The curves not carried

Suppose t is a train track on a surface S. Let C(t) be the set of isotopy classes of simple closed curves carried by t. Masur and Minsky prove that C(t) is quasi-convex inside the curve complex C(S). We prove that the complement, C(S) - C(t), is quasi-convex

Puzzling the 120-cell

We introduce Quintessence: a family of burr puzzles based on the geometry and combinatorics of the 120-cell. We discuss the regular polytopes, their symmetries, the dodecahedron as an important special case, the three-sphere, and the quaternions. We then construct the 120-cell, giving an illustrated survey of its geometry and combinatorics. This done, we describe the pieces out of which Quintessence is made. The design of our puzzle pieces uses a drawing technique of Leonardo da Vinci; the paper ends with a catalogue of new puzzles.Comment: 25 pages, many figures. Exposition and figures improved throughout. This is the long version of the shorter published versio