Saturday, August 03, 2019

[dgyjakxx] Abstract polytopes

Wikipedia has a definition of abstract polyhedra based on partially ordered sets.  However, it seems more natural to try to define them more geometrically, building up 1 dimension at a time:

Each face is "completely surrounded" by faces one dimension lower.  Each face participates exactly twice, once on each of its "sides", in the surrounding of a face one dimension higher.

Are the poset and geometric definitions of abstract polyhedra equivalent?

The tricky part, unresolved, is defining "surrounded" and (possibly) "sides".  In 2D to define a left and right side of a line segment, it needs to be a directed line segment.  But then in 3D, the concept of "sides" of a line segment disappears.  In 3D, we can order the vertices around a 2D face, then define sides of it by the left- or right-hand rule.

A regular 3D cube would have two 3D faces: its interior and its exterior, the latter having infinite volume, though volume is not a concept in abstract polyhedra.

Calculate the poset graphs for some well-known polyhedra.  For the larger ones, text representation of the graph is better.

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