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We provide a general set of rules for extracting the data defining a quivergauge theory from a given toric Calabi-Yau singularity. Our method combinesinformation from the geometry and topology of Sasaki-Einstein manifolds,AdS/CFT, dimers, and brane tilings. We explain how the field content, quantumnumbers, and superpotential of a superconformal gauge theory on D3-branesprobing a toric Calabi-Yau singularity can be deduced. The infinite family oftoric singularities with known horizon Sasaki-Einstein manifolds L^{a,b,c} isused to illustrate these ideas. We construct the corresponding quiver gaugetheories, which may be fully specified by giving a tiling of the plane byhexagons with certain gluing rules. As checks of this construction, we performa-maximisation as well as Z-minimisation to compute the exact R-charges of anarbitrary such quiver. We also examine a number of examples in detail,including the infinite subfamily L^{a,b,a}, whose smallest member is theSuspended Pinch Point.Comment: 46 pages, 21 figures, JHEP3. v2: minor changes and references added.  v3: typos and two figures correcte

Gauge theories from toric geometry and brane tilings