Science News - December 23, 2006

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 models 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 approxima- ON A PEDESTAL — This three-dimen- tions that mar the sional, simulated ice crystal resembles a resulting patterns— common, simple type of natural for instance, by gen- snowflake: a hexagonal column whose erating snowflake faces are indented because they grow shapes that are unre- more slowly than the column’s edges. alistically asymmetrical.

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 cur-
rently appear on U.S.
postage stamps.

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 1s and 0s 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.