Deltahedral Domes over Equiangular Polygons

TL;DR

This paper characterizes which equiangular polygons can be domed with convex polyiamonds on all faces except one, providing a complete classification based on the number of sides and edge lengths.

Contribution

It offers a complete characterization of equiangular polygons that can be domed with convex polyiamonds, extending the understanding of deltahedral structures.

Findings

Only polygons with 3, 4, 5, 6, 8, 10, 12 sides can be domed.

Conditions on edge lengths determine domability.

Provides a complete classification of domable equiangular polygons.

Abstract

A polyiamond is a polygon composed of unit equilateral triangles, and a generalized deltahedron is a convex polyhedron whose every face is a convex polyiamond. We study a variant where one face may be an exception. For a convex polygon P, if there is a convex polyhedron that has P as one face and all the other faces are convex polyiamonds, then we say that P can be domed. Our main result is a complete characterization of which equiangular n-gons can be domed: only if n is in {3, 4, 5, 6, 8, 10, 12}, and only with some conditions on the integer edge lengths.