  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.
