Physical Double Physical Double Bubbles in the Bubbles in the

Three-TorusThree-TorusStephen CarterStephen Carter

Department of MathematicsDepartment of Mathematics

Millersville University of Millersville University of PennsylvaniaPennsylvania

CoauthorsCoauthors Nicholas BrubakerNicholas Brubaker Sean EvansSean Evans Sherry LinnSherry Linn Ryan WalkerRyan Walker

Stephen CarterStephen Carter Daniel KravatzDaniel Kravatz Stephen PeurifoyStephen Peurifoy

Special Thanks to:Dr. Ron Umble

Millersville University

Dr. Frank MorganWilliams College

AbstractAbstract

In 2002 Cornelli, Alvarez, Walsh, and Beheshti conjectured and provided

computational evidence that there exist ten topological types of double bubbles providing the least-area way to enclose and separate two regions of prescribed

volume in the three-torus. We produced physical soap bubble models of all ten types in a plexiglass

box.

The Ten Conjectured The Ten Conjectured Double BubblesDouble Bubbles

Used by permission

TheoremTheorem

The ten conjectured surface area The ten conjectured surface area minimizing double bubbles physically minimizing double bubbles physically

exist.exist.

At least four of the ten conjectured surface At least four of the ten conjectured surface area minimizing double bubbles are area minimizing double bubbles are

physically stable.physically stable.

(If a bubble can be created without (If a bubble can be created without reflecting in sides of the box, then it is reflecting in sides of the box, then it is

stable.)stable.)

How to Construct a Two-How to Construct a Two-TorusTorus

Take a rectangle.Take a rectangle. Roll it into a tube.Roll it into a tube. Stretch the tube around and glue the Stretch the tube around and glue the

ends together.ends together. We applied the same idea to a We applied the same idea to a

rectangular box to create the three-rectangular box to create the three-torus.torus.

The Three-TorusThe Three-Torus

Identifying opposite sides of the box yields the three-torus.

What distinguishes bubbles What distinguishes bubbles in the Three-Torus?in the Three-Torus?

When two bubbles touch opposing When two bubbles touch opposing sides of the box directly across from sides of the box directly across from each other, they are part of the same each other, they are part of the same bubble.bubble.

For the ten examples, no two double For the ten examples, no two double bubbles have the same topological bubbles have the same topological type. type.

Pictures courtesy of John M. Sullivan, University of Illinois

Soap Bubble FormulaSoap Bubble Formula

One part Joy dish detergentOne part Joy dish detergent Two parts waterTwo parts water Two parts glycerinTwo parts glycerin

The Standard Double The Standard Double BubbleBubble

The Slab LensThe Slab Lens

The Slab CylinderThe Slab Cylinder

The Double SlabThe Double Slab

The Delauney ChainThe Delauney Chain

The Cylinder LensThe Cylinder Lens

The Cylinder CrossThe Cylinder Cross

The Double CylinderThe Double Cylinder

The Center BubbleThe Center Bubble

The Cylinder StringThe Cylinder String

ConclusionsConclusions

The ten conjectured surface area The ten conjectured surface area minimizing double bubbles minimizing double bubbles physically exist.physically exist.

The standard double bubble, slab The standard double bubble, slab lens, slab cylinder, and double slab lens, slab cylinder, and double slab are physically stable.are physically stable.

Open QuestionsOpen Questions

Are the ten conjectured surface area Are the ten conjectured surface area minimizing double bubbles the only minimizing double bubbles the only ones possible?ones possible?

If not, do any of the additional If not, do any of the additional double bubbles beat out one or more double bubbles beat out one or more of the ten?of the ten?

Are the Delauney chain, cylinder Are the Delauney chain, cylinder lens, cylinder cross, double cylinder, lens, cylinder cross, double cylinder, and cylinder string physically stable?and cylinder string physically stable?