Flowers of Iceâ€”Beauty, Symmetry, and Complexity: A Review of The

Flowers of Ice—Beauty,Symmetry, and Complexity:A Review of The Snowflake:Winter’s Secret BeautyReviewed by John A. Adam

402 NOTICES OF THE AMS VOLUME 52, NUMBER 4

Growing up as a child in southern Eng-land, my early memories of snow in-clude trudging home from school withmy father, gazing at the seemingly enor-mous snowdrifts that smoothed the

hedgerows, fields and bushes, while listening to thesoft “scrunch” of the snow under my Wellingtonboots. In the country, snow stretching as far as Icould see was not a particularly uncommon sight.The quietness of the land under a foot of snowseemed eerie. I cannot remember the first time Ilooked at snowflakes per se; my interests as asmall child were primarily in their spheroidallyshaped aggregates as they flew through the air.Many years later, as I cycled home from my officein Coleraine, Northern Ireland, I remember beingintrigued by a colored blotch of light to the westof the Sun at about the same elevation. Little did Iknow then that these two events of snowfall and“sundogs” (of which more anon) were intimatelyconnected. Since that time I have learned rathermore about meteorological optics, and this bookabout the beauty of snowflakes has challenged me to learn more of the physics and mathematicsbehind crystal formation in general and ice crys-tal formation in particular.

As in an earlier review [1], I will divide this review into two main sections; Libbrecht’s book

contains no mathematics, so the first section willaddress the qualitative features of snow andsnowflakes discussed by the author. I will draw on some of the general descriptions of snow and its properties both from this book and those by others mentioned below. I have decided to commenton every chapter individually, because each can be treated to some extent independently of theothers. The second section takes the form of amathematical appendix devoted to an outline ofsome of the mathematical aspects of crystal for-mation, with particular emphasis on ice crystals.In some sense, I hope, this will become a parallel“mathematical universe” to the first section. However, the growth of crystals is a complex combination of thermodynamics and statisticalphysics, and no review of this nature could do justice to the immense theoretical edifice that hasbeen erected on this topic.

This slim volume is informally yet clearly writ-ten, and the photographs (mostly taken by PatriciaRasmussen) are quite stunning. Obviously, it isaimed at a popular audience. In one sense, though,it is the latest in a disjoint set of popular books onpattern formation in nature: D’Arcy Thompson’s On Growth and Form [38]; Stevens’s Patterns in Nature [32]; Ball’s The Self-Made Tapestry [3];Stewart’s Nature’s Numbers [33] and What Shape Isa Snowflake? [34]; and finally, from a very differ-ent perspective, Bejan’s constructal theory as ex-pounded in Shape and Structure, from Engineeringto Nature [5]. With the exception of [34], none ofthese address the nature of the snowflake to anygreat extent. Some of Libbrecht’s low-key yet vivid

John A. Adam is a professor of mathematics at Old Dominion University. His email address is [email protected].

The Snowflake: Winter’s Secret Beauty, by KennethLibbrecht, with photographs by Patricia Rasmussen,Voyageur Press, Inc., 2003, hardcover, $20.00, ISBN 0-89658-630-8.

APRIL 2005 NOTICES OF THE AMS 403

descriptions captured my imagination, and I wouldcertainly recommend this book to any person wishing to “experience” something of the physicalintuition of a scientist.

As might be expected, the study of snowflakesis not new; no doubt people have been fascinatedby their beauty and symmetry since time im-memorial. According to [4], [16] the Chinese aware-ness of this was recorded in 135 B.C., while in Europe the Dominican scientist, philosopher, andtheologian, Albertus Magnus, studied them around1260 A.D. Not surprisingly, the astronomer andmathematician Johannes Kepler was intrigued bysnow crystals, writing a small treatise entitled Onthe Six-Cornered Snowflake. In 1611 he asked thefundamental question: There must be some definitecause why, whenever snow begins to fall, its initialformation invariably displays the shape of a six-cornered starlet. For if it happens by chance, whydo they not fall just as well with five corners orseven? In his treatise he compared their symmetrywith that of honeycombs and the seed arrangementinside pomegranates [4]. However, nothing wasknown in Kepler’s era of the molecular structureof water, which ultimately determines the hexag-onal shape of ice crystals, so Kepler was unable toexplain their shape in mechanical terms, though hedid attempt to do so using entirely reasonablepacking arguments. (Indeed, finding the densest(not necessarily periodic) packing of spheres isnow known as the Kepler problem [14].) In 1665 thescientist Robert Hooke publishedMicrographia, in which he described his observa-tions of snowflakes using a microscope. More his-torical details may be found in [4].

In the first chapter of The Snowflake (“The Cre-ative Genius”), the author asks the fundamentalquestions: How do crystals grow? Why do complexpatterns arise spontaneously in simple physical sys-tems? These are very profound questions, of course,and a considerable portion of past and present lit-erature in applied mathematics and theoreticalphysics is devoted to attempting to answer them.Obviously, the second question goes well beyondthe “mineral kingdom” of crystal formation into the“animal and vegetable” ones of patterns in (and on)living things. In particular, though, Libbrecht re-minds the reader that snowflakes are the productof a rich synthesis of physics, mathematics, andchemistry and that they are even fun to catch onone’s tongue! The range of such comments reflectsthe underlying parallel approaches of the book: onthe one hand, a good qualitative and nontechnicaldescription of the scientific aspects of snowflakeformation and, on the other, the sheer fun of doingscience. It is worth noting that in Libbrecht’s ter-minology, snow crystal to snowflake is as tulip toflower. In other words, a snowflake can be an individual snow crystal or collection of the same,

depending on context. The two words will be usedsynonymously here unless otherwise noted.

The short Chapter 2 is entitled “SnowflakeWatching”. In it, Libbrecht records a partial historyof snowflake watchers from Descartes to the present day. In particular, he makes mention of Wilson Bentley, a Vermont farmer who dedicatedmuch of his life to photographing snowflakes. Inthe late 1920s Bentley worked with physicist W. J.Humphreys to publish a book containing morethan 2,000 of his snow crystal images. (There is a1962 Dover edition of this book, Snow Crystals.) Thescientific discussion begins in earnest in the thirdchapter (“Snow Crystal Symmetry”). It starts withthe following quote from Richard P. Feynman (TheFeynman Lectures on Physics, 1963):

Poets say science takes away from thebeauty of the stars—mere globs of gasatoms. I too can see the stars on a desertnight, and feel them. But do I see lessor more?... What is the pattern, or themeaning, or the why? It does not doharm to the mystery to know a littleabout it. For far more marvelous is thetruth than any artists of the past imag-ined it.

A quote from Kepler in 1611 is included as hepondered the sixfold symmetry of snowflakes, andalthough Libbrecht makes no mention of it, a quotefrom D’Arcy Wentworth Thompson [38] is partic-ularly appropriate here:

The beauty of a snow-crystal dependson its mathematical regularity and sym-metry; but somehow the association of many variants of a single type, all related but no two the same, vastly increases our pleasure and admira-tion.…The snow-crystal is further complicated, and its beauty is notablyenhanced, by minute occluded bubblesof air or drops of water, whose sym-metrical form and arrangement are verycurious and not always easy to explain.Lastly, we are apt to see our snow crys-tals after a slight thaw has roundedtheir edges, and has heightened theirbeauty by softening their contours.

Returning to the question of complex patternsin nature, we may not be surprised to be remindedby Libbrecht that all crystals demonstrate the or-ganizational ability to self-assemble. Starting witha random collection of molecules, this is an exampleof spontaneous pattern formation. The best non-mathematical book I know on the broad spectrumof pattern formation is The Self-Made Tapestry [3],and the present book, much narrower in scope, isa good introduction to some of the underlying


concepts explained therein. For example: Libbrechtdoes a pretty good job of explaining to a nontech-nical readership the physics of crystal facets andtheir formation. One basic snow crystal shape is thehexagonal prism, which possesses two basal facets

and six prism facets, and de-pending on which of the twotypes grows faster, the prismcan become a long columnor a thin plate. The slowestmoving facets eventually de-fine the shape of the crystal.At this point the author asksa very significant question:How can molecular forces,operating only at thenanoscale, determine theshape of large crystals? An-swers to this and relatedquestions are hinted at in thefollowing two chapters.

And so to Chapter 4 (“Hi-eroglyphs from the Sky”), inwhich it is stated that the realpuzzle of snowflakes is theircombination of symmetry

and complexity and that major progress in solvingthis puzzle was made by the physicist Nakaya inthe 1930s. Eventually, he succeeded in growing in-dividual snow crystals in his laboratory under manydifferent humidity levels and temperatures, and itwas only a matter of time before he was able to clas-sify the symmetry and considerable variety dis-played by these crystals. He developed his famousmorphology diagram, demonstrating the remark-able sensitivity of the snow crystal form to its en-vironmental conditions. This diagram is repro-duced on page 45 of The Snowflake. Basically, atlow but fixed levels of supersaturation (degree ofhumidity), as the temperature decreases below 0◦Cto about –35◦C, snow crystals are essentially plates,then solid prisms, and then plates again. At highersupersaturation levels, the evolution is from den-drites to needles, hollow columns, sectored platesand dendrites, and then columns again. Essentially,the overall crystal shape, whether it is platelike orcolumnar, reveals something about the temperatureat which the crystal grew, and the complexity of thestructure indicates something about the humidity.However, each crystal falling on one’s nose is aproduct of the cumulative history it has under-gone as it has been wafted hither and yon by aircurrents through many different atmospheric con-ditions. In mathematical terms, we might think ofits shape being defined by a line integral over itspath through space and time. Generally, the lengthscale of variations of temperature and humidity willbe much larger than the dimension of the crystal,so each vertex or arm of the crystal experiences the

same conditions at eachmoment of time; theirsymmetry is evidently areflection of their sharedhistory.

It should be noted thatwhile Nakaya’s 1954 bookSnow Crystals: Naturaland Artificial is referredto on page 44, no men-tion is made of the im-

portant work by Furukawa, who studied underKuroda and Kobayashi, both of whom were studentsof Nakaya. Many valuable details and references onthe history and science of snow crystal research canbe found in Furukawa’s essay (http://www.lowtem.hokudai.ac.jp/~frkw/english/aletter.html).

In Chapter 5 (“Morphogenesis on Ice”) the searchfor an explanation of snow crystal complexity istaken yet further. Touch-ing on a point made ear-lier in this review, Lib-brecht temporarilybroadens his perspectiveby noting that while aflower is an example ofbiological morphogene-sis, even simpler physicalsystems exhibit this fea-ture. Thus whether it bewaves on the oceans orripples on snowdrifts and sand dunes, they are allrelatively simple pattern-forming systems in whichcomplexity arises spontaneously, but for Libbrechtthe snowflake is the poster child of morphogene-sis. Here as in earlier chapters he does a good jobof introducing the concept of self-organization inphysical (and, in passing, biological) systems. Thisis all to the good, given the preoccupation in somequarters with the concept of “intelligent design”(with all due respect to my fellow Christians). In a

sentence that I find veryappealing, Libbrecht notesthat instabilities like thosediscussed here are theheart of pattern formation,and nature is one unstablesystem heaped on top ofanother.

Scientifically, the realmeat of the book is to be

The three photographs of snowflakes (above)were provided by Yoshinori Furukawa of

Hokkaido University, who also provided thephotographs from which the background

snowflakes came. The ordinary symmetry ofsnowflake growth is really extraordinary.

http://www.lowtem.hokudai.ac.jp/~frkw/english/aletter.html




found in this chapter. There are several key com-ments made by Libbrecht that bear repeating aswritten: Growth is the key ingredient for the gen-eration of snow-crystal patterns. Left in isolation fora long time, an ice crystal will eventually turn intoa plain hexagonal prism.…Ornate patterns appearonly when a snow crystal is out of equilibrium, while it is growing. Such circumstances are often referred to as nonequilibrium conditions. Another important concept introduced is that of diffusion-limited growth. A snow crystal grows by assimilatingmolecules of water vapor into the existing ice lattice, provided the humidity is high enough. How-ever, continuing crystal growth gradually depletesthe vapor from layers of air adjacent to the crys-tal, and the remaining water vapor molecules mustdiffuse over increasingly larger distances. Sincethe travel time for diffusion (other things remain-ing unchanged) is proportional to the square of the distance traveled, it is clear that the growth rate of the crystal will be inhibited; the growth is now diffusion-limited.

An initial “seed” crystal consisting of severalhundred or thousand molecules has a molecularlyrough boundary. Depending on the crystallographicorientation, some orientations may grow morerapidly than others (under a spatially uniformgrowth drive), and thus a surface is created withboth positive and negative curvature. “Flatter” por-tions of the boundary are called facets, and whilea completely rough seed will not have facets, it maystill grow anisotropically. The curved regions “fillin” more readily than the facets, from which it isseen that ultimately the slower-growing facets define the crystal shape in the absence of signifi-cant branching (a mechanism discussed below).Combining the above two mechanisms, in diffusion-limited growth one might expect that the crystal ver-tices would “harvest” water vapor molecules fasterthan facets by virtue of their projection furtherinto the medium (if the mean free path of the vapor phase is of order the size of the system).This can induce a positive feedback loop known as a branching instability. As will be noted in thesecond section of this review, if this instability occurs, there is usually a self-limiting stabilizingmechanism, akin to surface tension, that eventuallybalances the destabilizing one. If this diffusion-limited branching instability is iterated over andover, a type of snow crystal known as a dendrite willdevelop. Ultimately, this may give rise to some kindof self-similar or fractal-like structure over severalorders of magnitude of scale, not unlike a fern leaf.This branching process is more pronounced in regions of higher vapor pressure, since then the diffusive transport of water vapor molecules ismore effective. Conversely, faceted growth is morelikely to occur in a lower vapor pressure environ-ment. Ultimately, the complex interplay between

faceting and branching is what determines theform of a given snow crystal in a kind of morpho-logical balancing act, and this in turn depends onthe temperature and humidity history of the grow-ing crystal and also its size. Libbrecht speculatesthat if snowflakes occur in other planetary atmos-pheres, they may well be different from the oneswe know and love.

Although he does not identify it as such, thebranching instability Libbrecht refers to is theMullins-Sekerka instability (see the second sectionof this article for more details). Nobody could faulthim for simplifying physical phenomena in a book of this kind, but as he implies elsewhere, thepicture is just not that straightforward. Indeed, Iam indebted to Professor J. S. Wettlaufer for thefollowing comment (in reference to a statementmade in [2]) that puts into perspective the com-plexity of the problem:

The initial question that needs to beasked is: How does a hexagon emergefrom a nucleus of some 1,000 mole-cules that, at the point of nucleation, donot necessarily reflect that symmetry?This has nothing whatever to do with thestructure of the background diffusionfield, but rather it is solely controlled bythe statistical mechanics of adsorbateson the seed. The seed is much smallerthan the mean free path in the vaporphase. The enhancement of the vaporfield at a corner, generically understoodas the “point effect of diffusion” or the“Berg-effect”, may or may not lead to aninstability. The conditions must be carefully determined; for example, isthe size of the seed much smaller thanthe characteristic scale of the diffusionfield? There is a plethora of outstand-ing physical and mathematical prob-lems related to the transition betweennucleation, interface controlled growthand diffusion limited shapes. Therefore,if and/or when a diffusive/Mullins-Sekerka instability occurs, understand-ing the habit of the crystal still requiresan understanding of the initial valueproblem and thus the evolution of acrystal from its birth.

In Chapter 6 (“Snowflake Weather”) Libbrechtmakes some statements about snow crystal quan-tities that might form the basis for several inter-esting estimation problems, but first it is necessaryto note some facts about snow. Consider first asmall ice cube 1 cm3 in volume. This has a massof one gram, and since the mass of a water mole-cule is about 3× 10−23g, there are about 3× 1022

molecules in this ice cube. A typical snow crystal


of mass 3× 10−5 g therefore may contain about1018 molecules. Depending on the type of crystal,temperature, wind speed, and other factors, in-cluding the density of packing, snow has consid-erable variation in its water content. An eight-footblanket of fresh snow may contain as little as 1 inchof water per unit of surface area or as much as 3feet. According to [16], the majority of U.S. snowshave a water-to-snow ratio in the range of 0.04 to0.10. I had to dig around for most of this infor-mation; I wish that facts such as these had beenincorporated into the book as an appendix.

In the beginning of this chapter we read:Snowflakes are being manufactured in the atmos-phere at an astounding rate—around a million billion crystals each second. Every ten minutes that’senough snow to make an unstoppable army of snow-men, one for every person in the world.... Let’s thinkabout this: on the basis of these numbers, after tenminutes there are about 1015 × 6× 102 = 6× 1017

crystals, and estimating a typical snowman to becomposed of two identical spheres of diameter 1.5ft. gives a volume of about 3.5 ft.3 or approximately3.5× (0.3)3 ≈ 0.1 m3 = 105 cm3. If we assume thatall the crystals formed make their way to the ground(a rather dubious assumption) and take an averagewater-to-snow ratio of 0.07 for U.S. snow (!), thenthe mass of crystals falling in a ten-minute inter-val is approximately 2× 1013g. Meanwhile, the massof our typical snowman is given by the product105 × 0.07 = 7× 103g. This means our army of

snowmen numbers about 3× 109, or about onehalf of the present world population. A factor oftwo is not a large discrepancy in this context, andin any case, that’s a lot of snowmen. (There is a de-lightful “Peanuts” cartoon showing a vast army ofsnowmen built on a cloudy day by that precociousgenius, Linus. He marches up and down declaringtheir military invincibility under any and all cir-cumstances. Then the Sun comes out. I use this par-ticular cartoon in mathematical modeling classesto emphasize the dangers of making false (or atbest, weak) assumptions.)

Later, another estimation beckons us. Libbrechtstates that the total global precipitation per day isequivalent to about 1015 liters of water, and eachof us typically exhales about 1 liter of water per dayinto the atmosphere. By simple proportion, there-fore, and using his figures, we infer that if ourcontribution to the water cycle were uniformly dis-tributed around the globe (seemingly yet anotherflawed assumption), then our average contribu-tion to the total water content of a snow crystal isabout 1018 ÷ 1015 = 103 molecules. On this num-ber, not surprisingly, we agree.

The chapter closes with a nice little sectionabout ice nucleation: how water molecules, likeadolescents, eventually learn to settle down and become very cool (or freeze, in the case of the former). There is also a rather intricate table classifying some of the many types of snow.

Chapter 7 (“A Field Guide to Falling Snow”) is byfar the longest in the book and is ac-companied by many beautiful pho-tographs. We are introduced to a verita-ble zoo of snow crystals: diamond dust(of which cirrus clouds are made), stellardendrites, sectored plates, columns andneedles, hollow columns, needles andbullets, capped columns, split stars andsplit plates, twinned crystals, twelve-sidedsnowflakes and double stars, chandeliercrystals, spatial dendrites, triangular crys-tals, and rimed snowflakes!

What type of ice crystal then was re-sponsible for the sundog I mentionedearlier (such patches of colored light arealso known as mock suns or parhelia)? Bil-lions of slowly falling horizontally ori-ented hexagonal plate like crystals (pre-sent in cirrus clouds) 30 microns or largerare the culprits. They behave like tinyprisms and refract light entering one ver-tical face (say face 1) and exiting face 3.Unlike rainbow formation, there is no re-flection involved, and so the red portionis always closest to the Sun. They occurat least 22◦ away from the Sun. There aremany other related phenomena one mayobserve, such as ice crystal halos, and

This photograph of atmospheric halos was taken at the South Pole, where conditions are frequently favorable. It was provided by Walter Tape, nowat the University of Alaska in Fairbanks. It is one of many photographs in

his book [35].


some of them very rare. A common example is the22◦ halo around the Sun (or the Moon, but it is muchfainter for obvious reasons). However, only small(less than about 20 microns in size) columnar crys-tals are responsible for this particular optical man-ifestion. Again, light is refracted between alternatefaces of the hexagonal columns, but the crystals aresmall enough to tumble, and so are randomly ori-ented as they fall. I see these halos, or parts of them,at least once a month on the average, and some-times they can last for an hour or two. There aremany other types of halos; details and further ref-erences may be found in [2] and [35]. A generalmathematical setting for halo theory is available in[36] (see also [18], in which the use of symmetryarguments is further explored, together with somespeculations on possible halo forms produced inthe atmosphere of Titan!)

In passing, Libbrecht briefly addresses the ques-tion: Why (not who) is snow white? To which we mayadd: Why does it seem so quiet outside after asnowfall? In fact, let us generalize the first ques-tion by asking why it does appear white, while ice(in sufficient quantities, as in a glacier) appearssomewhat blue in color. Snow is made of small crys-tals of ice, as we well know by now. There are myr-iads of tiny surfaces from which light is reflected,resulting in a very efficient scattering process inwhich very little absorption takes place. This highreflectivity, incidentally, is also the main reason whysnow is melted more by warm air than by direct sun-light. By contrast, ice is a continuous medium (onthe scale of ice crystals, at least), and so sunlightis more readily absorbed because of the longerpath-length between scattering centers. Blue lightis absorbed rather less efficiently than are longerwavelengths, but this is not obvious in somethingas small as an ice cube! Large quantities of iceallow an accumulation of the effects of scatteringand absorption by not only the ice molecules butthe plentiful supply of dust, air bubbles, and other

particulate matter. As to the question not asked in the book: sound waves are readily absorbed bya thick covering of fresh snow, because all the air pockets are prone to trapping the waves to acertain extent. As noted in [16], as snow ages, itchanges from being light and fluffy to smooth andhard, and in this state it can become an efficientreflector of sound waves, and sounds may seemclearer and be heard from greater distances.

Chapter 8 (“In Search of Identical Snowflakes”)addresses what might be termed “the question wehave all been waiting for”: Is every snowflakeunique? In a recently used phrase, the answer tothis question depends on what is meant by theword “is”. Less flippantly, the answer does dependon how powerful a “magnifying glass” one wishesto use to provide an answer. It is an axiom of fun-damental physics that all electrons are identical,so at this scale the components of all snowflakesare identical. However, since this applies to ele-phants and teapots as well, the concept is not par-ticularly helpful. Moving up in size from electrons,it is also true that “ordinary” (as opposed to“heavy”) water molecules are identical. Heavy wateris water composed of deuterium, a stable isotopeof hydrogen, occurring with a frequency of aboutone deuterium atom for every 5,000 of the lighterhydrogen atoms. Another stable isotope of oxygenhas a relative frequency of occurrence of aboutone in 500, so while a snow crystal might containwater molecules, about 1 in 500 water moleculeswill be different from the rest. The number of con-figurations containing these heavier oxygen atomsis combinatorially enormous, so the likelihood oftwo snowflakes being identical is equivalent tothat of the sustained existence of the proverbialsnowball in the proverbial very hot place. But as Lib-brecht reminds us, crystals contain impurity atoms,stacking faults, and other types of defects. And thisdoes not help matters, to say the least.

If we relax our requirement of mathematicaluniqueness at the atomic and molecular level, usinga good optical microscope with resolution down toabout 10−3 mm, it might well be possible to findtwo hexagonal plates that appear indistinguish-able, but the more complex crystals are very intri-cate, with all sorts of minor asymmetries, so theproblem gets worse again. Just as with the earlystages of male pattern baldness, it all depends onhow closely one looks.

The “afterword” by photographer Patricia Ras-mussen is beautiful, not least because of her obvious love for poetry. And with mention of the word beauty, it is appropriate to quote both Poincaré (as stated in Chapter 8 of the book) andKenneth Libbrecht himself as he ends this, his lastchapter.

This drawing of halos was made in Danzig,1661 by the astronomer Hevelius. It is one ofseveral early records described in the book onmeteorological optics by J. M. Pernter and F. M.Exner.


The scientist does not study nature be-cause it is useful; he studies it becausehe delights in it, and he delights in it because it is beautiful. (Jules Henri Poincaré, 1908)

There is great beauty in a large, sym-metrical stellar crystal. The beauty is en-hanced by the magnifying lens thatbrings out the fine structures in the ice.The beauty is enhanced still further byan understanding of the processes thatcreated it. (Kenneth G. Libbrecht, 2003)

A last word on the topic of snow crystals maybe of interest:

Thanks to the sharp eyes of a Minnesota man,it is possible that two identical snowflakes may fi-nally have been observed. While out snowmobiling,he noticed a snowflake that looked familiar to him.Searching his memory, he realized it was identicalto a snowflake he had seen as a child in Vermont.Weather experts, while excited, caution that thismay be difficult to verify.

See more about snow crystals on Kenneth Lib-brecht’s website www.snowcrystals.net.

Mathematical AppendixA snow crystal pattern is an interfacial phenome-non, occurring at the boundary between the solidphase and vapor phase of water. By contrast, an icecrystal grows at a solid/liquid boundary. There isa plethora of theoretical papers on the subject ofcrystal formation (for a sample, see the references),only a small proportion of these, understandably,being devoted to snow or ice crystals. I will at-tempt to provide a general overview of the topicand then address some of the particular models ofsnow and ice crystal formation.

As noted already several times, many nonbio-logical patterns in nature arise at the moving interface between two domains or phases, such as an ice-water vapor boundary, with competition between forces tending to stabilize and destabilizethe boundary. However, in contrast to complex biological systems, crystal growth perhaps repre-sents a conceptually simpler example of sponta-neous pattern formation and self-organization,based on the existing “laws” of thermodynamics,statistical mechanics, kinetics, and transport the-ory. Nevertheless, many of the theoretical problemsassociated with these phenomena are quite formi-dable mathematically.

A snowflake with its planar hexagonal symme-try is a good illustration of some of the questions

EquilibriumCrystalShape

0

n

rγ(n)-plot

Figure 1: The equilibrium crystal shape (for cubicsymmetry) formed from the Wulff construction; the

boundary is shown in bold. It is the interior envelope ofthe set of perpendiculars to radial rays intersecting the

γ(N ) (surface free energy) polar plot.(From [43].)

Is Seeing Believing?By searching through both old and

recent snowflake photographs witha computer search tool specially

designed for the task, candidates forindistinguishable flakes have been

found. Because of the poorer qualityof the picture on the left, however,some difficult work remains to be

done. These photographs weretaken on the first day of April, which

might account for the similarity ofthe flakes.

http://www.snowcrystals.net


that can be asked in a more general context. Theyare patterns that have emerged, apparently, froma structureless environment, and as has been noted previously, they are very sensitive to that environment. Depending on the history of each ice crystal as it moves through regions of differ-ent temperature, supersaturation, and airspeed,there will be formed several different types of icestructure; and since each history is (presumably)unique, so in principle is each snow crystal.

According to the review by Langer [19], regularlyfaceted crystals will form under a wide variety ofconditions when the molecules are tightly boundon crystallographic planes. At the other extreme,when the surface molecular binding is sufficientlyweak (as in many metals and alloys), growth isdominated by the mechanism of diffusion close tothe solidification front (and may be rapid). In suchcases, the fluid-solid interfaces are macroscopi-cally smooth but microscopically rough. Ice fallsbetween these two extremes: facets grow slowly parallel to the basal plane, but rapidly in the hexag-onal directions, and surfaces tend to be rounded.

There are two basic types of mechanism that contribute to the solidification process: diffusioncontrol (involving long-range processes) and in-terface control (involving local processes). In [42],models based on these mechanisms are referred torespectively as nongeometric and geometric growthmodels. In the latter, the interfacial growth veloc-ity is determined by local conditions only, andhence diffusion-driven morphological instabilitiesare absent. However, even this is something of anoversimplification, because a geometric model maybe used to examine diffusion-limited growth if the interfacial speed is sufficiently large. (In this context the appropriate length scale L is essentially the diffusivity divided by the interface speed; largegradients can exist, and the boundary layer ofthickness L effectively defines the interfacial re-gion.) A model is considered geometric if the normal velocity at an interfacial point dependsonly upon the shape and shape-dependent quan-tities of the interface.

By contrast, nongeometric models pertain togrowth on surfaces that are everywhere rough andare usually formulated as some type of free bound-ary problem. It transpires that such diffusion-controlled models of simple geometric shapes suchas planes, cylinders, and spheres are commonly unstable. For example, if a plane solid interface develops a small bump extending into the vaporphase or liquid phase, the temperature gradient islocally larger than in its immediate neighborhood(because the isotherms are closer at that point), latent heat diffuses away more rapidly, and thebump continues to grow, at least until the stabi-lizing effect of surface tension (via the curvatureof the interface) becomes comparable. How does

this stabilizing effect come about? The surfacemolecules on a bump with positive curvature havefewer nearest neighbors than do those on a planesurface and are thus more susceptible to being removed, and the bump will tend to move back tothe plane configuration. This corresponds to a reduction in the melting temperature of the bump.Similarly, molecules on the surface of a negativelycurved region have more nearest neighbors and aremore tightly bound; the melting temperature hasbeen raised.

Thus, a moving interfacial boundary is driven bya diffusion field gradient and inhibited by curvature-related forces. A common feature of such compe-tition between stabilizing and destabilizing influ-ences is the existence of a characteristic lengthscale for the resulting pattern. It is often difficultto predict the pattern selection principles that op-erate in these systems. Such patterns, as notedabove, are sensitive to the system geometry and ex-ternal conditions in general. A valuable review ofpattern formation models in this context can befound in [17]. Mathematically, the problem of amoving and developing interface between two dis-tinct media is a Stefan problem, wherein a nonlin-ear system evolves dynamically in time. As notedin [7], this problem is especially interesting whenthe interface motion is a nonequilibrium problem,where the configuration is initially such that the freeenergy of the system is not at an absolute minimum.

A very useful resource for mathematicians in-terested in the subject of crystal formation is thereview in [37]. Therein, an important convex set,known as the Wulff shape (or equilibrium crystalshape) is introduced. The equilibrium crystal shapeis the shape which minimizes the total surface freeenergy per unit area γ(N ) for the volume it en-closes, and its boundary Wγ is defined as

y

x

T

M

Pr

S

θφ

Figure 2: A portion of the crystal boundary S used inestablishing equations (4) for the total free energy F and the“area” of the two-dimensional crystal. (The notation of [9] isused in this figure.)


(1) Wγ = {r : r·N ≤ γ(N )∀N}.γ(N ) is orientation-dependent, N is the unit nor-mal vector, and r is the radius vector of any pointon the equilibrium crystal surface (see Figure 1).Wγ is determined uniquely by the Wulff Theorem(or construction), one proof of which is given below.Another important boundary is the steady-stategrowth shape WV defined by

(2) WV = {r : r·N ≤ V (N )∀N},

where V (N ) is the growth rate in the direction N .For further references to the Wulff construction,see [11].

Wulff’s TheoremIn a crystal at equilibrium, the distances of the facesfrom the centre of the crystal are proportional totheir surface free energies per unit area [9]. Theproof and notation therein for the two-dimensionalcase will be followed here. The fundamental ideais to relate the equilibrium shape of the “crystal”to the polar plot of the surface free energy. This is done via the Wulff construction (again, see Figure 1), which is the interior envelope of the setof perpendiculars to radial rays intersecting the latter [43]. The equilibrium shape is determined by the requirement that the total free energy for agiven crystal “area” is a minimum.

We consider a point P (x, y) on the boundary Sof the crystal; the polar coordinates of P are (r ,φ).Here the arc length parameter will be denoted byθ for consistency with [9], but also to emphasizethat we are dealing with rather general equilib-rium boundary curves as opposed to the circularones discussed in the model below. (Note that θwillbe used in two other ways in the models below.)The line segment OM (see Figure 2) is the perpen-dicular from the origin to the tangent line to S atP ; it has length p. Denoting the edge free energyper unit length of a boundary element by f (θ), thetotal free energy F and the area n (related to thenumber of molecules) are respectively given by

(3)F =

∫f (θ)ds =

∫f (θ)

(x2 + y2

)1/2 dθ

and n = 12

∫(xy − yx)dθ,

where x = dx/dθ, etc. From the figure and stan-dard algebra,

p = x cosθ + y sinθ.

The locus of M as the point P moves around theboundary curve S will be the pedal curve of S. Theexpression above and its derivative with respect toθ can be inverted to obtain

x = p cosθ − p sinθ and y = p sinθ + p cosθ.

Hence

(4) F =∫

(p+p)f (θ)dθ and n = 12

∫(p+p)p(θ)dθ.

The total free energy must be minimized subjectto the constraint n = constant. If the choice of Lagrange multiplier λ is made such that we can define the quantity

(5) Q(θ,p, p, p) = 12

(p + p)p − λ(p + p)f ,

then the appropriate Euler-Lagrange equation is

(6)∂Q∂p

− ddθ

(∂Q∂p

)+ d2

dθ2

(∂Q∂p

)= 0.

It follows that

(7) (p + p)− λ(f + f ) = 0,

with solution

(8) p(θ) = C sin (θ − θ0)+ λf (θ)

for arbitrary constants C and θ0. By choosing theorigin in such a way that the crystal has some rotational symmetry about it, the choice C = 0 ispermissible. Then

(9) p(θ) = λf (θ) ,

which establishes that the polar diagram of the edgefree energy is proportional to the pedal of the equi-librium shape of the crystal.

It is important to note that the steady-stateshape of a crystal is not as general as the Wulffshape for an equilibrium crystal, because it is notobtained from an initial value problem. Not all ini-tial shapes will in fact reach this steady-state shape,and neither does it indicate what class of geomet-ric models results in a convex form from an initialshape. Further details on this and other aspects ofthe problem may be found in [41].A “Toy” Model for Crystal GrowthAs a precursor to the more physically significantmodels of crystal growth discussed below, let usexamine a specific yet simple evolution equationfor the interface in a two-dimensional geometricalmodel and comment on its stability. The model thatfollows is not particularly realistic physically—only the lowest order curvature term is retained toexhibit the desired behavior of the boundary—butit must be emphasized that it is merely a mathe-matical simile; it cannot be derived from the kineticsof crystal growth for realistic crystals. In fact, amodel of this type is one in which the boundarylayer effectively defines the interface: as noted ear-lier, the characteristic length scale L is the ratio ofthe diffusion coefficient to the interfacial speed, andthis would be true in practice only if the crystalgrowth speeds were excessively high. Neverthe-less, for applied mathematicians, toy models are


the aesthetic equivalent of “back-of-the-envelope”calculations for engineers!

If x is the position vector of a point on the interface corresponding to outward unit normal N , then we assume that

(10) N·dxdt

= U (x; s, κ),

where the speed U is a function of the local sur-face geometry, being dependent on position, andimplicitly on arc length s and curvature κ. Whatform might U take? One choice is [7]:

(11) U = V (κ)+ γ ∂2κ∂s2

.

The second term corresponds to the stabilizationmechanism for short wavelengths when γ > 0.Heuristically we may choose the “curvature potential” V to take the form of a cubic for the following reasons. A planar interface cannot moveat all (except under very special circumstances), so V (0) = 0. The growth rate of a spherical crystal (or circular in two dimensions) behaves like R−1 forlarge R, so we would expect V ≈ ακ for small κ.Noting further that a solid with large curvaturewill contract due to forces of surface tension, Umust become negative for some value of κ. Hencethere must be a minimum “bubble” size for nu-cleation, and this corresponds to a term −βκ3 inV , where β is related to the minimum size for nu-cleation. A quadratic term, µκ2, is also included inthe expression for V to account for an asymmetryin the solidification process between the freezingand the remelting of the dynamic interface, so wewrite

(12) V (κ) = ακ + µκ2 − βκ3.

Stability CriteriaIn terms of the unit tangent vector T to the curveat a point with polar angle θ (s being arc lengthalong the boundary curve), the curvature vector κis variously defined as

κ = dT /ds = (dT /dθ)/(ds/dθ)

= (dT /dθ)/ |dx/dθ| ,

where x (θ) is the position vector. For a circle withtime-dependent radius r (t) and the special case ofU = V (κ) = κ (with the unit of length chosen to fixα = 1 [7], [8]), the equation

N·dxdt

= U (x; s, κ)

reduces to

(13)drdt= V

(1r

),

with solution

(14) r (t) = [2t + r2 (0)]1/2 .

Performing a linear stability analysis around thissolution, following [8] (see also [42]) we considerradial perturbations only, because tangential termscan be eliminated by a suitable gauge transforma-tion (equivalently, they can be shown to drop outof the linear stability analysis). Thus

(15) x = r [r (t)+ ε(t, θ)] .

Then

(16) κ = |κ| = 1r− 1r2

(∂2

∂θ2+ 1

)ε ≡ κ0 + κε,

where κ0 = r−1, and

(17)∂2κ∂s2

= − 1r4

∂2

∂θ2

( ∂∂θ2

+ 1)ε.

Since

V (κ0 + κε) = V (κ0)+ κεV ′ (κ0)+O(κ2ε ),

the linear perturbation ε(t, θ) may be shown to satisfy the equation

(18)

∂ε∂t=− V

′ (κ0)r2

(∂2

∂θ2+ 1

)

× ε − γr4

∂2

∂θ2

( ∂∂θ2

+ 1)ε.

Seeking eigenfunctions of the form

(19) ε(t, θ) = εmeσmt cosmθ,

we see that the linear growth rate

(20) σm =(m2 − 1

)r2

[V ′(

1r

)− γm

2

r2

].

The m = 1 mode corresponds to a uniform rota-tion of the circle and is a neutrally stable pertur-bation. As noted in [8], the first term is stable forall m (≠ 1) if V ′(1/r ) > 0, and since for sufficientlylarge radii V ∼ κ, this will occur even for nonzeroµ and β in the complete expression for V . This sim-ple result is the analog of the Mullins-Sekerka in-stability as discussed in [29], [30] (see also [10], [31],[46]). The second term is always stabilizing form > 1 (with γ playing a role analogous to that ofsurface tension), so this will act to limit the rangeof m-values that are in the unstable regime.

More Sophisticated ModelsBased on more detailed theoretical physics, howmight the interface evolution of a crystal be char-acterized mathematically? Following the approachin [42], we consider the boundary of a two-dimensional crystal to be represented by the closedcurve C [x (u, t) , y (u, t)] in the plane, with time de-pendent components parametrized by the variable


u. Let the arc length s be related to u via the rela-tion

(21) s (u, t) =∫ u

0

∣∣∣∣∂C(u′, t)∂u′

∣∣∣∣du′.Further, let W = |∂C(u, t)/∂u| = ds/du. θ is nowdefined to be the angle between the positive x-axisand the unit tangent vector T = (cosθ, sinθ) =W−1∂C(u, t)/∂u, and the unit normal vector N isinward pointing. The boundary evolves in terms ofthe normal velocity function V (θ,δµ) according to

(22)(∂C∂t

)u= −VN .

The initial-value problem with V defined as abovecan be solved exactly by the method of character-istics, since the normal velocity of the interface de-pends explicitly on the surface orientation alone.They are rays of the form x(t) = x0 + d (θ0) t , x0

being a point on the initial curve, and V (θ0; x0) isin the direction d (θ0) . The surface normal direc-tion is preserved along each characteristic, and sothe curve C at time t is defined by the set of allpoints x(t).

To discuss the curvature evolution, both globaland local, we require that the two appropriateFrenet equations for the unit tangent (T ) and nor-mal (N ) vectors at a point on the boundary curveare

∂T∂s

= κN and∂N∂s

= −κT ,

(κ = ∂θ/∂s). If arc length is used to parametrizethese quantities in terms of the closed interfacialboundary curve C [42], then

(23)T (s, t) = ∂C

∂s, κ(s, t) =

∣∣∣∣∣∂2C(s, t)∂s2

∣∣∣∣∣ ,and N (s, t) = κ−1(s, t)

∂2C(s, t)∂s2

.

Following [13] and also [42], we can prove the fol-lowing identities:

(24)

∂W∂t

= κVW ;[ ∂∂t,∂∂s

]= −κV ∂

∂s,

∂T∂s

= −∂V∂sN ;

∂θ∂t= −∂V

∂s.

Since W is essentially a measure of the length ofan infinitesimal displacement on the boundary,the differential equation for W represents its rateof dilation. The corresponding equations in T andθ describe the rotation rates of those quantities,as determined by the anisotropy of V. After somesubtle reformulation (involving gauge invariance ar-guments), a nonlocal integrodifferential equationfor the curvature evolution can be derived, namely:

(25)(∂κ (s, t)∂t

)s= −

(∂2V∂s2

+ κ2V)− ∂κ∂s

∫ s0κVds′.

As pointed out in [8], when the curve C is parame-trized by θ rather than by u or s , then the evolutionequation for κ becomes strictly local. This is a consequence of the fact that its rate of change atfixed θ is given by

(26)(∂κ∂t

)θ=(∂κ∂t

)u−(∂θ∂t

)(∂κ∂θ

)≡ ∂κ∂τ.

Thus

(27)∂κ∂τ

= −κ2V (θ) ,

where V = V + V ′′ is named the “velocity stiffness”in [42]. These results also can be derived by con-sidering the curve as a set of complex numbers inthe plane [8]. For finite V this equation has solu-tion

(28) κ = κ (0)1+ κ (0) Vτ

,

in terms of the initial curvature κ (0) . If V > 0, thecurvature will decrease monotonically in time at nonfaceted orientations for which κ (0) ≠ 0.For orientations in which V < 0, the curvature

diverges at finite time, τ = −(κ (0) V

)−1. In fact,

as discussed below, a shock (intersection of characteristics) develops before the minimum blow-up time.

In defining a local normal velocity with referenceto a weak “growth drive” δµ (the chemical poten-tial difference between the surface and externalphase), two different kinetic processes are mergedvia a convexity type of argument involving a tran-sition function ξ (θ) to form an expression for thelocal normal velocity of a crystal with n-gonal sym-metry. This is

(29)V (θ,δµ) = Vf (δµ)ξ (θ)+ Vr (θ,δµ)

(1− ξ (θ)

),

where

(30) Vf (δµ) = cf g (δµ) exp

(−πσ 2

kTδµ

),

and

(31) Vr (θ,δµ) = crδµ[1+ cosp

(nθ2

)],

where p is an even integer. Vf (for facet) is the nor-mal growth rate at facet orientations under the“growth drive” δµ, while Vr (for rough) describesthe normal interfacial motion at vicinal and mole-cularly rough orientations (due to surface migra-tion of adsorbed molecules away from facets).Since 0 ≤ ξ (θ) ≤ 1, this function is a measure of


the distribution between facetlike and roughlikegrowth. It is periodic in 2π/n , and ξ

(θf) = 1,

ξ(θf +π/n

) = 0 for a facet orientation θf . In [42]the choice of ξ (θ) = cosm (nθ/2) , m ≥ p beingeven, is made. If in particular θ = π/4 and n = 4in the definition of V , then for m = p = 2,V = crδµ + 8Vf , which implies that the rate of cur-vature decrease increases with δµ. In short, thepolygonalization of crystals corresponds to de-creasing curvature in rough orientations.

Thus, in general, a crystal has slowly growingmolecularly smooth faces (facets) and more rapidlygrowing faces that are rough at the molecular level.At this level, both structures are sensitive to tem-perature, as one might expect. Facet locations areassociated with minima of the surface free energy,and indeed a crystal shape may be only partiallyfaceted, but a major question for crystal growth isto find what processes are responsible for the evolution of an initial “seed crystal” towards afaceted asymptotic growth shape. This is calledglobal kinetic faceting and has been observed tooccur experimentally and numerically [26] and hasbeen predicted theoretically [42]. In this process,the rough orientations of partially faceted shapesgrow out of existence with decreasing curvature.Such a decrease implies the presence of disconti-nuities in surface slope (or a jump in the normalto the surface), so it is of considerable interest to be able to tie the time-dependence of the local curvature to the ultimate shape of the crystal boundary. Mathematically, discontinuities of this kind are associated with both singularities in the interfacial curvature and the intersection of characteristics [39]. The latter viewpoint, while complementary to the former, corresponds to the existence of shocks in the solution of the surfaceevolution equation. The dynamics of such shocks,when they exist, are of great importance in the interpretation of experimental data. Furthermore,it has been shown in [41] that if the initial seed crystal is convex, then convexity is preservedthroughout the whole growth process. Forms of V that possess cusps at faceted orientations also explain the formation of expanding facets separated from rough orientations by a corner,called shocking facets in [40]. Specifically, in placeof expression (29) for V (θ,δµ) , now

(32) V (θ,δµ) = Vf (δµ)ξ (θ)+ Vr (θ,δµ) ,

where

(33) Vr (θ,δµ) = crδµ(a |sin 2θ| − b sin2 2θ

),

where the coefficients a and b are related to cer-tain angular rates of change. It transpires in thiscase that models with the parameter range2b < a < 7/3b correspond to the formation and ex-pansion of facets accompanied by decreasing cur-

vature at rough orientations. Further, if t0 is the timeof corner formation, the facet collides with theshock at t ≈ 1.1t0 almost immediately after shockformation, consistent with observations [40].Green’s FunctionsThe literature on this subject abounds with exam-ples of the application of Green’s function tech-niques to the growth and development of crystalsand might well be a useful source of examples forgraduate classes. This particular analysis is basedon the approach in [40] (but see also [25]). In con-nection with shocking facets, the temperature fieldT around an expanding surface with a constant den-sity of heat sinks is computed using Green’s func-tions for the two-dimensional diffusion problem.The expression for the field at (x, z; t) is

(34)

u(x, z; t) = u0(x, z)

−∫ t

0dt ′

∫∞−∞

∫∞−∞G (x− x′, z − z′; t − t ′)

×Q(x′z′; t ′)dx′dz′,

where u = (T − T∞) /T∞ , T∞ being the melt tem-perature far from the interface. The temperaturefield before the formation of a shocking facet is u0,and the standard Green’s function for the diffusionproblem

(35)[ ∂∂t− k∇2

]G(x, z; t) = δ(x)δ(z)δ(t)

is

(36) G(x, z; t) = 14πkt

exp

(−x

2 + z2

4kt

),

k being the thermal diffusivity of the melt. Thesource term is

(37) Q(x, z; t) = qH(υt − x)H(x)δ(z),

in terms of the Heaviside step-function H and theDirac delta function δ (remember that distributiontheoretic distinctions and subtleties are usuallysuppressed in much of the scientific literature).The term q is a heat flux constant, proportional to the surface density of heat sinks. After some rearrangement

(38) u(x, z; t)− u0(x, z) = − q4k

∫ t0dt ′e−z2/4k(t−t′)

t − t ′ I,

where

I =√

4k(t − t ′)∫ −α+η−α

e−n2dn,

α = x/√

4k(t − t ′),

η = υt/√

4k(t − t ′).

Experimentally, the authors found that υt �√

4kt.Under the experimentally reasonable assumptionthat η� 1,


I ≈ η√

4k(t − t ′)e−a2;

whence

(39)

u(x, z; t)− u0(x, z) ≡ u(x, z; t)

≈ −qυt4k

∫∞(x2+z2)/4kt

e−λ

λdλ.

Noting that the exponential integral [6] is definedas

(40)

E1 (x) =∫∞θ

e−λ

λdλ

= − lnθ − γ +∞∑n=1

(−1)n+1 θn

n · n!,

where γ is the Euler-Mascheroni constant, it followsthat when (x2 + z2)/4kt � 1 ,

(41) u(x, z; t) ≈ −qυt4k

(−γ + ln

4ktx2 + z2

).

At the other extreme, for short times such that4kt � x2 + z2, u is exponentially small. Using ex-perimental data, Tsemekhman and Wettlaufer [40]estimate that the geometric model begins to fail att ≈ 10s . From this and related observations, theauthors conclude that geometric models of the en-tire interface are very accurate up to about t = 15s .Ultimately the model breaks down when q is nolonger constant, and nonlinear effects become sig-nificant. Studies such as this one confirm that geo-metric models are very useful in describing theearly stages of the evolution of a crystal surface.Some of these models predict the possible forma-tion of a shocking facet and are still applicable forsome time beyond this point. Indeed, according toWettlaufer [private communication], even thetimescales cited above do a great injustice to theutility of the geometric approach: for solid heliumgrowing from a superfluid, the geometric model isthe only model of significance, and it never breaksdown!

SummaryThere are many other mathematical aspects ofcrystal formation that have been investigated overthe last several decades. It is a field rich in theory,experiment, and applications (particularly inter-esting, for example, but outside the scope of thepresent review, is the asymptotic analysis found in[44], [45]). Interesting theoretical papers on dendriteformation are presented in the set [20]–[24], [27],[28]. In addition, many theoretical developments areto be found outside the “standard” applied math-ematical literature: some of the papers cited beloware published in the Journal of Crystal Growth,and Physical Review E, for example, so it is en-cumbent on those of us interested in entering a fieldsuch as this sometimes to walk in pastures new.

Personally, I have been both pleasantly surprisedand pleased to encounter such a broad swath ofthoughtfully written papers addressing crystalgrowth. I hope that this review will be a useful (butclearly not exhaustive) resource for readers wish-ing to pursue the subject further. However, a caveatis in order: a review of this nature must maintaina balance between material of interest to mathe-maticians while at the same time doing justice tothe physics of the problem. Inevitably, the simplemodels and oversimplified physical descriptionspresented here will not satisfy crystal theorists, butas in all topics on the interface between two mag-isterial disciplines, compromises must be made.This article is no exception, and I will end it witha warning: The mathematical physics of snow crys-tal formation is much more complicated than mightbe inferred from the discussion presented here!

AcknowledgmentsI am very grateful to Professor John S. Wettlaufer,who has not only provided me with several of hispapers and constructively reviewed an early draftof this manuscript but has also patiently, thought-fully, and enthusiastically responded to my ques-tions and comments during its preparation. I’malso grateful for Bill Casselman’s creative artwork.

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[46] E. YOKOYAMA and T. KURODA, Pattern formation insnow crystals occurring in the surface kinetic processand the diffusion process, Phys. Rev. A 41 (1990),2038–49.

SimulatingSnowflakesSnowflake formation is very complicated, and onewould not expect a purely theoretical account toexplain all that occurs. There have been many at-tempts to simulate the process in software. Evenhere, the complexities are overwhelming. The prin-cipal focus of effort has been to understand the ex-traordinary sixfold planar symmetry of snowflakescoexisting with a complicated dendritic structure.All simulation attempts seem to have been in a re-stricted two-dimensional environment.

The central problem is that the symmetry sug-gests deterministic growth, the dendritic structurerandomness. Of course some of the randomnessis contributed by a rapidly changing environment.But although the environment is uniform on thescale of a snowflake, the symmetry of realsnowflakes is not exact, and this also suggests thatlocal randomness on the snowflake is important.

Other considerations suggest the role of ran-domness. It is easy enough to construct dendritesin software that, at least locally, look somewhat likesnowflakes. The simplest process in which thishappens in a physical manner is diffusion-limitedaggregation, or DLA, in which particles wanderaround randomly and stick irreversibly onto a cen-tral, growing core whenever they hit it.

Diffusion-Limited AggregationParticles were initially randomly distributed on ahexagonal lattice with a single particle in the

central core. The paths show the motion of parti-cles wandering in randomly from the boundary ofthe region to attach to the core.

The characteristic feature of DLA is that bumpgrowth is unstable: high curvature means a highgradient in the distribution of particles, which aretherefore attracted to bumps. This instability com-bined with microscopic randomness presumablydoes play a role in snowflake growth.

Diffusion InstabilityBumps attract diffusing particles, which follow thegradient of a certain potential field. As it grows, thegradient increases, so a bump grows unstably.

The results of DLA do not look much likesnowflakes. One simple explanation for this is thatdendrites formed in a DLA process are fractal,

Diffusion Instability. Bumps attract diffusingparticles, which follow the gradient of a certain

potential field. As it grows, the gradientincreases, so a bump grows unstably.


which means that they possess a similar structureat all scales. In snowflakes surface tension forbidsthis and in any realistic simulation must be takeninto account. The combined effect of surface ten-sion and DLA is to establish a certain characteris-tic length involved in the growth process.

Up until about 1985 all attempts to simulatesnowflakes built the symmetry in by constructingone arm and then reflecting and rotating it. But thenNittman and Stanley showed that a truly randomprocess could in fact give rise to a good approxi-mate symmetry without artificial forcing. Theirprocess was an extension of DLA in which sites nextto the growing core freeze only when they en-counter several diffusing free particles. This re-quirement is already one that assures a certainstability in the process and leads to quasi-sym-metry but gives rise only to a very restricted rangeof snowflake shapes. In addition, they introduceda somewhat artificial parameter that generatesthicker dendritic arms. With this, quasi-symmetryis maintained and a wider range of shapes arises.Unfortunately, their construction seems to have lit-tle to do with physics.

A much more physically realistic account wasthat of Yokoyama and Kuroda (of Hokkaido Uni-versity). Their simulations involved a fairly de-tailed account of actual surface motion of moleculesin ice formation, as well as a detailed analysis ofhow diffusion behaves at corners. Although ex-tensive dendritic growth does not occur in theirmodels, it does not seem too far away.

One recent attempt at imitating snowflakes hasbeen done with a 2D cellular automaton by CliffordReiter. The only randomness built into his con-struction is in the initial state, however, and al-though some of his figures do resemble snowflakes,it is hard to see what this process has to do withreal life.

ReferencesP. BALL, The Self-Made Tapestry, Oxford Univ. Press, 1999,

pp. 110–127.J. NITTMANN and H. E. STANLEY, Nondeterministic approach

to anisotropic growth patterns with continuously tun-able morphology: The fractal properties of some realsnowflakes, J. Phys. A 20 (1987), L1185–91.

C. REITER, Chaos Solitons Fractals 23 (2005), 1111–1119.E. YOKOYAMA and T. KURODA, Phys. Rev. A 41 (1989),

2038–49.

—Bill Casselman, Graphics Editor

Diffusion-Limited Aggregation. Particles wereinitially randomly distributed on a hexagonal

lattice with a single particle in the central core.The paths show the motion of the particles

wandering in randomly from the boundary ofthe region to attach to the core.

Flowers of Iceâ€”Beauty, Symmetry, and Complexity: A Review of The

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