We study oriented connected closed polyhedral surfaces with non-degenerate triangular faces in three-dimensional Euclidean space, calling them polyhedra for short. A polyhedron is called flexible if its spatial shape can be changed continuously by changing its dihedral angles only.

We prove that, for every flexible polyhedron, some integer combination of its dihedral angles remains constant during the flex. The proof is based on a theorem by A. A. Gaifullin and L. S. Ignashchenko published in their article “Dehn invariant and scissors congruence of flexible polyhedra” Proc. Steklov Inst. Math. 302, 130–145 (2018).

As an application of the result obtained, we derive fundamentally new equations that are satisfied by first-order flexes of every flexible polyhedron. Moreover, we indicate two sources of such new equations. These sources are the Dehn invariants and rigidity matrix.

The talk is based on the following articles of the speaker:

Alexandrov, V.: A sufficient condition for a polyhedron to be rigid. J.
Geom. 110(2), 11 p. (2019).
https://eur02.safelinks.protection.outlook.com/?url=https%3A%2F%2Fdoi.or... (Paper No. 38)

Alexandrov, V.: Necessary conditions for the extendibility of a
first-order flex of a polyhedron to its flex. Beitr. Algebra Geom. (2020)
61:355-368. https://eur02.safelinks.protection.outlook.com/?url=https%3A%2F%2Fdoi.or...