computer graphics software developers

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.

Abstract: Autoregressive models have proven effective for triangular mesh generation by transforming it into predicting the next face, converting the non-differentiable connectivity prediction into a sequence of token predictions. However, generating quad-only meshes remains challenging because quad faces impose stricter topological constraints than triangles. The model must not only serialize quad faces into a valid token sequence, but also support unambiguous connectivity reconstruction during de-serialization. To address this, we propose Q-MAGIC, a novel autoregressive model for quad mesh generation. We first introduce a tailored serialization scheme for quad meshes, which requires at least two new vertices for the next face and is less flexible than triangles due to strict wiring constraints. To meet these requirements and reduce serialization length, we propose a singularity-aware serialization that prioritizes wiring from singular points, leading to more coherent traversals. Additionally, we propose a circular relative offset tokenization that encodes geometry mainly via local relative offsets. Built upon this representation, an autoregressive Transformer learns the underlying distribution of quad meshes via next-token prediction, enabling effective quad mesh generation. Extensive experiments demonstrate that Q-MAGIC produces more compact sequences while consistently improving geometric fidelity and mesh quality compared to state-of-the-art methods.

Abstract: Subdivision surfaces represent free-form geometry through a sparse control cage, but recovering that cage from a dense mesh is not merely a fitting problem: the system must infer where control is needed, which curves encode design features, and when an initial result should be revised. We present SubDGuide, a modeler-inspired agentic workflow for this task. Stage A reads aligned multiview geometric evidence and produces a compact plan for cage resolution, feature mapping, and broad-form fitting. Stage B inspects the resulting surface, requests targeted diagnostics, and chooses repair, rollback, or stopping actions. Geometry tools execute and verify every change; the planner never generates vertices or connectivity. On a fixed evaluation cohort, SubDGuide improves all five reported metrics over automatic remeshing: median Chamfer-L1 decreases from 0.641% to 0.443% of the target bounding-box diagonal, and F-score at a 1% tolerance increases from 82.00% to 92.78%. Stateful feedback improves four of five reported median metrics over one-shot planning, while verification retains the earlier checkpoint when a proposal is unhelpful. The same interface supports six multimodal planners, showing how semantic judgment can guide a practical, inspectable mesh-to-SubD workflow.