Steffen's Polyhedron

Steffen’s Polyhedron

by Greg Egan

Steffen’s polyhedron

changing shape Translucent version

seen from behind

Steffen’s polyhedron is a flexible polyhedron: a polyhedron that can change shape while maintaining the shapes of all its individual faces. It has been proved that no convex polyhedron can be flexible. The first non-self-intersecting flexible polyhedron with the topology of a sphere was found by Robert Connelly in 1977, but Klaus Steffen’s version has fewer faces, with just 14 triangles.

(For some time it was believed that the nine vertices of Steffen’s polyhedron was the minimum possible, but in 2024 a flexible polyhedron with just eight vertices was found by Gallet et al. Elvar Atlason, who kindly pointed out this new development to me, has publishing a detailed analysis of the problem, and constructed another solution with eight vertices.)

It’s not hard to see how Steffen’s polyhedron is constructed. First, consider the flexible shape we obtain by assembling four suitable triangles into a kind of pyramid:

This shape has one degree of freedom: we can alter, for example, the length of one of the diagonals of the non-planar quadrilateral ABCD that forms the base of the pyramid. We choose AB=CD=12, BC=AD=10, AP=10, BP=5, CP=12 and DP=11. (There’s nothing special about the particular numbers 5, 10, 11 and 12, but it is crucial that opposite edges of the quadrilateral have the same lengths.)

Now, suppose we attach two more triangles to this figure, along the edges AD and CD, with the two triangles sharing an edge, DE:

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