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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[edit | edit source]

Abstract polytopes[edit | edit source]

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

with the induced order from 𝓟.

Every section of a polytope is also a polytope.

Distinguished generators[edit | edit source]

For an polytope with distinguished generators then a section is a polytope generated by where .

Concrete[edit | edit source]

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