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ISIS Symmetry Congress 2001

Explore the use of symmetry and symmetry-breaking in creating artistic Escher spheres, spherical balls composed of identical tiles. Learn about the challenges of working with spherical symmetry and how to design and manufacture these unique art pieces. Also, discover the creation of pathways on a sphere inspired by Naum Gabo's sculptures.

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ISIS Symmetry Congress 2001

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  1. ISIS Symmetry Congress 2001 Symmetries on the Sphere Carlo H. Séquin University of California, Berkeley

  2. Outline 2 Tools to design / construct artistic artefacts: • “Escher Balls”: Spherical Escher Tilings • “Viae Globi”: Closed Curves on a Sphere • Discuss the use ofSymmetry • Discuss Symmetry-Breakingin order to obtain artistically more interesting results.

  3. Spherical Escher Tilings Jane YenCarlo SéquinUC Berkeley [1] M.C. Escher, His Life and Complete Graphic Work

  4. 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

  5. Spherical Tilings • Spherical Symmetry is difficult • Hard to understand • Hard to visualize • Hard to make the final object [1]

  6. 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

  7. [1] Interface Design • How can we make the system intuitive and easy to use? • What is the best way to communicate how spherical symmetry works?

  8. R3 R5 R3 R3 R5 R2 Spherical Symmetry • The Platonic Solids tetrahedron octahedron cube dodecahedron icosahedron

  9. 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)

  10. Escher Sphere Editor

  11. R3 R3 R2 R2 R2 R3 R3 R3 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 2 Tile 1

  12. Using an Initial Tiling Pattern • Easier to understand consequences of moving points • Guarantees proper tiling • Requires user to select the “right” initial tile [2] Tile 2 Tile 1 Tile 2

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

  14. Adding Bas-Relief • Stereographically project and triangulate: • Radial offsets can be given to points: • individually or in groups • separate mode from editing boundary points

  15. 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

  16. Fused Deposition ModelingZ-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 Several Fabrication Technologies • Both are layered manufacturing technologies

  17. Fused Deposition Modeling moving head inside the FDM machine support material

  18. 3D-Printing (Z-Corporation) infiltration de-powdering

  19. R3 R3 R2 R2 R2 R3 R3 R3 12 Lizard Tiles (FDM) Pattern 1 Pattern 2

  20. 12 Fish Tiles (4 colors) FDM Hollow, hand-assembled Z-Corp Solid monolithic ball

  21. 24 Bird Tiles FDM2-color tiling Z-Corp 4-color tiling

  22. Tiles Spanning Half the Sphere FDM4-color tiling Z-Corp 6-color tiling

  23. Hollow Structures FDM Hard to remove the support material Z-Corp Blow loose powder from eye holes

  24. Frame Structures FDM Support removal tricky, but sturdy end-product Z-Corp Colorful but fragile

  25. 60 Highly Interlocking Tiles 3D Printer Z-Corp.

  26. 60 Butterfly Tiles (FDM)

  27. PART 2: “Viae Globi”(Roads on a Sphere) • Symmetrical, closed curves on a sphere • Inspiration: Brent Collins’ “Pax Mundi”

  28. Sculptures by Naum Gabo Pathway on a sphere: Edge of surface is like seam of tennis ball; ==> 2-period Gabo curve.

  29. 2-period Gabo curve • Approximation with quartic B-splinewith 8 control points per period,but only 3 DOF are used.

  30. 3-period Gabo curve Same construction as for 2-period curve

  31. “Pax Mundi” Revisited • Can be seen as:Amplitude modulated, 4-period Gabo curve

  32. SLIDE-UI for “Pax Mundi” Shapes Good combination of interactive 3D graphicsand parameterizable procedural constructs.

  33. FDM Part with Support as it comes out of the machine

  34. “Viae Globi” Family (Roads on a Sphere) 2 3 4 5 periods

  35. 2-period Gabo sculpture • Looks more like a surface than a ribbon on a sphere.

  36. Via Globi 3 (Stone) Wilmin Martono

  37. Via Globi 5 (Wood) Wilmin Martono

  38. Via Globi 5 (Gold) Wilmin Martono

  39. 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.

  40. “Viae Globi” • Sometimes I started by sketching on a tennis ball !

  41. 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.

  42. “Circle-Splines” (SIGGRAPH 2001) Carlo Séquin Jane Yen On the plane -- and on the sphere

  43. 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 …

  44. Hamiltonian Paths Strictly realizable only on octahedron! Gabo-2 path. Pseudo Hamiltonian path (multiple vertex visits) Gabo-3 path.

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

  46. “Maloja” -- FDM part • A rather winding Swiss mountain pass road in the upper Engadin.

  47. “Stelvio” • An even more convoluted alpine pass in Italy.

  48. “Altamont” • Celebrating American multi-lane highways.

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

  50. 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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