Snowflakes or 3-D snowfakes?
Snowflakes have been puzzling mathematicians for about four centuries. Still, scientists have never been able to fully explain snowflake shapes. For example, is this true that their six-pointed structure reflect an underlying crystal structure? Now, two U.S. mathematicians have developed software that simulates 3-D snowflakes. And they discovered that even 'no two snowflakes are truly alike,' they're very similar to each other. In fact, the real mystery is why they are not more different from each other. But read more to discover beautiful images...
You can see above four examples of snowfakes created by two mathematics professors, David Griffeath of the University of Wisconsin-Madison and Janko Gravner, at UC Davis, but who got his PH.D. in Wisconsin. (Credit: Janko Gravner and David Griffeath) Here are two links to Janko Gravner's profile and to David Griffeath's Primordial Soup Kitchen website. Both researchers are focused on self-organization of random cellular automata.
Snowflakes are fragile and hard to look at because 'of the warmth in an observer's breath.' But how do they grow up in the first place? "Snowflakes grow from water vapor around some kind of nucleus, such as a bit of dust. The surface of the growing crystal is a complex, semi-liquid layer where water molecules from the surrounding vapor can attach or detach. Water molecules are more likely to attach at concavities in the crystal shape."
Simulating the growth of snowflakes on a computer is a good exercise, but how is it possible to know if the simulations are really generating snowflakes patterns? "The model built by Gravner and Griffeath of the University of Wisconsin-Madison takes the [above] factors, as well as temperature, atmospheric pressure and water vapor density, into account. By running the model under different conditions, the researchers were able to recreate a wide range of natural snowflake shapes. Rather than trying to model every water molecule, it divides the space into three-dimensional chunks one micrometer across. The program takes about 24 hours to produce one 'snowfake' on a modern desktop computer."
A full day on a desktop computer? Don't these researchers have access to clusters.
Anyway, if you're interested by snowflakes and crystal growth, you should read "What a Flake," published by Science News in December 2006. Here is how the author, Peter Weiss, described how cellular automata were used to simulate snowflakes. "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."
But Griffeath and Gravner thought that "snowflake patterns from such rudimentary cellular automata as being realistic." So they "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."
You can find images, movies and scientific papers on the Snowfakes. The images above have been extracted from a technical paper called "Modeling Snow Crystal Growth III: three-dimensional snowfakes" (PDF format, 40 pages, 9.59 MB). Be sure to read it because it contains many other beautiful images of "snowfakes.'
Sources: UC Davis News, January 16, 2008; Peter Weiss, Science News, Volume 170, Number 26/27, December 23, 2006, Page 408; and various websites
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