1. ISIS Symmetry Congress 2001
Florida 1999
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 of Symmetry
Discuss Symmetry-Breaking in order to obtain artistically more interesting results.

3. Spherical Escher Tilings
Jane Yen Carlo Séquin UC 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. Interface Design How can we make the system intuitive and easy to use?
What is the best way to communicate how spherical symmetry works?
[1]

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

9. Introduction to Tiling
Spherical Symmetry - defined by 7 groups
1) oriented tetrahedron elem: E, 8C3, 3C2
2) straight tetrahedron elem: E, 8C3, 3C2, 6S4, 6sd
3) double tetrahedron 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 elem: E, 20C3, 15C2, 12C5, 12C52
7) straight icosa/dodecah elem: E, 20C3, 15C2, 12C5, 12C52, i, 20S6, 12S10, S103, 15s
Platonic
Solids:
1,2)
4,5)
6,7)
With duals: 3)

10. Escher Sphere Editor

11. How the Program Works Choose symmetry based on a Platonic solid
Choose an initial tiling pattern to edit
= starting place
Example:
Tetrahedron R3 R2 R3 R2 R3 R2
Tile 1
Tile 2

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 1
Tile 2
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. Several Fabrication 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
Both are layered manufacturing technologies

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

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

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

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

21. 24 Bird Tiles
FDM 2-color tiling
Z-Corp
4-color tiling

22. Tiles Spanning Half the Sphere
FDM 4-color tiling
Z-Corp
6-color tiling

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

24. Support removal tricky, but sturdy end-product
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-spline with 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
Florida 1999
SLIDE-UI for “Pax Mundi” Shapes
Good combination of interactive 3D graphics and parameterizable procedural constructs.
Good combination of interactive 3D graphics and parameterizable procedural constructs.

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

34. “Viae Globi” Family (Roads on a Sphere)
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 paths on the edges of a Platonic solid, but had only moderate success.
Used free-hand sketching with C-splines,
then edited control vertices coordinates to 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 sections along 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
Florida 1999
Varying the Azimuth Parameter
Setting the orientation of the cross section …
The shape of the sweep curve on the sphere is just ONE aspect of the sculpture !
We can also play with the cross section
And with the orientation of the cross section around the sweep curve.
… by Frenet frame
… using torsion-minimization with
two different azimuth values

51. “Aurora” Path ~ Via Globi 2
Ribbon now lies perpendicular to sphere surface.
Reminded me of the bands in an Aurora Borrealis.

52. “Aurora - T” Same sweep path ~ Via Globi 2
Ribbon now lies tangential to sphere surface.

53. “Aurora – F” (views from 3 sides)
Still the same sweep path ~ Via Globi 2
Ribbon orientation now determined by Frenet frame.

54. “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.

55. 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.

56. We have come full circle …
QUESTIONS ?