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
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2
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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
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What is creativity ?
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Where do novel ideas come from ?
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Are there any truly novel ideas ?
Or are they evolutionary developments,
and just combinations of known ideas ?
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How do we evaluate open-ended designs ?
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What’s a good solution to a problem ?
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How do we know when we are done ?
“Do This in 3 D !”
What are the plausible constraints ?
n
n
n
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3D array of 2 x 2 x 2 vertices
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Visit all vertices exactly once
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Only nearest-neighbor connections
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Fill “local” neighborhood first
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Aim for self-similarity
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Recursive formulation (for arbitrary n)
Construction of 3D Hilbert Curve
Construction of 3D Hilbert Curve
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Use this element with proper orientation, mirroring.
Design Choices: 3D Hilbert Curve
What are the things one might optimize ?
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Maximal symmetry
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Overall closed loop
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No consecutive collinear segments
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No (3 or 4 ?) coplanar segment sequence
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others ... ?
 More than one acceptable solution !
Typical Early Student Solution
Design Flaws:
D. Garcia, and T. Eladi (1994)
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2 collinear segments
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less than maximal
symmetry
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4 coplanar segments
Jane Yen: “Hilbert Radiator Pipe” (2000)
Flaws
( from a sculptor’s
point of view ):
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4 coplanar segments
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Not a closed loop
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Broken symmetry
Plastic Model (from FDM) (1998)
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Support removal can be tedious, difficult !
Metal Sculpture at SIGGRAPH 2006
Design:
closed loop
maximal symmetry
at most 3 coplanar segments