Unfolding Overlaps of the Exceptional Regular Polytopes
2026-08-31 • Computational Geometry
Computational Geometry
AI summaryⓘ
The authors studied special 4-dimensional shapes called the 24-cell, 120-cell, and 600-cell. They tried to unfold these shapes by cutting along their ridges to make flat nets. However, they found that the flat pieces overlap and don't form proper nets. This means these shapes don't have the 'all-net property,' which is a known quality for simpler regular shapes. Their work completes the search for which regular shapes can be unfolded without overlaps.
4D polytopes24-cell120-cell600-cellridge unfoldingfacet overlapall-net propertyregular polytopesnet (polytope)
Authors
Satyan L. Devadoss, Matthew Harvey, David Richter
Abstract
We find explicit ridge unfoldings of the three exceptional 4D polytopes (24-cell, 120-cell, 600-cell) that result in overlaps of their facets. These failures bring an end to the full classification of regular polytopes with the all-net property.