In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, the Weaire–Phelan structure is a three-dimensional structure representing an idealised
foam
Foams are materials formed by trapping pockets of gas in a liquid or solid.
A bath sponge and the head on a glass of beer are examples of foams. In most foams, the volume of gas is large, with thin films of liquid or solid separating the reg ...
of equal-sized bubbles, with two different shapes. In 1993,
Denis Weaire
Denis Lawrence Weaire FRS (born 17 October 1942 in Dalhousie, Simla, India) is an Irish physicist and an emeritus professor of Trinity College Dublin (TCD).
Educated at the Belfast Royal Academy and Clare College, Cambridge, he held positions a ...
and Robert Phelan found that this structure was a better solution of the Kelvin problem of tiling space by equal volume cells of minimum surface area than the previous best-known solution, the Kelvin structure.
History and the Kelvin problem
![Truncated octahedra](https://upload.wikimedia.org/wikipedia/commons/9/91/Truncated_octahedra.png)
In two dimensions, the subdivision of the plane into cells of equal area with minimum average perimeter is given by the
hexagonal tiling, but although the first record of this
honeycomb conjecture
The honeycomb conjecture states that a regular hexagonal grid or honeycomb has the least total perimeter of any subdivision of the plane into regions of equal area. The conjecture was proven in 1999 by mathematician Thomas C. Hales.
Theorem
Le ...
goes back to the ancient Roman scholar
Marcus Terentius Varro
Marcus Terentius Varro (; 116–27 BC) was a Roman polymath and a prolific author. He is regarded as ancient Rome's greatest scholar, and was described by Petrarch as "the third great light of Rome" (after Vergil and Cicero). He is sometimes calle ...
, it was not proven until the work of
Thomas C. Hales
Thomas Callister Hales (born June 4, 1958) is an American mathematician working in the areas of representation theory, discrete geometry, and formal verification. In representation theory he is known for his work on the Langlands program and the p ...
in 1999.
In 1887,
Lord Kelvin
William Thomson, 1st Baron Kelvin, (26 June 182417 December 1907) was a British mathematician, Mathematical physics, mathematical physicist and engineer born in Belfast. Professor of Natural Philosophy (Glasgow), Professor of Natural Philoso ...
asked the corresponding question for three-dimensional space: how can space be partitioned into cells of equal volume with the least area of surface between them? Or, in short, what was the most efficient
soap bubble foam?
This problem has since been referred to as the Kelvin problem.
Kelvin proposed a foam called the Kelvin structure. His foam is based on the
bitruncated cubic honeycomb
The bitruncated cubic honeycomb is a space-filling tessellation (or honeycomb) in Euclidean 3-space made up of truncated octahedra (or, equivalently, bitruncated cubes). It has 4 truncated octahedra around each vertex. Being composed entirely of t ...
, a
convex uniform honeycomb
In geometry, a convex uniform honeycomb is a uniform polytope, uniform tessellation which fills three-dimensional Euclidean space with non-overlapping convex polyhedron, convex uniform polyhedron, uniform polyhedral cells.
Twenty-eight such honey ...
formed by the
truncated octahedron
In geometry, the truncated octahedron is the Archimedean solid that arises from a regular octahedron by removing six pyramids, one at each of the octahedron's vertices. The truncated octahedron has 14 faces (8 regular hexagon, hexagons and 6 Squa ...
, a space-filling
convex polyhedron
A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the wo ...
with 6 square faces and 8 hexagonal faces. However, this honeycomb does not satisfy
Plateau's laws
Plateau's laws describe the structure of soap films. These laws were formulated in the 19th century by the Belgian physicist Joseph Plateau from his experimental observations. Many patterns in nature are based on foams obeying these laws.
Laws ...
, formulated by Joseph Plateau in the 19th century, according to which minimal foam surfaces meet at
angles at their edges, with these edges meeting each other in sets of four with angles of
. The angles of the polyhedral structure are different; for instance, its edges meet at angles of
on square faces, or
on hexagonal faces. Therefore, Kelvin's proposed structure uses curvilinear edges and slightly warped
minimal surface
In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below).
The term "minimal surface" is used because these surfaces originally arose as surfaces that ...
s for its faces, obeying Plateau's laws and reducing the area of the structure by 0.2% compared with the corresponding polyhedral structure.
Although Kelvin did not state it explicitly as a conjecture,
[ write that it is "implicit rather than directly stated in Kelvin's original papers"] the idea that the foam of the bitruncated cubic honeycomb is the most efficient foam, and solves Kelvin's problem, became known as the Kelvin conjecture. It was widely believed, and no counter-example was known for more than 100 years. Finally, in 1993,
Trinity College Dublin
, name_Latin = Collegium Sanctae et Individuae Trinitatis Reginae Elizabethae juxta Dublin
, motto = ''Perpetuis futuris temporibus duraturam'' (Latin)
, motto_lang = la
, motto_English = It will last i ...
physicist
Denis Weaire
Denis Lawrence Weaire FRS (born 17 October 1942 in Dalhousie, Simla, India) is an Irish physicist and an emeritus professor of Trinity College Dublin (TCD).
Educated at the Belfast Royal Academy and Clare College, Cambridge, he held positions a ...
and his student Robert Phelan discovered the Weaire–Phelan structure through computer simulations of foam, and showed that it was more efficient, disproving the Kelvin conjecture.
Since the discovery of the Weaire–Phelan structure, other counterexamples to the Kelvin conjecture have been found, but the Weaire–Phelan structure continues to have the smallest known surface area per cell of these counterexamples. Although numerical experiments suggest that the Weaire–Phelan structure is optimal, this remains unproven. In general, it has been very difficult to prove the optimality of structures involving
minimal surface
In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below).
The term "minimal surface" is used because these surfaces originally arose as surfaces that ...
s. The minimality of the sphere as a surface enclosing a single volume was not proven until the 19th century, and the next simplest such problem, the
double bubble conjecture
In the mathematical theory of minimal surfaces, the double bubble theorem states that the shape that encloses and separates two given volumes and has the minimum possible surface area is a ''standard double bubble'': three spherical surfaces meet ...
on enclosing two volumes, remained open for over 100 years until being proven in 2002.
Description
The Weaire–Phelan structure differs from Kelvin's in that it uses two kinds of cells, although they have equal volume. Like the cells in Kelvin's structure, these cells are combinatorially equivalent to
convex polyhedra
A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the wo ...
. One is a
pyritohedron
In geometry, a dodecahedron (Greek , from ''dōdeka'' "twelve" + ''hédra'' "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagon ...
, an irregular
dodecahedron
In geometry, a dodecahedron (Greek , from ''dōdeka'' "twelve" + ''hédra'' "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagon ...
with pentagonal faces, possessing
tetrahedral symmetry
150px, A regular tetrahedron, an example of a solid with full tetrahedral symmetry
A regular tetrahedron has 12 rotational (or orientation-preserving) symmetries, and a symmetry order of 24 including transformations that combine a reflection a ...
(''T
h''). The second is a form of
truncated hexagonal trapezohedron
In geometry, the truncated hexagonal trapezohedron is the fourth in an infinite series of truncated trapezohedra. It has 12 pentagon and 2 hexagon faces. It can be constructed by taking a hexagonal trapezohedron and truncating the polar axis v ...
, a species of
tetrakaidecahedron with two hexagonal and twelve pentagonal faces, in this case only possessing two mirror planes and a
rotoreflection
In geometry, an improper rotation,. also called rotation-reflection, rotoreflection, rotary reflection,. or rotoinversion is an isometry in Euclidean space that is a combination of a rotation about an axis and a reflection in a plane perpendicul ...
symmetry. Like the hexagons in the Kelvin structure, the pentagons in both types of cells are slightly curved. The surface area of the Weaire–Phelan structure is 0.3% less than that of the Kelvin structure.
![Tetrastix](https://upload.wikimedia.org/wikipedia/commons/b/b3/Tetrastix.gif)
The tetrakaidecahedron cells, linked up in face-to-face chains of cells along their hexagonal faces, form chains in three perpendicular directions. A combinatorially equivalent structure to the Weaire–Phelan structure can be made as a tiling of space by unit cubes, lined up face-to-face into infinite square prisms in the same way to form a structure of interlocking prisms called
tetrastix
In geometry, it is possible to fill 3/4 of the volume of three-dimensional Euclidean space by three sets of infinitely-long square prisms aligned with the three coordinate axes, leaving cubical voids; John Horton Conway, Heidi Burgiel and Chaim Go ...
. These prisms surround cubical voids which form one fourth of the cells of the cubical tiling; the remaining three fourths of the cells fill the prisms, offset by half a unit from the integer grid aligned with the prism walls. Similarly, in the Weaire–Phelan structure itself, which has the same symmetries as the tetrastix structure, 1/4 of the cells are dodecahedra and 3/4 are tetrakaidecahedra.
The polyhedral
honeycomb
A honeycomb is a mass of Triangular prismatic honeycomb#Hexagonal prismatic honeycomb, hexagonal prismatic Beeswax, wax cells built by honey bees in their beehive, nests to contain their larvae and stores of honey and pollen.
beekeeping, Beekee ...
associated with the Weaire–Phelan structure (obtained by flattening the faces and straightening the edges) is also referred to loosely as the Weaire–Phelan structure. It was known well before the Weaire–Phelan structure was discovered, but the application to the Kelvin problem was overlooked.
Applications
In physical systems
![Patterned mold for ordered foams](https://upload.wikimedia.org/wikipedia/commons/b/b9/Patterned_mold_for_ordered_foams.jpg)
Experiments have shown that, with favorable
boundary conditions
In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to th ...
, equal-volume bubbles spontaneously
self-assemble into the Weaire–Phelan structure.
The associated polyhedral honeycomb is found in two related geometries of
crystal structure
In crystallography, crystal structure is a description of the ordered arrangement of atoms, ions or molecules in a crystal, crystalline material. Ordered structures occur from the intrinsic nature of the constituent particles to form symmetric pat ...
in
chemistry
Chemistry is the science, scientific study of the properties and behavior of matter. It is a natural science that covers the Chemical element, elements that make up matter to the chemical compound, compounds made of atoms, molecules and ions ...
. Where the components of the crystal lie at the centres of the polyhedra it forms one of the
Frank–Kasper phases
Topologically close pack (TCP) phases, also known as Frank-Kasper (FK) phases, are one of the largest groups of intermetallic compounds, known for their complex crystallographic structure and physical properties. Owing to their combination of ...
, the
A15 phase.
Where the components of the crystal lie at the corners of the polyhedra, it is known as the "Type I
clathrate
A clathrate is a chemical substance consisting of a lattice that traps or contains molecules. The word ''clathrate'' is derived from the Latin (), meaning ‘with bars, latticed’. Most clathrate compounds are polymeric and completely envelop t ...
structure".
Gas hydrate
Clathrate hydrates, or gas hydrates, clathrates, hydrates, etc., are crystalline water-based solids physically resembling ice, in which small non-polar molecules (typically gases) or polar molecules with large hydrophobic moieties are trapped ...
s formed by methane, propane and carbon dioxide at low temperatures have a structure in which
water
Water (chemical formula ) is an inorganic, transparent, tasteless, odorless, and nearly colorless chemical substance, which is the main constituent of Earth's hydrosphere and the fluids of all known living organisms (in which it acts as a ...
molecules lie at the nodes of the Weaire–Phelan structure and are
hydrogen bond
In chemistry, a hydrogen bond (or H-bond) is a primarily electrostatic force of attraction between a hydrogen (H) atom which is covalently bound to a more electronegative "donor" atom or group (Dn), and another electronegative atom bearing a ...
ed together, and the larger gas molecules are trapped in the polyhedral cages. Some
alkali metal
The alkali metals consist of the chemical elements lithium (Li), sodium (Na), potassium (K),The symbols Na and K for sodium and potassium are derived from their Latin names, ''natrium'' and ''kalium''; these are still the origins of the names ...
hydrides silicides
A silicide is a type of chemical compound that combines silicon and a (usually) more electropositive element.
Silicon is more electropositive than carbon. Silicides are structurally closer to borides than to carbides.
Similar to borides and carbi ...
and
germanides also form this structure, with silicon or germanium at nodes, and alkali metals in cages.
In architecture
![北京 水立方 - panoramio](https://upload.wikimedia.org/wikipedia/commons/f/f4/%E5%8C%97%E4%BA%AC_%E6%B0%B4%E7%AB%8B%E6%96%B9_-_panoramio.jpg)
The Weaire–Phelan structure is the inspiration for the design by
Tristram Carfrae
Tristram George Allen Carfrae, FRSA, FREng, FTSE, RDI (born 1 April 1959) is a British-Australian structural engineer and designer. He is currently Deputy Chair of Arup and an Arup Fellow.
Carfrae was awarded the Gold Medal of the Instituti ...
of the
Beijing National Aquatics Centre
The National Aquatics Centre (), and colloquially known as the Water Cube () and the Ice Cube (), is an aquatics center at the Olympic Green in Beijing, China.
The facility was originally constructed to host the aquatics competitions at the ...
, the 'Water Cube', for the
2008 Summer Olympics
The 2008 Summer Olympics (), officially the Games of the XXIX Olympiad () and also known as Beijing 2008 (), were an international multisport event held from 8 to 24 August 2008, in Beijing, China. A total of 10,942 athletes from 204 Na ...
.
See also
* ''
The Pursuit of Perfect Packing
''The Pursuit of Perfect Packing'' is a book on packing problems in geometry. It was written by physicists Tomaso Aste and Denis Weaire, and published in 2000 by Institute of Physics Publishing ( doi:10.1887/0750306483, ) with a second edition pub ...
'', a book by Weaire on this and related problems
References
External links
3D models of the Weaire–Phelan, Kelvin and P42a structures
page with illustrations and freely downloadable 'nets' for printing and making models.
"Weaire-Phelan Smart Modular Space Settlement" Alexandru Pintea, 2017, Individual First Priz
NASA Ames Space Settlement Contest:
{{DEFAULTSORT:Weaire-Phelan Structure
Minimal surfaces
Polyhedra
Honeycombs (geometry)
1994 introductions