Different Reflection Generated Polytopes with the Same Number of Vertices in 3D

In this paper, we study polytopes generated by three reflections groups. Each polytope is generated from a single point in the real Euclidean space, called a dominant point, by means of a series of reflections with respect to mirrors specifically defined for each of the reflection groups. It is shown that three different polytopes having the same number of vertices but with different dominant points may be synthesized for each of these groups.