ruggero gabbrielli- the kelvin problem: what space-filling arrangement of cells of equal volume has...

8/3/2019 Ruggero Gabbrielli- The Kelvin Problem: What space-filling arrangement of cells of equal volume has minimal surface area?

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The Kelvin ProblemThe Kelvin Problem

What space-filling arrangement of cells of equal volume

has minimal surface area?

Ruggero Gabbrielli, University of Bath


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Foam morphology

Andrew Kraynik (2003)


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Polyhedra in a foam

Andrew Kraynik (2003)


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A step back: 2D Kelvin problem

Consider partitions of the plane into equal-area regions: Each pattern has cells of same area

Which pattern has the least length per unit cell?

Is there a better pattern than C?

A B C


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The honeycomb conjecture

Thomas Hales (2001):

Any partition of theplane into regions of

equal area has

perimeter at least thatof the regular

hexagonal honeycomb

tiling.


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3D Single bubble

What is the shape of a single soap bubble in Euclidean 3-space?


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More bubbles

And if we have more than one?


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Arial size 20 Arial size 16


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Equal volume constraint

If all the bubbles have the same volume (not necessarily thesame shape)?

Two examples of a foams made of identical bubbles: Rhombic dodecahedron (Voronoi diagram of FCC)

Truncated octahedron (Kelvins partition)


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Rhombic dodecahderon

Voronoi diagram of FCC

Net name: flu

Non-simple polyhedron: not all the vertices are trivalent


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Truncated octahedron

Voronoi diagram of BCC

Net name: sod

Simple polyhedron: each vertex is trivalent


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Cost of a foam

The volume of each cell is fixed to be V

A is the average interface area per cell

A is half the boundary area of a typical cell, since each interface is shared by twocells

For example for a cube c=6, for a sphere c=4.84

It turns out that for sod c=5.306 and for flu c=5.345

32

V

Ac


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The most frequent cell in foams

13 faces: 1 quadrilateral, 10 pentagons and 2 hexagons

Ideally, a minimal foam should have an average number offaces of about 13.40 (R. Kusner)


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Pentagon content

Kelvin (sod)

Single polyhedron, 14 faces

Only hexagons and quadrilaterals

Weaire-Phelan (mep)

Two different polyhedra: 14 and 12 faces

Only pentagons and hexagons

Natural foams A wide range of polyhedra

11% quadrilaterals, 67% pentagons, 22% hexagons (E. Matzke)


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sod: Kelvin structure

Periodic unit One 14-hedron

Polyhedron 8 hexagons and 6

quadrilaterals

cost: 5.306


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mep: Weaire-Phelan structure

Periodic unit Six 14-hedra

Two 12-hedra

Polyhedra

12 pentagonsand 2 hexagons

12 pentagons

cost: 5.288


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Kelvin algorithm

The algorithm used has been developed by Olaf Delgado-

Friedrichs

It uses an idea introduced by Delaney and successively

developed by Dress to store the topological data of a tiling (or

the net carrying it) in a file.

Simple tilings containing polyhedra with 12 to 16 faces Containing only quadrilaterals, pentagons and hexagons

Search is made on vertex-k-transitive tilings

The algorithm looks for all the possible ways to tile tetrahedra

with given symmetries and constraints


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Results

Kelvin algorithm Simple tilings

4 to 6 sided faces

12 to 16 faced polyhedra

Euclidean 3-space

Minimal Foam Structures http://people.bath.ac.uk/rg247/javaview/start.html

http://people.bath.ac.uk/rg247/3d.html(best new result, cost: 5.313)


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Energies


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Swift-Hohenberg equation

A Matlab code has been used to generate point sets in three-

space The Voronoi partition has been created from each of these point

sets

The surface area has been minimized and calculated

Many of the partitions were non-simple. Simple modificatinos ofthese were produced with the Surface Evolver


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Body Centred Cubic (Kelvin)


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A15 (Phelan-Weaire)


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P42a (the new counter-example)


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Thank you!

Olaf Delgado-Friedrichs The Gavrog Project and the Kelvin algorithm

http://gavrog.sourceforge.net

Ken Brakke The Surface Evolver

http://www.susqu.edu/brakke/evolver/evolver.html

John Sullivan Vcs (Voronoi generator)

http://torus.math.uiuc.edu/jms/software

David Lloyd Swift-Hohenberg equation solver

http://personal.maths.surrey.ac.uk/st/D.J.Lloyd

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