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Saturday, January 19, 2008

Three-dimensional snowflakes

Math Models Snowflakes In Extraordinary Detail
Three-dimensional snowflakes can now be grown in a computer using a program developed by mathematicians at UC Davis and the University of Wisconsin-Madison.

No two snowflakes are truly alike, but they can be very similar to each other, said Janko Gravner, a mathematics professor at UC Davis. Why they are not more different from each other is a mystery, Gravner said. Being able to model the process might answer some of these questions.

Intricate, incredibly variable and beautiful, snowflakes have been puzzling mathematicians since at least 1611, when Johannes Kepler predicted that the six-pointed structure would reflect an underlying crystal structure.

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.

The model built by Gravner and David Griffeath of the University of Wisconsin-Madison takes these 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.

As in the real world, needles are the most common pattern of computer-generated snowflake. The classic six-pointed "dendritic" or feathery snowflake is relatively rare, both in the computer simulation and in nature.

Gravner and Griffeath also managed to generate some novel snowflakes, such as a "butterflake" that looks like three butterflies stuck together along the body. Gravner said there seemed to be no reason these shapes could not appear in nature, but they would be very fragile and unstable.

One surprise was that three-dimensional structure is often important, with complex structures often growing between two plates -- a feature that is difficult to see when observing actual snowflakes, but has been observed in careful studies of real snowflakes with electron microscopes.

Koch Snowflake
Snowflakes are amazing creations of nature. They seem to have intricate detail no matter how closely you look at them. One way to model a snowflake is to use a fractal which is any mathematical object showing "self-similarity" at all levels.

The Koch snowflake is constructed as follows. Start with a line segment. Divide it into 3 equal parts. Erase the middle part and substitute it by the top part of an equilateral triangle. Now, repeat this procedure for each of the 4 segments of this second stage. See Figure 1. If you continue repeating this procedure, the curve will never self-intersect, and in the limit you get a shape known as the Koch snowflake.

Amazingly, the Koch snowflake is a curve of infinite length!

And, if you start with an equilateral triangle and do this procedure to each side, you will get a snowflake, which has finite area, though infinite boundary!

Presentation Suggestions:
Draw pictures. If they like this Fun Fact, ask them: can you figure out how to construct a 3-dimensional example? [Hint: start with a regular tetrahedron. See Koch Tetrahedron for what happens.]

The Math Behind the Fact:
You can see that the boundary of the snowflake has infinite length by looking at the lengths at each stage of the process, which grows by 4/3 each time the process is repeated. On the other hand, the area inside the snowflake grows like an infinite series, which is geometric and converges to a finite area! You can learn about fractals in a course on dynamical systems.

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