75

The architecture of the world of atoms

76

Figure 8.1. Gaudin™s drawings of hypothetical molecular structures (1873).

Identifying what Kelvin called the ˜molecular tactics of a crystal™ remained a

hesitant and erroneous process until x-rays provided the means to determine these

Atoms and molecules: begging the question 77

Figure 8.2. The four elements and the Universe in Plato™s conception (from a drawing by

J Kepler).

arrangements reliably if not quite directly, early in the 20th century. Today we

can ¬nally see (or, more accurately, feel ) the individual atoms on the surface of a

crystal, using the scanning tunnelling or atomic force microscope.

8.2 Atoms and molecules: begging the question

Whether matter is discrete or continuous has been a subject of debate at least since

the time of the ancient Greek philosophers. The ¬rst detailed atomistic theory was

that of Plato (in the Timaeus) who described matter to be ˜One single Whole, with

all its parts perfect™. He associated the four ˜elements™”earth, water, ¬re and

air”with the form of four regular polyhedra”cube, icosahedron, tetrahedron

and octahedron”and with the dodecahedron he associated the Universe. Plato

attempted to match the properties of the elements with the shapes of the constitu-

tive atoms. For instance water, being ¬‚uid, was associated with the icosahedron,

which is the most spherical among the ¬ve regular solids. This theory was able

to offer an ad hoc explanation of phase transitions (for instance the transition

solid-to-liquid-to-vapour, which means earth-to-water-to-air and corresponds to

cube-to-icosahedron-to-octahedron).

The atomistic theory of Plato was dismissed by Aristotle. He argued that if

the elements are made up of these particles then the copies of each regular poly-

hedra must ¬ll the space around a point and this operation cannot be done with

the icosahedron and the octahedron (he thought erroneously that it was possible

to ¬ll space with regular tetrahedra).

The architecture of the world of atoms

78

In other periods atoms have been reduced to mere points, or considered to

be hard or soft spheres, or to have more exotic shapes endowed with speci¬c

properties inspired by chemistry. As Newton said, the invention of such ˜hooked

atoms™ by followers of Descartes often begged the question.

Christian Huygens in his Trait´ de la lumi` re (1690) suggested that the Ice-

e e

land spar (at that time very much studied for its birefringence) may be composed

of an array of slightly ¬‚attened spheroids (see ¬gure 8.3(c)). In this way he ex-

plained in one stroke the birefringence and the cleavage properties of these crys-

tals.

Ha¨ y™s celebrated constructions begged the question, in as much as he ex-

u

plained the external form of crystals as being due to the packing of small compo-

nents which were identical to the crystal itself. Ha¨ y™s 1784 Essai d™une theorie

u

sur la structure des crystaux was nevertheless the most perspicacious of early at-

tempts to make sense of crystals, in that he recognized that their angles are not

arbitrary but follow certain rules, still used today.

8.3 Atoms as points

The Newtonian vision was taken to an extreme by Roger Joseph Boscovitch, early

in the 18th century. He postulated that ˜matter is composed of perfectly indivisi-

ble, non-extended, discrete points™ which interacted with one another.

Boscovitch published his Theory of Natural Philosophy on 1758, when he

was a professor at the Collegium Romanum. He has been described as a philoso-

pher, astronomer, historian, engineer, architect, diplomat and man of the world.

Given all this, the book is a disappointingly dry exposition in which he attempts

to deduce much of physics from a ˜single law of forces™. This means a mutual

force between each pair of points. Boscovitch struggled to describe a possible

form for this, drawing illustrations which resemble modern graphs of interatomic

interactions.

Half a century later, French mathematicians such as Navier and Cauchy de-

veloped powerful theories of crystal elasticity based on this idea but independent

of any particular form for the interaction. Their elegant analysis of the effect of

crystal symmetry on properties has provided one of the enduring strands of phys-

ical mathematics. However, they were little concerned with the origins of crystal

structure itself.

Late in the 19th century, the Irish physicist Joseph Larmor reduced the role

of matter still further, to a mere mathematical singularity in the ether. His view

was enshrined in the book Aether and Matter, to which contemporaries jokingly

referred as ˜Aether and No Matter™. It was published in 1900, a date after which

Larmor declared that all progress in physical science had ceased. He was probably

not serious”it is, in fact, the date of the inception of quantum theory, which

¬nally told us what atoms are really like.

Today™s quantum mechanical picture of a nucleus surrounded by a cloud

Atoms as points 79

(a) (b)

(c)

Figure 8.3. Various atomic models for crystals: Ha¨ y (a), Wollaston (b), Huygens (c).

u

of electrons is a subtle one: it taxes the resources of the largest computers to

predict what happens when these clouds come into contact. The day has not yet

come when older, rough-and-ready descriptions are completely obsolete. It is still

useful for some purposes to picture atoms as hard balls with relatively weak forces

of attraction pulling them together. In particular the structures of many metals can

be understood in this way.

The architecture of the world of atoms

80

Figure 8.4. All the snow crystals have a common hexagonal pattern and most of them

show a hexagonal shape.

8.4 Playing hardball

Among the manifold older ideas about atoms, the elementary notion of a hard

sphere has endured as a useful one, even today.

Spherical ˜atoms™ were adopted by Kepler to explain the hexagonal shape of

snow¬‚akes (chapter 3). Kepler assumed that they were composed of tiny spheres

arranged in the plane in the triangular packing. He did not consider these spheres

to be atoms in the modern sense but more as the smallest particle of frozen water.

Of the later re-inventions of this type of theory, one of the most in¬‚uential

was that of William Barlow, writing in Nature in 1883.

Barlow is a prime example of the self-made scientist. His paper ˜On the

probable nature of the internal symmetry of crystals™ is remarkable for its total

lack of any reference to previous work. He happily ignored centuries of specula-

tions, particularly those of his compatriot William Wollaston (see ¬gure 8.3) early

in the same century. His in¬‚uence, particularly on and through Lord Kelvin, may

be attributed in part to his clear writing style, his choice of the simplest cases and

his use of attractive illustrations (¬gure 8.5).

Barlow™s method was to look for dense packings of spheres, with no attempt

at proof that they were the densest. This he left for mathematicians to consider

later. He mentioned the possibility that soft spheres might make more realistic

atoms but did little with this.

His down-to-earth commonsense approach might be regarded as a reaction

against the re¬ned mathematics of the French school, which many British natural

philosophers, such as Tait, had found rather indigestible. John Ruskin, venturing

into science in a manner which was then fashionable, had said in his Ethics of the

Dust (1865), which consisted of ˜ten lectures to little house wives on the elements

of crystallization™, that the ˜mathematical part of crystallography is quite beyond

girls™ strength™. One might suppose that it would be beyond Barlow™s strength as

Playing hardball 81

Figure 8.5. Some illustrations from Barlow™s papers.

well, but in fact he took it up avidly, and absorbed the full mathematical theory in

later years. Indeed he published in that area (albeit with some further disregard

for the precedents).

By 1897 he was ready to expound a more mature version of the theory in

a more erudite style. He described many possible stackings for hard spheres, of

equal or unequal sizes.

Barlow™s intuitive attack scored a number of notable hits, particularly in pre-

The architecture of the world of atoms

82

Figure 8.6. One of the earliest x-ray photographs (1896).

dicting the structures of the alkali halides. His place in the history of science was

then assured by a contemporary and apparently unrelated discovery.

On 8 November 1895 a professor of physics in W¨ rzburg realized that a new

u

type of ray was emanating from his discharge tube. The R¨ ntgen ray, named after

o

its discoverer but eventually to be called the x-ray, was an immediate popular

sensation, much to R¨ ntgen™s distaste. The potential for medical science and

o

the challenge to the modesty of Victorian ladies was clear”the implications for

physics were not. One of the most extraordinary of these, which took two decades

to emerge, was the determination of crystal structures using x-rays, vindicating

much of the guesswork of Barlow and others.

It should not be thought that all crystal structures are dense packings of balls.

In the structure of diamond, each atom has only four atoms as neighbours. And

this does not even qualify as a stable loose packing. This structure too was pre-

saged before it was observed”this time by Walter Nernst (1864“1941), better

known for his Heat Theorem (the Third Law of Thermodynamics).

8.5 Modern crystallography

In a crystal the structure repeats a local con¬guration of atoms as in a three-

dimensional wall paper. There are 14 ways to construct such periodic structures

in three dimensions, the Bravais lattices, but 230 different types of internal sym-

metry. This ˜crushingly high number of 230 possible orderings™, as Voigt called it,

was both challenging and depressing to the theorist, until x-ray diffraction offered

Crystalline packings 83

the means to use the theory in every detail.

When x-rays (electromagnetic radiation with a typical wavelength between

˚

0.1 and 10 Angstroms, i.e. between 0.000 000 01 and 0.000 001 mm) are incident

on a crystal, they are diffracted and form a pattern with sharp spots of high inten-

sity corresponding to speci¬c angular directions. On 8 June 1912 at the Bavarian

Science Academy of Munich, a study entitled ˜Interference effects with R¨ ntgen o

2

rays™ was presented . In this work Max von Laue developed a theory for the

diffraction of x-rays from a periodic packing of atoms, associating the spots of

intensity in the diffraction pattern with the regularity of the positions of the atoms

in the crystal structure. One year later Bragg reported the ¬rst determination of

crystal structures from x-ray diffraction for such systems as KCl, NaCl, KBr and

KI, con¬rming Barlow™s models 3 .

8.6 Crystalline packings

In many crystal structures one or several types of atom are in positions corre-

sponding to the centres of spheres in a sphere packing, as Barlow had supposed.

It has been noted by O™Keeffe and Hyde 4 , two experts in crystal structures, that ˜it

is hard to invent a simple symmetric sphere packing that does not occur in nature™.

Of these, the most important family is that of packings with the maximal

¼¼

density of which the face-centred cubic (fcc) is a member. This

maximal density can be realized in in¬nitely many ways, all of which are based

on the stacking of close-packed layers of spheres (¬gure 8.8), as practised by the

greengrocer.

Two possibilities present themselves for the relative location of the next

layer, if it is to ¬t snugly into the ¬rst one. Each successive layer offers a similar

choice, and only by following a particular rule will the fcc structure described by

Kepler result, with a cubic symmetry, which is not at all apparent in this building

procedure. A different rule produces the hexagonal close-packed (hcp) structure,

also described by Barlow, the next in order of simplicity. These two structures

occur widely among the structures formed by the elements of the Periodic Table.

More complex members of the same family”for example the double hexagonal

close-packed structure”are found in alloys.

The original appeal of crystals lay in their external shapes, and these pro-

vided clues to their internal order. However, the precise shape of a crystal in

equilibrium cannot be deduced from this order alone. According to a principle

enunciated by Gibbs and Curie in 1875, the external shape of a crystal minimizes

¾

Friedrich W, Knipping P and Laue M 1912 Interferenz-Erscheinungen bei Roentgenstrahlen S. B.

Bayer. Akad. Wiss. pp 303“22.

¿

Bragg W L 1913 The structure of some crystals as indicated by their diffaction Proc. Roy. Soc. A

89 248“77.

O™Keefe M and Hyde G B 1996 Crystal Structures (Washington, DC: Mineralogical Society of

America).

The architecture of the world of atoms

84

(a)

(b)

Figure 8.7. X-ray diffraction pattern (a) and the relevant crystalline structure (b) for NaCl,

as reported by Bragg (1913).

the total surface energy. This is made up of contributions from each facet but

different types of facet are more expensive in terms of energy.

Some of the observed shapes can be realized by a rule developed by Bravais:

the largest facets have the densest packing of atoms (which might be expected to

have the lowest surface energy).

If the packing fraction is decreased a little from its maximum value, allowing

the hard spheres some room to move, and they are given some kinetic energy, what

can be said about the competition between these structures? This is a very delicate

question for thermodynamics, and it has only been settled recently by extensive

computations. The winner (as the more stable structure) is fcc 5 .

Some natural elements which have an fcc or hcp crystalline structure are

given in table 8.1.

Woodcock L V 1997 Entropy difference between the face-centred cubic and hexagonal close-

packed crystal structures Nature 385 141“3.

Tetrahedral packing 85

(a) (b)

(c)

Figure 8.8. The hexagonal closed-packed (hcp) (a) and cubic closed-packed (fcc) (b)

structures. These lattices are generated by a sequence of layers of spheres in the triangular

packing con¬guration (c). Suppose that the ¬rst layer of spheres has centres in position

A, the second layer can be placed in position B (or equivalently C), and for the third layer

we have two alternatives: (i) placing the centres of the spheres in position A generating

the sequence ABABAB. . . (which corresponds to the hcp lattice); (ii) placing the centres

in position C generating the sequence ABCABCABC. . . (which corresponds to the fcc

lattice).

8.7 Tetrahedral packing

Regular tetrahedra cannot pack together to ¬ll space but irregular ones may do

so. Of special interest for crystal chemistry are packings in which neighbouring

atoms are on the vertices of such a system of closely packed tetrahedra. These

structures are called ˜tetrahedrally packed™.

A very important tetrahedrally-packed structure is the body-centred cubic

lattice (bcc). This is the crystalline structure of many chemical elements. The

bcc structure is the only tetrahedrally packed structure where all tetrahedra are

identical.

The architecture of the world of atoms

86

Table 8.1. Some natural elements which have an fcc or hcp crystalline structure. These are

at room temperature unless otherwise indicated.

fcc hcp

Al Be

Ag Cd

Ar (20 K) Co

Au He (He , 2 K)

Ca Gd

Ce Mg

¬-Co Re

Cu Ti

Ir Zn

Kr (58 K)

La

Ne (20 K)

Ni

Pb

Pd

Pt

Æ-Pu

Rh

Sr

Th

Xe (92 K)

Yb

A special class of structures consists of those in which the packing is re-

stricted to con¬gurations in which ¬ve or six tetrahedra meet at an edge. These

are the crystal structures of some of the more important intermetallic phases. Such

structures are known as tetrahedrally close packed (tcp) and were described in the

1950s by Frank and Kasper 6 . These two eminent crystallographers were inspired

to produce their classi¬cation of complex alloy structures by a visit to Toledo,

where the Moorish tiling patterns incorporate subtle mixtures of coordination

and symmetry, particularly ¬vefold features such as the regular pentagon. In the

Frank“Kasper structures the packed spheres have various combinations of 12, 14,

15 and 16 neighbours, and the average is between 13.33. . . and 13.5.

Each tetrahedron of a tetrahedrally-packed structure has four other tetrahe-

dra sharing its faces, so the dual structure, derived by placing the vertices in the

centre of the tetrahedra (i.e. in the holes between spheres), forms a four-connected

network. Such a network can be regarded as the packing of polyhedra which are

Frank F C and Kasper J S 1958 Acta Crystallogr. 11 184; 1959 Acta Crystallogr. 12 483.

Quasicrystals 87

restricted to pentagonal and hexagonal faces only. These networks also represent

signi¬cant structures in chemistry”the clathrates (section 7.8).

8.8 Quasicrystals

We have seen in the previous paragraphs how, starting from such clues as the regu-

lar angular shapes of crystals, scientists constructed a theory”crystallography”

in which the intrinsic structure of crystals was described as a periodic assembly

of atoms, in an apparently complete system of ordered structures in the solid

state. But in November 1984 a revolution took place: Shechtman, Blech, Gratias

and Chan identi¬ed, in rapidly solidi¬ed AlMn alloys, an apparently new state

of condensed matter, ordered but not periodic. The scienti¬c journal Physics To-

day headlined ˜Puzzling Crystals Plunge Scientists into Uncertainty™. Marjorie

Senechal, a mathematician who has been one of the protagonists of this revolu-

tion, described this climate of astonishment in her book Quasicrystals and Geom-

etry7 :

It was evident almost immediately after the November 1984 announce-

ment of the discovery of crystals with icosahedral symmetry that new

areas of research had been opened in mathematics as well as in solid

state science. For nearly 200 years it had been axiomatic that the in-

ternal structure of a crystal was periodic, like a three-dimensional wall-

paper pattern. Together with this axiom, generations of students had

learned its corollary: icosahedral symmetry is incompatible with peri-

odicity and is therefore impossible for crystals. Over the years, an ele-

gant and far-reaching mathematical theory had been developed to inter-

pret these ˜facts™. But suddenly”in the words of the poet W B Yeats”

all is changed, changed utterly.

What terrible beauty was born? These new solids, with diffraction patterns

which exhibit symmetries which are forbidden by the crystallographic restric-

tions, have been called quasicrystals. The internal structure of a quasicrystal is

an ordered packing of identical local con¬gurations with non-periodic positions

in space.

With this perspective, it is a shock to realize that the seed of quasi-crystallin-

ity was already there in Kepler™s work 8 . In his book Harmonices Mundi (1619)

Kepler described a repetitive structure with the ˜forbidden™ ¬vefold symmetry, but

with ˜certain irregularities™.

If you really wish to continue the pattern, certain irregularities must be

admitted, (. . . ) as it progresses this ¬ve-cornered pattern continually

introduces something new. The structure is very elaborate and intricate.

Senechal M 1995 Quasicrystals and Geometry (Cambridge: Cambridge University Press).

Or even older instances in Islamic architecture: see Makovicky E 1992 Fivefold Symmetry ed I Har-

gittai (Singapore: World Scienti¬c) p 67.

The architecture of the world of atoms

88

(a) (b)

Figure 8.9. Two non-periodic tilings: that proposed by Kepler in 1619 (a) and that pro-

posed by Penrose in 1974 (b). Note that Kepler™s one is ¬nite whereas Penrose™s can be

continued on the whole plane.

Hundreds of years later, in 1974, Roger Penrose produced a tiling that can

be considered the realization of the one described by Kepler (see ¬gure 8.9) 9.

The Penrose tiling covers the entire plane; it is non-periodic but repetitive.

Non-periodic means that if one takes two identical copies of the structure there

is only one position where these two structures superimpose perfectly. In other

words, sitting in a given position, the landscape around, up to an in¬nite distance,

is unique and cannot be seen from any other point. Repetitive means that any local

part of the structure is repeated an in¬nite number of times in the whole structure.

Exactly ten years after Penrose™s work, this pattern was ¬rst observed in

nature.

What most surprised the researchers after the discovery of quasicrystals was

that these structures have diffraction patterns with well-de¬ned sharp peaks that

were previously considered to be the signature of periodicity. How can aperiodic

structures have sharp diffraction peaks? Let us just say that the condition to have

diffraction is associated with a strong form of repetitiveness, which is typical of

these quasicrystalline structures.

Mathematically, these quasiperiodic patterns can be constructed from a crys-

talline periodic structure in a high-dimensional hyperspace by cutting it with a

plane oriented with an irrational angle in respect to the crystalline axis. This con-

Penrose R 1974 Pentaplexity Bull. Inst. Math. Appl. 10 266“71.

Amorphous solids 89

struction explains the existence of diffraction peaks but does not give any physical

understanding.

Why has nature decided to pack atoms in these non-periodic but repetitive

quasicrystalline con¬gurations? This is a matter of debate, unlikely to be quickly

resolved10.

As a consequence of this revolution it was necessary for the scienti¬c com-

munity to ask: what is to be considered to be a crystal?

The International Union of Crystallography established a commission on

Aperiodic Crystals that, in 1992, proposed for ˜crystal™ the following de¬nition:

A crystal is any solid with an essentially discrete diffraction diagram.

According to this edict, the de¬nition of a crystal which we have given earlier is

too restrictive but the new one will, alas, be a mystery to many.

8.9 Amorphous solids

Since ancient times, the distinction has been made between crystals and non-

crystalline or amorphous (shapeless) solids. But the ¬rst category was reserved

for large crystals such as gemstones. It was not realized until the present century

that most inanimate materials (and quite a few biological ones as well) consist of

¬ne crystalline grains, invisible to the eye and not easy to recognize even under a

microscope. Hence they were wrongly classi¬ed as amorphous. Despite impor-

tant clues, such as the fracture surface of typical metals, crystallinity was regarded

as the exception rather than the rule 11 .

It remains dif¬cult to distinguish an aggregate of very ¬ne crystals from an

amorphous solid using x-ray diffraction, and careers have been founded on an

increasingly meaningless distinction. Today such amorphous materials as window

glass are accepted as having a random structure, just as John Tyndall suggested

with typical lyricism in Heat”A Mode of Motion (1863):

To many persons here present a block of ice may seem of no more in-

terest and beauty than a block of glass; but in reality it bears the same

relation to glass that orchestral harmony does to the cries of the mar-

ketplace. The ice is music, the glass is noise; the ice is order, the glass

is confusion. In the glass, molecular forces constitute an inextricably

entangled skein; in ice they are woven to a symmetric texture (. . . )

Amorphous metals, usually obtained by very rapid cooling from the liquid

state, also have a random structure, in this case approximated by Bernal™s random

½¼

See, for instance, Steinhardt P, Jeong H-C, Saitoh K, Tanaka M, Abe E and Tsai A P 1998 Ex-

perimental veri¬cation of the quasi-unit-cell model of quasicrystal structure Nature 396 55“7; 1999

Nature 399 84.

½½

See for general reference: Lines M E 1994 On the Shoulders of Giants (Bristol: Institute of Physics

Publishing).

The architecture of the world of atoms

90

sphere packings (chapter 3). This structure gives them exceptional properties,

useful in magnetic devices or as a coating on razor blades and in the manufacture

of state-of-the-art golf clubs. What was once a mere academic curiosity now

caresses the chin of the aspiring executive and adds several yards to his drive into

the bushes from the ¬rst tee.

8.10 Crystal nonsense

Old errors cast long shadows in our conception of the natural world. Astrology

still commands copious column-inches in the daily papers, spiritualists ply a busy

trade, and books abound on the mystical power of crystals. A suitable crystal,

we are told, can radiate its energy to us and in¬‚uence our aura, in a harmonious

vibration of happiness. If only it were so easy. . . . Solid state physicists would go

around with permanent smiles on their faces.

This strange attribution of a hidden potency is derived from the time when

crystals were regarded as rare exceptions to the general disorder of inanimate

nature. Their strange perfection of form must have led primitive man to wonder

at them: it is known that Peking Man collected rock crystals. Perhaps in modern

times the renewed fascination with ˜crystal energy™ also derives from the period

of early radio when ˜crystal sets™, consisting of little besides a point contact to a

crystal, acting as a recti¬er, could be used to listen to radio broadcasts. Magic

indeed!

The more sophisticated may have been aware that the details of the process

of growth of a crystal proved intractable to any convincing explanation for some

time. The problem was, for example, mentioned by A E H Tutton (1924) The

Natural History of Crystals:

One of the most deeply interesting aspects of a crystal (. . . ) concerns

the mysterious process of its growth from a solution (. . . ). The story of

the elucidation, as far as it has yet been accomplished, of the nature of

crystallization from solution in water is one of the most romantic which

the whole history of science can furnish.

It is, by its very nature, a delicate problem of surface science, where minute

amounts of impurities, or defects, can do strange things to help or hinder growth.

But, by and large, crystallization is no longer such a mystery”otherwise the semi-

conductor industry could hardly be engaged in making huge silicon crystals of

extraordinary perfection, every day.

The progress of physics did not, however, de¬‚ect Rupert Sheldrake from us-

ing the assumed intractability of the explanation of crystallization as a pretext for

his provocative theory of ˜morphic resonance™, according to which crystallization

is guided by a memory from the past. As a botanist of impeccable pedigree, he

was then able to generalize his principle to the living world, creating a consider-

able following and almost unbearable irritation in the orthodox scienti¬c commu-

nity.

Chapter 9

Apollonius and concrete

9.1 Mixing concrete

Any builder knows that to obtain compact packings in granular mixtures such as

the ˜aggregate™ used to make concrete, the size of the particles must vary over

a wide range. The reason is evident: small particles ¬t into the interstices of

larger ones, leaving smaller interstices to be ¬lled, and so on. A typical recipe for

very dense mixtures starts with grains of a given size and mixes them with grains

of smaller and smaller sizes in prescribed ratios of size and quantity, as already

mentioned in chapter 2. The resulting mixture has a density that can approach

unity.

Such recursive packing was already imagined around the 200 BC by Apol-

lonius of Perga (269“190 BC), a mathematician of the Alexandrine school. He

is classi¬ed with Euclid and Archimedes among the great mathematicians of the

Greek era. His principal legacy is the theory of those curves known as conic sec-

tions (ellipse, parabola, hyperbola). He brought it to such perfection that 1800

years passed before Descartes recast it in terms of his new methods.

The method of recursive packing reappeared more recently in a letter by

G W Leibniz (1646“1716) to Brosses:

Imagine a circle; inscribe within it three other circles congruent to each

other and of maximum radius; proceed similarly within each of these

circles and within each interval between them, and imagine that the

process continues to in¬nity. (See ¬gure 9.1.)

Something similar also arose in the work of the Polish mathematician, Wa-

claw Sierpi´ ski (1887“1969), who wrote a paper in 1915 on what has come to

n

be called the ˜Sierpi´ ski Gasket™, and well known as a good example of a fractal

n

91

Apollonius and concrete

92

Figure 9.1. Apollonian packing.

Figure 9.2. Sierpi´ ski Gasket.

n

structure (see ¬gure 9.2). Ian Stewart has called it the ˜incarnation of recursive

geometry™.

The aim of Sierpi´ ski was to provide an example of a curve that crosses itself

n

at every point, ˜a curve simultaneously Cartesian and Jordanian of which every

point is a point of rami¬cation™. Clearly this curve is a fractal, but this word was

Apollonian packing 93

coined by Benoˆt Mandelbrot 1 only in 1975.

±

Sierpi´ ski was exceptionally proli¬c”he published 720 papers and more

n

than 60 books. He called himself an ˜Explorer of the In¬nite™.

9.2 Apollonian packing

In the packing procedure known as ˜Apollonian Packing™, one starts with three

mutually touching circles and puts in the hole between them a fourth circle which

touches all three. Then the same procedure is iterated.

Apollonius studied the problem of ¬nding the circle that is tangent to three

given objects (each of which may be a point, line or circle). Euclid had already

solved the two easiest cases in his Elements, and the other (apart from the three-

circle problem) appeared in the Tangencies of Apollonius. The three-circle prob-

lem (or the kissing-circle problem) was ¬nally solved by Vi` te (1540“1603) and

e

the solutions are called Apollonian circles. A formula for ¬nding the radius (– )

of the fourth circle which touches three mutually tangent circles of radii (– ½ , –¾

and –¿ ) was given by Ren´ Descartes in a letter in November 1643 to Princess

e

Elisabeth of Bohemia:

½ · –½ · –½ · –½

¾ ¾ ¾ ¾

¾ –½ ¾ ¿

½·½·½·½ ¾

(9.1)

– – – –

½ ¾ ¿

This formula was rediscovered in 1936 by the physicist Sir Frederick Soddy

who expressed it in the form of a poem, ˜The Kiss Precise™ 2 :

Four pairs of lips to kiss maybe

Involves no trigonometry.

™Tis not so when four circles kiss

Each one the other three.

To bring this off the four must be

As three in one or one in three.

If one in three, beyond a doubt

Each gets three kisses from without.

If three in one, then is that one

Thrice kissed internally.

Four circles to the kissing come,

The smaller are the benter,

The bend is just the inverse of

The distance from the centre.

½

Mandelbrot B B 1977 The Fractal Geometry of Nature (New York: Freeman).

¾

Soddy F 1936 The kiss precise Nature 137 1021.

Apollonius and concrete

94

Though their intrigue left Euclid dumb.

There™s now no need for the rule of thumb.

Since zero bends a straight line

And concave bends have minus sign,

The sum of the squares of all four bends

Is half the square of their sum.

To spy out spherical affairs

An oscular surveyor

Might ¬nd the task laborious,

The sphere is much the gayer,

And now besides the pair of pairs

A ¬fth sphere in the kissing shares.

Yet, signs and zero as before,

For each to kiss the other four

The square of the sum of all ¬ve bends

Is thrice the sum of their squares

In the Apollonian procedure, the size of the circles inserted inside the holes

become smaller and smaller and the packing fraction approaches unity in the in¬-

nite limit. For example, one can start from three equal tangent unit circles which

pack with a density of 0.907. . . . Inside the hole one can insert a circle with ra-

dius ½ ¼ ½ and the density becomes 0.95. . . . Now there are three

holes where one can insert three circles with radius ½ ½ ¼ ¼ ¿ and

the density rises to ¼ .

9.3 Packing fraction and fractal dimension

Pursued inde¬nitely, Apollonian packing leads to a dense system with packing

½. But how is this limit reached? Suppose for instance that we

fraction

start with circles of radii –Ð – and stop the sequence when the radii arrive at the

minimum value – —‘ ÐÐ . The packing fraction is

´¾ µ

–—‘

½ ÐÐ

(9.2)

–Ð –

where is the fractal dimension. Indeed the Apollonian packing is a clas-

sical example of a fractal, in which the structure is composed of many simi-

lar components with sizes that scale over an in¬nite range. Numerical simu-

½ ¿¼ 3 . The analytical determination of is a surpris-

lations give

ingly dif¬cult problem. Exact bounds have been calculated by Boyd who found

½ ¿¼¼ ½ ½ ¿½ ¿ 4. The fractal dimension can be calculated in the two

¿

Manna S S and Hermann H J 1991 Precise determination of the fractal dimension of Apollonian

packing and space-¬lling bearings J. Phys. A: Math. Gen. 24 L481“90.

Boyd D W 1973 Mathematica 20 170.

Packing fraction in granular aggregate 95

Figure 9.3. ˜So we may image similar rings of spheres above and below (. . . ) and then

being all over again to ¬ll up the remaining spaces and so on ad in¬nitum, every sphere

added increasing the number that have to be added to ¬ll it up!™ [Soddy F 1937 The bowl

of integers and the hexlet Nature 139 77“9.]

½

interesting models for packings with triangles and hexagons which have

½ respectively5.

and

9.4 Packing fraction in granular aggregate

The Apollonian packing procedure can be extended to three dimensions. In this

case, four spheres are closely packed touching each other and a ¬fth one is in-

serted in the hole between them. Then the procedure continues in¬nitely as in

two dimensions.

Descartes™ theorem (equation (9.1)) was extended to three dimensions by

Soddy in the third verse of his poem and to dimensions by Gosset in another

poem also entitled ˜The Kiss Precise™ 6 .

And let us not con¬ne our cares

To simple circles, planes and spheres,

But rise to hyper ¬‚ats and bends

Where kissing multiple appears.

In ’-ic space the kissing pairs

Are hyperspheres, and Truth declares”

As ’ · ¾ such osculate

Bidaux R, Boccara N, Sarma G, de Seze L, de Gennes P G and Parodi O 1973 Statistical properties

of focal conic textures in smetic liquid crystals J. Physique 34 661“72; Eggleton A 1953 Proc. Camb.

Phil. Soc. 49 26.

Gosset T 1937 The kiss precise Nature 139 62.

Apollonius and concrete

96

Each with ’ · ½ fold mate

The square of the sum of all the bends

Is ’ times the sum of their squares.

Gosset™s equation replaces the factor 2 in front of equation (9.1) with a factor

(the space dimension, ’ in Gosset™s poem). In three dimensions it gives, for

example, a radius of 0.2247. . . for the maximum sphere inside the hole between

four touching spheres.

The relation for the packing density can also be extended to three and higher

dimensions7 .

´ µ

–—‘

½ ÐÐ

(9.3)

–Ð –

This expression is not limited to the Apollonian case but is valid for any fractal

–—‘ ÐÐ ) with power law

packing in the limit of very wide polydispersity (– Ð –

in the size distribution.

The fractal dimension for Apollonian sphere packings has been less studied

than in the two-dimensional case. However, the two models of packings with

hexagons and triangles can be easily extended to three dimensions giving

½ ¾ and ¾.

Engineers have long known 8 that the porosity ” of the grain mixture is a

function of the ratio between radii of the smallest and the largest grains utilized.

They empirically use the equation

½

–—‘

½ ÐÐ

” (9.4)

–Ð –

¿ ½ ¾

A comparison with equation (9.3) gives the fractal dimension as .

Herrmann H J, Mantica G and Bessis D 1990 Space-¬lling bearings Phys. Rev. Lett. 65 3223“6.

Caquot M A 1937 Le role des mat´ riaux inerts dans le b´ ton Soci´ t´ des Ingeni´ urs Civils de France

e e ee e

pp 563“82.

Chapter 10

The Giant™s Causeway

10.1 Worth seeing?

The Giant™s Causeway is a columnar basalt formation on the north coast of Ire-

land. It has been an object of admiration for many centuries and the subject of

continual scienti¬c debate 1 . Although it still draws tourists from afar, Dr John-

son™s acid remark that it was ˜worth seeing but not worth going to see™, echoed

by the irony of Thackeray™s account of a visit, may be justi¬ed, since similar ge-

ological features occur throughout the world. Among the more notable examples

are those of the Auvergne in France, Staffa in Scotland and the Devil™s Postpile in

the Sierra Nevada of California.

The primary historical importance of the debate on the origins of the Cause-

way lies in it being a focus of the intellectual battle between the Neptunists and

the Vulcanists in the 18th century. The history of geology delights in giving such

titles to its warring sects: others have been classed as Plutonists, Catastrophists

and Uniformitarians.

To the convinced Neptunist the origin of rocks lay in the sedimentary pro-

cesses of the sea, while a Vulcanist would argue for volcanic action. As in most

good arguments, both sides were right in certain cases. But in the case of basalt

the Neptunists were seriously wrong.

Why should this concern us here? Simply because the story is intertwined

with many of the strands of ideas on packing and crystallization in the understand-

ing of materials which emerged over the same period.

What was so fascinating about the Causeway?

½

Herries Davies G L 1981 A Geology of Ireland ed C H Holland (Edinburg: Scottish Academic).

97

The Giant™s Causeway

98

Figure 10.1. Sketch of the Giant™s Causeway (from Philosophical Transactions of the

Royal Society 1694).

10.2 Idealization oversteps again

As with the bee™s cell, but even more so, sedentary commentators on the Cause-

way have generally overstated the perfection of order which is to be seen in the

densely packed basalt columns. They have been described as ˜hexagonal™, im-

plying that the pattern is a perfect honeycomb. This is not at all the case (see

¬gure 10.2).

Eyewitness reports, especially by unscienti¬c visitors, were generally more

accurate. The Percy family of Boston reported in their Visit to Ireland (1859):

Do they not all look alike?

Yes, just as the leaves are alike in general construction, but endlessly

diverse, just as all human faces are alike, but all of them possessed of

an individual identity.

But as the story was passed around, idealization constantly reasserted a reg-

The ¬rst of¬cial report 99

Figure 10.2. Distribution of the number of sides of basalt columns in two of the most

famous sites where these occur (after Spry A 1962 The origin of columnar jointing, partic-

ularly in basalt ¬‚ans J. Geol. Soc. Australia 8 191).

ular hexagonal pattern for the Causeway, instead of the elegant random pattern

in which less than half of the polygons are six-sided, as one can verify from ¬g-

ure 10.2.

˜Hexagonal™ was an evocative word, calling to mind the form of many crys-

tals. It was natural therefore to call the Causeway ˜crystalline™; and see the

columns as huge crystals, even though their surfaces were rough in appearance.

Alternatively, it was suggested that the columns are formed by compaction or by

cracking. The latter was the choice of the Vulcanists, who saw the basalt as slowly

cooling and contracting, until it cracked.

That random cracking should have this effect seems almost as unlikely as

crystallization, at ¬rst glance, yet it has come to be accepted, as we explain later.

This has not stopped the continued generation of wild theories, even late in the

20th century.

10.3 The ¬rst of¬cial report

In 1693 Sir Richard Bulkeley made a report of the Causeway to the Royal So-

ciety in London. Like many who were to follow, he offered an account of the

phenomenon without troubling himself to go to see it. He relayed the news from

a scholar and traveller well known to him, that it ˜consists all of pillars of perpen-

dicular cylinders, Hexagones and Pentagones, about 18 to 20 inches in diameter™.

While offering no promise to make a visit himself, he offered to answer

any queries. In the following year Samuel Foley published answers to questions

forwarded by Bulkeley. Already the similarity to crystalline forms was noted. A

more scholarly and verbose account by Thomas Molyneux of Dublin followed,

The Giant™s Causeway

100

complete with classical references, and containing some intemperate criticisms of

the original reports. Despite his superior tone, the author confesses that ˜I have

never as yet been upon the place myself™. He noted a similarity to certain fossils

described by Lister but found the difference of scale dif¬cult to explain away.

For a time the arguments lapsed, but a number of ¬ne engravings with de-

tailed notes were published. Art served science well, in providing an inspiring

and accurate picture for the armchair theorists of geology. The correspondence

resumed around 1750. Richard Pockock had spent a week at the Causeway, and

settled on a Neptunist mechanism of precipitation.

In 1771, N Desmarest published a memoir which was to be central to the

Vulcanist/Neptunist dispute. In his view the ˜regular forms of basalt are the re-

sult of the uniform contraction undergone by the fused material as it cooled and

congealed™. This was countered by James Keir, who reasserted the crystalline

hypothesis, drawing on observations of the recrystallization of glass. Admittedly

there was a great difference of scale, but ˜no more than is proportionate to the

difference observed between the little works of art and the magni¬cent operations

of nature™. Here ˜art™ means craft or industry, for recrystallization was a matter of

intense commercial interest in the attempt to reproduce oriental porcelain.

The Reverend William Hamilton added further support to crystallization

with published letters and a dreadful 100-page poem (˜Come lonely Genius of

my natal shore. . . ™), published in 1811. This and vitriolic rebuttals of Desmarest

by Kirwan and Richardson did not succeed in reversing the advance of the Vul-

canist hypothesis. It was, said Richardson, an ˜anti-Christian and anti-monarchist

conspiracy™, since it set out to ˜impeach the chronology of Moses™. He favoured

a model of compression of spheroidal masses to form columns, with some labo-

ratory experiments to back it up.

It was probably fair comment when Robert Mallett summarized the state of

play in 1875 by saying that ˜no consistent or even clearly intelligible theory of the

production of columnar structure can be found™.

Mallett was an early geophysicist, who invented the term ˜seismology™. The

name of his engineering works still adorns the railings of Trinity College Dublin.

He undertook a thorough review of the basalt question and attempted to publish

it in the Proceedings of the Royal Society. After ¬ve months and four referees he

was told that ˜it was not deemed expedient to print it at present™”a splendidly

diplomatic refusal to publish the work of a Fellow of the Society.

This reversal may well have stemmed from his trenchant criticism of ˜very

crude and ill-thought-out notions™ and a ˜bad or imperfect experiment inaccurately

reasoned upon and falsely applied™ by his predecessors, who were ˜blinded by a

preconceived and falsely based hypothesis™. His own advocacy was directed in

support of contraction and cracking. He gave credit for this to James Thomson

(the Glasgow professor who was the father of Lord Kelvin) together with the

French school of Desmarest.

He ¬nally succeeded in publishing his article in the lesser (and less conser-

vative) journal Philosophical Magazine.

Mallett™s model 101

Figure 10.3. Polyhedral basaltic columns in the Giant™s Causeway.

10.4 Mallett™s model

Mallett proposed to attack the problem ˜in a somewhat more determinate manner™.

By this he meant that his approach would be mathematical and quantitative, in

contrast to the hand-waving of the other geologists. This more modern style has

made the article in¬‚uential ever since.

One of his principal interests was in the energy liberated in volcanic erup-

tions, so it was natural for him to think in terms of the total energy of the system

of cracks rather than their precise mechanism of formation. At a time when there

was a general tendency to express the laws of physics as minimal principles, he

appealed rather vaguely to the ˜principle of least action™ and ˜the minimum ex-

penditure of work™. What crack pattern would minimize energy?

This question makes little sense unless some constraint ¬xes the size of

the cells of the pattern, but this was somehow ignored, and he triumphantly an-

nounced that the hexagonal pattern was best, by comparison with other simple

cases.

The Giant™s Causeway

102

10.5 A modern view

D™Arcy Wentworth Thompson recognized that cracks which proceed explosively

from isolated centres could never form such a harmonious pattern. He failed to

see the possibility that crack patterns ¬rst formed deep within a lava ¬‚ow could

propagate very slowly outwards as it cooled. A careful reading of Mallett™s pa-

per shows that he had already recognized this, and this part of his description is

impeccable.

In their slow motion the cracks migrate until they form a balanced network

which propagates unchanged. To have this property it need not be ordered: the

best analogy is the arrangement of atoms when a liquid becomes a glass on cool-

ing. There is local order only.

When precisely the realization dawned more generally that it must be so is

not clear, but certainly Cyril Stanley Smith gave this explanation in 1981, when

preparing the published version of a lecture to geologists. It appears very natural

to the modern mind. Until full computer simulations are performed, one hesitates

to state it emphatically and risk joining such a long list of unwarranted claims.

10.6 Lost city?

In October 1998, an article in the British press reported the possible discovery of

a lost city by a documentary ¬lm-maker in Nicaragua. The evidence consisted

of 62 polygonal basalt columns. Troops had been dispatched to guard the ¬nd

against looters. Geologists had expressed some scepticism. . . .

It seems that the columnar basalt story, begun in 1693, is destined to run and

run.

Chapter 11

Soccer balls, golf balls and

buckyballs

11.1 Soccer balls

A favourite close-up of the television sports director shows a soccer ball distend-

ing the net. This offers the opportunity to compare the two: for the net is nowa-

days made in the form of the hexagonal honeycomb, and the surface of the ball

looks roughly similar. Closer examination reveals the presence of 12 pentagons

among the hexagons on the ball.

The problem of the soccer ball designer was to produce a convenient poly-

hedral form which is a good approximation to a sphere. The presently favoured

design replaces a traditional one and we are not aware of the precise arguments

that brought this about. Certainly it is more aesthetic, on account of its high sym-

metry, which is of¬cially described as icosahedral. The simplest design of this

type would be that of the pentagonal dodecahedron (¬gure 5.6), but this was per-

haps not suf¬ciently close to a sphere. Instead, 32 faces are used, of which 12 are

pentagons and the rest are hexagons.

By a curious coincidence, this icon of modern sport has cropped up in a

prominent role in modern science as well, as we shall see shortly. But ¬rst let us

switch sports and examine the golf ball.

11.2 Golf balls

The dynamics of a sphere immersed in a viscous ¬‚uid presents one of the classic

set-piece problems of physics and engineering, dating back to the work of Newton

(applied, in particular, to the motion of the Earth through the ether), and tidied

103

Soccer balls, golf balls and buckyballs

104

Figure 11.1. A soccer ball.

Figure 11.2. A golf ball.

up in some respects by Sir George Gabriel Stokes more than 100 years ago. In

modern times it can be safely assumed that there has been a large investment in

a better understanding of the motion of a sphere in air, since it commands the

attention of many important people on the golf courses of the world.

The extraordinary control of the golf ball™s ¬‚ight which is exercised by

(some) golfers owes much to the special effects imparted by the spin which is

imposed on the ball by the angled blade of the club. This can be as much as

10 000 revolutions per minute. Because of its backspin the ball is subject to an

upward force, associated with the de¬‚ection of its turbulent wake. This is the

Magnus Effect, also responsible for the swerve of a soccer ball. It causes the ball

to continue to rise steeply until, both velocity and spin having diminished, it drops

almost vertically onto its target.

It was found that marking the surface of the ball enhanced the effect, and

a dimple pattern evolved over many years. Today such patterns typically consist

of 300“500 dimples. All are consistent with the incorporation of a ˜parting line™

where two hemispherical moulds meet to impress the shape. Locally the dimples

are usually close-packed on the surface of the ball.

Buckyballs 105

We know of no physical theory which would justify any particular arrange-

ment. Many are used, whether motivated by whim or the respect for the intellec-

tual property of established designs. Titleist has favoured an essentially icosahe-

dral ball derived from the pentagonal dodecahedron by adding hexagons, just as

for the soccer ball. Note that here we are dealing with a ˜dual™ structure: each

dimple lies at the centre of one of the polygons. A simple topological rule gov-

erns all such patterns. Indeed, to wrap a honeycomb on a sphere or other closed

surface is not possible, without introducing other kinds of rings. According to

Euler™s theorem the minimum price to be paid is 12 pentagons. One may take the

pentagonal dodecahedron, beloved of the Greeks, and expand it by the addition

of any number of hexagons (except, as it happens, one). In some cases the result

is an elegantly symmetric structure.

11.3 Buckyballs

The modern science of materials has matured to the point at which progress seems

barred in many directions. One cannot envisage, for example, magnetic solids

which are much more powerful than the best of today™s products, because they

are close to very basic theoretical limits. (The scientist who says this type of

thing is always in danger of following in the footsteps of the very great men who

denounced the aeroplane, the space ship and the exploitation of nuclear energy as

patent impossibilities.)

Despite this sense of convergence to a state in which optimization rather than

discovery is the goal, new materials continue to make dramatic entrances. Some-

times they arise because certain assortments of many elements have not previ-

ously been tried in chemical combination. There is one great exception to this

trend towards combinatorial research. The sensational advance in carbon chem-

istry which goes by the affectionate nickname of ˜buckyballs™ has raised more

eyebrows and opened more doors than anything else of late, with the possible

exception of high-temperature superconductors.

This is not a case of a single, momentous revelation: rather one of steadily

increasing knowledge and decreasing incredulity over many years, from the ¬rst

tentative clues to the establishment of major research programmes throughout the

world. The story up to 1994 has been compellingly recounted by Jim Baggott in

Perfect Symmetry1 .

At its conclusion Baggott wondered which of the four central personalities

of his tale would be rewarded by the Nobel Prize, which is limited to a trio. The

answer came in 1996: the Royal Swedish Academy of Science awarded the Nobel

Prize in Chemistry to Robert F Curl, Harold W Kroto and Richard E Smalley.

The carbon atom has long been renowned as the most versatile performer in

the periodic table. It is willing to join forces with other atoms either three or four

½

Baggott J 1994 Perfect Symmetry: The Accidental Discovery of Buckminsterfullerene (Oxford: Ox-

ford University Press).

Soccer balls, golf balls and buckyballs

106

Figure 11.3. The C buckyball.

¼

at a time. Pure carbon with fourfold bonding is diamond, whereas graphite con-

sists of sheets with the honeycomb structure in which there is threefold bonding.

The two solids have vastly different properties; one is hard, the other soft; one

is transparent, the other opaque; one is horribly expensive, the other very cheap.

Nothing could better illustrate the falsity of the ancient idea that all the properties

of elements spring directly from the individual atoms.

For pure carbon, that was supposed to be the end of the story, give or take

a few other forms to be found under extremely high pressures. Yet we now rec-

ognize that graphite-like sheets can be wrapped to form a spherical molecule of

60 atoms”the buckyball”and buckyballs can be assembled to form an entirely

new type of carbon crystal, with startling properties. And further possibilities

continue to emerge in the laboratory or the fevered imagination of molecule de-

signers: other, larger molecules, concentric molecules like onions, tubular forms

called nanotubes, which may be key components of future nanoengineering. The

buckyball belongs to an in¬nite family of possibilities which comprise the new

subject of fullerene chemistry.

The buckyball has a great future, a fascinating history and even an intriguing

prehistory. The existence of molecules like this had been teasingly conjectured

by a columnist in the New Scientist, and Buckminster Fuller built most of his rep-

utation on the architectural applications of such structures (generally containing

many more hexagons than the buckyball). Hence the C ¼ molecule was ¬rst bap-

tised buckminsterfullerene in his honour. This is by no means a large mouthful by

Buckminster Fuller 107

chemical standards, but the snappier ˜buckyball™ has steadily gained currency at

its expense2 .

11.4 Buckminster Fuller

Buckminster Fuller3 has been described by an admirer as a ˜protean maverick™.

For many he is the prime source of insight into many of the structures which we

have pondered in this book. Over several decades he poured forth a torrent of

ideas and assertions which combined the Greek faith in geometry as lying at the

heart of all nature with vague and super¬cially impressive notions of energy and

synergy. The resulting pot-pourri is inspiring or bewildering, according to taste.

When he ventures into fundamental descriptions of nature, it is reminiscent of the

speculations of those 19th-century ether theorists.

One of Fuller™s assertions was that all nature is ˜tetrahedronally coordinated™.

Here he was thinking of close-packing of spheres (although this, in part, con-

tradicts the statement”not all arrangements are tetrahedral). He seems to have

claimed to have discovered the ideal close-packed arrangement, only to ¬nd it

in the work of Sir William Bragg, who he then supposed to have independently

found it around 1924!

It is the practical outcome of Fuller™s ruminations, in the form of the geodesic

dome, that we remember today. Here again there is some question of priority.

Tony Rothman, in Science a la Mode, has pointed out that such structures were

`

patented by the Carl Zeiss company, for the construction of planetariums, in the

1920s. Fuller™s patent is dated 1954. Rothman generously gives the protean mav-

erick the bene¬t of the doubt. . . .

11.5 The Thomson problem

Another problem which involved placing points on a sphere was posed by J J

Thomson in 1904 in the context of speculations about classical models of the

atom, which were soon to be rendered out-of-date by quantum mechanics. But as

with many other cases in this book, the problem has survived in its abstract math-

ematical form, and continues to intrigue mathematicians and challenge computer

scientists. It is simply this: what is the arrangement of Æ point electrical charges

on a sphere, which minimizes the energy associated with their interactions? This

is just the sum of ½ – over all pairs of points, where – is their separation. Gen-

erally speaking the solution is neither the best packing nor the most symmetric

arrangement of points. Beginning with L Foppl, around 1910, mathematicians

¾

It has been suggested that, since the polyhedral structure of the buckyball and soccer ball date back

to Archimides, ˜archiball™ might be more appropriate.

¿

See, for reference, Marks R W 1960 The Dymaxion World of Buckminster Fuller (Reinhold); Roth-

man T 1989 Science a la Mode, Physical Fashions and Fictions (Princeton, NJ: Princeton University

`

Press).

Soccer balls, golf balls and buckyballs

108

have accepted Thomson™s challenge. Kusner and Sullivan 4 analysed the cases

¾¼, discovering that there is only one stable structure when Æ is smaller

Æ

than 16.

The target of most researchers is to ¬nd structures of low energy using com-

putational search procedures and identify the one which is lowest among these,

for large values of Æ 5 . Some of the structures which crop up here are similar to

those of Buckminster Fuller constructions and golf balls.

This computational quest is an ideal testing-ground for new software ideas,

such as simulated annealing (see section 13.11).

The minimal energy structures for small Æ are surprising in some cases: for

Æ we do not ¬nd the obvious arrangement in which the charges are at the

corners of a cube, but rather a twisted version of this.

½¾ the familiar icosahedral structure is found, with the charges at

For Æ

the corners of the pentagonal dodecahedron (¬gure 5.6). Each charge has ¬ve

nearest neighbours. Thereafter this structure is elaborated to accommodate more

charges. For all Æ which satis¬es

½¼´‘¾ · ’¾ · ‘’µ · ¾

Æ (11.1)

with ‘ and ’ being positive integers; this can be done very neatly as in ¬g-

ure 11.4.

All the additional charges have six neighbours. In special cases these corre-

spond to the buckyball or soccer ball structure which we have already admired.

That is not the end of the story. Eventually at high Æ these structures can

be improved by modi¬cations which introduce more charges with ¬ve or seven

neighbours.

11.6 The Tammes problem

Many pollen grains are spheroidal and have exit points distributed on the surface.

The pollen comes out from these points during fertilization. The position of the

exit points is rather regular and the number of them varies from species to species.

In 1930 the biologist Tammes described the number and the arrangement of the

exit points in pollen grains of many species. He found that the preferred numbers

are 4, 6, 8, 12, while 5 never appears. The numbers 7, 9 and 10 are quite rare and

11 is almost never found. He also found that the distance between the exit points

is approximately constant, and the number of these points is proportional to the

surface of the sphere 6 .

Tammes posed the following question: given a minimal distance between

them, how many points can be put on the sphere? We can think of the points as

Kusner R and Sullivan J 1997 Geometric Topology ed W H Kazez (International Press).

Altschuler E L et al 1997 Possible global minimum lattice con¬gurations for Thomson™s problem

of charges on a sphere Phys. Rev. Lett. 78 2681.

Tammes P M L 1930 On the origin of number and arrangement of the places of exit on pollen grains

Diss. Groningen.

The Tammes problem 109

Figure 11.4. Three low-energy structures for charges on a sphere as depicted by Altschuler

et al.

associated with (curved) discs of a certain size, which are not allowed to overlap.

Tammes attacked the problem in an empirical way by taking a rubber sphere and

drawing circles on it with a compass. He found, for instance, that when the space

is enough for ¬ve circles then an extra circle can always be inserted. In this

case the six circles are located at the vertices of an octahedron. In this way the

preference for 4 and 6 and the aversion for 5 in pollen grains may be explained.

Tammes also found that when 11 points ¬nd enough space then 12 can also be

placed at the vertices of an icosahedron.

The ¬rst of these results has been mathematically proved to be valid for the

surface of a sphere in three dimensions and it has been extended to any dimension.

If on the surface of a sphere in -dimensional space more than · ½ discs can be

placed then ¾ such discs can be placed at the extremities of the coordinate axes.

The Tammes problem is the subject of an enormous amount of literature.

Soccer balls, golf balls and buckyballs

110

Mathematically the questions raised by Tammes can be expressed as follows:

what is the largest diameter Æ of Æ equal circles that can be placed on the

surface of a unit sphere without overlap? How must the circles be arranged, and

is there a unique arrangement?

½¾ and Æ ¾ . These and other

Exact solutions are known only for Æ

solutions are shown in table 11.1 (taken from Croft 7 ).

Table 11.1.

Æ Æ Arrangement

½ ¼Æ

2 Opposite ends of a diameter

½¾¼Æ

3 Equilateral triangle in the equator plane

½¼ Æ 28™

4 Regular tetrahedron

¼Æ

5 Regular octahedron less one point, not unique con¬guration

¼Æ

6 Regular octahedron

Æ 52™

7 Unique con¬guration

Æ 52™

8 Square anti-prism

¼Æ 32™

9 Unique con¬guration

Æ 9™

10 Unique con¬guration

¿Æ 26™

11 Icosahedron less one point, not unique con¬guration

¿Æ 26™

12 Icosahedron

¿Æ 41™

24 Snub cube

Æ ½ which is the Tammes empirical

Note that for Æ and 12, Æ

result.

A bound for the minimum distance between any pair of points on the sur-

face of the unit sphere, was given in 1943 by Fejes T´ th

o

—

Æ

cosec ¾

´Æ ¾µ (11.2)

¿

with the limit exact for Æ and 12 8 .

11.7 Helical packings

The packing of spheres around a cylinder results in helical patterns, which may of-

ten be seen in street festivals when balloons are used to decorate lamposts. These

attractive structures have an interesting history because they are found in many

plants. Botanists have long been fascinated by the way in which branches or

Croft H T, Falconer K J and Guy R K 1991 Unsolved Problems in Geometry (New York: Springer)

p 108.

Ogilvy C S 1994 Excursions in Mathematics (New York: Dover) p 99.

Helical packings 111

Figure 11.5. Carbon crystals make tubular structures called ˜nanotubes™. In the tubular

part carbons make a network with rings of six atoms, whereas the positive curvature is

induced by rings of ¬ve atoms.

leaves are disposed along a stem, or petals in ¬‚ower. They have found many dif-

ferent helical arrangements, but almost all have a strange mathematical property,

which is the main preoccupation of much of the extensive literature on this sub-

ject, at times acquiring a mystical ¬‚avour. This dates back (at least) to Leonardo.

In the 19th century, the Bravais brothers, and later Airy and Tait began a more

modern study, which surprisingly continues today. The reviewer of a recent book

by Roger Jean, said that it ˜remains one of the most striking phenomena of biol-

ogy™.

Recently the subject has recurred in an exciting new context”the creation

of nanotubes, which are tiny tubes with walls which are a single layer of carbon

atoms. They are ¬rst cousins to the ˜buckyballs™ (see section 11.3). They were

¬rst made in the 1970s by Morinobu Endo, a PhD student at the University of

Orleans.

Biologists call this subject ˜phyllotaxis™, and obscure it further with terms

such as ˜parastichies™, which are rather repellent to the ¬rst-time reader. But

these helical structures are really quite simple things. If we are to place spheres

or points on the surface of a cylinder, it is much the same as placing them on

a plane. We therefore expect to ¬nd the close-packed triangular arrangement of

chapter 2, which is optimal under a variety of conditions. The difference lies in

the fact that the surface is wrapped around the cylinder and joins with itself. It is

like wallpapering a large pillar”the packing must continue smoothly around the

cylinder, without interruption.

We could think of cutting out a strip from the triangular planar packing and

wrapping it around the cylinder. We might have to make some adjustments to

avoid a bad ¬t where the edges come together. We can displace the two edges

with respect to each other and/or uniformly deform the original pattern (not easy

with wallpaper!), in order to have a good ¬t.

Let us reverse this train of thought: roll the cylinder across a plane, ˜printing™

its surface pattern again and again. We expect to get the close-packed pattern, or a

Soccer balls, golf balls and buckyballs

112

Figure 11.6. Bubbles packed in a cylinder show the familiar hexagonal honeycomb

wrapped on a cylinder. This simulation by G Bradley shows surface patterns and the inte-

rior for one of these structures.

strained version of it. The three directions of this pattern, in which the points line

up, correspond to helices on the cylinder. Taking one such direction, how many

helices do you need to complete the pattern? This de¬nes an integer, and the three

integers , Ð, ‘ corresponding to the three directions can be used to distinguish

this cylindrical pattern from all others. It is easily seen that two of these must add

to give the third integer.

This is the notation of phyllotaxis, applied to plants, to nanotubes and to

bubble packings within cylinders. In the last case, the surface structure is always

of this close-packed form.

Chapter 12

Packings and kisses in high

dimensions

12.1 Packing in many dimensions

The world of mathematics is not con¬ned to the three dimensions of the space that

we inhabit. Mathematicians study sphere-packing problems in spaces of arbitrary

dimension. Geometrical puzzles can be posed and solved in such spaces. Some

practical challenges end up in such a form.

With this chapter we present a short excursion into some of the relevant

points concerning packings of spheres in high dimensions and their applications.

The topic is explored in a very comprehensive way by Conway and Sloane in their

book Sphere Packings, Lattices and Groups 1 , which is considered the ˜bible™ of

this subject.

Packings in many dimensions ¬nd applications in number theory, numeri-

cal solutions of integrals, string theory, theoretical physics and digital commu-

nications. In particular, some problems in the theory of communications, with

a bearing on the optimal design of codes, can be expressed as the packing of -

dimensional spheres. Indeed, in signal processing it is convenient to divide the

whole information into uniform pieces and associate each piece with a point in

a -dimensional space (a point in a -dimensional space is simply a string of

real numbers ™ ½ ™¾ ™¿ ™ ). To transmit and recover the information in

the presence of noise one must ensure that these points are separated by a dis-

tance larger than that at which the additional noise would corrupt the signal. Each

point (encoded information) can be seen as surrounded by a ¬nite volume, a -

dimensional ball with a diameter larger than the additional noise. The encoded

½

Conway J H and Sloane N J A 1988 Sphere Packings, Lattices and Groups (Berlin: Springer).

113

Packings and kisses in high dimensions

114