# 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