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Physical Double Bubbles in the ThreeTorus
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How to Construct a Two-Torus. Take a rectangle. Roll it into a tube. ... sides of the box yields the three-torus. What distinguishes bubbles in the Three-Torus? ... – PowerPoint PPT presentation
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Title: 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?
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