local con¬guration considered is now the union of the Delaunay tetrahedra with a

common vertex in the centre of a given sphere (this is called the Delaunay star).

This was the kind of decomposition ¬rst adopted by Hales in his programme. For

instance, the Delaunay decomposition succeeds in the icosahedral case, giving a

score of 7.999 98. But there exists at least one local con¬guration with a higher

score. This is an assembly of 13 spheres around a central one, which is obtained

by taking 12 spheres centred at the vertices of an icosahedron and distorting the

arrangement by pressing the 13th sphere into one of the faces. This con¬guration

¼ ½ . Another nasty con¬gura-

scores 8.34 and has local packing fraction

tion which has a score dangerously close to eight is the ˜pentahedral prism™. This

is an assembly of 12 spheres around a central one, shown in ¬gure 4.1.

The Vorono¨ and Delaunay decompositions can be mixed in in¬nitely many

±

ways. This is what Hales attempted by decomposing space in ˜Q-systems™ and

associated stars. This decomposition was successful to establish the score of the

pentahedral prism at 7.9997. However, new and nasty con¬gurations remained to

be considered. . . .

At last? 39

4.3 At last?

On 10 August 1998, as one of the authors of this book was picking his ¬shing

tackle out of the back of his car, his eye fell on a headline in a British newspaper.

All thoughts of angling were dismissed for a while.

KEPLER™S ORANGE STACKING PROBLEM QUASHED

In a short report Simon Singh announced Thomas Hales™ success, and quoted

John Conway, a leading expert and mentor of Hales: ˜For the last decade Hales™s

work on sphere packings has been painstaking and credible. If he says he™s done

it, then he™s quite probably right™.

Back at the of¬ce an email message had been received, which must have

disturbed quite a few other summer holidays as well.

From Thomas Hales

Date: Sun, 9 Aug 1998 09:54:56

To:

Subject: Kepler conjecture

Dear colleagues,

I have started to distribute copies of a series of papers

giving a solution to the Kepler conjecture, the oldest

problem in discrete geometry. These results are still

preliminary in the sense that they have not been refereed

and have not even been submitted for publication, but the

proofs are to the best of my knowledge correct and

complete.

Nearly four hundred years ago, Kepler asserted that no

packing of congruent spheres can have a density greater

than the density of the face-centred cubic packing. This

assertion has come to be known as the Kepler conjecture.

In 1900, Hilbert included the Kepler conjecture in

his famous list of mathematical problems.

In a paper published last year in the journal ˜Discrete

and Computational Geometry™, (DCG), I published a

detailed plan describing how the Kepler conjecture might

be proved. This approach differs significantly from

earlier approaches to this problem by making extensive

use of computers. (L. Fejes Toth was the first to suggest

the use of computers.) The proof relies extensively on

methods from the theory of global optimization, linear

programming, and interval arithmetic.

Proof positive?

40

The full proof appears in a series of papers totalling

well over 250 pages. The computer files containing the

computer code and data files for combinatorics, interval

arithmetic, and linear programs require over 3 gigabytes

of space for storage.

Samuel P. Ferguson, who finished his Ph.D. last year at

the University of Michigan under my direction, has

contributed significantly to this project.

The papers containing the proof are:

An Overview of the Kepler Conjecture, Thomas C. Hales

A Formulation of the Kepler Conjecture, Samuel P.

Ferguson and Thomas C. Hales

Sphere Packings I, Thomas C. Hales

(published in DCG, 1997)

Sphere Packings II, Thomas C. Hales

(published in DCG, 1997)

Sphere Packings III, Thomas C. Hales

Sphere Packings IV, Thomas C. Hales

Sphere Packings V, Samuel P. Ferguson

The Kepler Conjecture (Sphere Packings VI),

Thomas C. Hales

Postscript versions of the papers and more information

about this project can be found at

http://www.math.lsa.umich.edu/˜hales

Tom Hales

A month later, after Hales had enjoyed his own holiday, he kindly con-

sented to answer a few emailed questions that might illuminate this account of

his achievement. His answers were as follows, with some minor editing:

Date: Wed, 30 Sep 1998 08:42:12

From: Tom Hales

Reply-To: Tom Hales

To:

cc:

Subject: Re: your mail

> Dear Thomas

>

At last? 41

> - when were you first attracted to the problem?

In 1982, I took a course from John Conway on groups and

geometry.

> - what was the hardest part?

The problem starts out as an optimization problem in an

infinite number of variables. The original problem must

be replaced by an optimization in a finite number of

variables. It was extremely difficult to find a

finite-dimensional formulation that was simple enough

for computers to handle.

> - in what your method is different from the

> previous approaches?

This approach makes extensive use of computers,

especially interval arithmetic and linear programming

methods. Most previous work was based on the Voronoi

cells. This approach creates a hybrid of Voronoi cells

and Delaunay simplices.

> - did you follow the lead/style of anyone in

> particular?

My greatest source of inspiration on this problem was

L. Fejes Toth. He was the first to propose an

optimization problem in a finite number of variables

and the first to propose the use of computers. But my

proof differs from the program he originally proposed.

> - is it correct to say that this is a "traditional"

> proof with no significant elements of

> computer-based proof?

Not at all. The computer calculations are an essential

part of the proof.

> - what was your first reaction when Hsiang claimed

> a proof?

I have followed his work closely from the very start.

My doubts about his work go back to a long discussion we

had in Princeton in 1990. I discuss my reaction to his

work further in my Intelligencer article.

Proof positive?

42

> - have you always been confident in success?

In the fall of 1994, I found how to make the hybrid

decompositions work, combining the Voronoi and Delaunay

approaches. I have been optimistic since then.

> - your proof fill 10 papers and about 250 pages,

> why does it need so much?

This is a constrained nonlinear optimization problem

involving up to 150 variables with many local maxima that

come uncomfortably close to the global maxima. Rigorous

approaches to problems of this complexity as generally

regarded as hopelessly difficult.

> - do you think that in the future a different

> approach might be able to reduce the size and

> complexity of this proof?

This is not an optimal proof. I have concrete ideas about

how the proof might be simplified. Although I™m quite

certain the proof can be simplified, it will require

substantial research to carried this out. There could also

be other proofs along completely different lines, but I do

not have any definite ideas here.

> - can you calculate the maximum size of a cluster

> of spheres with a given density

> (larger than the Kepler one)?

This is an interesting question that a number of people

would like to understand. I would be curious to know whether

my methods might lead to something here.

As things stand in 2000, it would appear that the credit for solving this

centuries-old question may well go to Hales. But only time (and much toil by

colleagues) will yield the eventual guarantee of proof. Or will it start another

chapter of debate?

4.4 Who cares?

This is a fair question, often addressed to startled scientists and mathematicians

by puzzled journalists. Particularly in this case, why does proof matter, if we

know the truth anyway?

One answer is that some people did admit to a tiny sliver of doubt insinuating

itself into the certainty of their conviction. These were usually not physical scien-

The problem of proof 43

tists, who would assert that if there was a better structure they would have spotted

it by now, written somewhere in the book of nature. This may be insuf¬ciently

humble: surprises do occur from time to time, even in crystallography. They of-

ten result in Nobel Prizes. As Alfred North Whitehead said, ˜in creative thought

common sense is a bad master. Its sole criterion for judgement is that the new

ideas shall look like the old ones, in other words it can only act by suppressing

originality.™

Another answer is that mathematicians have to prove things, as birds ¬‚y and

¬sh swim. It is silly to ask why, since it is in the very nature of their profession to

create these elegant cultural artefacts. It is not that mathematics consists entirely

of theorems. There is a rough texture of conjecture and useful approximation, held

up here and there by a rigorous proof, serving the same purpose as the concrete

framework of a building. The more of these the better, to stop the whole thing

collapsing under the weight of loose speculation.

Rigorous proofs and exact results are like the gold bars in the vaults of the

Federal Reserve, guaranteeing the otherwise unreliable monetary transactions of

the world. They are apparently useless since they are not put to any direct use,

and yet they have real value. (Since the ¬rst draft of this book was written there

has been much debate on the abandonment of gold reserves, so the simile may

soon be a feeble one. The value of a proof is more durable.)

4.5 The problem of proof

In 1967 the announcement in the New York Times of a computer proof of the four-

colour conjecture by Kenneth Appel and Wolfgang Haken sparked off a lively

debate on the acceptability of such methods. Thirty years on they are much less

controversial, as we continue to accommodate ourselves to the ever greater role

of computers in our lives. It would be more provocative to debate, say, the limits

of arti¬cial intelligence or the possibilities of computer consciousness than to

dispute the validity of computer proofs. Nevertheless, since we have encountered

them in packing theorems, particularly in the work of Hales, let us rehearse the

arguments as presented by, for example, Davis and Hersh in The Mathematical

Experience (1981).

The traditional view has held that it must be possible to check every step in

a proof, in order for it to be acceptable. In some cases, such as that of the proof of

Fermat™s last theorem by Wiles, few will ever be able to accomplish this. Those

that do so (who may include the referees for publication, if they are conscientious)

will be the guarantors upon which the rest of us can rely. But modern computer

proofs generate such a multitude of logical operations within the machine that it

is beyond any human capacity to follow them, even if they are made manifest.

There is no getting away from the necessity to rely on the correct function-

ing of the machine and its software, in response to the programmer™s instructions.

They can be checked only at the level of programming, or in terms of the repro-

Proof positive?

44

ducibility of the result on a quite different machine, rather as some experimental

results are tested. If these precautions are taken, today™s generation is quite happy

to accord the same status to the proof as in days of old. They will say: There was

always a tiny element of uncertainty in any elaborate traditional proof. Human

beings are susceptible to error, as are computers. Rather more so, perhaps?

Some of us would still sigh. We point to economy, elegance and transparency

as cardinal virtues of a good mathematical proof, award low scores to the new

methodology on those criteria, and call for renewed efforts to be more explicit 2 .

4.6 The power of thought

In one of many important articles on polyhedra and packings, the eminent Cana-

dian mathematician H S M Coxeter chose to begin with a poem by Charles

Mackay:

Cannonballs may aid the truth,

But thought™s a weapon stronger,

We™ll win our battles by its aid;

Wait a little longer.

To this we might add a verse to bring it up to date:

If you fail to reach your end,

your siege has come to nought,

call on your electronic friend,

to spare a microsecond™s thought.

Whether computers can really ˜think™ is a deeper problem than the merely

technical one of proof-checking. We may someday accept an even stranger role

for these machines in mathematics. Will they write books as well?

¾

For a further exploration of this ongoing debate, see the article by: Horgan J 1993 The death of

proof Sci. Am. October 75, and the responses and discussion that followed in later issues.

Chapter 5

Peas and pips

5.1 Vegetable staticks

When soft objects are tightly packed, they change their shapes to eliminate the

wasted interstitial space. Even if they begin as spheres they will develop into

polyhedra. The question is: which types of polyhedra will be formed? The forma-

tion of foam by bubbles is an example of such a process; another is to be found in

the familiar insulating material of polystyrene, formed by causing small spheres

to expand and ¬ll a mould. But the most famous early experiment was carried out

with peas.

The Reverend Stephen Hales performed this classic experiment, in an age

when science was practiced as much in the parlour or the kitchen as in the labo-

ratory. Hales compressed a large quantity of peas (or rather expanded them under

pressure by absorption of water) and described what he observed in a book with

the charming title of Vegetable Staticks.

I compressed several fresh parcels of Pease in the same Pot, with a

force equal to 1600, 800, and 400 pounds; in which Experiments, tho™

the Pease dilated, yet they did not raise the lever, because what they

increased in bulk was, by great incumbent weight, pressed into the in-

terstices of the Pease, which they adequately ¬lled up, being thereby

formed into pretty regular Dodecahedrons 1.

½

Published in 1727 by Stephen Hales, with the title ˜Vegetable Staticks: or An Account of some

Statistical Experiments on the Sap Vegetables: being an essay towards a Natural History of Vegetation.

Also, a Specimen of An Attempt to Analyse the Air, By great Variety of Chymio-Statical Experiments;

Which were read at several Meetings before the Royal Society™ and dedicated to His Royal Highness

George Prince of Wales.

45

Peas and pips

46

Figure 5.1. The experimental apparatus used by Hales to demonstrate the force exerted by

dilating peas. When the lid was loaded with a weight, the dilated peas ¬ll the interstices,

developing polyhedral forms.

At this point the reverend gentleman™s report is misleading. We will see later

in this chapter that when soft objects are closely packed in a disordered fashion,

they form polyhedral grains with irregular shapes. Only a few of them have 12

faces. Moreover, regular dodecahedral cells 2 cannot ¬ll three-dimensional space,

so Hales purported to observe something which is plainly impossible. But this

mistake probably has a simple explanation: in such packings the majority of faces

are pentagons and the dodecahedron is the regular solid made with pentagonal

faces. The history of science is full of such cases in which the observer tries to

draw a neat conclusion from complex and variable data. Too determined a search

for a simple conclusion can lead to an erroneously idealized one.

Rob Kusner (a mathematician currently at the University of Massachusetts

at Amherst) tells us that a modern version of Hales™ experiment with peas was

carried out in New England a few summers ago by a group of undergraduates

using water-balloons, greased with vegetable oil and stuffed into a large chest

freezer. Some balloons burst from being pierced by sharp ice crystals, and others

did not freeze, but it was still possible to see the packing patterns since the low

temperature folding on the balloon rubber left permanent marking (lighter colour)

at the folds.

¾

The reader unfamiliar with their polyhedra should turn to ¬gure 5.6.

Stephen Hales 47

Figure 5.2. Stephen Hales (1677“1761).

5.2 Stephen Hales

At ¬rst sight, the pea-packing experiments of Stephen Hales might seem to be

the dabblings of a dilettante. Not so. Hales was a signi¬cant ¬gure in the rapid

development of science after Newton. He is much mentioned, for example, in

the work of Lavoisier, and has been the subject of several biographies and many

portraits.

Belonging to the gentry of the south of England, he had no dif¬culty in gain-

ing access to undemanding positions as a clergyman, many of which were in the

gift of members of his class. The research that he carried out over many years

forms part of the foundation of today™s science of the physiology of plants and

animals”the study of function as opposed to mere form, as in anatomy and tax-

onomy. He was also something of a technologist, being credited with the inven-

tion of forced ventilation. It has not always been self-evident that fresh air is good

for us! The navy, in particular, enthusiastically adopted his recommendations in

an attempt to improve the health of sailors.

It was in a physiological spirit that Hales performed his experiment with

dried peas. The pressure associated with their uptake of water was at issue: what

we would today call ˜osmotic pressure™. The fact that the peas, swelling while un-

der pressure, were compressed into polyhedral forms, was really incidental. This

may be offered as an excuse for the unfortunate inaccuracy of Hales™ description

of their polyhedral form.

It would be pleasing to establish some family connection between this man

and his modern namesake (chapter 4). Alas, Thomas Hales reports that none has

been established, and reminds us that Stephen died without issue.

Peas and pips

48

5.3 Pomegranate pips

About one century before Hales, Kepler had studied the shape of the pips inside

a pomegranate. These seeds are soft juicy grains with polyhedral shapes. Kepler

was trying to understand the origin of the hexagonal shapes of snow¬‚akes, so

the pomegranate grains were used as an example of the spontaneous formation of

regular geometrical polyhedral shapes in a packing. Kepler observed that these

grains have rhomboidal faces and he remembered that the same rhomboidal faces

are present in the bottom of bees™ cells and make up the interface between the two

opposite layers of cells in the honeycomb (see chapter 6).

Attracted by these rhombes, I started to search in the geometry if a body

similar to the ¬ve regular solids and to the fourteen Archimedean could

have been constituted uniquely of rhombes. I found two (. . . ). The

¬rst is constituted of twelve rhombes, (. . . ) this geometrical ¬gure, the

closest possible to the regularity, ¬ll the solid space, as the hexagon, the

square and the triangle ¬ll the plane.

(. . . ) if one opens a rather large-sized pomegranate, one will see most

of its loculi squeezed into the same shape, unless they are impeded by

the peduncles that take food to them.

(. . . ) What agent creates the rhomboid shape in the cells of the honey-

comb and in the loculi of the pomegranate?

Kepler argues that the key to the honeycomb and pomegranate is the problem

of packing equal-sized spherical objects into the smallest possible space; and he

¬nds the answer in a conjecture for the closest sphere packing (chapter 3). Indeed,

if one takes this packing and expands the spheres, grains with 12 rhombic faces

are obtained.

Kepler carefully warns the readers that one must take a ˜rather large-sized

pomegranate™ to ¬nd many rhombohedral grains inside. Indeed, in a typical

pomegranate the rhombohedral grain shape is not so common. The packing is

not so perfect and the grains take different shapes. Just like Stephen Hales, Ke-

pler was oversimplifying his conclusions.

5.4 The improbable seed

An example will be useful to show why the shape observed by Kepler in the

pomegranate seeds is quite unusual in disordered packings.

Take a world atlas and put your ¬nger on a continent, then follow the border

between two states. This line follows mountains, rivers or other features for ge-

ological or historical reasons. The set of states in the continent can be seen as a

two-dimensional packing, or a jigsaw puzzle, where each state is a tile with an ir-

regular complex shape different from the others. These tiles ¬t perfectly together

¬lling the continent without leaving any empty space. This is an irregular packing

The improbable seed 49

Figure 5.3. A pomegranate.

Figure 5.4. Vorono¨ partitions are made here from four points in the plane. The whole

±

space is subdivided into four domains which are the regions of space closest to each point.

The full lines mark the boundary between these four domains. These lines intersect in

one common four-connected vertex only when the points are symmetrically placed on the

vertices of a rectangle. Any other con¬guration generates a couple of three-connected

vertices.

of very different elements but with an important common property: everywhere

only three states meet in a common vertex. In other words, in such a packing the

vertex connectivity is equal to three. This is true all around the world for national

states or provinces, with”as for any good rule”at least one exception that we

will leave the reader to ¬nd in the world atlas.

If there is no special symmetry, the threefold vertex is the type of intersection

that is automatically generated in two-dimensional space-¬lling packings. Con-

sider, for example, four cells in a Vorono¨ partition as shown in ¬gure 5.4. The

±

cells meet at the same four-connected vertex only in the case when the centres of

the cells are on the vertices of a rectangle; in all the other”in¬nite”possibilities

the cells meet on two three-connected vertices. Only an in¬nitesimal displace-

ment of the original points of the Vorono¨ construction is needed to split the four-

±

Peas and pips

50

fold vertex into two threefold ones.

Patterns with vertex connectivity three are not only found in the political

division of territory but are also present in a very large class of natural packings

and patterns. Some examples are given in ¬gure 5.5.

In three dimensions the most probable con¬guration has four cells that meet

at a common vertex. This is the minimal vertex connectivity and”again”con¬g-

urations with higher connectivity can be split into two or more minimally con-

nected ones by in¬nitesimal rearrangements.

The Kepler rhombic dodecahedra pack together in such a way as to produce

vertices of connectivity six. This is therefore an improbable con¬guration and it

will not be naturally formed in disordered packings. A Vorono¨ partition from the

±

centres of the spheres in the Kepler packing generates a space-¬lling assembly of

rhomboid dodecahedra, but when the position of the spheres is slightly deformed

by in¬nitesimal displacements the vertex connectivity becomes four and the aver-

age number of faces in the Vorono¨ cells increases to a value of around 14.

±

5.5 Biological cells, lead shot and soap bubbles

From the beginning of microscopy, anatomists were impressed by the similari-

ties between the shape of biological cells in undifferentiated tissues and that of

bubbles in foam. Robert Hooke includes in his Micrographia (1665) an obser-

vation on the ˜Schematism or Texture of Cork, and the Cells and Pores of some

other such frothy Bodies™ and describes the pith of a plant as ˜congeries of very

small bubbles™. For centuries, with almost no exceptions, cells in undifferentiated

tissues were described as regular dodecahedra (the impossible peas of Hales) or

as rhombic dodecahedra (the improbable pomegranate seed of Kepler). Then, at

the end of the 19th century a new type of cell with 14 faces emerged. It was the

tetrakaidecahedron, a polyhedron that Lord Kelvin proposed in 1887 as the struc-

ture that divides space ˜with minimum partitional area™. (We will follow Kelvin™s

line of thought in chapter 7.)

It was only at the beginning of this century that careful and extensive studies

of the shape of bubbles in foams and cells in tissues were undertaken. In partic-

ular, a large series of biological tissues was meticulously studied by F T Lewis

of Harvard University between 1923 and 1950 3 . He concluded that cells in un-

differentiated tissues have polyhedral shapes with about 14 faces on average and

he inferred that they tend to be approximated by Kelvin™s ideal polyhedron. Fur-

ther investigation now indicates that the cells have about 14 faces on average but

a large variety of shapes contribute to form the cellular structure. The Kelvin

polyhedron is rarely observed 4.

¿

See, for example, Lewis F T 1950 Reciprocal cell division in epidermal and subepidermal cells Am.

J. Bot. 37 715“21.

Dormer K J 1980 Fundamental Tissue Geometry for Biologists (Cambridge: Cambridge University

Press).

Biological cells, lead shot and soap bubbles 51

Figure 5.5. Two-dimensional cellular patterns (from Weaire D and Rivier N 1984 Soap,

cells and statistics”random patterns in two dimensions Contemp. Phys. 25 59).

Peas and pips

52

Figure 5.6. A pentagonal dodecahedron (a); a rhombic dodecahedron (b) and a

tetrakaidecahedron (c).

The experimental study of the form of biological cells inspired experiments

in which cellular structures were created by compressing together soft spheres

to ¬ll all the space (as Reverend Hales did with peas). In this way, the resulting

structure can be disassembled and the shapes of the individual cells easily studied.

A classical experiment of this kind is the one by Marvin 5 , who compressed 730

pellets of lead shot in a steel cylinder. When the spheres are carefully packed layer

by layer in the closest way, then”not surprisingly”the cells take Kepler™s rhom-

bic dodecahedral shape. Totally different shapes are observed when the spheres

are packed in a disordered way, for instance by putting the shot in the container at

random or by shaking it before compression. In this case, the cellular structures

have polyhedral cells with faces that vary from triangles to octagons with the great

majority being pentagons and, less abundantly, squares and hexagons. The aver-

age number of neighbours was reported to be 14.17 in a set of 624 internal lead

Marvin J W 1939 The shape of compressed lead shot and its relation to cell shape Am. J. Bot. 26

280“8.

Biological cells, lead shot and soap bubbles 53

pellets. No Kepler cells were observed in these experiments.

One may look for similar structures in deformed bubbles of foam, but look-

ing at bubbles inside a foam is quite dif¬cult, if we wish to observe the properties

of internal bubbles. In 1946, Matzke 6 reported the study of the shapes of 600

central cells (which he claimed to be of equal volume) in a soap froth. This re-

search is still the most extensive study of the structure of foam bubbles up to the

present time. He found an average number of faces per bubble of 13.7 and a pre-

dominance of pentagonal faces, with 99.6% of all faces being either quadrilateral,

pentagonal or hexagonal. Kepler™s cell never appeared, nor was that of Kelvin

observed. This disappointing tale is told in more detail in chapter 7.

Biological cells, peas, lead shot and soap bubbles are quite different sys-

tems, although all consist of polyhedral cells packed together to ¬ll the whole

space. A biological tissue is generated by growth and mitosis (division) of cells.

The shape of a cell is therefore continuously changing following the mitotic cycle.

In contrast to this, when lead shot is compressed the shapes of the cells are mostly

determined by the environment of the packed spheres. Indeed, during the com-

pression, rearrangements are very rare. In foams the structure is strictly related

to the interfacial energy and bubbles assume shapes that minimize the global sur-

face area. The great similarity in the polyhedral shapes of the cells in these very

different systems can therefore be attributed only to the inescapable geometrical

condition of ¬lling space.

Matzke E B 1946 The three-dimensional shape of bubbles in foam”an analysis of the role of

surface forces in three-dimensional cell shape determination Am. J. Bot. 33 58“80.

Chapter 6

Enthusiastic admiration: the

honeycomb

6.1 The honeycomb problem

We have encountered various cases of cellular structures, which divide space into

cells. How can this be done most economically, in terms of the surface area of

the cells? It is not clear that this has any relevance to the squashed peas of Hales

or the lead shot of Marvin, but it certainly is the guiding principle for foams (the

subject of the next chapter), for which the cell interfaces cost energy. The bubble

packing which we call a foam is not alone in minimizing surface area. Emulsions,

such as that of oil and vinegar shaken to make a salad dressing, conform to the

same principle.

For centuries this principle has also been supposed to govern the construction

of the honeycomb by the bee. The bee, it has been said, needs to make an array

of equal cells in two dimensions, using a minimum of wax, and hence requires a

pattern with the minimum perimeter per cell.

Although the perfection of the honeycomb is a very proper object for admi-

ration, it may be naive to impute to the bee the single mathematical motive of

saving of wax, just as it can hardly be said that the greengrocer cares much about

the maximum density of oranges. There really aren™t many reasonable alternatives

to the two-dimensional hexagonal structure for the honeycomb. Other consider-

ations surely impose themselves, such as simplicity and mechanical stability, in

the evolutionary optimization of the hive.

A full account of the arguments that have raged over the shape of the bee™s

cell would read like a history of Western thought. We can ¬nd one of the ¬rst at-

tempts at an explanation in Pliny (Naturalis Historia) who associated the hexago-

54

The honeycomb problem 55

Figure 6.1. The Hungarian mathematician L Fejes T´ th has been a leader in the mathe-

o

matics of packings for many years, and his son G Fejes T´ th now follows in his footsteps.

o

nal shape of the cell with the fact that bees have six legs. Among the other notable

minds that have been brought to bear on it, we must count at least those of: Pappus

of Alexandria (Fifth Book), Buffon (Histoire Naturelle), Kepler, Koenig, Maraldi,

R´ amur, Lord Brougham, Maurice Maeterlinck (La Vie des Abeilles), Samuel

e

Haughton, Colin MacLaurin, Jules Michelet (L™insecte), and Charles Darwin (The

Origin of Species). ˜He must be a dull man™ said Darwin, who could contemplate

this subject ˜without enthusiastic admiration™. Not surprisingly, he attributed ˜the

most wonderful of all known instincts™ to ˜numerous, successive, slight modi¬ca-

tions of simpler instincts™.

Darwin™s account of the process by which the honeybee achieves its precise

constructions, by forming rough walls and re¬ning them, is instructive, but he is

not quite correct in saying that ˜they are absolutely perfect in economizing wax™

as we shall shortly see, when we turn to the three-dimensional aspect of the hive,

that is, the structure of the interface between the two opposed honeycombs. For

the moment we address only its two-dimensional aspect, the arrangement of the

elongated cells which is visible on the surface. Does this two-dimensional pattern

of cells of equal area have the least possible perimeter?

Of course it does: it is well known that this pattern is the best. What has

remained hidden from general appreciation is that this proposition has not until

now been fully proved! This was rarely stated, probably because most authors

cannot believe there is no proof hidden somewhere in the unfathomable depths

of the technical literature. We saw in chapter 2 that a proof exists for the closely

related problem of optimal packing of equal discs, but this should not be confused

with the question posed by the honeybee.

This should take its rightful place alongside Kepler™s problem as a notable

frustration for the mathematician, both in the personal sense and in the technical

one (a circle being the best form for a single cell if we ignore the rest). Frank

Enthusiastic admiration: the honeycomb

56

Morgan has drawn attention to the problem 1.

There does exist a proof of a lesser theorem, once again attributed to Fejes

T´ th. It imposes certain restrictions, of which the most important is the require-

o

ment that all the sides of the cells are straight. This follows from the convexity

principle which was described in chapter 2. But in general it is very natural for

them to be curved, so this is a much weaker result than one would like.

At the time of writing, Thomas Hales has informally announced that he will

shortly publish such a full proof. If con¬rmed this will complete, together with

his analysis of the Kepler conjecture (chapter 4), a remarkable double.

Date: Mon, 7 Jun 1999 12:37:27 -0400 (EDT)

From: Tom Hales

To: Denis Weaire

cc:

Subject: honeycombs

Dear Denis Weaire,

If all goes well, I™ll announce a solution to the honeycomb

conjecture in a few days. (I don™t make any assumptions

about the convexity or topology of the cells.) I™ve shown

it to Frank Morgan and John Sullivan, and they didn™t find

any obvious problems with my proof.

I hope you don™t mind that in my acknowledgements, I quote

from your email message to me from October, ˜Given its

celebrated history, it seems worth a try...™. Thanks for

attracting my attention to the problem.

Of course, I™m fascinated by the Kelvin problem too, but I

don™t think that will be solved anytime soon...

Best,

Tom

6.2 What the bees do not know

As we have warned, it turns out that bees (even Hungarian ones) are not so smart

after all, when the three-dimensional aspect of their construction is considered.

½

Morgan F 1999 The hexagonal honeycomb conjecture Trans. Am. Math. Soc. 351 1733. See also

Morgan F 2000 Geometric Measure Theory: A Beginners Guide, 3rd Edition (New York: Academic).

What the bees do not know 57

Figure 6.2. The beehive.

The enthusiastic admiration of the scienti¬c community for the pattern of the

beehive has often extended to the internal structure of the honeycomb. It has two

sides from which access is possible, with a partition wall in the middle.

A ¬‚at wall would be wasteful of wax. Instead the bee chooses a faceted wall

which neatly ¬ts the two halves of the honeycomb, when the cells are staggered

with respect to each other. The edges of the facets meet each other and the side

walls of the hexagonal cells at the same angle (the Maraldi angle) of approxi-

mately ½¼ Æ that, as we will see in the next chapter, Plateau recognized in foams

as a consequence of the minimization of surface energy. So the parsimony of the

bee apparently extends even to this internal structure. (Nowadays the bee has no

real choice, since the wall is provided by the keeper as a preformed ˜foundation™.)

There is a close connection between this strategy and the successive opti-

mal stacking of close-packed planes of spheres. The Vorono¨ construction (sec-

±

tion 2.5) applied to two such layers gives, for the partition between them, precisely

the form of the bee™s wall.

Indeed, the bee™s faceted wall can be shown to minimize area and hence the

expenditure of wax, in the limited sense that any small change will increase the

area. The occurrence of the Maraldi angle signals this: indeed it gained its name

in this context.

Learned academies have sung the praises of the bees for basing their con-

struction on the Maraldi angle and, in so doing, have exaggerated its precision.

This has caused some speculation as to whether the Almighty has endowed these

creatures with an understanding of advanced mathematics.

The whole story is told in detail in D™A W Thompson™s On Growth and

Form2 . Dismissing earlier follies (including Darwin™s) he is driven to the extreme

conclusion that ˜the bee makes no economies; and whatever economies lie in the

¾

Thomson D™A W 1942 On Growth and Form 2nd edn (Cambridge).

Enthusiastic admiration: the honeycomb

58

Figure 6.3. The alternative of Fejes T´ th (a) to the bee™s design (b).

o

theoretical construction, the bee™s handiwork is not ¬ne or accurate enough to

take advantage of them™. Where Darwin had invoked slow changes in response

to a marginal advantage of design, Thompson looked for physical forces at work,

supposing the thin wax ¬lm of the hive to be more or less ¬‚uid at the same time

in their construction, and so forming the angles dictated by surface tension.

Where precisely the truth lies in this old and muddled dispute about angles

we do not know, but Thompson would clearly have been delighted to learn of

Fejes T´ th™s startling conclusion, many years later: there is an entirely different

o

arrangement which is better than the design of the bee! This was published in a

charming paper entitled ˜What the Bees Know and What They Do Not Know™ 3 .

The bee™s design can be improved, with a saving of 0.4% of the surface area

of the wall, by using a different arrangement of facets, again constructed with

the angles dictated by surface tension. The alternative presented by Fejes T´ th o

is shown in ¬gure 6.3 and is closely related to the Kelvin structure described in

chapter 7.

Bees do sometimes create the Hungarian mathematician™s design locally,

whenever they are left to build the wall themselves, and then make a mistake,

so that the two parts of the honeycomb are misaligned.

¿

Fejes T´ th L 1964 Bull. Am. Math. Soc. 70 468“81.

o

Chapter 7

Toils and troubles with

bubbles

7.1 Playing with bubbles

Foams and bubbles have fascinated scientists of all ages, in all ages. Most have

devoted some time to admiration of what Robert Boyle called ˜the soap bubbles

that boys are wont to play with™. Part of their charm and mystery lies in the colours

produced by the interference of light in thin ¬lms. Small clusters of bubbles, or the

extended ones we call foams, have elegant structures which call for explanation.

The poet and philosopher may point to the ephemeral nature of these things

as a metaphor for our own mortality or the transience of fame and fortune, but the

¬rst interest of the scientist is in making relatively stable foams. This is not dif¬-

cult with a little ordinary detergent solution shaken in a sealed container. Looking

into it, one can see that, though disordered, it shows clear evidence of some prin-

ciples of equilibrium at work. What are they?

When spherical bubbles pack together to form a foam (as when they rise out

of a glass of beer) they are forced into polyhedral shapes as gravity extracts most

of the liquid from their interstices. What began as a sphere packing, in conformity

to the rules of previous chapter, now presents a different paradigm for pattern in

nature. In this case the density is ¬xed and surface area is to be minimized 1.

½

See for general reference: Sadoc J F and Rivier N (ed) 1999 Foams and Emulsions (Dordrecht:

Kluwer); Ball P 1999 The Self-made Tapestry: Pattern Formation in Nature (Oxford: Oxford Univer-

sity Press); Hildebrandt S and Tromba A 1996 The Parsimonious Universe (Berlin: Springer).

59

Toils and troubles with bubbles

60

Figure 7.1. Playing with soap bubbles.

7.2 A blind man in the kingdom of the sighted

The man who most clearly saw the principles of bubble-packing was blind. Joseph

Antoine Ferdinand Plateau caused irrecoverable damage to his eyes by staring at

the sun in an experiment on the retention of vision. The 1999 eclipse brought

many public reminders of the extreme danger of doing so. He began to go blind

in 1841 and had lost all vision by 1844.

Michael Faraday wrote consolingly and prophetically to him:

Well may you and your friends rejoice that though, in the body, you

have met with a heavy blow and great discouragement, still the spirit

makes great compensation, and shines with glorious light across the

bodily darkness.

Today Plateau is remembered for his later researches, undertaken with the help of

family, friends and students, leading to his great work Statique experimentale et

th´ orique des liquides soumis aux seules forces mol eculaires (1873).

e ´

As a hero of Belgian science, he was elevated to the rank of ˜Chevalier™ in

the Order of L´ opold. His extraordinary dedication to science did not preclude a

e

happy family life but he was often preoccupied. It is recorded that he disappeared

for six hours while on honeymoon in Paris, returning eventually to his distraught

bride, to say that he had forgotten that he had just been married. A similar case is

to be found in the Irish mathematician George Gabriel Stokes, whose love-letters

contained too much mathematical physics to be fully effective.

At the heart of Plateau™s classic text were those experiments with wire frames

dipped in soap solution which are still commonly used in lecture demonstrations.

They were popularized as such by C V Boys in his Soap-Bubbles, their Colours

and the Forces which mould them, being the substance of many lectures delivered

to juvenile and popular audiences, published by the Society for Promoting Chris-

A blind man in the kingdom of the sighted 61

Figure 7.2. A Plateau frame.

tian Knowledge in 1911 2 . The tradition of Plateau™s experiments continues today

in the hands of Cyril Isenberg 3 and others, and one can purchase the frames at

modest cost from Beevers Molecular Models.

Plateau™s book was well received. J C Maxwell reviewed it in Nature, ¬rst

asking ironically”Can the poetry of bubbles survive this?”then replying with

this encomium:

Which, now, is the more poetical idea”the Etruscan boy blowing bub-

bles for himself, or the blind man of science teaching his friends to blow

them, and making out by a tedious process of question and answer the

condition of the forms and tints which he can never see?

The meaning of the book™s title is not self-evident. It may be taken to mean

the laws of equilibrium of liquids under surface tension, when gravity is negligi-

ble. Or, plainly put, what are the shapes and connections of soap ¬lms?

Plateau™s laws which answer this question are as follows:

(1) Films can only meet three at a time and they do so symmetrically, so that the

angles between them are ½¾¼ Æ .

(2) The lines along which they meet are themselves joined in vertices at which

only four lines (or six ¬lms) can meet. Again they are symmetric, so that

the angle between the lines has the value “— ½ ´ ½ µ or approximately ½¼ Æ

¿

(the tetrahedral, or Maraldi, angle).

(3) The ¬lms and the lines are curved in general: the average amount by which

the ¬lms are bowed in or out is determined by the difference in pressure

between the gas on either side (Laplace™s law).

Note that the third law does not dictate that a zero pressure difference implies

a ¬‚at ¬lm. Saddle-shaped surfaces can have zero mean curvature.

¾

A later edition is Boys C V 1959 Soap Bubbles (New York: Dover).

¿

Isenberg C 1992 The Science of Soap Films and Soap Bubbles (New York: Dover).

Toils and troubles with bubbles

62

Figure 7.3. Soap foam (courtesy of J Cilliers, UMIST).

If soap ¬lms are trapped between two glass plates in order to create a two-

dimensional structure, only ¬rst and third laws are needed.

These laws were inspired by observation, but most of them are easily ratio-

nalized by theory. Soap ¬lms have energies proportional to their surface area, and

they therefore tend to contract and pull with a force (the surface tension) on their

boundaries. Plateau™s laws express the conditions for stable equilibrium of these

forces and the gas pressures which act on the ¬lms.

Proving Plateau 63

Figure 7.4. Different equilibrium vertex con¬gurations, in terms of geodesics. (Redrawn

from Almgren F Jr and Taylor J 1976 Sci. Am. 235 82“93.)

7.3 Proving Plateau

Although he was not averse to mathematical analysis, Joseph Plateau™s essential

method was that of generalization from observation”he was content to leave it

to others to ¬nd theoretical justi¬cations of his laws.

Ernest Lamarle, a Belgian mathematician who was expert in differential

geometry, provided some of this mathematical underpinning in the 1860s. He

showed that the principle of minimal area implied everything that Plateau had re-

ported. In particular he gave a justi¬cation of the rule that states that only four

soap ¬lms can meet at a point. A foam cannot have stable vertices formed by

more than this number of ¬lms.

The proof runs to many pages. It begins with the classi¬cation of every

possible form of vertex, consistent with the balance of the surface tension forces

acting in the adjoining ¬lms. This in itself does not guarantee stability: most of

these turn out to be unstable equilibria.

One might expect to encounter a great dif¬culty at this point, if there are

in¬nitely many possibilities for such a vertex. But it turns out that there are only

a few.

A small sphere centred on the vertex must have intersections with the soap

¬lms which form a pattern on its surface as follows:

(1) The lines are geodesics: that is, each lies in a plane which cuts the sphere in

half.

(2) The lines intersect at ½¾¼ Æ , three at a time.

The possible patterns of this type are highly reminiscent of the earliest stages

in embryonic development.

Lamarle proceeded to devise ways in which each of the more complicated

vertices could be deformed and dissociated into combinations of the elementary

one, while lowering the total surface area. These de¬ne modes of instability of

the vertex, hence disqualifying it as a possible stable con¬guration.

Toils and troubles with bubbles

64

For serious mathematicians, the story does not end here. In particular, it

reappeared on the agenda of Fred Almgren, 100 years after Lamarle.

Just as Lamarle had performed, at one level, a clean-up operation on

Plateau™s arguments, so Fred Almgren set out to perfect the proofs of Lamarle

and others in the study of minimal surfaces 4 . Their work contained hidden or

explicit assumptions of smoothness in the minimal structures that they purported

to describe.

Mathematicians can conceive all sorts of strange entities with surfaces which

are the opposite of smooth”not just rough, but perhaps in¬nitely so. In recent

years Benoit Mandelbrot has taught us that these monstrous constructions are not

really alien to our world. Previously, we have projected on to nature a vision of

rounded smoothness and continuity which is only one extreme of reality. Much

of nature is better described as rough and ragged, jagged and jerky. The romantic

landscapes and seascapes, which are the subject of so much of our art, display

such fractal forms, but scientists somehow ignored them for rather too long. Cyril

Stanley Smith (chapter 14) was a rare exception. He complained that ˜scientists

have tended to overlook the form of the world and concentrate on forces™, and

attacked the smooth, periodic idealizations of material science in his time.

This said, it can hardly be maintained that much real doubt should be enter-

tained about the assumption of smoothness in the particular case of soap ¬lms.

Almgren™s quest was for completeness and rigour, to eliminate nasty possibilities,

however hypothetical, from the proof. Much of his fastidious work in this vein

over a 35-year career at Princeton was compiled in a 1720-page paper, unpub-

lished at the time of his death in 1997.

Another part of Almgren™s legacy is to be found in successive generations

of graduate students. One of these was Jean Taylor, who became his wife. It fell

to her to construct the new version of Lamarle™s proof, which she published in

19765. It is of comparable length to the 19th century paper, but couched in the

inscrutable language of geometric measure theory.

Almgren and his successors have leavened their rather impenetrable studies

with a lively sense of the more accessible and practical facets of their subject.

In particular, Ken Brakke emerged from that school to write the Surface Evolver

software, the fruits of which we will see later in this chapter.

Almgren F Jr 1976 Existence and regularity almost everywhere of solutions to elliptic variational

problems with constraints Mem. Am. Math. Soc. 165; Almgren F Jr and Taylor J 1976 Sci. Am. 235

82“93.

Taylor J 1976 The structure of singularities in soap bubbles-like and soap-¬lm-like minimal surfaces

Ann. Math. 103 489“539.

Foam and ether 65

Minimal surfaces

The theory of minimal surfaces has continued to be an active focus of

research in this century. Several Fields Medals (a particularly prestigious

mathematical prize) have acknowledged great achievements in that area.

In particular, Jesse Douglas received the ¬rst medal in 1936 for his con-

tribution to the ˜Plateau problem™, which is concerned with a single soap

¬lm spanning a loop of wire of arbitrary shape.

7.4 Foam and ether

Plateau™s rules apply to any foam in equilibrium. They place restrictions on, but

do not de¬ne in full, the answer to our question: which structure is best? This we

have already answered for two dimensions; and, indeed, experiments with foams

of equal bubbles do reproduce the honeycomb.

But just as ball bearings are uncooperative in the search for ideal structures,

so are soap bubbles in three dimensions. A foam of equal-sized bubbles remains

disordered, in practice. This did not stop Sir William Thomson (later Lord Kelvin)

attacking the theoretical problem of the ideal ordered foam, in 1887. Indeed he

does not seem to have tried to make such a ˜monodisperse™ foam, despite his credo

(stated later) that theory must be anchored in reality.

At that time Kelvin was the pre-eminent, if ageing, ¬gure of British science.

He still had a strong appetite for scienti¬c endeavour”in the end he published

over 600 papers, a score worthy of the most competitive (and repetitive) of to-

day™s careerists. They stretch across the spectrum of physics from telegraphy and

electrical technology to the second law of thermodynamics, for which we honour

his name in the scienti¬c unit of temperature. Quite a man.

Ether

Apollonius of Tyana is said to have asked the Brahmins of what they

supposed the cosmos to be composed.

˜Of the ¬ve elements™.

˜How can there be a ¬fth™ demanded Apollonius ˜beside water and air and

earth and ¬re?™

˜There is the ether™ replied the Brahmin ˜which we must regard as the

element of which the gods are made; for just as all mortal creatures inhale

the air, so do immortal and divine natures inhale the ether.™ (From Sir

Oliver Lodge 1925 Ether and Reality (London: Hodder and Stoughton)

p 35.)

Toils and troubles with bubbles

66

Figure 7.5. Sir William Thomson (Lord Kelvin) (1824“1907).

One of the great quests of Victorian science was the search for a physical

model for light waves. In the centuries-old debate between the advocates of par-

ticle and wave interpretations of light, the wave enthusiasts had gained the upper

hand by the middle of the 19th century. It remained to specify the substance”the

ether”the vibrations of which, like those of sound in air, constituted the light

waves. We have already encountered the fanciful ideas of Osborne Reynolds,

concerning the nature of the ether (chapter 3). The word itself came down to us

from the Greeks, for whom it represented a ¬ery heaven into which souls were

received.

British natural philosophers were determinedly realistic in their outlook, al-

ways trying to relate the world of microscopic and invisible phenomena to ev-

eryday experience. This may be the reason that neither relativity nor quantum

mechanics, both of which con¬‚ict with everyday experience, can be listed among

the achievements of that school.

Lord Kelvin, together with P G Tait, wrote in the preface to their textbook:

Nothing can be more fatal to progress than too con¬dent reliance on

mathematical symbols; for the student is only too apt to take the easier

course, and consider the formula and not the fact as the physical reality.

In common with others, Kelvin was not easily impressed by formalism and ab-

straction, despite being a ¬rst-rate mathematician. He did not even join the grow-

ing band of Maxwellians who fully accepted the theory of James Clerk Maxwell,

in which light emerges not as a mechanical vibration but rather as a variation in

electric and magnetic ¬elds. The implication of the equations of Maxwell, which

Foam and ether 67

Figure 7.6. Kelvin™s palatial residence on the Scottish coast.

required a whole generation of debate to clarify, was triumphantly vindicated in

1887 by the experiment of Heinrich Hertz in Germany. In this climactic moment

of the history of science, man ˜won the battle lost by the giants of old, has snatched

the thunderbolt from Jove himself and enslaved the all-pervading ether™. These

are the words of George Francis Fitzgerald, addressing the British Association.

It was a little too late to convert Kelvin, who went on cooking up material

ether models until he died, insisting that the elusive substance was ˜a real thing™.

˜Nothing™ said Fitzgerald later ˜will cure Sir William Thomson, short of the com-

plete overthrow of the whole idea™. Nothing indeed was going to de¬‚ect this

gallant knight from tilting at a favourite windmill.

On 29 September 1887, Kelvin woke up, sat up, and wrote in his notebook

˜rigidity of foam™. He had conceived the notion that the ether might be a foam,

a wild idea that Gibbs politely called ˜the audacity of genius™, after complain-

ing about the proliferation of published speculations on the ether. (Amazingly,

the latest speculation on the nature of space-time now seems headed in the same

direction.)

Kelvin turned to Plateau™s book for inspiration in trying to decide what struc-

ture the foam should have, and was soon playing with wire frames. His niece

Agnes King wrote on 5 November:

When I arrived here yesterday Uncle William and Aunt Fanny met me

at the door, Uncle William armed with a vessel of soap and glycerine

prepared for blowing soap bubbles, and a tray with a number of math-

ematical ¬gures made of wire. These he dips into the soap mixture

and a ¬lm forms or adheres to the wires very beautifully and perfectly

regularly. With some scienti¬c end in view he is studying these ¬lms.

By then Kelvin had already solved the problem that he had set himself, i.e. to

de¬ne the ideal structure of equal bubbles: what partitioning of space into equal

volumes minimizes their surface area? Or rather, he had come up with a reason-

Toils and troubles with bubbles

68

able conjecture, a masterful design which he thought nature must be compelled to

follow. He recorded it in his notebook on 4 November.

Since Kelvin™s ¬rst words on the subject”rigidity of foam”were written

in early September at 7.15am while in bed, we may presume that his foam ether

model was conceived during the night. This is a common enough phenomenon.

The scientist who retires, his brain feverishly obsessed with a single problem, is

likely enough to spend the night attacking it in that semi-conscious state which

is ideal for unbridled yet directed thought. Helmholtz gave as the requirement

for mathematical reasoning that ˜the mind should remain concentrated on a single

point, undisturbed by collateral ideas on the one hand, and by wishes and hopes

on the other™.

Maxwell described such an experience in a poem;

What though Dreams be wandering fancies,

By some lawless force entwined,

Empty bubbles, ¬‚oating upwards

Through the current of the mind,

There are powers and thoughts within us (. . . ).

Kelvin™s thought indeed consisted of empty bubbles, for it was a foam with-

out gas that he saw as a possible ether model. Such a thing cannot be stable but he

convinced himself otherwise. So, when he had succeeded in describing an appro-

priate structure for the ether foam, he rushed into print with it in the Philosophical

Magazine. The speed of this publication (not much more than a month later) rivals

or exceeds that of the printed journals of today. It may have helped that the great

man himself was editor of the journal, but one presumes that reviewers would

hardly have questioned his insights.

7.5 The Kelvin cell

The cell described by Kelvin may be described as a modi¬ed form of a truncated

octahedron, a term used by Kepler. Kelvin chose to call it the ˜tetrakaidecahe-

dron™. Coxeter has called this name ˜outrageous™; and it does seem unnecessary. It

was one of the 13 Archimedean solids and was familiar, for example, to Leonardo.

It plays an important role in modern solid state theory and crystallography, but it

was not so well known to physicists in the 1880s as it is today 6. Kelvin™s ability

to visualize it derived in part from his contributions to crystallography, to which

he applied his characteristically down-to-earth approach, advising students to buy

1000 wooden balls and study their possible arrangements. He saw that this poly-

hedron can be packed to ¬ll all space. Furthermore, only a little curvature of the

hexagonal faces is necessary to bring it into complete conformity with Plateau™s

requirements (Maraldi angles, etc). He adroitly calculated this subtle curvature

See: Weaire D (ed) 1997 The Kelvin Problem (London: Taylor and Francis).

The Kelvin cell 69

(a) (b)

Figure 7.7. The Kelvin cell (a). ˜Kelvin™s Bedspring™ (b) (courtesy of the University of

Glasgow).

but had dif¬culties in drawing it. True to form, he made a wire model so that it

could be seen in concrete form. This survives in the University of Glasgow and

is known as ˜Kelvin™s bed-spring™. Visitors to what remains of the Lisbon EXPO

can admire the large Kelvin cells at the centre of a framework of cables, designed

as a climbing frame for children.

Kelvin™s various attempts to bring forth an acceptable model for the mate-

rial ether were greeted with limited and diminishing enthusiasm. This particular

version was called ˜utterly frothy™ by a Cambridge don. When Fitzgerald tried

to be conciliatory by suggesting that Kelvin™s models were at best allegories, he

received a spirited retort in the words of Sheridan™s Mrs Malaprop: ˜certainly not

an allegory on the banks of the Nile™.

It is curious that Kelvin said nothing about a direct experimental test, al-

though he described a way of making his structure with wire frames (which begs

the question). Perhaps he was unaware of the ease with which equal bubbles can

be made, simply by blowing air through a thin nozzle immersed in a soap solution.

Unfortunately his uncon¬rmed conjecture was accepted too readily as the estab-

lished truth by others. It was as if, as John Ziman once said of another theoretical

model, ˜the Word had been made Flesh™. Uncritically accepted, his conjecture

remained unchallenged for quite some time.

In one unfortunate sequel, a Russian mathematician posed the same prob-

Toils and troubles with bubbles

70

lem as Kelvin, and developed the same conjecture, in 1992. He was apparently

oblivious of the work of his illustrious predecessor.

Most beautiful and regular

There has never been much doubt that Kelvin™s is the correct solution if all foam

cells are restricted to have an identical shape and orientation. The doubt arises

when they are given more freedom than this, as they have in nature, while main-

taining equal volumes.

It is part of the physicist™s faith that things are simple. (But not too simple,

as Einstein warned.) There is always a provisional prejudice in favour of a neat

solution rather than a complicated one. Or, put in grander language by D™Arcy

Wentworth Thompson, ˜the perfection of mathematical beauty is such (as Colin

MacLaurin learned of the bee), that whatsoever is most beautiful and regular is

also found to be most useful and excellent™.

Such a precept would be greatly improved by the addition of ˜generally

speaking™ but the imperious sweep of this superlative prose stylist would not

admit it. The Kelvin problem was for him solved ˜in the twinkling of an eye™,

presumably that of Kelvin. Thus was the Word made Flesh, at least in the mind

of the succeeding generation.

Kelvin™s favourite polyhedron also played a large role in his research on

space-¬lling structures in relation to general crystallography. In 1893 he wrote to

Rayleigh about this, in the midst of ¬ling patents, worrying about Home Rule and

other preoccupations. Referring decorously to his wife as ˜Lady Kelvin™, as be¬ts

a correspondence between two members of the House of Lords, he said that she

had begun to make a tetrakaidecahedral pin-cushion, which ˜will make all clear™.

Kelvin always brought his work home.

7.6 The twinkling of an eye

The American botanist Edwin Matzke used Thompson™s phrase as the title of a

lecture he gave to the botanical club at Columbia University in 1950. He poured

scorn on the widespread acceptance of Kelvin™s conjecture, together with the erro-

neous conclusions of Buffon and Hales, also repeated uncritically by Thompson.

They were relegated to the ˜limbo of quaint and forgotten dreams™.

It had led him and other biologists to undertake fruitless searches for the

Kelvin cell in natural cellular structures. It was nowhere to be found. He had

been driven to perform the experiment so long overlooked, by making an actual

foam of equal bubbles.

He did so in a manner which now looks foolish. It should have been known to

him that such bubbles may be created simply by blowing air steadily through a ¬ne

nozzle beneath the surface of a soap solution. Instead he and his assistants blew

every bubble individually with a syringe and added it carefully to the foam. This

Simulated soap 71

was a great labour and probably a labour of Sysiphus, because foam experiments

must be completed quickly to avoid effects due to diffusion of gas between cells.

˜Is this an indictment of twinkling eyes?™, asked Matzke in the end, after

failing to ¬nd a single Kelvin cell, and generously answered ˜no™ but not before

condemning the naivety of his predecessors.

Despite reservations about the validity of Matzke™s methods, there is nothing

wrong with his general conclusion: the cells remain disordered in an equilibrium

structure which is not optimal (like the bag of ball bearings), and there are no

Kelvin cells to be seen within the foam.

All this heavy labour and even heavier irony discouraged others, but the

mathematical question remained. Was Kelvin™s solution the best, at least in prin-

ciple? Some authorities, for example Hermann Weyl, were inclined to believe that

it was, while others had an open mind. Fred Almgren of Princeton said in 1982

that ˜despite the claims of various authors to the contrary it seems an open ques-

tion™. A few mathematicians 7 and especially the Almgren group kept the Kelvin

problem alive. One of Almgren™s graduate students (John Sullivan) has said: ˜I

think most people assumed this partition was best. But Fred alone was convinced

it could be beaten™.

Rob Kusner maintains that many people were also convinced that it could

be beaten. Indeed he was among them. In 1992 he proved that an assembly of

equal-pressure bubbles can minimize the interfacial surface area in a foam with

an average number of faces greater or equal to ½¿ ¿ ¿¿ 8 . The inequality

did not exclude the Kelvin solution, but Kusner (following ideas from Coxeter

and Bernal) was inclined to think that a minimal partition must have an average

number of faces close to 13.5.

7.7 Simulated soap

Today™s mathematicians can ease their frustration with the solution of dif¬cult

problems and with the illustration of abstract results by the use of powerful com-

puter simulations and advanced graphics. A leading exponent of such techniques

is Ken Brakke of Susquehanna University in Pennsylvania. A former student

of the Princeton school of Fred Almgren, he set out to develop a new and ¬‚ex-

ible computer code for producing surfaces of minimum area (Brakke™s Surface

Evolver). Once completed, it was generously offered to the world at large 9 . It has

been continuously updated ever since, and used for many things, from the shape

of a pendant liquid drop to the modelling of solder connections in semiconductor

circuits. It is, in particular, ideal for the Kelvin problem and it was set to work on

it around 1990.

Choe J 1989 On the existence and regularity of fundamental domains with least boundary area J.

Diff. Geom. 29 623“63.

Kusner R 1992 The number of faces in a minimal foam Proc. R. Soc. A 439 683“6.

http://www.susqu.edu/FacStaff/b/brakke/

Toils and troubles with bubbles

72

(a) (b)

Figure 7.8. (a) The Weaire“Phelan structure. (b) The observation of these cells in a real

form.

Kelvin™s conjecture at ¬rst survived this ¬rst onslaught by modern technol-

ogy: no better structure could be found.

7.8 A discovery in Dublin

In late 1993 Robert Phelan began his research at Trinity College Dublin. His task

was to explore the Kelvin problem and variations upon that theme, using Brakke™s

Surface Evolver.

Phelan had joined a computational physics group which had a broad back-

ground in solid state and materials science. Hence there was no question of a

blind search for an alternative structure. An idea was formed of the type of struc-

ture which might be competitive with that of Kelvin, essentially one with a lot of

pentagonal faces. What structures in nature have such a form?

There is one class of chemical compounds in which covalent bonds cre-

ate suitable structures: the bonds are tetrahedral. The compounds are called

clathrates, a reference to the fact that they are made up of polyhedral cages of

bonds. Usually they form because they create convenient homes for guest atoms

or molecules. Gas pipelines in the Arctic are sometimes clogged up with clathrate

crystals of ice.

The cages of bonds can be visualized as foam cells. Most of the rings of

A discovery in Dublin 73

Figure 7.9. The Weaire“Phelan structure in a computer-generated image by J M Sullivan.

bonds on the sides of the cages are ¬vefold, creating pentagonal faces, as seemed

to be required. The suggestion therefore was to explore the clathrates, particularly

the two simplest ones, Clathrate I and II. The ¬rst of these was fed into the Evolver

as a foam structure with equal cell volumes. Such a structure is a regular assembly

of two types of cell with, respectively, 12 and 14 faces and results in a foam with

13.5 faces in average.

When the ¬rst output emerged, it was immediately evident that it was going

to defeat Kelvin. When fully equilibrated, it turned out to have a surface area 0.3%

less than that of the venerable conjecture. This does not sound like very much but

endeavours in optimization always have to strive for quite small differences, in

horse-races and elsewhere. The margin of success in this case was recognized as

quite large10 .

Date: 9 December 1993 03:25:03.13

From: "brakke@geom.umn.edu"

Reply-To: "brakke@geom.umn.edu"

To: "dweaire@vax1.tcd.ie", "rphelan@alice.phy.tcd.ie"

cc:

Subject: kelvin

I confirm your results. I got the area down to 21.156.

As soon as I saw the picture on the screen, I was sure

you had it. I always figured the way to beat Kelvin was

to use lots of pentagons, since pentagon vertex angles

½¼

Later, Almgren, Kusner and Sullivan provided a rigorous proof that the Weaire“Phelan structure is

of lower energy than that of Kelvin™s.

Toils and troubles with bubbles

74

are very close to the tetrahedral angle. So when I saw

you had almost all pentagons, I knew you™d done it.

Congratulations.

Ken Brakke

It was, according to Almgren, a ˜glorious day for surface minimization the-

ory™ but, strictly speaking, it provides no more than a counter-example to Kelvin™s

conjecture and hence a replacement for it. The problem of proof that it is opti-

mal in an absolute sense remains, though con¬dence grows that this structure will

not in turn be surpassed: many related candidates have already been tried, mostly

from the wide class of clathrate structures 11 .

½½

See: Weaire D (ed) 1997 The Kelvin Problem (London: Taylor and Francis); Rivier N 1994 Kelvin™s

conjecture on minimal froths and the counter-example of Weaire and Phelan Phil. Mag. Lett. 69 297“

303.

Chapter 8

The architecture of the world

of atoms

8.1 Molecular tactics

Another book which appeared in the same year as that of Plateau (1873) was

L™architecture du monde des atomes by Marc-Antoine Gaudin, from which we

borrow the title for this chapter on the role of packing ideas in crystallography.

Gaudin sought to reconcile the laws of chemistry with the ¬ndings of early

crystallography, the experimental part of which comprised the study of the ex-

ternal forms of crystals. He constructed molecules of various shapes, consistent

with the symmetry of the corresponding crystal. The molecules were composed

of atoms in the required proportions. These atoms were, in turn, considered to

be made up of particles of ether but it suf¬ced for his purpose that they were

assumed to be roughly spherical and packed together with roughly constant sepa-

rations. His illustrations were delightful (¬gure 8.1) but his speculations were no

more than a shot in the dark, one small chapter in a confused story not concluded

until the early 20th century.

That crystals owe their beautiful angular forms to regular arrangements of

atoms or molecules was a very old hypothesis, but only in the late 19th century

was it at last pursued with rigour and related to properties: this was the birth of

solid state physics, which grew to be the dominant sector of modern physics, at

least in terms of the active population of researchers 1 .

½

See for general reference: Smith C S 1981 A Search for Structure (Cambridge, MA: MIT Press);

Burke J G 1966 Origin of the Science of Crystals (University of California Press); Weaire D L and

Windsor C G (ed) 1987 Solid State Science, Past, Present and Predicted (Bristol: Institute of Physics