Artist’s Sketch, SIGGRAPH 2006, Boston, Hilbert Cube 512 Hilbert Cube 512

The 2D Hilbert Curve (1891) A plane-filling Peano curve Fall 1983: CS Graduate Course: “Creative Geometric Modeling” Do This In 3 D !

Artist’s Use of the Hilbert Curve Helaman Ferguson, “Umbilic Torus NC” Silicon bronze, 27 x 27 x 9 in., SIGGRAPH’86

Construction of the 2D Hilbert Curve 1 2 3

This is one example from a talk presented at Bridges 1999 Analogies from 2D to 3D Exercises in Disciplined Creativity Carlo H. Séquin University of California, Berkeley

Motivation: CS 285 Design Exercises u What is creativity ? u Where do novel ideas come from ? u Are there any truly novel ideas ? Or are they evolutionary developments, and just combinations of known ideas ? u How do we evaluate open-ended designs ? u What’s a good solution to a problem ? u How do we know when we are done ?

“Do This in 3 D !” What are the plausible constraints ? u 3D array of 2 n x 2 n x 2 n vertices u Visit all vertices exactly once u Only nearest-neighbor connections u Fill “local” neighborhood first u Aim for self-similarity u Recursive formulation (for arbitrary n)

Construction of 3D Hilbert Curve

u Use this element with proper orientation, mirroring.

Design Choices: 3D Hilbert Curve What are the things one might optimize ? u Maximal symmetry u Overall closed loop u No consecutive collinear segments u No (3 or 4 ?) coplanar segment sequence u others... ?  More than one acceptable solution !

Typical Early Student Solution Design Flaws: u 2 collinear segments u less than maximal symmetry u 4 coplanar segments D. Garcia, and T. Eladi (1994)

Jane Yen: “Hilbert Radiator Pipe” (2000) Flaws ( from a sculptor’s point of view ): u 4 coplanar segments u Not a closed loop u Broken symmetry

Plastic Model (from FDM) (1998) u Support removal can be tedious, difficult !

Metal Sculpture at SIGGRAPH 2006 Design: u closed loop u maximal symmetry u at most 3 coplanar segments