# Are equilateral polyhedra with triangular faces rigid?

For the purposes of this question, a polyhedron has triangular facets.

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 equilateral flexible polyhedra or are all equilateral polyhedra rigid?

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.

• FYI: A polyhedron with equilateral-triangle faces is a Deltahedron.
– Blue
Jun 17, 2022 at 1:00
• Take two regular tetrahedra with only an edge in common: there's your flexible deltahedron. Jun 17, 2022 at 17:02
• @Intelligentipauca this wouldn’t meet the common definition of a polyhedron. Jun 19, 2022 at 1:03
• If your edges are not perfectly rigid, you can consider a equilateral pentagonal Siamese dipyramid. It is not mathematically flexible but you can continuous deform it with relative variation of edge lengths within $0.5\%$. (see this). Jun 19, 2022 at 1:50
• Now posted to MO, mathoverflow.net/questions/427453/… Jul 28, 2022 at 23:33

As noted in a comment, a polyhedron with congruent equilateral triangles for faces is called a "deltahedron". This reference lists "Bellows deltahedra" as flexible deltahedra that seem to meet OP's criteria. It shows two distinct examples, and also links to another (possibly semi-defunct) website with additional images and descriptions of these objects. If I'm understanding the images correctly, the simplest example has $$48$$ faces and a $$6$$-fold rotational symmetry.