“… despite the connotations of the word ‘twin’, a dodecahedron actually has 5 twins. […] But here’s something deeper that Ocneanu claims to have proved, in unpublished work. Suppose you take one of these twins. It, too, will have 5 twins. One of these will be the dodecahedron you started with. But the other 4 will be new dodecahedra: that is, dodecahedra rotated in new ways. […] How many different dodecahedra can you get by continuing to take twins? Infinitely many! This image by Roice Nelson shows the vertices of a dodecahedron, its twins, the twins of its twins, the twins of the twins of its twins, the twins of the twins of the twins of its twins, and the twins of the twins of the twins of the twins of its twins …”