What a flake: computers get the hang of ice-crystal growth.

With a camera-equipped microscope of his own making, Kenneth G. Libbrecht shoots some of the world's most stunning photographs of snowflakes. Since October, four of the physicist's images have adorned U.S. postage stamps. Each stamp displays an exquisitely intricate, burst-shaped crystal that, because of the photographer's distinctive lighting, glows like polished metal.

Scientists have pondered such enchanting patterns of snowflakes for at least 400 years. In the early 1600s astronomer Johannes Kepler wrote a short article in which he puzzled over the extraordinary six-sided symmetry of the crystals. Soon afterward, mathematician-philosopher Rene Descartes jotted down some of the first written descriptions of snowflakes, also expressing awe at their perfect symmetry. Robert Hooke, one of the first scientists to use a microscope, observed and drew many snowflakes.

The fascination continues. Some scientists are examining snowflakes to learn about the physics of ice crystallization, while others are investigating the relationships between ice-crystal growth and properties of clouds and snow. Despite centuries of scrutiny, however, no one can fully explain snowflake shapes.

Besides hunting for exquisite snowflakes at locations around the world, Libbrecht grows snowflake crystals under controlled conditions in his laboratory at the California Institute of Technology in Pasadena. Such experiments are extraordinarily tricky, says Libbrecht, whose research over 2 decades has ranged from solar physics to gravitational waves.

Other researchers have lately made strides in replicating snowflake growth, using computers to simulate water vapor diffusion and other processes that control ice crystallization. In some of the new work, a team of mathematicians has simulated the stages of snowflake growth photographed by Libbrecht and other experimenters.

Despite the authentic look of the computer-generated patterns, scientists can't yet predict the effects of temperature, humidity, and other factors on the shape of a specific snowflake. "We all have guesses, but no one has come up with a theory that's robust and accounts for all the possibilities" says meteorologist Dennis Lamb of Pennsylvania State University in University Park.

Nonetheless, the new patterns' verisimilitude suggests that scientists are on-target. "Once you find the right paradigm," says mathematician David S. Griffeath of the University of Wisconsin-Madison, "modeling has the potential to explain things that have eluded people for a long time."

MYSTERIOUS BEAUTY The road to snowflake knowledge has always been slippery, whether scientists were working to measure snowflakes or to simulate their alluring patterns electronically.

Fragile, too tiny to be easily manipulated, and likely to melt because of the warmth in an observer's breath, ice crystals make less-than-ideal experimental subjects. What's more, nearby surfaces or crystals alter the flakes' patterns of growth, as do slight changes in many factors, including temperature, humidity, wind, impurities, motion, and sunlight.

Scientists have had to devise elaborate means to study snowflake growth without perturbing it. In experiments started in the 1930s, for instance, Japanese physicist Ukichiro Nakaya grew crystals on rabbit hairs, strands of spider-web, or other filaments. He observed that slightly different temperatures and humidities lead to radically different outcomes, such as long needles instead of thin flakes.

More recently, Libbrecht and his colleagues have used electric fields to make ice needes whose tips, when the voltage was then shut off, could serve as platforms for nearly pristine-snowflake growth (SN: 7/11/98, p. 23).

Some investigators of cloud physics, such as Lamb, suspend developing flakes in their labs by means of air currents or electrostatic forces.

Other researchers fly through clouds, catching new crystals in tubes jutting from aircraft. Cloud researchers examine not only the ultimate crystal shapes but also how rapidly snowflakes grow--a factor that affects the balance among frozen, liquid, and gaseous forms of water within clouds. That balance, in turn, plays a role in how clouds influence climate and in the likelihood that aircraft flying through clouds will accumulate dangerous ice coatings on their wings and other surfaces.

SIMPLE SIMULATIONS Investigators simulating crystal growth on computers face the challenge of determining whether their programs are generating authentic snowflake patterns.

The software entrepreneur and scientific maverick Stephen Wolfram recently reasserted claims made by him and others in the 1980s that simple computer algorithms, called cellular automata, can create realistic snowflake shapes.

A cellular automaton generates a pattern by coloring each location on a grid according to a rule that takes into account the colors of neighboring locations. For snowflake simulations, such computer programs operate on a honeycomb because ice crystals, considered at the molecular level, are made up of water molecules arranged in hexagons.

Elementary rules can make authentic snowflake patterns on a honeycomb grid, Wolfram says. One rule, for example, states that a location should be made black when it has one and only one neighboring location that's already black. With such simple rules, "it is actually quite easy to reproduce the basic features of the overall behavior that occurs in real snowflakes,' Wolfram said in A New Kind of Science (2002, Wolfram Media). In that book, he promoted cellular automata as an alternative to conventional mathematical tools for a wide range of scientific problems (SN: 8/16/03, p. 106).

However, some snowflake-simulation specialists don't accept the snowflake patterns from such rudimentary cellular automata as being realistic. Although the results are "snowflakelike," the models ignore nearly all the underlying physics, says mathematician Clifford A. Reiter of Lafayette College in Easton, Pa.

Griffeath, too, dismisses the authenticity of such patterns. "We came to the conclusion that these cellular-automata models had nothing to do with the way snowflakes grow" he says, referring to a recent review that he performed with fellow mathematician Janko Gravner of the University of California, Davis.

Since the early 1990s, researchers have also created computer nodels of snowflake growth that use partial differential equations to represent physical processes. However, those models have hit snags, Griffeath says.

In some models, the computations produce simplistic crystals that lack the intricate features that are typical of so many snowflakes, such as bristly, elaborate side branching. In others, the equations inadequately represent the physical processes or require approximations that mar the resulting patterns--for instance, by generating snowflake shapes that are unrealistically asymmetrical.

GET REAL Snowflake simulations recently entered a new phase. A few years ago, Reiter began devising a way to mimic ice-crystal growth by means of cellular automata that use ranges of numbers, rather than just the Is and Os typical of simpler cellular automata, to characterize grid cells. He reports using such "fuzzy" automata to simulate snowflake growth around hexagonal seed crystals. Replicating a process that occurs in clouds, the diffusion of water vapor controlled where and when new hexagons of ice would be added to the growing crystal.

Reiter described his method in the February 2005 Chaos, Solitons, and Fractals. (To see animations of such snowflake growth, go to ww2.lafayette.edu/~reiterc/mvp/sfn/.) The new approach has fared extraordinarily well at replicating the look of some types of snowflakes, Griffeath says.

Building on Reiter's innovation, Griffeath and Gravner have now used yet another variation on cellular automata, known as a coupled-lattice map, to model snowflakes. The approach avoids the breakdowns that plague models based on partial differential equations, Griffeath says.

The latest algorithm uses more-complex rules for choosing when to add ice to the crystal than the prior automaton models did. Consequently, it simulates an array of physical processes affecting ice crystals, not just the water vapor diffusion that Reiter included.

Before deciding whether the water vapor at a location should add another morsel of ice to the expanding flake, the algorithm deduces from the pattern on the grid, for example, whether a location on the ice crystal's edge sits in a pit, on a protrusion, or at a straight boundary. In doing so, it incorporates the delicate balance observed in real snowflakes between growth processes that create branches and processes that preserve the expanding crystal's smooth edges.

"It's the tension or battle between those two forces that makes for all [the variety in flake] morphology" Griffeath says.

Among other realistic touches, the new algorithm includes a process for tracking reversible conversions between ice and vapor, which take place in the evolution of bona fide snowflakes.

To overcome the gaps in knowledge about snowflake formation, the model includes seven adjustable settings that enable the researchers to control the rates and thresholds in their simulated processes. One setting introduces a little randomness into crystal growth, causing flakes to be slightly imperfect--as real ones are.

Information on the snowflake simulations by Griffeath and Gravner, including software for the model and animations that show it in action, are available at psoup.math.wisc.edu/Snowfakes.htm.

The new model's inventors had previously simulated the growth of idealized crystals of an unspecified material. When they began working on snow, they turned to Libbrecht to learn about aspects of ice-crystal physics specific for the natural flakes.

"These guys are making some real headway," Libbrecht says. "They get things that look remarkably like real snowflakes"

The most striking signs of that authenticity, Griffeath says, show up in comparisons between intermediate stages of simulated snowflake growth and of lab-grown crystals filmed by Libbrecht and others. The similarity suggests that the simulations replicate not just the ultimate appearances of the snowflakes but also the processes that yield those final shapes.

The model represents snowflakes as two-dimensional objects.

The next frontier for the model is the third dimension, Griffeath says.

Reiter has already expanded his snowflake model into three dimensions. The 3-D simulations produce snowcrystal forms like those observed in nature, including some that hadn't turned up in the two-dimensional models, report Reiter and Chen Ning of Shenyang Jianzhu University in China in an upcoming Computers and Graphics.

Among the newly produced forms are bars with no branches or other daborations. However, when the scientists tried to make the model yet truer to the physics of ice crystallization, the simulations generated patterns that didn't grow, but rather oscillated between two forms,

In preliminary versions of their 3-D model, Griffeath and Gravner also report seeing elongated shapes without branches. The team is still working to produce images of branched snowflakes, and Griffeath says that initial results look promising.

The 2-D model "turned out better than we could possibly have hoped for" he says. Given that the 3-D snowflake model is "closer to the physics," Griffeath adds, it should"do even better at showing how these things grow."

MATHEMATICAL LIKENESSES--A new type of computer simulation of snowflake growth generates patterns (at top on facing page and below on this page) that closely match the shapes of two-actual, photographed snowflakes (at bottom on facing page and right on this page). The model that produced the mathematical structures predicts whether water vapor will freeze at any given location, The photos of real flakes currently appear on U.S. postage stamps.

ON A PEDESTAL--This three-dimensional, simulated ice crystal resembles a common, simple type of natural snowflake: a hexagonal column whose faces are indented because they grow more slowly than the column's edges.

UNDER CONSTRUCTION--AS a real ice crystal grows (left to right in top row), competing factors promote branching or build smooth edges. A virtual crystal created by an algorithm that simulates those factors exhibits comparable stages of development (left to right in bottom row).