A particularly interesting 3D symmetry group is this pyritohedral group, also called 3*2 in the Conway/Thurston orbifold notation or m3 in crystallography. It is the unique polyhedral point group that is neither a rotation group nor a reflection group.

The motion that we are watching was popularized by Buckminster Fuller as the "jitterbug"; it unfolds an octahedron to a cuboctahedron, keeping pyritohedral symmetry. In the middle, there is one additional stage with extra symmetry – an icosahedron!

Highlighting the six extra edges of the icosahedron, we can let them follow along in the jitterbug. They become face diagonals of the cuboctahedron and then internal diagonals of the next icosahedron. This configuration can be built as a rigid tensegrity framework, with struts along the green diagonals and springs or cables along the red triangle edges.

If we add back in the missing icosahedral edges, the green lines join to form three perpendicular golden rectangles.

These six green edges of the icosahedron are parallel or perpendicular to each other. They form part of a symmetric five-coloring of all thirty edges. A rotation of order five along the diagonal of one of the golden rectangles finds the five cosets of the pyritohedral group within the full icosahedral symmetry group.

Returning to the picture of three golden rectangles, and focusing just on their boundaries, we see an interesting link of three components, which we will now shade in three distinct colors.

This link, known as the Borromean rings, has the amazing property that if any one component is removed, the other two can separate. But all three taken together are truly linked.

This so-called Brunnian property has led the rings to be used over many centuries in many cultures as a symbol of interconnectedness or of strength through unity. Most often, the Borromean rings are drawn as if made from three round circles, but this is known to be geometrically impossible.

What is then a nice shape for the Borromean rings? Starting from the golden rectangles, we might round off the corners to form three stadium curves – semicircles connected by straight segments. We keep the pyritohedral symmetry, with the components in perpendicular planes.

Very close to this is the minimizing configuration for a Coulomb-style repulsive-charge energy. The minimizer has many possible euclidean shapes, however, because the energy is Möbius-invariant. One shape is made of interwoven rings, while another has the pyritohedral symmetry.

Physically more natural, perhaps, is the search for tight knots and links, made of rope with a constant circular cross-section, pulled tight to minimize the length of the core.

The critical configuration we obtain for the Borromean rings is just a tenth of a percent shorter than a very similar one built from convex and concave circular arcs. Its more inticrate geometry can be described explicitly using elliptic integrals, thanks to a balance criterion that says how a tight link must balance its elastic force with contact forces.

This tight configuration of the Borromean rings still has pyritohedral symmetry. A view along the 3-fold symmetry axis has been selected for the new logo of the International Mathematical Union. The rings represent the interconnectedness not only of the various fields of mathematics, but also of the mathematical community around the world.

‹
›
×