Construction properties of combinatorial deltahedra

Abstract A deltahedron is defined as an ordered triple (V, E, T) of sets whose members are called vertices, edges and triangles respectively, and which obeys certain axioms based on the properties of geometrical polyhedra all of whose faces are triangles. An operation of adding an extra vertex is defined and it is shown that not every deltahedron can be obtained from a tetrahedron by a sequence of such additions. Operations of transferring certain vertices from one part of the deltahedron to another, and of replacing one edge by another are described and it is shown that any deltahedron can be transformed into any other on the same number of vertices by a sequence of such operations. Reference is made to the plant layout problem, the investigation of which led to these results.