Continuous Flattening of α-Trapezoidal Polyhedra

Abstract

It was proved that any orthogonal polyhedron is continuously flattened by using a property of a rhombus. We investigated the method precisely, and found that there are infinitely many ways to flatten such polyhedra. We prove that the infimum of the area of moving creases is zero for α-trapezoidal polyhedra, which is a generalization of semi-orthogonal polyhedra. Also we prove that, for any integer n, there exists a continuous flattening motion whose area of moving creases is arbitrarily small for any n-gonal pyramid with a circumscribed base and a top vertex being just above the incenter of the base. As a by-product we provide a continuous flattening motion whose area of moving creases is arbitrarily small for more general types of polyhedra.