1. Physical Double Bubbles in the Three-Torus Stephen Carter Department of Mathematics Millersville University of Pennsylvania

2. Coauthors Nicholas Brubaker Nicholas Brubaker Sean Evans Sean Evans Sherry Linn Sherry Linn Ryan Walker Ryan Walker Stephen Carter Daniel Kravatz Stephen Peurifoy Special Thanks to: Dr. Ron Umble Millersville University Dr. Frank Morgan Williams College

3. Abstract 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.

4. The Ten Conjectured Double Bubbles Used by permission

5. Theorem The ten conjectured surface area minimizing double bubbles physically exist. At least four of the ten conjectured surface area minimizing double bubbles are physically stable. (If a bubble can be created without reflecting in sides of the box, then it is stable.)

6. How to Construct a Two-Torus Take a rectangle. Take a rectangle. Roll it into a tube. Roll it into a tube. Stretch the tube around and glue the ends together. Stretch the tube around and glue the ends together. We applied the same idea to a rectangular box to create the three-torus. We applied the same idea to a rectangular box to create the three-torus.

7. The Three-Torus Identifying opposite sides of the box yields the three-torus.

8. What distinguishes bubbles in the Three-Torus? When two bubbles touch opposing sides of the box directly across from each other, they are part of the same bubble. When two bubbles touch opposing sides of the box directly across from each other, they are part of the same bubble. For the ten examples, no two double bubbles have the same topological type. For the ten examples, no two double bubbles have the same topological type. Pictures courtesy of John M. Sullivan, University of Illinois

9. Soap Bubble Formula One part Joy dish detergent One part Joy dish detergent Two parts water Two parts water Two parts glycerin Two parts glycerin

10. The Standard Double Bubble

11. The Slab Lens

12. The Slab Cylinder

13. The Double Slab

14. The Delauney Chain

15. The Cylinder Lens

16. The Cylinder Cross

17. The Double Cylinder

18. The Center Bubble

19. The Cylinder String

20. Conclusions The ten conjectured surface area minimizing double bubbles physically exist. The ten conjectured surface area minimizing double bubbles physically exist. The standard double bubble, slab lens, slab cylinder, and double slab are physically stable. The standard double bubble, slab lens, slab cylinder, and double slab are physically stable.

21. Open Questions Are the ten conjectured surface area minimizing double bubbles the only ones possible? Are the ten conjectured surface area minimizing double bubbles the only ones possible? If not, do any of the additional double bubbles beat out one or more of the ten? If not, do any of the additional double bubbles beat out one or more of the ten? Are the Delauney chain, cylinder lens, cylinder cross, double cylinder, and cylinder string physically stable? Are the Delauney chain, cylinder lens, cylinder cross, double cylinder, and cylinder string physically stable?