Physical Double Bubbles in the ThreeTorus

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Physical Double Bubbles in the Three-Torus

• Daniel Kravatz
• Department of Mathematics
• Millersville University of Pennsylvania

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Coauthors

• Nicolas Brubaker
• Sean Evans
• Stephen Peurifoy

• Stephen Carter
• Sherry Linn
• Ryan Walker

Special Thanks to
Dr. Ron Umble Millersville University Dr. Frank
Morgan Williams College
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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.

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The Ten Conjectured Double Bubbles
Used by permission
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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.)

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How to Construct a Two-Torus

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

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The Three-Torus
Identifying opposite sides of the box yields the
three-torus.
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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.
• For the ten examples, no two double bubbles have
  the same topological type.

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Soap Bubble Formula

• One part Joy dish detergent
• Two parts water
• Two parts glycerin

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The Standard Double Bubble
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The Delauney Chain
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The Cylinder Lens
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The Cylinder Cross
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The Double Cylinder
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The Slab Lens
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The Center Bubble
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The Cylinder String
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The Slab Cylinder
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The Double Slab
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Conclusions

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

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Open Questions

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