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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