# Hales's fan associated with a polyhedron

In Hales's book (cited below), he associates what he calls a fan with any convex polyhedron in $\mathbb{R}^3$. I will not define his notion of fan, but let his figure (p.137) serve as a definition:

My question is: Has this natural object been defined and used previously in other contexts and perhaps under another name?

Thomas Hales, Dense Sphere Packings: A Blueprint for Formal Proofs. 2012. (Cambridge link)

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The fan Hales is using is called the "face fan" of the polytope.

In toric varieties, one mainly considers the outer normal fan of a polytope, which has a ray for each facet (perpendicular to it). The face fan, on the other hand, has a ray for each vertex. There is an inclusion-reversing map between the poset of faces of these two fans (if we omit the zero-faces). So that is probably what Hales means by saying that the two notions bear some resemblance but are not the same.

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Thank you, Hugh! This likely explains his comments about the notation. – Joseph O'Rourke Aug 9 '13 at 1:30

Yes, fans have long been used in the theory of toric varieties. I believe they may have been introduced (as "éventails") by Demazure (1970). I don't know who first spelled out the relation with convex polyhedra, but that can be found in such books as Oda (1988) or Audin (2004).

I'm assuming this is the same notion you're after. In Audin's words, "The idea one must have in mind is that a fan is what is left from a convex polyhedron when the "sizes" of its faces are forgotten."

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No doubt this is the reason he calls it a "fan." Hales does say, "a fan in our sense bears some relation to a toric variety fan." But he also says they are not the same. I'll have to puzzle this through. Thanks! – Joseph O'Rourke Aug 5 '13 at 23:24