Exploring Möbius Bands and Single-Sided Surfaces

1. CS 39R Carlo H. Séquin Single-Sided Surfaces EECS Computer Science Division University of California, Berkeley

2. Twisting a ribbon into a Möbius band Making a Single-Sided Surface

3. Simple Möbius Bands • A single-sided surface with a single edge: A closed ribbon with a 180°flip. A closed ribbon with a 540°flip.

4. More Möbius Bands Max Bill’s sculpture of a Möbius band. The “Sue-Dan-ese” M.B., a “bottle” with circular rim.

5. A Möbius Band Transfromation Widen the bottom of the band by pulling upwards its two sides,  get a Möbius basket, and then a Sudanese Möbius band.

6. Many Different Möbius Shapes Left-twisting versions shown – can be smoothly transformed into one another • Topologically, these are all equivalent:They all are single-sided,They all have ONE rim,They all are of genus ONE, and E.C. = 0. • Each shape is chiral: its mirror image differs from the original.

7. These are NOT Möbius Bands ! • What you may find on the Web under “Möbius band” (1):

8. These are NOT Möbius Bands ! • What you may find on the Web under “Möbius band” (2):

9. These are NOT Möbius Bands ! • What you may find on the Web under “Möbius band” (3):

10. TWO Möbius Bands ! • Two Möbius bands that eventually get fused together:

11. Topological Surface Classification The distinguishing characteristics: • Is it two-sided, orientable – or single-sided, non-orientable? • Does it have rims? – How many separate closed curves? • What is its genus? – How many handles or tunnels? • Is it smooth – or does it have singularities (e.g. creases)? Can we make a single-sided surface with NO rims?

12. Classical “Inverted-Sock” Klein Bottle

13. Can We Do Something Even Simpler? • Yes, we can! • Close off the rim of any of those Möbius bands with a suitably warped patch (a topological disk). • The result is known as the Projective Plane.

14. The Projective Plane -- Equator projects to infinity. -- Walk off to infinity -- and beyond …come back from opposite direction: mirrored, upside-down !

15. The Projective Plane is a Cool Thing! • It is single-sided:Flood-fill paint flows to both faces of the plane. • It is non-orientable:Shapes passing through infinity get mirrored. • A straight line does not cut it apart!One can always get to the other side of that line by traveling through infinity. • It is infinitely large! (somewhat impractical)It would be nice to have a finite model with the same topological properties . . .

16. Trying to Make a Finite Model • Let’s represent the infinite plane with a very large square. • Points at infinity in opposite directions are the same and should be merged. • Thus we must glue both opposing edge pairs with a 180º twist. Can we physically achieve this in 3D ?

17. Possible “Rectangle Universes” • 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

18. Cross-Surface Construction

19. Wood / Gauze Model of Projective Plane Cross-Surface = “Cross-Cap” + punctured sphere

20. Cross-Cap Imperfections • Has 2 singular points with infinite curvature. • Can this be avoided?

21. Steiner Surface (Tetrahedral Symmetry) Plaster model by T. Kohono A gridded model by Sequin Can singularities be avoided ?

22. Can Singularities be Avoided ? Werner Boy, a student of Hilbert,was asked to prove that it cannot be done. But he found a solution in 1901 ! • It has 3 self-intersection loops. • It has one triple point, where 3 surface branches cross. • It may be modeled with 3-fold symmetry.

23. Various Models of Boy’s Surface

24. Main Characteristics of Boy’s Surface Key Features: • Smooth everywhere! • One triple point, • 3 intersection loops emerging from it.

25. Projective Plane With a Puncture The projective plane minus a disk is: • a Möbius band; • or a cross-cap; • or a Boy cap. • This makes a versatile building block! • with an open rim by which it can be grafted onto other surfaces.

26. Another Way to Make a Boy Cap Frame the hole with 3 opposite stick-pairs and 6 connector loops: Similar to the way we made a cross cap from a 4-stick hole:

27. Mӧbius Band into Boy Cap • Credit: Bryant-Kusner

28. Geometrical Surface Elements “Cross-Cap”“Boy Cap”“Boy Cup” • Single-sided surface patches with one rim. • Topologically equivalent to a Möbius band. • “Plug-ins” that can make any surface single-sided. • “Building blocks” for making non-orientable surfaces. • Inspirational design shapes for consumer products, etc.

29. Boy Cap + Disk = Boy Surface Mӧbius Band + Disk = Projective Plane Genus = 1; E.C. = 1 • In summary: • And: TWO Mӧbius Bands = Klein Bottle Genus = 2; E.C. = 0 See: 

30. KOJ = MR + ML 2 Möbius Bands Make a Klein Bottle

31. Classical Klein Bottle from 2 Boy-Caps BcL BcR BcL + BcR = KOJ “Inverted Sock” Klein bottle:

32. Klein Bottle with S6 Symmetry • Take two complementary Boy caps. • Rotate left and right halves 180°against each other to obtain 3-fold glide symmetry, or S6 overall.

33. Klein Bottle from 2 Identical Boy-Caps BcL BcR BcR + BcR = K8R • There is more than one type of Klein bottle ! Twisted Figure-8 Klein Bottle:

34. Model the Shape with Subdivision • Start with a polyhedral model . . . Level 0 Level 1 Level 2

35. Make a Gridded Sculpture!

36. Increase the Grid Density

37. Actual Sculpture Model

38. S6 Klein Bottle Rendered by C. Mouradian http://netcyborg.free.fr/

39. Fusing Two Identical Boy Surfaces • Both shapes have D3 symmetry; • They differ by a 60°rotation between the 2 Boy caps.

40. Building Blocks To Make Any Surface • A sphere to start with; • A hole-punchto make punctures:Each decreases Euler Characteristic by one. • We can fill these holes again with: • Disks:Increases Euler Characteristic by one. • Cross-Caps: Makes surface single-sided. • Boy-caps: Makes surface single-sided. • Handles (btw. 2 holes): Orientability unchanged. • Cross-Handles (btw. 2 holes): Makes surface single-sided. Genus changes; E.C. unchanged.

41. Constructing a Surface with χ = 2 ‒ h • Punch h holes into a sphere and close them up with: cross-caps or Boy caps or Closing two holes at the same time: handles or cross-handles

42. Single-sided Genus-3 Surfaces Renderings by C.H. Séquin Sculptures by H. Ferguson

43. Concept of a Genus-4 Surface 4 Boy caps grafted onto a sphere with tetra symmetry.

44. Genus-4 Surface Using 4 Boy-Caps Employ tetrahedral symmetry! ( 0°rotation between neighbors) (60°rotation between neighbors)