INVOLUTE presents sporadic groups (and other notable finite simple groups) through twisty-puzzle-inspired grids; solving them is an interactive way to build intuition for group theory structures. Move the pieces, inspect cycles, and design algorithms with the help of built-in analysis tools.

Each layout is the product of a long computational search. Elements drawn from a collection of every possible group involution are combined into thousands of 3D graph candidates by letting the pieces arrange themselves into their natural geometric positions (check them out in the app's 3D view). Considering only the least tangled and most symmetric results, I flattened the best of them by hand into playable 2D grids.

The group structures expressed by these puzzles were discovered, rather than designed, and yet brilliant intricacies seem to emerge out of nowhere: positions you can't quite reach, pieces trapped in hidden orbits, organized behaviors among a multitude of points. The grids seem to fight back against your best efforts to rearrange them. As you attempt to solve these puzzles, I hope that you might rediscover the beauty in the exceptional groups that have captivated mathematicians.