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.

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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.

The researchers' findings reveal a previously unknown mathematical relationship

"Ground states" are the lowest energy states of matter. They are created in real life by taking a liquid and slowly cooling it until you reach absolute zero temperature. The resulting arrangement of molecules or particles is a "ground state," which often is an ordered crystal structure. This image shows an ordered ground state (called the face-centered cubic lattice), which is the end result of the slow cooling process

Materials' Crystal Properties Illuminated By Mathematical 'Lighthouse'
A deeper fundamental understanding of complex materials may now be possible, thanks to a pair of Princeton scientists who have uncovered a new insight into how crystals form.
The researchers' findings reveal a previously unknown mathematical relationship between the different arrangements that interacting particles can take while freezing. The discovery could give scientists insight into the essential behaviors of materials such as polymers, which are the basis of plastics.

Molecules in a material cooled to absolute zero can take on a multitude of different configurations. Historically, scientists' difficulty with identifying crystallized molecules' spatial arrangements from this high number of possible configurations has blocked theoretical efforts to understand these materials' qualities, but the new findings could offer the tool that science needs.

"We believe our 'duality relations' will be a useful theoretical tool to understand how individual particles come together to form a crystal," said Salvatore Torquato, a professor of chemistry who co-wrote the paper with senior chemist Frank Stillinger. "If we can tune the interactions among particles that form a crystal, we might be able to create materials that respond to light or mechanical stress in novel ways."

A material that maintains its exact size and shape through extremes in temperature, for example, might be valuable in the manufacture of orbiting space telescopes, whose mirrors need to retain their shape as they pass from sunlight into the Earth's shadow.

A crystal is the state of matter that is easiest to analyze because its frozen molecules are motionless and often regularly organized. A crystal's properties -- its ability to bend light, for example -- generally reveal valuable information about how its constituent molecules will behave at higher temperatures, such as when they become a liquid.

The challenge is that many complex materials can crystallize into a multitude of different structures. When a substance is cooled to nearly absolute zero, and it can take on an enormously large number of possible "ground states" -- the term for the molecular arrangement with the lowest possible energy. Scientists seek to determine the true ground state because it provides a fundamental understanding of matter in the solid state and its possible uses. However, determining which molecular pattern is the true ground state requires mathematical proof that is hard to come by.

"We resort to approximations," said Christos Likos, a professor of theoretical physics at the University of Dusseldorf in Germany. "They help us produce meaningful results sometimes, but we need to have a lighthouse occasionally to show us we're on the right path. Such lighthouses are rare in this business, but Sal and Frank have found one."

Torquato and Stillinger's findings explore particles' behavior as they attract and repel each other over varying distances. By analyzing this behavior, the scientists were able to conceive a precise mathematical correspondence -- called duality relations -- between possible arrangements of particles. The work will enable the researchers to draw important conclusions about how particles at very low temperatures interact over great distances, a situation that is very difficult to treat theoretically.

"Once ground states can be determined and controlled with certainty, scientists might create materials with properties virtually unknown in nature," Torquato said.


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Crystal quality and optical properties for Nd:GdVO4 single crystals by a floating zone method
Abstract
Single crystals of neodymium-doped gadolinium orthovanadate (Nd:GdVO4) were successfully grown by the floating zone method (FZ) with an infrared convergence type heater. These crystal quality and optical properties are sufficient for application to a laser host. Distribution of (Nd/Nd + Gd) concentration by 50 mm in the growth direction was 0.20 ± 0.02 against a target concentration of 0.20. Distribution of extinction ratio at 633 nm by 50 mm in the growth direction was 51 ± 2 dB. Laser performance for these crystals with a-cut were measured. The maximum slope efficiency was 65% and the maximum output power was 3.4 W with continuous wave at 808 nm.