ISIS Symmetry Congress 2001

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ISIS Symmetry Congress 2001

• Symmetries on the Sphere
• Carlo H. Séquin
• University of California, Berkeley

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Outline

• 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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Spherical Escher Tilings
Jane YenCarlo SéquinUC Berkeley
1 M.C. Escher, His Life and Complete Graphic
Work
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Introduction

• 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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Spherical Tilings

• Spherical Symmetry is difficult
• Hard to understand
• Hard to visualize
• Hard to make the final object

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Our 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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Interface Design

• How can we make the system intuitive and easy to
  use?
• What is the best way to communicate how spherical
  symmetry works?

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Spherical Symmetry

• The Platonic Solids

tetrahedron
octahedron
cube
dodecahedron
icosahedron
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Introduction 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
1,2)
4,5)
6,7)
With duals
3)
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Escher Sphere Editor
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How the Program Works

• Choose symmetry based on a Platonic solid
• Choose an initial tiling pattern to edit
• starting place
• Example
• Tetrahedron

R3
R2
Tile 1
Tile 2
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Using an Initial Tiling Pattern

• Easier to understand consequences of moving
  points
• Guarantees proper tiling
• Requires user to select the right initial
  tile

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Tile 2
Tile 1
Tile 2
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Modifying the Tile

• Insert and move boundary points
• system automatically updates the tile based on
  symmetry
• Add interior detail points

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Adding 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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Creating 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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Several Fabrication Technologies

• Both are layered manufacturing technologies

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Fused Deposition Modeling
moving head
inside the FDM machine
support material
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3D-Printing (Z-Corporation)
infiltration
de-powdering
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12 Lizard Tiles (FDM)
Pattern 1
Pattern 2
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12 Fish Tiles (4 colors)
FDM Hollow, hand-assembled
Z-Corp Solid monolithic ball
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24 Bird Tiles
FDM2-color tiling
Z-Corp 4-color tiling
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Tiles Spanning Half the Sphere
FDM4-color tiling
Z-Corp 6-color tiling
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Hollow Structures
FDM Hard to remove the support material
Z-Corp Blow loose powder from eye holes
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Frame Structures
FDM Support removal tricky, but sturdy
end-product
Z-Corp Colorful but fragile
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60 Highly Interlocking Tiles
3D Printer Z-Corp.
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60 Butterfly Tiles (FDM)
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PART 2 Viae Globi (Roads on a Sphere)

• Symmetrical, closed curves on a sphere
• Inspiration Brent Collins Pax Mundi

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Sculptures by Naum Gabo

• Pathway on a sphere
• Edge of surface is like seam of tennis ball
• gt 2-period Gabo curve.

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2-period Gabo curve

• Approximation with quartic B-splinewith 8
  control points per period,but only 3 DOF are
  used.

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3-period Gabo curve

• Same construction as for 2-period curve

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Pax Mundi Revisited

• Can be seen as Amplitude modulated, 4-period
  Gabo curve

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SLIDE-UI for Pax Mundi Shapes
Good combination of interactive 3D graphicsand
parameterizable procedural constructs.
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FDM Part with Support

• as it comes out of the machine

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Viae Globi Family (Roads on a Sphere)
2 3 4
5 periods
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2-period Gabo sculpture

• Looks more like a surface than a ribbon on a
  sphere.

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Via Globi 3 (Stone)
Wilmin Martono
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Via Globi 5 (Wood)
Wilmin Martono
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Via Globi 5 (Gold)
Wilmin Martono
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More 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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Viae Globi

• Sometimes I started by sketching on a tennis ball
  !

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A Better CAD Tool is Needed !

• A way to make nice curvy paths on the surface of
  a spheregt C-splines.
• A way to sweep interesting cross sectionsalong
  these spherical pathsgt SLIDE.
• A way to fabricate the resulting designsgt Our
  FDM machine.

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Circle-Splines (SIGGRAPH 2001)
Carlo Séquin Jane Yen
On the plane -- and on the sphere
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Defining 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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Hamiltonian Paths
Pseudo Hamiltonian path (multiple vertex
visits)? Gabo-3 path.

• Strictly realizable only on octahedron!? Gabo-2
  path.

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Another Conceptual Approach

• Start from a closed curve, e.g., the equator
• And gradually deform it by introducing twisting
  vortex movements

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Maloja -- FDM part

• A rather winding Swiss mountain pass road in the
  upper Engadin.

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Stelvio

• An even more convoluted alpine pass in Italy.

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Altamont

• Celebrating American multi-lane highways.

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Lombard

• A very famous crooked street in San Francisco
• Note that I switched to a flat ribbon.

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Varying the Azimuth Parameter

• Setting the orientation of the cross section

using torsion-minimization with two different
azimuth values
by Frenet frame
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Aurora

• Path Via Globi 2
• Ribbon now lies perpendicular to sphere surface.
• Reminded me ofthe bands in anAurora Borrealis.

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Aurora - T

• Same sweep path Via Globi 2
• Ribbon now lies tangential to sphere surface.

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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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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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Conclusions

• 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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We have come full circle
QUESTIONS ?