Pythagoras tree built from dodecahedrons avoids self-collisions
A Pythagoras Tree Grown from a Dodecahedron
Computational Geometry
Summary
A Pythagoras tree is a fractal usually made of squares or cubes. In this work, the authors show how to build such a tree using 3D shapes called dodecahedra, carefully placing smaller copies on its edges so they don't collide. They provide exact mathematical details, prove no overlapping occurs between unrelated parts, and reveal special scaling and twisting values that make the tree’s surface area and geometry mathematically elegant. They also explore whether the tree can grow on all faces simultaneously but find limitations based on shape and scale.
What this means in practice
- •For computer graphics artists: Create intricate 3D fractal tree structures with precise mathematical guarantees against self-overlap for use in visual art and animations.
- •For computer graphics software developers: Implement fractal generation tools incorporating dodecahedron-based Pythagoras trees with proven collision-free geometry to expand modeling capabilities.
Authors
Miguel Esteban Pérez Ibarra
Abstract
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.