MOSAIC, Seattle, Aug. 2000

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

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

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

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

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

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

7. What is Art ?

8. Five Important Two-Manifolds X=0 X=0X=0 X=0 X=1G=1 G=2 G=1 cylinder Möbius band torus Klein bottle cross-cap

9. Deforming a Rectangle • 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 John Goodman Max Bill

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 splitby M.C. Escher And a maquette made by Solid Free-form Fabrication

17. Torus Construction • Glue together both pairs of opposite edges on rectangle • Surface has no edges • 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 Note: Surface is representedby a loose set of bands ==> yields transparency “Rhythm of Life” by John Robinson, emulated by Nick Mee at Virtual Image Publishing, Ltd.

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

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

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

26. Klein Mugs Klein bottle in glassby 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. Klein Bottle Skeleton (FDM) 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 • Cannot be smoothly deformed into the classical Klein Bottle • Still single sided -- no edges

34. Figure-8 Klein Bottle • Woven byCarlo Séquin,16’’, 1997

35. Triply Twisted Fig.-8 Klein Bottle

36. Triply Twisted Fig.-8 Klein Bottle

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

38. Single-thread Figure-8 Klein Bottle Modelingwith 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 • This is the last remaining rectangle warping case. • 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 • < 4-finger exercise >What is this beast ? • A model of the Projective Plane • An infinitely large flat plane. • Closed through infinity, i.e., lines come back from opposite direction. • 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 PROJECTIVE PLANE C -- 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 • Both, Klein bottle and projective planeare single-sided, have no edges.(They differ in genus, i.e., connectivity) • The cross cap on a torusmodels a Klein bottle. • The cross cap on a spheremodels the projective plane,but has some undesirable singularities. • Can we avoid these singularities ? • Can we get more symmetry ?

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