Steffen's flexible polyhedron is embedded. A proof via symbolic computations

TL;DR

This paper proves that Steffen's flexible polyhedron is embedded, using computer symbolic calculations, confirming a long-standing assumption in the mathematical theory of flexible polyhedra.

Contribution

The authors provide the first rigorous proof that Steffen's flexible polyhedron is embedded, resolving a longstanding open question in the field.

Findings

Steffen's polyhedron is confirmed to be embedded.

The proof was achieved through computer symbolic calculations.

This result settles a long-standing assumption in the theory of flexible polyhedra.

Abstract

A polyhedron is flexible if it can be continuously deformed preserving the shape and dimensions of every its face. In the late 1970's Klaus Steffen constructed a sphere-homeomorphic embedded flexible polyhedron with triangular faces and with 9 vertices only, which is well-known in the theory of flexible polyhedra. At about the same time, a hypothesis was formulated that the Steffen polyhedron has the least possible number of vertices among all embedded flexible polyhedra without boundary. A counterexample to this hypothesis was constructed by Matteo Gallet, Georg Grasegger, Jan Legersk{\'y}, and Josef Schicho in 2024 only. Surprisingly, until now, no proof has been published in the mathematical literature that the Steffen polyhedron is embedded. Probably, this fact was considered obvious to everyone who made a cardboard model of this polyhedron. In this article, we prove this fact using computer symbolic calculations.