A Pythagoras Tree Grown from a Dodecahedron

Overview & Synopsis

The Pythagoras tree of A. E. Bosman (1942) is drawn with squares in the plane and, in the three-dimensional versions we know of, with cubes. This article documents a tree built instead from regular dodecahedra: each dodecahedron carries five smaller copies, one on each edge of a single pentagonal "crown" face, scaled by $g \approx 0.4346$ and twisted by $45^\circ$ about the edge. The construction first existed only as a pair of rendered images and two undocumented C++ programs. We recover its exact parameters from the original Blender scene, reformulate it as an iterated function system with condensation, and prove with a short computer-assisted argument that it never collides with itself: two of its dodecahedra overlap only when one is the parent of the other. A Pythagorean identity for the six face axes of the dodecahedron singles out the scale $1/\sqrt{5}$ and the twist $\arccos(1/\sqrt{5})$. With these values each child rests flat against a face of its parent, the five children together have exactly the parent's surface area, and the crown of the tree is a fractal of Hausdorff dimension exactly 2. Finally we ask whether the tree can grow on all twelve faces at once. A parity argument shows that at the artwork's proportions it cannot, but smaller scale factors admit certified clash-free all-faces trees.