This is the story of a simple “subdivision rule” — a rule for breaking a pentagon into six pentagonal pieces — which leads us on a circuitous route through a lot of mathematical territory. The journey starts with a simple application of circle packing to create some pleasing “almost round” embeddings, leading to an infinite ‘regular’ conformal tiling of the plane, and then coming across a rational map and its iterates, a Grothendieck “dessin d'enfant” and its infinite cousin, associated with a Konigsfunction, and finally to a “pentagonal” number. This will be a largely visual tour representing work by the speaker and Phil Bower, Jim Cannon, Bill Floyd, Walter Parry, and Rick Kenyon.