Bridges, Coimbra, 2011

1. Bridges, Coimbra, 2011 Tori Story Carlo H. Séquin EECS Computer Science Division University of California, Berkeley

2. Art  Math “Tubular Sculptures” Solstice by Charles Perry, Tampa, Florida (1985) (3,2)-Torus-Knot

3. Math  Art This is a “topology” talk ! • In principle, geometric shape is irrelevant;it is all about “connectivity.” • But good shapes can help to make visible and understandable important connectivity issues. • Here I try to make these shapes not only clear, but also as “beautiful” as possible. • Perhaps some may lead to future sculptures.

4. Topology • Shape does not matter -- only connectivity. • Surfaces can be deformed continuously.

5. Smoothly Deforming Surfaces • Surface may pass through itself. • It cannot be cut or torn; it cannot change connectivity. • It must never form any sharp creases or points of infinitely sharp curvature. OK

6. (Regular) Homotopy Two shapes are called homotopic, if they can be transformed into one anotherwith a continuous smooth deformation(with no kinks or singularities). Such shapes are then said to be:in the same homotopy class.

7. Optiverse Sphere Eversion J. M. Sullivan, G. Francis, S. Levy (1998) You may have seen this at previous conferences

8. Bad Torus Eversion macbuse: Torus Eversion http://youtu.be/S4ddRPvwcZI

9. Illegal Torus Eversion • Moving the torus through a puncture is not legal. ( If this were legal, then everting a sphere would be trivial! ) NO !

10. Legal Torus Eversion

11. End of Story ? • These two tori cannot be morphed into one another!

12. Tori Can Be Parameterized • Surface decorations (grid lines) are relevant. • We want to maintain them during all transformations. Orthogonalgrid lines: These 3 tori cannot be morphed into one another!

13. What is a Torus? • Step (1): roll rectangle into a tube. • Step (2): bend tube into a loop. magenta “meridians”, yellow “parallels”, green “diagonals”must all close onto themselves! (1) (2)

14. How to Construct a Torus, Step (1): • Step (1): Roll a “tube”,join up meridians.

15. How to Construct a Torus, Step (2): • Step 2: Loop:join up parallels.

16. Surface Decoration, Parameterization • Parameter lines must close onto themselves. • Thus when closing the toroidal loop, twist may be added only in increments of ±360° +360° 0° –720° –1080° Meridial twist = M-twist

17. Various Fancy Tori

18. A bottle with an internal knotted passage An Even Fancier Torus

19. Super-Fancy Knotted Torus “Dragon Fly”by Andrew Lee, CS-184, Spring 2011

20. Tori Story: Main Message • Regardless of any contorted way in which one might form a decorated torus, all possible tori fall into exactly four regular homotopy classes.[ J. Hass & J. Hughes, Topology Vol.24, No.1, (1985) ] • All tori in the same class can be transformed into each other with smooth homotopy-preserving motions. • But what do these tori look like ? • I have not seen a side-by-side depiction of 4 generic representatives of the 4 classes.

21. 4 Generic Representatives of Tori • For the 4 different regular homotopy classes: OO O8 8O 88 Characterized by: PROFILE / SWEEP

22. Torus Classification ? = ? = ? Of which type are these tori ?

23. Unraveling a Trefoil Knot Simulation of a torsion-resistant material Animation by Avik Das

24. Twisted Parameterization How do we get rid of unwanted twist ?

25. (Cut) Tube, with Zero Torsion Cut Note the end-to-end mismatch in the rainbow-colored stripes

26. Figure-8 Warp Introduces Twist If tube-ends are glued together, twisting will occur

27. Twist Is Counted Modulo 720° • We can add or remove twist in a ±720° increment with a “Figure-8 Cross-over Move”. Push the yellow / green ribbon-crossing down through the Figure-8 cross-over point

28. Un-warping a Circle with 720° Twist Simulation of a torsion-resistant material Animation by Avik Das

29. Dealing with a Twist of 360° Take a regular torus of type “OO”, and introduce meridial twist of 360°, What torus type do we get? “OO” + 360°M-twist warp thru 3D  representative “O8”

30. Other Tori Transformations ? Eversions: • Does the Cheritat operation work for all four types? Twisting: • Twist may be applied in the meridial direction or in the equatorial direction. • Forcefully adding 360 twist may change the torus type. Parameter Swap: • Switching roles of meridians and parallels

31. Transformation Map

32. Legal Torus Eversion

33. Torus Eversion: Lower Half-Slice Arnaud Cheritat, Torus Eversion: Video on YouTube

34. Torus Eversion Schematic Shown are two equatorials. Dashed lines have been everted.

35. A Different Kind of Move • Start with a triple-fold on a self-intersecting figure-8 torus; • Undo the figure-8 by moving branches through each other; • The result is somewhat unexpected:  Circular Path, Fig.-8 Profile, Swapped Parameterization!

36. New: We need to un-twist a lobe;  movement through 3D space: adds E-twist ! Parameter Swap Move Comparison

37. Trying to Swap Parameters This is the goal: Focus on the area where the tori touch, and try to find a move that flips the surface from one torus to the other.

38. A Handle / Tunnel Combination: View along purple arrow

39. Two Views of the “Handle / Tunnel”

40. Flip roles by closing surface above or below the disk “Handle / Tunnel” on a Disk

41. ParameterSwap(Conceptual) fixed central saddle point illegal pinch-off points

42. Flipping the Closing Membrane • Use a classical sphere-eversion process to get the membrane from top to bottom position! Starting Sphere Everted Sphere

43. Sphere Eversion S. Levy, D. Maxwell, D. Munzner: Outside-In (1994)

44. Dirac Belt Trick Unwinding a loop results in 360° of twist

45. Outside-In Sphere Eversion S. Levy, D. Maxwell, D. Munzner: Outside-In (1994)

46. Undo unwanted eversion: A Legal Handle / Tunnel Swap Let the handle-tunnel ride this process !

47. Sphere Eversion Half-Way Point Morin surface

48. Torus Eversion Half-Way Point This would make a nice constructivist sculpture ! What is the most direct move back to an ordinary torus ?

49. World of Wild and Wonderful Tori

50. Another Sculpture ? Torus with triangular profile, making two loops, with 360° twist