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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.

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    $\begingroup$ FYI: A polyhedron with equilateral-triangle faces is a Deltahedron. $\endgroup$
    – Blue
    Jun 17, 2022 at 1:00
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    $\begingroup$ Take two regular tetrahedra with only an edge in common: there's your flexible deltahedron. $\endgroup$
    – Intelligenti pauca
    Jun 17, 2022 at 17:02
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    $\begingroup$ @Intelligentipauca this wouldn’t meet the common definition of a polyhedron. $\endgroup$
    – Hussein
    Jun 19, 2022 at 1:03
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    $\begingroup$ 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). $\endgroup$
    – achille hui
    Jun 19, 2022 at 1:50
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    $\begingroup$ Now posted to MO, mathoverflow.net/questions/427453/… $\endgroup$
    – Gerry Myerson
    Jul 28, 2022 at 23:33

1 Answer 1

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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.

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    $\begingroup$ The semi-defunct website says that these tetrahedra are rigid since the equatorial angle determines the mouth opening. $\endgroup$
    – Hussein
    Jun 23, 2022 at 23:55

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