Graphics Lunch, Oct. 27, 2011 “Tori Story” ( Torus Homotopies ) EECS Computer Science Division University of California, Berkeley Carlo H. Séquin

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

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

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

“Optiverse” Sphere Eversion Turning a sphere inside-out in an “energy”-efficient way. J. M. Sullivan, G. Francis, S. Levy (1998)

Bad Torus Eversion macbuse: Torus Eversion

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

Legal Torus Eversion

End of Story ? … No ! u These two tori cannot be morphed into one another! Circular cross-section Figure-8 cross-section

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

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

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

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

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

Various Fancy Tori

An Even Fancier Torus A bottle with an internal knotted passage

Tori Story: Main Message u 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) ] Oriented surfaces of genus g fall into 4 g homotopy classes. u All tori in the same class can be deformed into each other with smooth homotopy-preserving motions. u I have not seen a side-by-side depiction of 4 generic representatives of the 4 classes.

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

Figure-8 Warp Introduces Twist!

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

Twist Is Counted Modulo 720° u 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

Twisted Parameterization How do we get rid of unwanted twist ?

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

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

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

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

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

Transformation Map

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

A Handle / Tunnel Combination: View along purple arrow

Two Views of the “Handle / Tunnel”

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

Parameter Swap (Conceptual) illegal pinch-off points fixed central saddle point

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

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

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

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

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

Sphere Eversion Half-Way Point Morin surface

Torus with Knotted Tunnel

Analyzing the Twist in the Ribbons The meridial circles are clearly not twisted.

Analyzing the Twist in the Ribbons The knotted lines are harder to analyze  Use a paper strip!

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

u Just 4 Tori-Classes! u Four Representatives: u Any possible torus fits into one of those four classes! u An arsenal of possible moves. u Open challenges: to find the most efficent / most elegant trafo (for eversion and parameter swap). u A glimpse of some wild and wonderful tori promising intriguing constructivist sculptures. u Ways to analyze and classify such weird tori. Conclusions

Q U E S T I O N S ? Thanks: u John Sullivan, Craig Kaplan, Matthias Goerner; Avik Das. u Our sponsor: NSF #CMMI (EDI) More Info: u UCB: Tech Report EECS html Next Year: u Klein bottles.

World of Wild and Wonderful Tori

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

Doubly-Looped Tori Step 1: Un-warping the double loop into a figure-8 No change in twist !

Movie: Un-warping a Double Loop Simulation of a material with strong twist penalty “Dbl. Loop with 360° Twist” by Avik Das

Mystery Solved ! Dbl. loop, 360° twist  Fig.8, 360° twist  Untwisted circle

Doubly-Rolled Torus

Double Roll  Double Loop u Reuse a previous figure, but now with double walls: Switching parameterization: u Double roll turns into a double loop; u The 180° lobe-flip removes the 360° twist; u Profile changes to figure-8 shape; u Unfold double loop into figure-8 path.  Type 88

Mystery Solved ! Doubly-rolled torus w. 360° twist  Untwisted type 88 torus

Tori with Collars Torus may have more than one collar !

Turning a Collar into 360° Twist Use the move from “Outside-In” based on the Dirac Belt Trick,

Legal Torus Eversion

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

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

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

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