In Grothendieck's Sketch of a Programme he spends a few pages discussing polyhedra over arbitrary rings and concludes with some intriguing remarks on specializing polyhedra over their "most singular characteristics". I am having trouble understanding what he means or finding other references.
Here is a link to the version of Sketch I was reading. Sketch of a Programme
More specifically, Grothendieck begins (p. 252):
In 1977 and 1978, in parallel with two C4 courses on the geometry of the cube and that of the icosahedron, I began to become interested in regular polyhedra, which then appeared to me as particularly concrete “geometric realizations” of combinatorial maps, the vertices, edges and faces being realised as points, lines and plans respectively in a suitable 3-dimensional affine space, and respecting incidence relations. This notion of a geometric realisation of a combinatorial map keeps its meaning over an arbitrary base field, and even over an arbitrary base ring. It also keeps its meaning …
My understanding is that he is viewing the polyhedron as a configuration of affine subspaces—what does he mean by "combinatorial maps"? This seems to be the key to understanding the last line, on the concept making sense over an arbitrary ring.
The most curious remark comes at the end of this section where he writes (p. 255):
… on the already known cases. Thus, examining the Pythagorean polyhedra one after the other, I saw that the same small miracle was repeated each time, which I called the combinatorial paradigm [underlined in original] of the polyhedra under consideration. Roughly speaking, it can be described by saying that when we consider the specialisation of the polyhedra in the or one of the most singular characteristic(s) (namely characteristics 2 and 5 for the icosahedron, characteristic 2 for the octahedron), we read off from the geometric regular polyhedron over the finite field ($\mathbb F_4$ and $\mathbb F_5$ for the icosahedron, $\mathbb F_2$ for the octahedron) a particularly elegant (and unexpected) description of the combinatorics of the polyhedron. It seems to me that I perceived there a principle of great generality, which I believed I found again for example in a later reflection on the combinatorics of the system of 27 lines on a cubic surface, and its relations with the root system $E_7$. Whether it happens that such a principle really exists, and even that we succeed in uncovering it from its cloak of fog, or that it recedes as we pursue it and ends up vanishing like a Fata Morgana, I find in it for my part a force of motivation, a rare fascination, perhaps similar to that of dreams. No doubt that following such an unformulated call, the unformulated seeking form, from an elusive glimpse which seems to take pleasure in simultaneously hiding and revealing itself — can only lead far, although no one could predict where…
How is he determining the "most singular characteristics" of these polyhedra? If I can understand the combinatorial maps comment above, it may make more sense how he is considering the specializations of icosahedra over finite fields. What is the elegant description of their combinatorics Grothendieck refers to?
If anyone can explain Grothendieck's comments or point to other references, I would be appreciative.
