CS 39R

1. CS 39R Simple 2-Manifolds Carlo H. Séquin University of California, Berkeley

2. Deforming a Rectangle • 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 surface

3. Five Important Two-Manifolds X=0 X=0 G=0 G=1X=0 X=0 X=1G=1 G=2 G=1 Cylinder Möbius band Torus Klein bottle Cross surface

4. Cylinder Construction

5. Möbius Band Construction

6. Cylinders as Sculptures John Goodman Max Bill

7. The Cylinder in Architecture Chapel

8. Möbius Sculpture by Max Bill

9. Möbius Sculptures by Keizo Ushio

10. More Split Möbius Bands Split Moebius band by M.C. Escher And a maquette made by Solid Free-form Fabrication

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

12. “Bonds of Friendship” J. Robinson 1979

13. Torus Sculpture by Max Bill

14. 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.

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

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

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

18. Klein Mugs Klein bottle in glassby Cliff Stoll, ACME Fill it with beer --> “Klein Stein”

19. 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.”

20. Klein Bottle Skeleton (FDM)

21. Klein Bottle Skeleton (FDM) Struts don’t intersect !

22. Another Type of Klein Bottle • Cannot be smoothly deformed into the classical Klein Bottle • Still single sided -- no boundaries

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

24. Triply Twisted Fig.-8 Klein Bottle

25. Triply Twisted Fig.-8 Klein Bottle

26. 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 !

27. Single-thread Figure-8 Klein Bottle Modelingwith SLIDE

28. Zooming into the FDM Machine

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

30. Single-thread Figure-8 Klein Bottle

31. 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 ?

32. Cross-cap Construction

33. 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.

34. 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.

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

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

37. “Four Canoes” by Helaman Ferguson

38. 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 Dyck’s surface (genus3). • The cross cap on a spheremodels the projective plane,but has some undesirable singularities. • Can we avoid these singularities ? • Can we get more symmetry ?

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

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

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

42. Again: Alleviate Self-intersections Strut passesthrough hole

43. Skeleton of a Steiner Surface

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

45. 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 ! • 3-fold symmetry • based on hexagonal domain

46. Model of Boy Surface Computer graphics by François Apéry

47. Model of Boy Surface Computer graphics by John Sullivan

48. Model of Boy Surface Computer graphics by John Sullivan

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

50. Double Covering of Boy Surface • Wire model byCharles Pugh • Decorated by C. H. Séquin: • Equator • 3 Meridians, 120º apart