On the problem of the finiteness of the number of the combinatorial types of infinite convex polyhedra with equiangular faces

Abstract

An infinite (complete) convex polyhedron with equiangular faces, that is, such that all the angles of each of its faces are equal to one another, is called irreducible if the number of monogonal faces belonging to it cannot be decreased (by identifying their sides) while preserving the equiangularity of all of the other faces and the convexity of the polyhedron itself (a lack of conditional edges). Any infinite convex polyhedron with equiangular faces can be obtained from the corresponding irreducible one by pasting in the missing number of monogons. It is proved that the number of combinatorially different irreducible polyhedra is finite, not counting three infinite series of frusta of cones with finite or infinite bases and right prisms with infinite bases. It is also established that, without exception, all infinite convex polyhedra with equiangular faces and total curvature 2πare the derivatives of closed convex polyhedra with equiangular faces. The proof is carried out with the help of A. D. Milka's method from Ross. Zh. Mat., 1988, 3A830.