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Dodechameleon !

This is a response to a recent Azimuth blog post by John Baez.

John Baez writes:

The mathematics of viruses with 5-fold symmetry is fascinating. Just today, I learned of Reidun Twarock‘s recent discoveries in this area:

• Reidun Twarock, Mathematical virology: a novel approach to the structure and assembly of viruses, Phil. Trans. R. Soc. A 364 (2006), 3357-3373.
To understand these more unusual viruses, Twarock needed to use some very clever math:

• Thomas Keef and Reidun Twarock, Affine extensions of the icosahedral group with applications to the three-dimensional organisation of simple viruses, J. Math. Biol. 59 (2009), 287-313.

This is a fascinating paper, thanks for passing it on. It prompts me to document a fun fact that I haven’t seen documented anywhere. It has at least in common with their work to suggest looking into 4D analogues – in turn a fun idea by itself, that mathematics would now allow to dream up 4D viruses in non-trivial logical detail !

I believe the right concise wording for my fun fact – and I would welcome rectification – is to say that the surface of the regular platonic dodecahedron
Image of regular dodecahedron
is isometric to the outer surface or shell of both Poinsot’s great dodecahedron
image of Poinsot's great dodecahedron
and to that of the first stellation of the icosahedron (aka small triambic icosahedron).
image of small triambic isosahedron
All three have icosahedral symmetry. The latter two are self-intersecting polyhedra so that a distinction needs to be made between their surface and the part of it that’s apparent to the outside; what I call the outer shell and what is usually the only surface considered when making paper models. To restate my fun fact in simpler words,

if you imagine these surfaces made out of paper, it is in principle possible to refold any one of the three without any cutting, to obtain any of the other two.

Easier in practice would be to cut up the surface enough to lay it out flat and then refold it and glue together the lips of the cuts.

This property has been well known to paper modelists for a long time in the case of the two self-intersecting polyhedra, but AFAIK it hadn’t yet been remarked that both are similarly related in turn to the regular dodecahedron. The reason this is not obvious is that the mapping from or to the regular dodecahedron preserves no edge or face, while in the case of the other two polyhedra the mapping relates both faces and edges one-to-one (when viewing the outer shells as polyhedra in their own right, or else we probably need to speak of “folds” for edges and “flat polygons” for faces).

On the left is a flat surface from which either one of the three polyhedra displayed can be constructed. A single pentagon from the regular dodecahedron is shown in red, while in green is shown a single triangular face of the outer shell of either a great dodecahedron or a small triambic icosahedron.