We introduce Quintessence: a family of burr puzzles based on the geometry and combinatorics of the 120-cell. Written for a broad, mathematically-minded audience, our paper discusses 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 catalog of new puzzles.

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