Flowers of Ice—Beauty, Symmetry, and Complexity: a Review of the Snowflake: Winter's Secret Beauty, Volume 52, Number 4

Flowers of Ice—Beauty, Symmetry, and Complexity: A Review of The Snowflake: Winter’s Secret Beauty

Reviewed by John A. Adam

rowing up as a child in southern Eng- contains no mathematics, so the first section will land, my early memories of snow in- address the qualitative features of snow and clude trudging home from school with snowflakes discussed by the author. I will draw my father, gazing at the seemingly enor- on some of the general descriptions of snow and Gmous snowdrifts that smoothed the its properties both from this book and those by hedgerows, fields and bushes, while listening to the others mentioned below. I have decided to comment soft “scrunch” of the snow under my Wellington on every chapter individually, because each can boots. In the country, snow stretching as far as I be treated to some extent independently of the could see was not a particularly uncommon sight. others. The second section takes the form of a The quietness of the land under a foot of snow mathematical appendix devoted to an outline of seemed eerie. I cannot remember the first time I some of the mathematical aspects of crystal for- looked at snowflakes per se; my interests as a mation, with particular emphasis on ice crystals. small child were primarily in their spheroidally In some sense, I hope, this will become a parallel shaped aggregates as they flew through the air. “mathematical universe” to the first section. Many years later, as I cycled home from my office However, the growth of crystals is a complex in Coleraine, Northern Ireland, I remember being combination of thermodynamics and statistical intrigued by a colored blotch of light to the west physics, and no review of this nature could do of the Sun at about the same elevation. Little did I justice to the immense theoretical edifice that has know then that these two events of snowfall and been erected on this topic. “sundogs” (of which more anon) were intimately This slim volume is informally yet clearly writ- connected. Since that time I have learned rather ten, and the photographs (mostly taken by Patricia more about meteorological optics, and this book Rasmussen) are quite stunning. Obviously, it is about the beauty of snowflakes has challenged aimed at a popular audience. In one sense, though, me to learn more of the physics and mathematics it is the latest in a disjoint set of popular books on behind crystal formation in general and ice crys- pattern formation in nature: D’Arcy Thompson’s tal formation in particular. On Growth and Form [38]; Stevens’s Patterns in As in an earlier review [1], I will divide this Nature [32]; Ball’s The Self-Made Tapestry [3]; review into two main sections; Libbrecht’s book Stewart’s Nature’s Numbers [33] and What Shape Is a Snowflake? [34]; and finally, from a very differ- John A. Adam is a professor of mathematics at Old ent perspective, Bejan’s constructal theory as ex- Dominion University. His email address is [email protected]. pounded in Shape and Structure, from Engineering The Snowflake: Winter’s Secret Beauty, by Kenneth Libbrecht, with photographs by Patricia Rasmussen, to Nature [5]. With the exception of [34], none of Voyageur Press, Inc., 2003, hardcover, $20.00, ISBN these address the nature of the snowflake to any 0-89658-630-8. great extent. Some of Libbrecht’s low-key yet vivid

402 NOTICES OF THE AMS VOLUME 52, NUMBER 4 descriptions captured my imagination, and I would depending on context. The two words will be used certainly recommend this book to any person synonymously here unless otherwise noted. wishing to “experience” something of the physical The short Chapter 2 is entitled “Snowflake intuition of a scientist. Watching”. In it, Libbrecht records a partial history As might be expected, the study of snowflakes of snowflake watchers from Descartes to the is not new; no doubt people have been fascinated present day. In particular, he makes mention of by their beauty and symmetry since time im- Wilson Bentley, a Vermont farmer who dedicated memorial. According to [4], [16] the Chinese aware- much of his life to photographing snowflakes. In ness of this was recorded in 135 B.C., while in the late 1920s Bentley worked with physicist W. J. Europe the Dominican scientist, philosopher, and Humphreys to publish a book containing more theologian, Albertus Magnus, studied them around than 2,000 of his snow crystal images. (There is a 1260 A.D. Not surprisingly, the astronomer and 1962 Dover edition of this book, Snow Crystals.) The mathematician Johannes Kepler was intrigued by scientific discussion begins in earnest in the third snow crystals, writing a small treatise entitled On chapter (“Snow Crystal Symmetry”). It starts with the Six-Cornered Snowflake. In 1611 he asked the the following quote from Richard P. Feynman (The fundamental question: There must be some definite Feynman Lectures on Physics, 1963): cause why, whenever snow begins to fall, its initial Poets say science takes away from the formation invariably displays the shape of a six- beauty of the stars—mere globs of gas cornered starlet. For if it happens by chance, why atoms. I too can see the stars on a desert do they not fall just as well with five corners or night, and feel them. But do I see less seven? In his treatise he compared their symmetry or more?... What is the pattern, or the with that of honeycombs and the seed arrangement meaning, or the why? It does not do inside pomegranates [4]. However, nothing was harm to the mystery to know a little known in Kepler’s era of the molecular structure about it. For far more marvelous is the of water, which ultimately determines the hexag- truth than any artists of the past imag- onal shape of ice crystals, so Kepler was unable to ined it. explain their shape in mechanical terms, though he did attempt to do so using entirely reasonable A quote from Kepler in 1611 is included as he packing arguments. (Indeed, finding the densest pondered the sixfold symmetry of snowflakes, and (not necessarily periodic) packing of spheres is although Libbrecht makes no mention of it, a quote now known as the Kepler problem [14].) In 1665 the from D’Arcy Wentworth Thompson [38] is partic- scientist Robert Hooke published ularly appropriate here: Micrographia, in which he described his observa- The beauty of a snow-crystal depends tions of snowflakes using a microscope. More his- on its mathematical regularity and sym- torical details may be found in [4]. metry; but somehow the association In the first chapter of The Snowflake (“The Cre- of many variants of a single type, all ative Genius”), the author asks the fundamental related but no two the same, vastly questions: How do crystals grow? Why do complex increases our pleasure and admira- patterns arise spontaneously in simple physical sys- tion.…The snow-crystal is further tems? These are very profound questions, of course, complicated, and its beauty is notably and a considerable portion of past and present lit- enhanced, by minute occluded bubbles erature in applied mathematics and theoretical of air or drops of water, whose sym- physics is devoted to attempting to answer them. metrical form and arrangement are very Obviously, the second question goes well beyond curious and not always easy to explain. the “mineral kingdom” of crystal formation into the Lastly, we are apt to see our snow crys- “animal and vegetable” ones of patterns in (and on) tals after a slight thaw has rounded living things. In particular, though, Libbrecht re- their edges, and has heightened their minds the reader that snowflakes are the product beauty by softening their contours. of a rich synthesis of physics, mathematics, and chemistry and that they are even fun to catch on Returning to the question of complex patterns one’s tongue! The range of such comments reflects in nature, we may not be surprised to be reminded the underlying parallel approaches of the book: on by Libbrecht that all crystals demonstrate the or- the one hand, a good qualitative and nontechnical ganizational ability to self-assemble. Starting with description of the scientific aspects of snowflake a random collection of molecules, this is an example formation and, on the other, the sheer fun of doing of spontaneous pattern formation. The best non- science. It is worth noting that in Libbrecht’s ter- mathematical book I know on the broad spectrum minology, snow crystal to snowflake is as tulip to of pattern formation is The Self-Made Tapestry [3], flower. In other words, a snowflake can be an and the present book, much narrower in scope, is individual snow crystal or collection of the same, a good introduction to some of the underlying

APRIL 2005 NOTICES OF THE AMS 403 concepts explained therein. For example: Libbrecht same conditions at each does a pretty good job of explaining to a nontech- moment of time; their nical readership the physics of crystal facets and symmetry is evidently a their formation. One basic snow crystal shape is the reflection of their shared hexagonal prism, which possesses two basal facets history. and six prism facets, and de- It should be noted that pending on which of the two while Nakaya’s 1954 book types grows faster, the prism Snow Crystals: Natural can become a long column and Artificial is referred or a thin plate. The slowest to on page 44, no men- moving facets eventually de- tion is made of the im- fine the shape of the crystal. portant work by Furukawa, who studied under At this point the author asks Kuroda and Kobayashi, both of whom were students a very significant question: of Nakaya. Many valuable details and references on How can molecular forces, the history and science of snow crystal research can operating only at the be found in Furukawa’s essay (http://www. nanoscale, determine the lowtem.hokudai.ac.jp/~frkw/english/ shape of large crystals? An- aletter.html). swers to this and related In Chapter 5 (“Morphogenesis on Ice”) the search questions are hinted at in the for an explanation of snow crystal complexity is following two chapters. taken yet further. Touch- And so to Chapter 4 (“Hi- ing on a point made ear- eroglyphs from the Sky”), in lier in this review, Lib- which it is stated that the real brecht temporarily puzzle of snowflakes is their broadens his perspective combination of symmetry by noting that while a and complexity and that major progress in solving flower is an example of this puzzle was made by the physicist Nakaya in biological morphogene- the 1930s. Eventually, he succeeded in growing in- sis, even simpler physical dividual snow crystals in his laboratory under many systems exhibit this fea- different humidity levels and temperatures, and it ture. Thus whether it be was only a matter of time before he was able to clas- waves on the oceans or sify the symmetry and considerable variety dis- ripples on snowdrifts and sand dunes, they are all played by these crystals. He developed his famous relatively simple pattern-forming systems in which morphology diagram, demonstrating the remark- complexity arises spontaneously, but for Libbrecht able sensitivity of the snow crystal form to its en- the snowflake is the poster child of morphogene- vironmental conditions. This diagram is repro- sis. Here as in earlier chapters he does a good job duced on page 45 of The Snowflake. Basically, at of introducing the concept of self-organization in low but fixed levels of supersaturation (degree of physical (and, in passing, biological) systems. This humidity), as the temperature decreases below 0◦C is all to the good, given the preoccupation in some to about –35◦C, snow crystals are essentially plates, quarters with the concept of “intelligent design” then solid prisms, and then plates again. At higher (with all due respect to my fellow Christians). In a supersaturation levels, the evolution is from den- sentence that I find very drites to needles, hollow columns, sectored plates appealing, Libbrecht notes and dendrites, and then columns again. Essentially, that instabilities like those the overall crystal shape, whether it is platelike or discussed here are the columnar, reveals something about the temperature heart of pattern formation, at which the crystal grew, and the complexity of the and nature is one unstable structure indicates something about the humidity. system heaped on top of However, each crystal falling on one’s nose is a another. product of the cumulative history it has under- Scientifically, the real gone as it has been wafted hither and yon by air meat of the book is to be currents through many different atmospheric conditions. In mathematical terms, we might think of The three photographs of snowflakes (above) its shape being defined by a line integral over its were provided by Yoshinori Furukawa of path through space and time. Generally, the length Hokkaido University, who also provided the scale of variations of temperature and humidity will photographs from which the background be much larger than the dimension of the crystal, snowflakes came. The ordinary symmetry of so each vertex or arm of the crystal experiences the snowflake growth is really extraordinary.

404 NOTICES OF THE AMS VOLUME 52, NUMBER 4 found in this chapter. There are several key com- faceting and branching is what determines the ments made by Libbrecht that bear repeating as form of a given snow crystal in a kind of morpho- written: Growth is the key ingredient for the gen- logical balancing act, and this in turn depends on eration of snow-crystal patterns. Left in isolation for the temperature and humidity history of the grow- a long time, an ice crystal will eventually turn into ing crystal and also its size. Libbrecht speculates a plain hexagonal prism.…Ornate patterns appear that if snowflakes occur in other planetary atmos- only when a snow crystal is out of equilibrium, pheres, they may well be different from the ones while it is growing. Such circumstances are often we know and love. referred to as nonequilibrium conditions. Another Although he does not identify it as such, the important concept introduced is that of diffusion- branching instability Libbrecht refers to is the limited growth. A snow crystal grows by assimilating Mullins-Sekerka instability (see the second section molecules of water vapor into the existing ice of this article for more details). Nobody could fault lattice, provided the humidity is high enough. How- him for simplifying physical phenomena in a ever, continuing crystal growth gradually depletes book of this kind, but as he implies elsewhere, the the vapor from layers of air adjacent to the crys- picture is just not that straightforward. Indeed, I tal, and the remaining water vapor molecules must am indebted to Professor J. S. Wettlaufer for the diffuse over increasingly larger distances. Since following comment (in reference to a statement the travel time for diffusion (other things remain- made in [2]) that puts into perspective the com- ing unchanged) is proportional to the square of plexity of the problem: the distance traveled, it is clear that the growth rate The initial question that needs to be of the crystal will be inhibited; the growth is now asked is: How does a hexagon emerge diffusion-limited. from a nucleus of some 1,000 mole- An initial “seed” crystal consisting of several cules that, at the point of nucleation, do hundred or thousand molecules has a molecularly not necessarily reflect that symmetry? rough boundary. Depending on the crystallographic This has nothing whatever to do with the orientation, some orientations may grow more structure of the background diffusion rapidly than others (under a spatially uniform field, but rather it is solely controlled by growth drive), and thus a surface is created with the statistical mechanics of adsorbates both positive and negative curvature. “Flatter” por- on the seed. The seed is much smaller tions of the boundary are called facets, and while than the mean free path in the vapor a completely rough seed will not have facets, it may phase. The enhancement of the vapor still grow anisotropically. The curved regions “fill field at a corner, generically understood in” more readily than the facets, from which it is as the “point effect of diffusion” or the seen that ultimately the slower-growing facets “Berg-effect”, may or may not lead to an define the crystal shape in the absence of signifi- instability. The conditions must be cant branching (a mechanism discussed below). carefully determined; for example, is Combining the above two mechanisms, in diffusion- the size of the seed much smaller than limited growth one might expect that the crystal ver- the characteristic scale of the diffusion tices would “harvest” water vapor molecules faster field? There is a plethora of outstand- than facets by virtue of their projection further ing physical and mathematical prob- into the medium (if the mean free path of the lems related to the transition between vapor phase is of order the size of the system). nucleation, interface controlled growth This can induce a positive feedback loop known and diffusion limited shapes. Therefore, as a branching instability. As will be noted in the if and/or when a diffusive/Mullins- second section of this review, if this instability Sekerka instability occurs, understand- occurs, there is usually a self-limiting stabilizing ing the habit of the crystal still requires mechanism, akin to surface tension, that eventually an understanding of the initial value balances the destabilizing one. If this diffusion- problem and thus the evolution of a limited branching instability is iterated over and crystal from its birth. over, a type of snow crystal known as a dendrite will develop. Ultimately, this may give rise to some kind In Chapter 6 (“Snowflake Weather”) Libbrecht of self-similar or fractal-like structure over several makes some statements about snow crystal quan- orders of magnitude of scale, not unlike a fern leaf. tities that might form the basis for several inter- This branching process is more pronounced in esting estimation problems, but first it is necessary regions of higher vapor pressure, since then the to note some facts about snow. Consider first a diffusive transport of water vapor molecules is small ice cube 1 cm3 in volume. This has a mass more effective. Conversely, faceted growth is more of one gram, and since the mass of a water mole- likely to occur in a lower vapor pressure environ- cule is about 3 × 10−23g, there are about 3 × 1022 ment. Ultimately, the complex interplay between molecules in this ice cube. A typical snow crystal

APRIL 2005 NOTICES OF THE AMS 405 of mass 3 × 10−5 g therefore may contain about snowmen numbers about 3 × 109 , or about one 1018 molecules. Depending on the type of crystal, half of the present world population. A factor of temperature, wind speed, and other factors, in- two is not a large discrepancy in this context, and cluding the density of packing, snow has consid- in any case, that’s a lot of snowmen. (There is a de- erable variation in its water content. An eight-foot lightful “Peanuts” cartoon showing a vast army of blanket of fresh snow may contain as little as 1 inch snowmen built on a cloudy day by that precocious of water per unit of surface area or as much as 3 genius, Linus. He marches up and down declaring feet. According to [16], the majority of U.S. snows their military invincibility under any and all cir- have a water-to-snow ratio in the range of 0.04 to cumstances. Then the Sun comes out. I use this par- 0.10. I had to dig around for most of this infor- ticular cartoon in mathematical modeling classes mation; I wish that facts such as these had been to emphasize the dangers of making false (or at incorporated into the book as an appendix. best, weak) assumptions.) In the beginning of this chapter we read: Later, another estimation beckons us. Libbrecht Snowflakes are being manufactured in the atmos- states that the total global precipitation per day is 15 phere at an astounding rate—around a million equivalent to about 10 liters of water, and each billion crystals each second. Every ten minutes that’s of us typically exhales about 1 liter of water per day enough snow to make an unstoppable army of snow- into the atmosphere. By simple proportion, there- men, one for every person in the world.... Let’s think fore, and using his figures, we infer that if our about this: on the basis of these numbers, after ten contribution to the water cycle were uniformly dis- minutes there are about 1015 × 6 × 102 = 6 × 1017 tributed around the globe (seemingly yet another crystals, and estimating a typical snowman to be flawed assumption), then our average contribu- composed of two identical spheres of diameter 1.5 tion to the total water content of a snow crystal is 18 ÷ 15 = 3 ft. gives a volume of about 3.5 ft.3 or approximately about 10 10 10 molecules. On this num- 3.5 × (0.3)3 ≈ 0.1 m3 = 105 cm3. If we assume that ber, not surprisingly, we agree. The chapter closes with a nice little section all the crystals formed make their way to the ground about ice nucleation: how water molecules, like (a rather dubious assumption) and take an average adolescents, eventually learn to settle down and water-to-snow ratio of 0.07 for U.S. snow (!), then become very cool (or freeze, in the case of the the mass of crystals falling in a ten-minute inter- former). There is also a rather intricate table val is approximately 2 × 1013g. Meanwhile, the mass classifying some of the many types of snow. of our typical snowman is given by the product Chapter 7 (“A Field Guide to Falling Snow”) is by 105 × 0.07 = 7 × 103 g. This means our army of far the longest in the book and is ac- companied by many beautiful photographs. We are introduced to a verita- ble zoo of snow crystals: diamond dust (of which cirrus clouds are made), stellar dendrites, sectored plates, columns and needles, hollow columns, needles and bullets, capped columns, split stars and split plates, twinned crystals, twelve-sided snowflakes and double stars, chandelier crystals, spatial dendrites, triangular crystals, and rimed snowflakes! What type of ice crystal then was responsible for the sundog I mentioned earlier (such patches of colored light are also known as mock suns or parhelia)? Bil- lions of slowly falling horizontally ori- ented hexagonal plate like crystals (present in cirrus clouds) 30 microns or larger are the culprits. They behave like tiny prisms and refract light entering one ver- tical face (say face 1) and exiting face 3. Unlike rainbow formation, there is no reflection involved, and so the red portion This photograph of atmospheric halos was taken at the South Pole, where is always closest to the Sun. They occur ◦ conditions are frequently favorable. It was provided by Walter Tape, now at least 22 away from the Sun. There are at the University of Alaska in Fairbanks. It is one of many photographs in many other related phenomena one may his book [35]. observe, such as ice crystal halos, and

406 NOTICES OF THE AMS VOLUME 52, NUMBER 4 particulate matter. As to the question not asked in the book: sound waves are readily absorbed by a thick covering of fresh snow, because all the air pockets are prone to trapping the waves to a certain extent. As noted in [16], as snow ages, it changes from being light and fluffy to smooth and hard, and in this state it can become an efficient reflector of sound waves, and sounds may seem clearer and be heard from greater distances. Chapter 8 (“In Search of Identical Snowflakes”) addresses what might be termed “the question we This drawing of halos was made in Danzig, have all been waiting for”: Is every snowflake 1661 by the astronomer Hevelius. It is one of unique? In a recently used phrase, the answer to several early records described in the book on this question depends on what is meant by the meteorological optics by J. M. Pernter and F. M. word “is”. Less flippantly, the answer does depend Exner. on how powerful a “magnifying glass” one wishes to use to provide an answer. It is an axiom of fun- some of them very rare. A common example is the damental physics that all electrons are identical, ◦ 22 halo around the Sun (or the Moon, but it is much so at this scale the components of all snowflakes fainter for obvious reasons). However, only small are identical. However, since this applies to ele- (less than about 20 microns in size) columnar crys- phants and teapots as well, the concept is not par- tals are responsible for this particular optical man- ticularly helpful. Moving up in size from electrons, ifestion. Again, light is refracted between alternate it is also true that “ordinary” (as opposed to faces of the hexagonal columns, but the crystals are “heavy”) water molecules are identical. Heavy water small enough to tumble, and so are randomly ori- is water composed of deuterium, a stable isotope ented as they fall. I see these halos, or parts of them, of hydrogen, occurring with a frequency of about at least once a month on the average, and some- one deuterium atom for every 5,000 of the lighter times they can last for an hour or two. There are hydrogen atoms. Another stable isotope of oxygen many other types of halos; details and further ref- has a relative frequency of occurrence of about erences may be found in [2] and [35]. A general one in 500, so while a snow crystal might contain mathematical setting for halo theory is available in water molecules, about 1 in 500 water molecules [36] (see also [18], in which the use of symmetry will be different from the rest. The number of con- arguments is further explored, together with some figurations containing these heavier oxygen atoms speculations on possible halo forms produced in is combinatorially enormous, so the likelihood of the atmosphere of Titan!) two snowflakes being identical is equivalent to In passing, Libbrecht briefly addresses the ques- that of the sustained existence of the proverbial tion: Why (not who) is snow white? To which we may snowball in the proverbial very hot place. But as Lib- add: Why does it seem so quiet outside after a snowfall? In fact, let us generalize the first ques- brecht reminds us, crystals contain impurity atoms, tion by asking why it does appear white, while ice stacking faults, and other types of defects. And this (in sufficient quantities, as in a glacier) appears does not help matters, to say the least. somewhat blue in color. Snow is made of small crys- If we relax our requirement of mathematical tals of ice, as we well know by now. There are myr- uniqueness at the atomic and molecular level, using iads of tiny surfaces from which light is reflected, a good optical microscope with resolution down to −3 resulting in a very efficient scattering process in about 10 mm, it might well be possible to find which very little absorption takes place. This high two hexagonal plates that appear indistinguish- reflectivity, incidentally, is also the main reason why able, but the more complex crystals are very intri- snow is melted more by warm air than by direct sun- cate, with all sorts of minor asymmetries, so the light. By contrast, ice is a continuous medium (on problem gets worse again. Just as with the early the scale of ice crystals, at least), and so sunlight stages of male pattern baldness, it all depends on is more readily absorbed because of the longer how closely one looks. path-length between scattering centers. Blue light The “afterword” by photographer Patricia Ras- is absorbed rather less efficiently than are longer mussen is beautiful, not least because of her wavelengths, but this is not obvious in something obvious love for poetry. And with mention of as small as an ice cube! Large quantities of ice the word beauty, it is appropriate to quote both allow an accumulation of the effects of scattering Poincaré (as stated in Chapter 8 of the book) and and absorption by not only the ice molecules but Kenneth Libbrecht himself as he ends this, his last the plentiful supply of dust, air bubbles, and other chapter.

APRIL 2005 NOTICES OF THE AMS 407 n

A last word on the topic of snow crystals may be of interest: Thanks to the sharp eyes of a Minnesota man, it is possible that two identical snowflakes may finally have been observed. While out snowmobiling, he noticed a snowflake that looked familiar to him. Searching his memory, he realized it was identical γ(n)-plot r to a snowflake he had seen as a child in Vermont. Weather experts, while excited, caution that this may be difficult to verify. 0 See more about snow crystals on Kenneth Lib- brecht’s website www.snowcrystals.net.

Equilibrium Mathematical Appendix Crystal A snow crystal pattern is an interfacial phenome- Shape non, occurring at the boundary between the solid phase and vapor phase of water. By contrast, an ice crystal grows at a solid/liquid boundary. There is a plethora of theoretical papers on the subject of crystal formation (for a sample, see the references), only a small proportion of these, understandably, Figure 1: The equilibrium crystal shape (for cubic being devoted to snow or ice crystals. I will at- symmetry) formed from the Wulff construction; the tempt to provide a general overview of the topic boundary is shown in bold. It is the interior envelope of and then address some of the particular models of the set of perpendiculars to radial rays intersecting the snow and ice crystal formation. γ(N ) (surface free energy) polar plot. As noted already several times, many nonbio- (From [43].) logical patterns in nature arise at the moving The scientist does not study nature be- interface between two domains or phases, such as an ice-water vapor boundary, with competition cause it is useful; he studies it because between forces tending to stabilize and destabilize he delights in it, and he delights in it the boundary. However, in contrast to complex because it is beautiful. (Jules Henri biological systems, crystal growth perhaps repre- Poincaré, 1908) sents a conceptually simpler example of spontaneous pattern formation and self-organization, There is great beauty in a large, sym- based on the existing “laws” of thermodynamics, metrical stellar crystal. The beauty is en- statistical mechanics, kinetics, and transport the- hanced by the magnifying lens that ory. Nevertheless, many of the theoretical problems brings out the fine structures in the ice. associated with these phenomena are quite formi- The beauty is enhanced still further by dable mathematically. an understanding of the processes that A snowflake with its planar hexagonal symme- created it. (Kenneth G. Libbrecht, 2003) try is a good illustration of some of the questions

Is Seeing Believing? By searching through both old and recent snowflake photographs with a computer search tool specially designed for the task, candidates for indistinguishable flakes have been found. Because of the poorer quality of the picture on the left, however, some difficult work remains to be done. These photographs were taken on the first day of April, which might account for the similarity of the flakes.

408 NOTICES OF THE AMS VOLUME 52, NUMBER 4 y that can be asked in a more general context. They are patterns that have emerged, apparently, from a structureless environment, and as has been T noted previously, they are very sensitive to that environment. Depending on the history of each M ice crystal as it moves through regions of differ- S ent temperature, supersaturation, and airspeed, there will be formed several different types of ice r structure; and since each history is (presumably) unique, so in principle is each snow crystal. P According to the review by Langer [19], regularly faceted crystals will form under a wide variety of φ θ conditions when the molecules are tightly bound x on crystallographic planes. At the other extreme, when the surface molecular binding is sufficiently S weak (as in many metals and alloys), growth is Figure 2: A portion of the crystal boundary used in F dominated by the mechanism of diffusion close to establishing equations (4) for the total free energy and the the solidification front (and may be rapid). In such “area” of the two-dimensional crystal. (The notation of [9] is cases, the fluid-solid interfaces are macroscopi- used in this figure.) cally smooth but microscopically rough. Ice falls this stabilizing effect come about? The surface between these two extremes: facets grow slowly molecules on a bump with positive curvature have parallel to the basal plane, but rapidly in the hexag- fewer nearest neighbors than do those on a plane onal directions, and surfaces tend to be rounded. surface and are thus more susceptible to being There are two basic types of mechanism that removed, and the bump will tend to move back to contribute to the solidification process: diffusion the plane configuration. This corresponds to a control (involving long-range processes) and in- reduction in the melting temperature of the bump. terface control (involving local processes). In [42], models based on these mechanisms are referred to Similarly, molecules on the surface of a negatively respectively as nongeometric and geometric growth curved region have more nearest neighbors and are models. In the latter, the interfacial growth veloc- more tightly bound; the melting temperature has ity is determined by local conditions only, and been raised. hence diffusion-driven morphological instabilities Thus, a moving interfacial boundary is driven by are absent. However, even this is something of an a diffusion field gradient and inhibited by curvature- oversimplification, because a geometric model may related forces. A common feature of such compe- be used to examine diffusion-limited growth if tition between stabilizing and destabilizing influ- the interfacial speed is sufficiently large. (In this ences is the existence of a characteristic length context the appropriate length scale L is essentially scale for the resulting pattern. It is often difficult the diffusivity divided by the interface speed; large to predict the pattern selection principles that op- gradients can exist, and the boundary layer of erate in these systems. Such patterns, as noted thickness L effectively defines the interfacial re- above, are sensitive to the system geometry and ex- gion.) A model is considered geometric if the ternal conditions in general. A valuable review of normal velocity at an interfacial point depends pattern formation models in this context can be only upon the shape and shape-dependent quan- found in [17]. Mathematically, the problem of a tities of the interface. moving and developing interface between two dis- By contrast, nongeometric models pertain to tinct media is a Stefan problem, wherein a nonlin- growth on surfaces that are everywhere rough and ear system evolves dynamically in time. As noted are usually formulated as some type of free bound- in [7], this problem is especially interesting when ary problem. It transpires that such diffusion- the interface motion is a nonequilibrium problem, controlled models of simple geometric shapes such where the configuration is initially such that the free as planes, cylinders, and spheres are commonly energy of the system is not at an absolute minimum. unstable. For example, if a plane solid interface A very useful resource for mathematicians in- develops a small bump extending into the vapor phase or liquid phase, the temperature gradient is terested in the subject of crystal formation is the locally larger than in its immediate neighborhood review in [37]. Therein, an important convex set, (because the isotherms are closer at that point), known as the Wulff shape (or equilibrium crystal latent heat diffuses away more rapidly, and the shape) is introduced. The equilibrium crystal shape bump continues to grow, at least until the stabi- is the shape which minimizes the total surface free lizing effect of surface tension (via the curvature energy per unit area γ(N ) for the volume it en- of the interface) becomes comparable. How does closes, and its boundary Wγ is defined as

APRIL 2005 NOTICES OF THE AMS 409 (1) W ={r : r·N ≤ γ(N )∀N }. 1 γ (4) F = (p+p¨)f (θ)dθ and n = (p+p¨)p(θ)dθ. 2 γ(N ) is orientation-dependent, N is the unit normal vector, and r is the radius vector of any point The total free energy must be minimized subject on the equilibrium crystal surface (see Figure 1). to the constraint n = constant. If the choice of

Wγ is determined uniquely by the Wulff Theorem Lagrange multiplier λ is made such that we can (or construction), one proof of which is given below. define the quantity Another important boundary is the steady-state 1 growth shape W defined by (5) Q(θ,p,p,˙ p¨) = (p + p¨)p − λ(p + p¨)f , V 2 (2) WV ={r : r·N ≤ V(N )∀N }, then the appropriate Euler-Lagrange equation is where V(N ) is the growth rate in the direction N . ∂Q d ∂Q d2 ∂Q For further references to the Wulff construction, (6) − + = 0. see [11]. ∂p dθ ∂p˙ dθ2 ∂p¨ Wulff’s Theorem It follows that In a crystal at equilibrium, the distances of the faces from the centre of the crystal are proportional to (7) (p + p¨) − λ(f + f¨) = 0, their surface free energies per unit area [9]. The proof and notation therein for the two-dimensional with solution case will be followed here. The fundamental idea (8) p(θ) = C sin (θ − θ0) + λf (θ) is to relate the equilibrium shape of the “crystal”

to the polar plot of the surface free energy. This for arbitrary constants C and θ0. By choosing the is done via the Wulff construction (again, see origin in such a way that the crystal has some Figure 1), which is the interior envelope of the set rotational symmetry about it, the choice C = 0 is of perpendiculars to radial rays intersecting the permissible. Then latter [43]. The equilibrium shape is determined = by the requirement that the total free energy for a (9) p(θ) λf (θ) , given crystal “area” is a minimum. which establishes that the polar diagram of the edge We consider a point P(x, y) on the boundary S free energy is proportional to the pedal of the equi- of the crystal; the polar coordinates of P are (r,φ). librium shape of the crystal. Here the arc length parameter will be denoted by It is important to note that the steady-state θ for consistency with [9], but also to emphasize shape of a crystal is not as general as the Wulff that we are dealing with rather general equilib- shape for an equilibrium crystal, because it is not rium boundary curves as opposed to the circular obtained from an initial value problem. Not all ini- ones discussed in the model below. (Note that θ will tial shapes will in fact reach this steady-state shape, be used in two other ways in the models below.) and neither does it indicate what class of geomet- The line segment OM (see Figure 2) is the perpen- ric models results in a convex form from an initial dicular from the origin to the tangent line to S at shape. Further details on this and other aspects of P; it has length p. Denoting the edge free energy the problem may be found in [41]. per unit length of a boundary element by f (θ), the A “Toy” Model for Crystal Growth total free energy F and the area n (related to the As a precursor to the more physically significant number of molecules) are respectively given by models of crystal growth discussed below, let us examine a specific yet simple evolution equation 2 2 1/2 F = f (θ) ds = f (θ) x˙ + y˙ dθ for the interface in a two-dimensional geometrical (3) model and comment on its stability. The model that 1 and n = (xy˙ − yx˙) dθ, follows is not particularly realistic physically— 2 only the lowest order curvature term is retained to where x˙ = dx/dθ, etc. From the figure and stan- exhibit the desired behavior of the boundary—but dard algebra, it must be emphasized that it is merely a mathe- p = x cos θ + y sin θ. matical simile; it cannot be derived from the kinetics of crystal growth for realistic crystals. In fact, a The locus of M as the point P moves around the model of this type is one in which the boundary boundary curve S will be the pedal curve of S. The layer effectively defines the interface: as noted ear- expression above and its derivative with respect to lier, the characteristic length scale L is the ratio of θ can be inverted to obtain the diffusion coefficient to the interfacial speed, and x = p cos θ − p˙sin θ and y = p sin θ + p˙cos θ. this would be true in practice only if the crystal growth speeds were excessively high. Neverthe- Hence less, for applied mathematicians, toy models are

410 NOTICES OF THE AMS VOLUME 52, NUMBER 4 1/2 the aesthetic equivalent of “back-of-the-envelope” (14) r (t) = 2t + r 2 (0) . calculations for engineers! If x is the position vector of a point on the Performing a linear stability analysis around this interface corresponding to outward unit normal solution, following [8] (see also [42]) we consider N , then we assume that radial perturbations only, because tangential terms dx can be eliminated by a suitable gauge transforma- (10) N· = U(x; s,κ), tion (equivalently, they can be shown to drop out dt of the linear stability analysis). Thus where the speed U is a function of the local surface geometry, being dependent on position, and (15) x = ˆr [r(t) + ε(t,θ)] . implicitly on arc length s and curvature κ. What Then form might U take? One choice is [7]: 1 1 ∂2 ∂2κ (16) κ =|κ|= − + 1 ε ≡ κ0 + κε, (11) U = V(κ) + γ . r r 2 ∂θ2 ∂s2 where κ = r −1, and The second term corresponds to the stabilization 0 mechanism for short wavelengths when γ>0. ∂2κ 1 ∂2 ∂ (17) =− + 1 ε. Heuristically we may choose the “curvature ∂s2 r 4 ∂θ2 ∂θ2 potential” V to take the form of a cubic for the following reasons. A planar interface cannot move Since at all (except under very special circumstances), V (κ + κ ) = V (κ ) + κ V (κ ) + O(κ2), so V(0) = 0. The growth rate of a spherical crystal 0 ε 0 ε 0 ε −1 (or circular in two dimensions) behaves like R for the linear perturbation ε(t,θ) may be shown to large R, so we would expect V ≈ ακ for small κ. satisfy the equation Noting further that a solid with large curvature ∂ε V (κ ) ∂2 will contract due to forces of surface tension, U =− 0 + 1 must become negative for some value of κ. Hence ∂t r 2 ∂θ2 (18) there must be a minimum “bubble” size for nu- γ ∂2 ∂ − 3 × ε − + 1 ε. cleation, and this corresponds to a term βκ in r 4 ∂θ2 ∂θ2 V, where β is related to the minimum size for nucleation. A quadratic term, µκ2, is also included in Seeking eigenfunctions of the form the expression for V to account for an asymmetry σmt (19) ε(t,θ) = εme cos mθ, in the solidification process between the freezing and the remelting of the dynamic interface, so we we see that the linear growth rate write m2 − 1 1 γm2 (12) V(κ) = ακ + µκ2 − βκ3. (20) σ = V − . m r 2 r r 2 Stability Criteria The m = 1 mode corresponds to a uniform rota- In terms of the unit tangent vector T to the curve tion of the circle and is a neutrally stable pertur- at a point with polar angle θ (s being arc length bation. As noted in [8], the first term is stable for along the boundary curve), the curvature vector κ all m (≠ 1) if V (1/r) > 0, and since for sufficiently is variously defined as large radii V ∼ κ, this will occur even for nonzero κ = dT /ds = (dT /dθ)/(ds/dθ) µ and β in the complete expression for V. This simple result is the analog of the Mullins-Sekerka in- = (dT /dθ)/ |dx/dθ| , stability as discussed in [29], [30] (see also [10], [31], where x (θ) is the position vector. For a circle with [46]). The second term is always stabilizing for time-dependent radius r(t) and the special case of m>1 (with γ playing a role analogous to that of U = V(κ) = κ (with the unit of length chosen to fix surface tension), so this will act to limit the range α = 1 [7], [8]), the equation of m-values that are in the unstable regime. More Sophisticated Models dx N· = U(x; s,κ) Based on more detailed theoretical physics, how dt might the interface evolution of a crystal be char- reduces to acterized mathematically? Following the approach dr 1 in [42], we consider the boundary of a two- (13) = V , dt r dimensional crystal to be represented by the closed curve C [x (u, t) ,y(u, t)] in the plane, with time de- with solution pendent components parametrized by the variable

APRIL 2005 NOTICES OF THE AMS 411 u. Let the arc length s be related to u via the rela- (25) tion ∂κ (s,t) ∂2V ∂κ s =− + κ2V − κVds. 2 u ∂C(u,t) ∂t s ∂s ∂s 0 = (21) s (u, t) du . 0 ∂u As pointed out in [8], when the curve C is parametrized by θ rather than by u or s, then the evolution Further, let W =|∂C(u, t)/∂u|=ds/du. θ is now equation for κ becomes strictly local. This is a defined to be the angle between the positive x-axis consequence of the fact that its rate of change at and the unit tangent vector T=(cos θ,sin θ) = fixed θ is given by −1 C N W ∂ (u, t)/∂u, and the unit normal vector is inward pointing. The boundary evolves in terms of ∂κ ∂κ ∂θ ∂κ ∂κ (26) = − ≡ . the normal velocity function V (θ,δµ) according to ∂t θ ∂t u ∂t ∂θ ∂τ ∂C (22) =−VN . Thus ∂t u ∂κ (27) =−κ2V˜ (θ) , The initial-value problem with V defined as above ∂τ can be solved exactly by the method of character- ˜ = + istics, since the normal velocity of the interface de- where V V V is named the “velocity stiffness” pends explicitly on the surface orientation alone. in [42]. These results also can be derived by con- sidering the curve as a set of complex numbers in They are rays of the form x(t) = x0 + d (θ0) t , x0 the plane [8]. For finite V˜ this equation has solu- being a point on the initial curve, and V (θ0; x0) is tion in the direction d (θ0) . The surface normal direc- κ (0) tion is preserved along each characteristic, and so (28) κ = , the curve C at time t is defined by the set of all 1 + κ (0) Vτ˜ points x(t). ˜ To discuss the curvature evolution, both global in terms of the initial curvature κ (0) . If V>0, the and local, we require that the two appropriate curvature will decrease monotonically in time ≠ Frenet equations for the unit tangent (T ) and nor- at nonfaceted orientations for which κ (0) 0. ˜ mal (N ) vectors at a point on the boundary curve For orientations in which V< 0, the curvature −1 are diverges at finite time, τ =− κ (0) V˜ . In fact, ∂T ∂N as discussed below, a shock (intersection of = κN and =−κT , ∂s ∂s characteristics) develops before the minimum blow-up time. (κ = ∂θ/∂s). If arc length is used to parametrize In defining a local normal velocity with reference these quantities in terms of the closed interfacial to a weak “growth drive” δµ (the chemical poten- boundary curve C [42], then tial difference between the surface and external ∂C ∂2C(s,t) phase), two different kinetic processes are merged T (s,t) = ,κ(s,t) = , ∂s ∂s2 via a convexity type of argument involving a tran- (23) sition function ξ (θ) to form an expression for the ∂2C(s,t) and N (s,t) = κ−1(s,t) . local normal velocity of a crystal with n-gonal sym- ∂s2 metry. This is Following [13] and also [42], we can prove the fol- (29) lowing identities: V (θ,δµ) = Vf (δµ) ξ (θ) + Vr (θ,δµ) 1 − ξ (θ) , ∂W ∂ ∂ ∂ = κVW; , =−κV , where ∂t ∂t ∂s ∂s −πσ2 (30) V (δµ) = c g (δµ) exp , (24) f f kT δµ ∂T ∂V ∂θ ∂V =− N ; =− . and ∂s ∂s ∂t ∂s nθ (31) V (θ,δµ) = c δµ 1 + cosp , Since W is essentially a measure of the length of r r 2 an infinitesimal displacement on the boundary, the differential equation for W represents its rate where p is an even integer. Vf (for facet) is the nor- of dilation. The corresponding equations in T and mal growth rate at facet orientations under the θ describe the rotation rates of those quantities, “growth drive” δµ, while Vr (for rough) describes as determined by the anisotropy of V. After some the normal interfacial motion at vicinal and mole- subtle reformulation (involving gauge invariance ar- cularly rough orientations (due to surface migra- guments), a nonlocal integrodifferential equation tion of adsorbed molecules away from facets). for the curvature evolution can be derived, namely: Since 0 ≤ ξ (θ) ≤ 1, this function is a measure of

412 NOTICES OF THE AMS VOLUME 52, NUMBER 4 the distribution between facetlike and roughlike vature at rough orientations. Further, if t0 is the time = growth. It is periodic in 2π/n, and ξ θf 1, of corner formation, the facet collides with the ξ θf + π/n = 0 for a facet orientation θf . In [42] shock at t ≈ 1.1t0 almost immediately after shock the choice of ξ (θ) = cosm (nθ/2) , m ≥ p being formation, consistent with observations [40]. even, is made. If in particular θ = π/4 and n = 4 Green’s Functions in the definition of V˜ , then for m = p = 2, The literature on this subject abounds with exam- V˜ = cr δµ + 8Vf , which implies that the rate of cur- ples of the application of Green’s function tech- vature decrease increases with δµ. In short, the niques to the growth and development of crystals polygonalization of crystals corresponds to de- and might well be a useful source of examples for creasing curvature in rough orientations. graduate classes. This particular analysis is based Thus, in general, a crystal has slowly growing on the approach in [40] (but see also [25]). In con- molecularly smooth faces (facets) and more rapidly nection with shocking facets, the temperature field growing faces that are rough at the molecular level. T around an expanding surface with a constant den- At this level, both structures are sensitive to tem- sity of heat sinks is computed using Green’s func- perature, as one might expect. Facet locations are tions for the two-dimensional diffusion problem. associated with minima of the surface free energy, The expression for the field at (x, z; t) is and indeed a crystal shape may be only partially u(x, z; t) = u0(x, z) faceted, but a major question for crystal growth is t ∞ ∞ to find what processes are responsible for the (34) − dt G (x − x ,z− z ; t − t ) evolution of an initial “seed crystal” towards a 0 −∞ −∞ faceted asymptotic growth shape. This is called ×Q(xz; t)dxdz, global kinetic faceting and has been observed to occur experimentally and numerically [26] and has where u = (T − T∞) /T∞ , T∞ being the melt tem- been predicted theoretically [42]. In this process, perature far from the interface. The temperature the rough orientations of partially faceted shapes field before the formation of a shocking facet is u0, grow out of existence with decreasing curvature. and the standard Green’s function for the diffusion Such a decrease implies the presence of disconti- problem nuities in surface slope (or a jump in the normal ∂ to the surface), so it is of considerable interest (35) − k∇2 G(x, z; t) = δ(x)δ(z)δ(t) to be able to tie the time-dependence of the local ∂t curvature to the ultimate shape of the crystal is boundary. Mathematically, discontinuities of this 1 x2 + z2 kind are associated with both singularities in (36) G(x, z; t) = exp − , 4πkt 4kt the interfacial curvature and the intersection of characteristics [39]. The latter viewpoint, while k being the thermal diffusivity of the melt. The complementary to the former, corresponds to the source term is existence of shocks in the solution of the surface evolution equation. The dynamics of such shocks, (37) Q(x, z; t) = qH(υt − x)H(x)δ(z), when they exist, are of great importance in the in terms of the Heaviside step-function H and the interpretation of experimental data. Furthermore, Dirac delta function δ (remember that distribution it has been shown in [41] that if the initial seed theoretic distinctions and subtleties are usually crystal is convex, then convexity is preserved suppressed in much of the scientific literature). throughout the whole growth process. Forms of The term q is a heat flux constant, proportional V that possess cusps at faceted orientations to the surface density of heat sinks. After some also explain the formation of expanding facets rearrangement separated from rough orientations by a corner, q t e−z2/4k(t−t ) called shocking facets in [40]. Specifically, in place − =− (38) u(x, z; t) u0(x, z) dt I, of expression (29) for V (θ,δµ) , now 4k 0 t − t where = + (32) V (θ,δµ) Vf (δµ) ξ (θ) Vr (θ,δµ) , −α+η = − −n2 where I 4k(t t ) e dn, −α 2 (33) Vr (θ,δµ) = cr δµ a |sin 2θ|−b sin 2θ , α = x/ 4k(t − t ), where the coefficients a and b are related to cer- η = υt/ 4k(t − t). tain angular rates of change. It transpires in this √ case that models with the parameter range Experimentally, the authors found that υt 4kt. 2b

APRIL 2005 NOTICES OF THE AMS 413 − 2 Personally, I have been both pleasantly surprised I ≈ η 4k(t − t)e a ; and pleased to encounter such a broad swath of whence thoughtfully written papers addressing crystal growth. I hope that this review will be a useful (but clearly not exhaustive) resource for readers wish- u(x, z; t) − u0(x, z) ≡ u˜(x, z; t) ing to pursue the subject further. However, a caveat (39) qυt ∞ e−λ ≈− dλ. is in order: a review of this nature must maintain 4k (x2+z2)/4kt λ a balance between material of interest to mathe- Noting that the exponential integral [6] is defined maticians while at the same time doing justice to as the physics of the problem. Inevitably, the simple ∞ e−λ models and oversimplified physical descriptions E1 (x) = dλ θ λ presented here will not satisfy crystal theorists, but (40) ∞ + (−1)n 1 θn as in all topics on the interface between two mag- =−ln θ − γ + , n · n! isterial disciplines, compromises must be made. n=1 This article is no exception, and I will end it with where γ is the Euler-Mascheroni constant, it follows a warning: The mathematical physics of snow crys- that when (x2 + z2)/4kt 1 , tal formation is much more complicated than might be inferred from the discussion presented here! qυt 4kt (41) u˜(x, z; t) ≈− −γ + ln . 4k x2 + z2 Acknowledgments I am very grateful to Professor John S. Wettlaufer, At the other extreme, for short times such that who has not only provided me with several of his 4kt x2 + z2, u˜ is exponentially small. Using ex- papers and constructively reviewed an early draft perimental data, Tsemekhman and Wettlaufer [40] of this manuscript but has also patiently, thought- estimate that the geometric model begins to fail at fully, and enthusiastically responded to my ques- t ≈ 10s . From this and related observations, the authors conclude that geometric models of the en- tions and comments during its preparation. I’m tire interface are very accurate up to about t = 15s. also grateful for Bill Casselman’s creative artwork. Ultimately the model breaks down when q is no References longer constant, and nonlinear effects become significant. Studies such as this one confirm that geo- [1] J. A. ADAM, Like a bridge over colored water: A math- metric models are very useful in describing the ematical review of The Rainbow Bridge: Rainbows in Art, Myth and Science, Notices of the AMS 49 (2002), early stages of the evolution of a crystal surface. 1360–71. Some of these models predict the possible forma- [2] ——— , Mathematics in Nature: Modeling Patterns in the tion of a shocking facet and are still applicable for Natural World, Princeton University Press, Princeton, some time beyond this point. Indeed, according to NJ, 2003. Wettlaufer [private communication], even the [3] P. BALL, The Self-Made Tapestry, Oxford University timescales cited above do a great injustice to the Press, Oxford, 1999. utility of the geometric approach: for solid helium [4]——— , H2O: A Biography of Water, Phoenix, London, growing from a superfluid, the geometric model is 1999. the only model of significance, and it never breaks [5] A. BEJAN, Shape and Structure, from Engineering to down! Nature, Cambridge University Press, Cambridge, 2000. [6] N. BLEISTEIN and R. A. HANDELSMAN, Asymptotic Expan- Summary sions of Integrals, Dover Publications, Inc., New York, There are many other mathematical aspects of 1986. crystal formation that have been investigated over [7] R. C. BROWER, D. A. KESSLER, J. KOPLIK, and H. LEVINE, Geo- the last several decades. It is a field rich in theory, metrical approach to moving interface dynamics, Phys. experiment, and applications (particularly inter- Rev. Lett. 51 (1983), 1111–4. esting, for example, but outside the scope of the [8] ——— , Geometrical models of interface evolution, Phys. Rev. A 29 (1984), 1335–42. present review, is the asymptotic analysis found in [9] W. K. BURTON, N. CABRERA, and F. C. FRANK, The growth [44], [45]). Interesting theoretical papers on dendrite of crystals and the equilibrium structure of their sur- formation are presented in the set [20]–[24], [27], faces, Phil. Trans. R. Soc. A 243 (1951), 299–358. [28]. In addition, many theoretical developments are [10] A. A. CHERNOV, Stability of faceted shapes, J. Cryst. to be found outside the “standard” applied math- Growth 24/25 (1974), 11–31. ematical literature: some of the papers cited below [11] M. ELBAUM and J. S. WETTLAUFER, Relation of growth and are published in the Journal of Crystal Growth, equilibrium crystal shapes, Phys. Rev. E 48 (1993), and Physical Review E, for example, so it is en- 3180–3. cumbent on those of us interested in entering a field [12] T. FUJIOKA and R. F. SEKERKA, Morphological stability such as this sometimes to walk in pastures new. of disc crystals, J. Cryst. Growth 24/25 (1974), 84–93.

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STEVENS, Patterns in Nature, Atlantic-Little, Brown, Other considerations suggest the role of ran- Boston, 1974. domness. It is easy enough to construct dendrites [33] I. STEWART, Nature’s Numbers, Basic Books, New York, in software that, at least locally, look somewhat like 1995. snowflakes. The simplest process in which this [34] ——— , What Shape Is a Snowflake? W. H. Freeman & happens in a physical manner is diffusion-limited Co., New York, 2001. aggregation, or DLA, in which particles wander [35] W. TAPE, Folds, pleats and halos, Amer. Scientist around randomly and stick irreversibly onto a cen- (1982), 467–74. tral, growing core whenever they hit it. [36] W. TAPE and G. P. KÖNNEN, A general setting for halo theory, Appl. Opt. 38 (1999), 1552–625. Diffusion-Limited Aggregation [37] J. E. TAYLOR, J. W. CAHN, and C. A. HANDWERKER, Geo- metric models of crystal growth, Acta Metall. Mater. Particles were initially randomly distributed on a 40 (1992), 1443–74. hexagonal lattice with a single particle in the

APRIL 2005 NOTICES OF THE AMS 415 which means that they possess a similar structure at all scales. In snowflakes surface tension forbids this and in any realistic simulation must be taken into account. The combined effect of surface tension and DLA is to establish a certain characteristic length involved in the growth process. Up until about 1985 all attempts to simulate snowflakes built the symmetry in by constructing one arm and then reflecting and rotating it. But then Nittman and Stanley showed that a truly random process could in fact give rise to a good approxi- mate symmetry without artificial forcing. Their process was an extension of DLA in which sites next to the growing core freeze only when they encounter several diffusing free particles. This requirement is already one that assures a certain stability in the process and leads to quasi-symmetry but gives rise only to a very restricted range of snowflake shapes. In addition, they introduced Diffusion-Limited Aggregation. Particles were a somewhat artificial parameter that generates initially randomly distributed on a hexagonal thicker dendritic arms. With this, quasi-symmetry lattice with a single particle in the central core. is maintained and a wider range of shapes arises. The paths show the motion of the particles Unfortunately, their construction seems to have lit- wandering in randomly from the boundary of tle to do with physics. the region to attach to the core. A much more physically realistic account was central core. The paths show the motion of parti- that of Yokoyama and Kuroda (of Hokkaido Uni- cles wandering in randomly from the boundary of versity). Their simulations involved a fairly de- the region to attach to the core. tailed account of actual surface motion of molecules The characteristic feature of DLA is that bump in ice formation, as well as a detailed analysis of growth is unstable: high curvature means a high how diffusion behaves at corners. Although ex- gradient in the distribution of particles, which are tensive dendritic growth does not occur in their therefore attracted to bumps. This instability com- models, it does not seem too far away. bined with microscopic randomness presumably One recent attempt at imitating snowflakes has does play a role in snowflake growth. been done with a 2D cellular automaton by Clifford Reiter. The only randomness built into his con- Diffusion Instability struction is in the initial state, however, and al- Bumps attract diffusing particles, which follow the though some of his figures do resemble snowflakes, gradient of a certain potential field. As it grows, the it is hard to see what this process has to do with gradient increases, so a bump grows unstably. real life. References P. 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 real snowflakes, 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 Instability. Bumps attract diffusing particles, which follow the gradient of a certain potential field. As it grows, the gradient increases, so a bump grows unstably.

The results of DLA do not look much like snowflakes. One simple explanation for this is that dendrites formed in a DLA process are fractal,

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