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CHSCHSUCBUCB ISIS Symmetry Congress 2001ISIS Symmetry Congress 2001
Symmetries on the Sphere
Carlo H. Séquin
University of California, Berkeley
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CHSCHSUCBUCB OutlineOutline
2 Tools to design / construct artistic artefacts:
“Escher Balls”: Spherical Escher Tilings
“Viae Globi”: Closed Curves on a Sphere
Discuss the use of Symmetry
Discuss Symmetry-Breaking in order to obtain artistically more interesting results.
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CHSCHSUCBUCB
Jane YenJane YenCarlo SCarlo Sééquinquin
UC BerkeleyUC Berkeley
[1] M.C. Escher, His Life and Complete Graphic Work
Spherical Escher TilingsSpherical Escher Tilings
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CHSCHSUCBUCB IntroductionIntroduction
M.C. Escher
graphic artist & print maker
myriad of famous planar tilings
why so few 3D designs?
[2] M.C. Escher: Visions of Symmetry
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CHSCHSUCBUCB Spherical TilingsSpherical Tilings
Spherical Symmetry is difficult
Hard to understand
Hard to visualize
Hard to make the final object
[1]
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CHSCHSUCBUCB Our GoalOur Goal
Develop a system to easily design and manufacture “Escher spheres” = spherical balls composed of identical tiles.
Provide visual feedback
Guarantee that the tiles join properly
Allow for bas-relief decorations
Output for manufacturing of physical models
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CHSCHSUCBUCB Interface DesignInterface Design
How can we make the system intuitive and easy to use?
What is the best way to communicate how spherical symmetry works?
[1]
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CHSCHSUCBUCB Spherical SymmetrySpherical Symmetry
The Platonic Solids
tetrahedron octahedron cube dodecahedron icosahedron
R3 R5 R5R3 R3 R2
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CHSCHSUCBUCB Introduction to TilingIntroduction to Tiling
Spherical Symmetry - defined by 7 groups
1) oriented tetrahedron 12 elem: E, 8C3, 3C2
2) straight tetrahedron 24 elem: E, 8C3, 3C2, 6S4, 6sd
3) double tetrahedron 24 elem: E, 8C3, 3C2, i, 8S4, 3sd
4) oriented octahedron/cube 24 elem: E, 8C3, 6C2, 6C4, 3C42
5) straight octahedron/cube 48 elem: E, 8C3, 6C2, 6C4, 3C42, i, 8S6, 6S4, 6sd, 6sd
6) oriented icosa/dodecah. 60 elem: E, 20C3, 15C2, 12C5, 12C52
7) straight icosa/dodecah. 120 elem: E, 20C3, 15C2, 12C5, 12C52, i, 20S6, 12S10,
12S103, 15s
Platonic Solids:
With duals:
1,2)
3)
6,7)4,5)
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CHSCHSUCBUCB
Escher Sphere EditorEscher Sphere Editor
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CHSCHSUCBUCB How the Program WorksHow the Program Works
Choose symmetry based on a Platonic solid
Choose an initial tiling pattern to edit
= starting place
Example:
Tetrahedron
R3
R2R3
R2
R3
R3
R3
R2
Tile 1 Tile 2
R3
R2
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CHSCHSUCBUCB Using an Initial Tiling PatternUsing an Initial Tiling Pattern
• Easier to understand consequences of moving points• Guarantees proper tiling• Requires user to select the “right” initial tile
[2]
Tile 1 Tile 2 Tile 2
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CHSCHSUCBUCB Modifying the TileModifying the Tile
Insert and move boundary points
system automatically updates the tile based on symmetry
Add interior detail points
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CHSCHSUCBUCB
Adding Bas-ReliefAdding Bas-Relief
Stereographically project and triangulate:
Radial offsets can be given to points: individually or in groups separate mode from editing boundary points
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CHSCHSUCBUCB
Creating a SolidCreating a Solid
The surface is extruded radially inward or outward extrusion, spherical or detailed base
Output in a format for free-form fabrication individual tiles or entire ball
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CHSCHSUCBUCB Several Fabrication TechnologiesSeveral Fabrication Technologies
Both are layered manufacturing technologies
Fused Deposition Modeling Z-Corp 3D Color Printer
- parts are made of plastic - starch powder glued together - each part is a solid color - parts can have multiple colors => assembly
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CHSCHSUCBUCB Fused Deposition ModelingFused Deposition Modeling
supportmaterial
movinghead
inside the FDM machine
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CHSCHSUCBUCB 3D-Printing (Z-Corporation)3D-Printing (Z-Corporation)
infiltrationde-powdering
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CHSCHSUCBUCB 12 Lizard Tiles (FDM)12 Lizard Tiles (FDM)
Pattern 1R3
R2R3
R2
R3
R3
R3
R2
Pattern 2
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CHSCHSUCBUCB 12 Fish Tiles (4 colors)12 Fish Tiles (4 colors)
Z-CorpSolid monolithic ball
FDM Hollow, hand-assembled
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CHSCHSUCBUCB 24 Bird Tiles24 Bird Tiles
Z-Corp4-color tiling
FDM2-color tiling
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CHSCHSUCBUCB Tiles Spanning Half the SphereTiles Spanning Half the Sphere
Z-Corp6-color tiling
FDM4-color tiling
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CHSCHSUCBUCB Hollow StructuresHollow Structures
Z-CorpBlow loose powder
from eye holes
FDM Hard to remove the
support material
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CHSCHSUCBUCB Frame StructuresFrame Structures
Z-CorpColorful but fragile
FDM Support removal tricky,but sturdy end-product
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CHSCHSUCBUCB
3D PrinterZ-Corp.
60 Highly Interlocking Tiles60 Highly Interlocking Tiles
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CHSCHSUCBUCB 60 Butterfly Tiles (FDM)60 Butterfly Tiles (FDM)
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CHSCHSUCBUCB PART 2: “Viae Globi”PART 2: “Viae Globi” (Roads on a Sphere)(Roads on a Sphere)
• Symmetrical, Symmetrical, closed closed curvescurves on a sphereon a sphere
• Inspiration: Inspiration:
Brent Collins’ Brent Collins’ “Pax Mundi”“Pax Mundi”
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CHSCHSUCBUCB Sculptures by Naum GaboSculptures by Naum Gabo
Pathway on a sphere:
Edge of surface is like seam of tennis ball;
==> 2-period Gabo curve.
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CHSCHSUCBUCB 2-period Gabo curve2-period Gabo curve
Approximation with quartic B-spline
with 8 control points per period,
but only 3 DOF are used.
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CHSCHSUCBUCB 3-period Gabo curve3-period Gabo curve
Same construction as for 2-period curve
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CHSCHSUCBUCB ““Pax Mundi” RevisitedPax Mundi” Revisited
Can be seen as:
Amplitude modulated, 4-period Gabo curve
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CHSCHSUCBUCB SLIDE-UI for “Pax Mundi” ShapesSLIDE-UI for “Pax Mundi” Shapes
Good combination of interactive 3D graphicsand parameterizable procedural constructs.
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CHSCHSUCBUCB FDM Part with SupportFDM Part with Support
as it comes out of the machine
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CHSCHSUCBUCB ““Viae Globi” Family Viae Globi” Family (Roads on a Sphere)(Roads on a Sphere)
2 3 4 5 periods
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CHSCHSUCBUCB 2-period Gabo sculpture2-period Gabo sculpture
Looks more like a surface than a ribbon on a sphere.
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CHSCHSUCBUCB Via Globi 3 (Stone)Via Globi 3 (Stone)
Wilmin Martono
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CHSCHSUCBUCB Via Globi 5 (Wood)Via Globi 5 (Wood)
Wilmin Martono
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CHSCHSUCBUCB Via Globi 5 (Gold)Via Globi 5 (Gold)
Wilmin Martono
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CHSCHSUCBUCB More Complex PathwaysMore Complex Pathways
Tried to maintain high degree of symmetry,
but wanted higly convoluted paths …
Not as easy as I thought !
Tried to work with Hamiltonian pathson the edges of a Platonic solid,but had only moderate success.
Used free-hand sketching with C-splines,
then edited control vertices coordinatesto adhere to desired symmetry group.
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CHSCHSUCBUCB ““Viae Globi”Viae Globi”
Sometimes I started by sketching on a tennis ball !
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CHSCHSUCBUCB A Better CAD Tool is Needed !A Better CAD Tool is Needed !
A way to make nice curvy paths on the surface of a sphere:==> C-splines.
A way to sweep interesting cross sectionsalong these spherical paths:==> SLIDE.
A way to fabricate the resulting designs:==> Our FDM machine.
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CHSCHSUCBUCB ““Circle-Splines” (SIGGRAPH 2001)Circle-Splines” (SIGGRAPH 2001)
Carlo SéquinJane Yen
On the plane -- and on the sphere
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CHSCHSUCBUCB Defining the Basic Path ShapesDefining the Basic Path Shapes
Use Platonic or Archimedean solids as “guides”:
Place control points of an approximating spline at the vertices,
or place control points of an interpolating spline at edge-midpoints.
Spline formalism will do the smoothing.
Maintain some desirable degree of symmetry,
and make sure that curve closes – difficult !
Often leads to the same basic shapes again …
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CHSCHSUCBUCB Hamiltonian PathsHamiltonian Paths
Strictly realizable only on octahedron! Gabo-2 path.
Pseudo Hamiltonian path (multiple vertex visits) Gabo-3 path.
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CHSCHSUCBUCB Another Conceptual ApproachAnother Conceptual Approach
Start from a closed curve, e.g., the equator
And gradually deform it by introducing twisting vortex movements:
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CHSCHSUCBUCB ““Maloja” -- FDM partMaloja” -- FDM part
A rather winding Swiss mountain pass road in the upper Engadin.
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CHSCHSUCBUCB ““Stelvio”Stelvio”
An even more convoluted alpine pass in Italy.
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CHSCHSUCBUCB ““Altamont”Altamont”
Celebrating American multi-lane highways.
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CHSCHSUCBUCB ““Lombard”Lombard”
A very famous crooked street in San Francisco
Note that I switched to a flat ribbon.
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CHSCHSUCBUCB Varying the Azimuth ParameterVarying the Azimuth Parameter
Setting the orientation of the cross section …
… by Frenet frame … using torsion-minimization withtwo different azimuth values
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CHSCHSUCBUCB ““Aurora”Aurora”
Path ~ Via Globi 2
Ribbon now lies perpendicular to sphere surface.
Reminded me ofthe bands in anAurora Borrealis.
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CHSCHSUCBUCB ““Aurora - T”Aurora - T”
Same sweep path ~ Via Globi 2
Ribbon now lies tangential to sphere surface.
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CHSCHSUCBUCB ““Aurora – F” (views from 3 sides)Aurora – F” (views from 3 sides)
Still the same sweep path ~ Via Globi 2
Ribbon orientation now determined by Frenet frame.
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CHSCHSUCBUCB ““Aurora-M”Aurora-M”
Same path on sphere,
but more play with the swept cross section.
This is a Moebius band.
It is morphed from a concave shape at the bottom to a flat ribbon at the top of the flower.
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CHSCHSUCBUCB ConclusionsConclusions
Focus on spherical symmetries to make artistic artefacts.
Undecorated Platonic solids are artistically not too interesting (too much symmetry).
Breaking the mirror symmetries leads to more interesting shapes (snubcube)
use tiles with rotational symmetries, or asymmetrical wiggles on Gabo curves.
Can also break symmetry with a varying orientation of the swept cross section.
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CHSCHSUCBUCB We have come full circle …We have come full circle …