1. MOSAIC, Seattle, Aug. 2000 Turning Mathematical Models into Sculptures Carlo H. Séquin University of California, Berkeley

2. Boy Surface in Oberwolfach ä Sculpture constructed by Mercedes Benz ä Photo from John Sullivan

3. Boy Surface by Helaman Ferguson ä Marble ä From: “Mathematics in Stone and Bronze” by Claire Ferguson

4. Boy Surface by Benno Artmann ä From home page of Prof. Artmann, TU-Darmstadt ä after a sketch by George Francis.

5. Samples of Mathematical Sculpture Questions that may arise: ä Are the previous sculptures really all depicting the same object ? ä What is a “Boy surface” anyhow ?

6. The Gist of my Talk Topology 101: ä Study five elementary 2-manifolds (which can all be formed from a rectangle) Art-Math 201: ä The appearance of these shapes as artwork (when do math models become art ? )

7. What is Art ? What is Art ?

8. Five Important Two-Manifolds cylinderMöbius band torusKlein bottlecross-cap X=0 X=0 X=0 X=0 X=1 G=1 G=2 G=1

9. Deforming a Rectangle ä All five manifolds can be constructed by starting with a simple rectangular domain and then deforming it and gluing together some of its edges in different ways. cylinder Möbius band torus Klein bottle cross-cap

10. Cylinder Construction

11. Möbius Band Construction

12. Cylinders as Sculptures Max Bill John Goodman

13. The Cylinder in Architecture Chapel

14. Möbius Sculpture by Max Bill

15. Möbius Sculptures by Keizo Ushio

16. More Split Möbius Bands Typical lateral split by M.C. Escher And a maquette made by Solid Free-form Fabrication

17. Torus Construction ä Glue together both pairs of opposite edges on rectangle ä Surface has no edges ä Double-sided surface

18. Torus Sculpture by Max Bill

19. “Bonds of Friendship” J. Robinson 1979

20. Proposed Torus “Sculpture” “Torus! Torus!” inflatable structure by Joseph Huberman

21. “Rhythm of Life” by John Robinson “DNA spinning within the Universe” 1982

22. Virtual Torus Sculpture “Rhythm of Life” by John Robinson, emulated by Nick Mee at Virtual Image Publishing, Ltd. Note: Surface is represented by a loose set of bands ==> yields transparency

23. Klein Bottle -- “Classical” ä Connect one pair of edges straight and the other with a twist ä Single-sided surface -- (no edges)

24. Klein Bottles -- virtual and real Computer graphics by John Sullivan Klein bottle in glass by Cliff Stoll, ACME

25. Many More Klein Bottle Shapes ! Klein bottles in glass by Cliff Stoll, ACME

26. Klein Mugs Klein bottle in glass by Cliff Stoll, ACME Fill it with beer --> “Klein Stein”

27. Dealing with Self-intersections Different surfaces branches should “ignore” one another ! One is not allowed to step from one branch of the surface to another. ==> Make perforated surfaces and interlace their grids. ==> Also gives nice transparency if one must use opaque materials. ==> “Skeleton of a Klein Bottle.”

28. Klein Bottle Skeleton (FDM)

29. Struts don’t intersect !

30. Fused Deposition Modeling

31. Looking into the FDM Machine

32. Layered Fabrication of Klein Bottle Support material

33. Another Type of Klein Bottle ä Cannot be smoothly deformed into the classical Klein Bottle ä Still single sided -- no edges

34. ä Woven by Carlo Séquin, 16’’, 1997 Figure-8 Klein Bottle

35. Triply Twisted Fig.-8 Klein Bottle

37. Avoiding Self-intersections ä Avoid self-intersections at the crossover line of the swept fig.-8 cross section. ä This structure is regular enough so that this can be done procedurally as part of the generation process. ä Arrange pattern on the rectangle domain as shown on the left. ä After the fig.-8 - fold, struts pass smoothly through one another. ä Can be done with a single thread for red and green !

38. Single-thread Figure-8 Klein Bottle Modeling with SLIDE

39. Zooming into the FDM Machine

40. Single-thread Figure-8 Klein Bottle As it comes out of the FDM machine

41. Single-thread Figure-8 Klein Bottle

42. The Doubly Twisted Rectangle Case ä This is the last remaining rectangle warping case. ä We must glue both opposing edge pairs with a 180º twist. Can we physically achieve this in 3D ?

43. Cross-cap Construction

44. Significance of Cross-cap ä What is this beast ? ä A model of the Projective Plane ä An infinitely large flat plane. ä Closed through infinity, i.e., lines come back from opposite direction. ä But all those different lines do NOT meet at the same point in infinity; their “infinity points” form another infinitely long line.

45. The Projective Plane C PROJECTIVE PLANE -- Walk off to infinity -- and beyond … come back upside-down from opposite direction. Projective Plane is single-sided; has no edges.

46. Cross-cap on a Sphere Wood and gauze model of projective plane

47. “Torus with Crosscap” Helaman Ferguson ( Torus with Crosscap = Klein Bottle with Crosscap )

48. “Four Canoes” by Helaman Ferguson

49. Other Models of the Projective Plane ä Both, Klein bottle and projective plane are single-sided, have no edges. (They differ in genus, i.e., connectivity) ä The cross cap on a torus models a Klein bottle. ä The cross cap on a sphere models the projective plane, but has some undesirable singularities. ä Can we avoid these singularities ? ä Can we get more symmetry ?

50. Steiner Surface (Tetrahedral Symmetry) ä Plaster Model by T. Kohono

51. Construction of Steiner Surface ä Start with three orthonormal squares … … connect the edges (smoothly). --> forms 6 “Whitney Umbrellas” (pinch points with infinite curvature)

52. Steiner Surface Parametrization ä Steiner surface can best be built from a hexagonal domain. Glue opposite edges with a 180º twist.

53. Again: Alleviate Self-intersections Strut passes through hole

54. Skeleton of a Steiner Surface

55. Steiner Surface ä has more symmetry; ä but still has singularities (pinch points). Can such singularities be avoided ? (Hilbert)

56. Can Singularities be Avoided ? Werner Boy, a student of Hilbert, was asked to prove that it cannot be done. But found a solution in 1901 ! ä 3-fold symmetry ä based on hexagonal domain

57. Model of Boy Surface Computer graphics by François Apéry

58. Model of Boy Surface Computer graphics by John Sullivan

59. Model of Boy Surface Computer graphics by John Sullivan

60. Quick Surprise Test ä Draw a Boy surface (worth 100% of score points)...

61. Another “Map” of the “Boy Planet” ä From book by Jean Pierre Petit “Le Topologicon” (Belin & Herscher)

62. Double Covering of Boy Surface ä Wire model by Charles Pugh ä Decorated by C. H. Séquin: ä Equator ä 3 Meridians, 120º apart

63. Revisit Boy Surface Sculptures Helaman Ferguson - Mathematics in Stone and Bronze

64. Boy Surface by Benno Artmann ä Windows carved into surface reveal what is going on inside. (Inspired by George Francis)

65. Boy Surface in Oberwolfach ä Note: parametrization indicated by metal bands; singling out “north pole”. ä Sculpture constructed by Mercedes Benz ä Photo by John Sullivan

66. Boy Surface Skeleton Shape defined by elastic properties of wooden slats.

67. Boy Surface Skeleton (again)

68. Goal: A “Regular” Tessellation ä “Regular” Tessellation of the Sphere (Buckminster Fuller Domes.)

69. “Ideal” Sphere Parametrization Buckminster Fuller Dome: almost all equal sized triangle tiles.

70. “Ideal” Sphere Parametrization Epcot Center Sphere

71. Tessellation from Surface Evolver ä Triangulation from start polyhedron. ä Subdivision and merging to avoid large, small, and skinny triangles. ä Mesh dualization. ä Strut thickening. ä FDM fabrication. ä Quad facet ! ä Intersecting struts.

72. Paper Model with Regular Tiles ä Only meshes with 5, 6, or 7 sides. ä Struts pass through holes. ä Only vertices where 3 meshes join. --> Permits the use of a modular component...

73. The Tri-connector

74. Tri-connector Constructions

75. Tri-connector Ball (20 Parts)

76. Expectations ä Tri-connector surface will be evenly bent, with no sharp kinks. ä It will have intersections that demonstrate the independence of the two branches. ä Result should be a pleasing model in itself. ä But also provides a nice loose model of the Boy surface on which I can study various parametrizations, geodesic lines...

77. Hopes ä This may lead to even better models of the Boy surface: ä e.g., by using the geodesic lines to define ribbons that describe the surface ä (this surface will keep me busy for a while yet !)

78. Conclusions ä There is no clear line that separates mathematical models and art work. ä Good models are pieces of art in themselves. ä Much artwork inspired by such models is no longer a good model for understanding these more complicated surfaces. ä My goal is to make a few great models that are appreciated as good geometric art, and that also serve as instructional models.

79. End of Talk

80. === spares ===

81. Rotating Torus

82. Looking into the FDM Machine