leq z\leq y\}$

with the induced order from 𝓟.

Every section of a polytope is also a polytope.

Distinguished generators

For an polytope with distinguished generators $\langle \rho _{0},\rho _{1},\dots ,\rho _{n}\rangle$ then a section is a polytope generated by $\langle \rho _{i},\rho _{i+1},\dots ,\rho _{j-1},\rho _{j}\rangle$ where $0\leq i\leq j\leq n$.

Concrete

A section of a polytope is a figure of an element. This includes the figures and elements of the original polytope themselves.