# Envelope Polyhedra

## TL;DR

This paper introduces envelope polyhedra, a new class of regular polyhedra with unique arrangements of polygons and dihedral angles, leading to novel finite and infinite multiply connected networks with complex topologies.

## Contribution

It defines envelope polyhedra, expanding the classification of regular polyhedra by allowing variable dihedral angles and complex topologies, including infinite networks.

## Key findings

- Introduces envelope polyhedra with variable dihedral angles.
- Describes finite polyhedra with toroidal topology.
- Presents infinite multiply connected polygon networks.

## Abstract

This paper presents an additional class of regular polyhedra--envelope polyhedra--made of regular polygons, where the arrangement of polygons (creating a single surface) around each vertex is identical; but dihedral angles between faces need not be identical, and some of the dihedral angles are 0 degrees (i.e., some polygons are placed back to back). For example, squares, 6 around a point, is produced by deleting the triangles from the rhombicuboctahedron, creating a hollow polyhedron of genus 7 with triangular holes connecting 18 interior and 18 exterior square faces. An empty cube missing its top and bottom faces becomes an envelope polyhedron, squares, 4 around a point, with a toroidal topology. This definition leads to many interesting finite and infinite multiply connected regular polygon networks, including one infinite network with squares, 14 around a point, and another with triangles, 18 around a point. These are introduced just over 50 years after my related paper on infinite spongelike pseudopolyhedra in American Mathematical Monthly (Gott, 1967).