This is a review of three articles: [Bellissard et al. 2005] (Jean Bel-lissard, Riccardo Benedetti and Jean-Marc Gambaudo, “Spaces of tilings, finite telescopic approximations, and gap-labeling”, to appear in Communications in Mathematical Physics), [Benameur and Oyono-Oyono 2003] (“Gap-labelling for quasi-crystals”, pp. 11–22 in Operator algebras and mathematical physics, Theta Foundation, Bucharest, 2003), and [Kaminker and Putnam 2003] (“A proof of the gap labeling conjecture”, Michigan Mathematical Journal 51 (2003), 537–546). It first appeared as a Featured Review in Mathematical Reviews, and is reprinted here by permission, with slight modifications. The three reviewed articles are herein referred to as BBG, BO and KP.

The Gap Labeling Theorem was originally conjectured in [Bellissard et al. 2000]. The problem arises in a mathematical version of solid state physics in the context of aperiodic tilings. Its three proofs, discovered independently by the authors above, all lie in K-theory. Here is the core result of these papers.