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Graphics Lunch, Oct. 27, 2011. “ Tori Story” ( Torus Homotopies ). Carlo H. Séquin. EECS Computer Science Division University of California, Berkeley. Topology. Shape does not matter -- only connectivity. Surfaces can be deformed continuously. (Regular) Homotopy.

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graphics lunch oct 27 2011
Graphics Lunch, Oct. 27, 2011

“Tori Story”(Torus Homotopies )

Carlo H. Séquin

EECS Computer Science Division

University of California, Berkeley

  • Shape does not matter -- only connectivity.
  • Surfaces can be deformed continuously.
regular homotopy
(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.

smoothly deforming surfaces
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.


optiverse sphere eversion
“Optiverse” Sphere Eversion

J. M. Sullivan, G. Francis, S. Levy (1998)

Turning a sphere inside-out in an “energy”-efficient way.

bad torus eversion
Bad Torus Eversion

macbuse: Torus Eversion

illegal torus eversion
Illegal Torus Eversion
  • Moving the torus through a puncture is not legal.

( If this were legal, then everting a sphere would be trivial! )

NO !

end of story no
End of Story ? … No !

Circular cross-section Figure-8 cross-section

  • These two tori cannot be morphed into one another!
tori can be parameterized
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!

what is a torus
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)

how to construct a torus step 1
How to Construct a Torus, Step (1):
  • Step (1): Roll a “tube”,join up red meridians.
how to construct a torus step 2
How to Construct a Torus, Step (2):
  • Step 2: Loop:join up yellowparallels.
surface decoration parameterization
Surface Decoration, Parameterization
  • Parameter grid 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 , or “M-twist”

tori story main message
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) ]Oriented surfaces of genus g fall into 4g homotopy classes.
  • All tori in the same class can be deformed into each other with smooth homotopy-preserving motions.
  • I have not seen a side-by-side depiction of 4 generic representatives of the 4 classes.
4 generic representatives of tori
4 Generic Representatives of Tori
  • For the 4 different regular homotopy classes:


OO O8 8O 88

Characterized by: PROFILE / SWEEP

cut tube with zero torsion
(Cut) Tube, with Zero Torsion


Note the end-to-end mismatch in the rainbow-colored stripes

twist is counted modulo 720
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

twisted parameterization
Twisted Parameterization

How do we get rid of unwanted twist ?

dealing with a twist of 360
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”

torus classification
Torus Classification ?

= ?

= ?

Of which type are these tori ?

un warping a circle with 720 twist
Un-warping a Circle with 720° Twist

Simulation of a torsion-resistant material

Animation by Avik Das

unraveling a trefoil knot
Unraveling a Trefoil Knot

Simulation of a torsion-resistant material

Animation by Avik Das

other tori transformations
Other Tori Transformations ?


  • Does the Cheritat operation work for all four types?


  • 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
trying to swap parameters
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.

a handle tunnel combination
A Handle / Tunnel Combination:

View along purple arrow

parameter swap conceptual



saddle point

illegal pinch-off points

flipping the closing membrane
Flipping the Closing Membrane
  • Use a classical sphere-eversion process to get the membrane from top to bottom position!

Starting Sphere

Everted Sphere

sphere eversion
Sphere Eversion

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

dirac belt trick
Dirac Belt Trick

Unwinding a loop results in 360° of twist

outside in sphere eversion
Outside-In Sphere Eversion

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

a legal handle tunnel swap

Undo unwanted eversion:

A Legal Handle / Tunnel Swap

Let the handle-tunnel ride this process !

analyzing the twist in the ribbons
Analyzing the Twist in the Ribbons

The meridial circles are clearly not twisted.

analyzing the twist in the ribbons1
Analyzing the Twist in the Ribbons

The knotted lines are harder to analyze  Use a paper strip!

torus eversion half way point
Torus Eversion Half-Way Point

This would make a nice constructivist sculpture !

What is the most direct move back to an ordinary torus ?

  • Just 4 Tori-Classes!
  • Four Representatives:
  • Any possible torus fits into one of those four classes!
  • An arsenal of possible moves.
  • Open challenges: to find the most efficent / most elegant trafo(for eversion and parameter swap).
  • A glimpse of some wild and wonderful toripromising intriguing constructivist sculptures.
  • Ways to analyze and classify such weird tori.
q u e s t i o n s
Q U E S T I O N S ?


  • John Sullivan, Craig Kaplan, Matthias Goerner;Avik Das.
  • Our sponsor: NSF #CMMI-1029662 (EDI)

More Info:

  • UCB: Tech Report EECS-2011-83.html

Next Year:

  • Klein bottles.
another sculpture
Another Sculpture ?

Torus with triangular profile, making two loops, with 360° twist

doubly looped tori
Doubly-Looped Tori

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

movie un warping a double loop
Movie: Un-warping a Double Loop

Simulation of a material with strong twist penalty

“Dbl. Loop with 360° Twist” by Avik Das

mystery solved
Mystery Solved !

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

double roll double loop
Double Roll  Double Loop
  • Reuse a previous figure, but now with double walls:
  • Switching parameterization:
  • Double roll turns into a double loop;
  • The 180° lobe-flip removes the 360° twist;
  • Profile changes to figure-8 shape;
  • Unfold double loop into figure-8 path.  Type 88
mystery solved1
Mystery Solved !

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

tori with collars
Tori with Collars

Torus may have more than one collar !

turning a collar into 360 twist
Turning a Collar into 360° Twist

Use the move from “Outside-In” based on the Dirac Belt Trick,

torus eversion lower half slice
Torus Eversion: Lower Half-Slice

Arnaud Cheritat, Torus Eversion: Video on YouTube

torus eversion schematic
Torus Eversion Schematic

Shown are two equatorials. Dashed lines have been everted.

a different kind of move
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!