The set of flexible nondegenerate polyhedra of a prescribed combinatorial structure is not always algebraic

TL;DR

The paper demonstrates that the set of flexible nondegenerate polyhedra with a given combinatorial structure is not always algebraic by constructing specific examples of nonflexible polyhedra as limits of flexible ones.

Contribution

It provides explicit examples showing that the collection of flexible polyhedra with a fixed combinatorial type is not necessarily an algebraic set.

Findings

Constructed nondegenerate nonflexible polyhedra as limits of flexible ones

Showed the non-algebraic nature of the set of flexible polyhedra

Provided counterexamples to previous assumptions about algebraic sets of polyhedra

Abstract

We construct some example of a closed nondegenerate nonflexible polyhedron $P$ in Euclidean 3-space that is the limit of a sequence of nondegenerate flexible polyhedra each of which is combinatorially equivalent to $P$. This implies that the set of flexible nondegenerate polyhedra combinatorially equivalent to $P$ is not algebraic.