# The icosahedra of edge length 1

## TL;DR

This paper explores variations of the regular icosahedron with fixed edge lengths, identifying all rigid classes with symmetries and discovering a flexible form, supported by visualizations and explicit data.

## Contribution

It classifies all rigid icosahedra with fixed edge length and symmetry, and finds a new flexible icosahedron variant.

## Key findings

- All rigid equivalence classes identified
- One continuous family of flexible icosahedra found
- Visualizations and data provided

## Abstract

Retaining the combinatorial Euclidean structure of a regular icosahedron, namely the 20 equiangular (planar) triangles, the 30 edges of length 1, and the 12 different vertices together with the incidence structure, we investigate variations of the regular icosahedron admitting self-intersections of faces. We determine all rigid equivalence classes of these icosahedra with non-trivial automorphism group and find one curve of flexible icosahedra. Visualisations and explicit data for this paper are available under http://algebra.data.rwth-aachen.de/Icosahedra/visualplusdata.html.