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Artist’s Sketch, SIGGRAPH 2006, Boston, 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 ? n n n u 3D array of 2 x 2 x 2 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 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: D. Garcia, and T. Eladi (1994) u 2 collinear segments u less than maximal symmetry u 4 coplanar segments 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: closed loop maximal symmetry at most 3 coplanar segments