physical double bubbles in the three-torus stephen carter department of mathematics millersville...
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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?