For this week’s SUMS talk we’re lucky to have Associate Professor Saul Schleimer from the University of Warwick, speaking about the 120-cell and the interlocking burr puzzles it inspires. He has asked that anyone who owns any of these puzzles to bring them along to the talk. Abstract: (Joint work with Henry Segerman.) We introduce Quintessence: a new family of 3D-printed interlocking puzzles based on the geometry and combinatorics of the 120-cell. I’ll review spheres, polytopes and stereographic projection in dimensions two and three before generalizing to dimension four. We’ll briefly discuss the quaternions, the three-sphere, isometries of three-space, polytopes, and particularly the four-dimensional 120-cell and its combinatorial Hopf fibration on our way to building the puzzles.