Current researches

[guy]

A forum for discussing current researches into polyhedra seems sorely lacking these days. So I thought I'd see if one here gathers any interest.

Several people are working actively on abstract polytope theory, in which the abstraction is a set-theoretic construct which captures the connectivity or incidences of vertices, edges, faces, cells, etc. This may them be "realized" as a geometric figure by mapping it into real space.

I have found abstract theory very elegant and powerful, but it still has something missing. I split realization into two stages. The first stage interprets the abstract poset as a "rubber-sheet" polyhedron of the kind traditionally used by topologists, which has a structure and a defined overall topology but no hard shape. I call this a morphic polyhedron. The second stage then concretizes the morphic figure as a familiar geometric polyhedron. Things get more complicated in higher dimensions, as abstract theory is more general than the traditional models. For more, see my web pages on the General theory of polytopes and polyhedra.

I am currently applying this theory to stellation and facetting, where duality plays an important role. See my pages on Stellation and facetting.

Is anybody else actively working on polyhedron theory?