A Pythagoras Tree Grown from a Dodecahedron
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 ≈ 0.4346 and twisted by 45∘ 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/√5 and the twist arccos(1/√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.
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BibTeX
@misc{ibarra2026pythagoras,
title = {{A Pythagoras Tree Grown from a Dodecahedron}},
author = {Miguel Esteban Pérez Ibarra},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33975},
eprint = {2610.33975},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33975},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - A Pythagoras Tree Grown from a Dodecahedron
AU - Miguel Esteban Pérez Ibarra
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.33975
UR - https://arxiv.org/abs/2610.33975
AB - 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 ≈ 0.4346 and twisted by 45^∘ 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/√5 and the twist arccos(1/√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.
ER -