Convex polyhedra are rigid by Cauchy’s theorem. Steffen’s polyhedron is an example of a non-convex polyhedron that is flexible (i.e., non-rigid). However, it appears to have edges of different lengths. My question: are there flexible polyhedra with equilateral triangular faces? I am interested in both finite and (non-trivial) infinitesimal flexibility.


Motivation: I have a bars-and-balls magnetic construction set and I would like to build a flexible polyhedron. But all the bars I have are equal in length.


Cross-posted on [math.stackexchange](https://math.stackexchange.com/questions/4474324/are-equilateral-polyhedra-with-triangular-faces-rigid)



Are polyhedra with equilateral triangular faces rigid?
This depends on how do you define a "polyhedron". If you accept a doubly covered lozenge (two copies of two adjacent equilateral triangles), then no. But under reasonable nondegeneracy conditions the answer is mostly likely yes. We worked on this problem in our recent [Domes over curves](https://arxiv.org/abs/2005.02555) paper, joint with Alexey Glazyrin (see $\S$5.2). If there was a flexible polyhedron of this type this would disprove some conjectures in the area that we believe are true. Anyway, your question is formalized as Conjecture 5.5 where we also have plenty of context on the subject.


P.S. Both Alexey and I gave several Zoom talks on this paper. You can find the links [on my website](https://www.math.ucla.edu/%7Epak/papers/research.htm#gc).


