Snowflakes six out of---several algorithms generated by the snowflake image, and wish everyone Christmas Eve and Happy Xmas

Today and Tomorrow is Christmas Eve, and a few pictures of snowflakes should be under the festival. It was intended to write an academic paper on algorithm-generating snowflakes, but found it difficult to write down. The main is not to find a raise the mathematical formula of the paper, so send a blog it. There are 10 PNG images and 3 gif animated images, all of which are two-valued images in black and white.

Snowflake is a typical chaotic fractal object, which can embody the unification of rules and randomness. The rule is that every snowflake is hexagonal, at least hexagonal in shape. And every molecule that makes up a snowflake is randomly chaotic. There is no way to determine where a water molecule is located in a snowflake after a snowflake is generated.

Snowflakes are the crystallization of water molecules, and water molecules are made up of an oxygen atom and two hydrogen atoms. The strict internal arrangement rules make the snowflake form the geometric pattern of the rule. And the nature of this great mathematician and artist, but also in the rules of the infinite variety of novel changes, so that the snowflake slices different, blossoming beautiful. Each piece of snow is both axis symmetry and center symmetry. Snowflakes are a natural blend of mathematics and art, one of the most common and simple, but most beautiful things.

Snowflakes have a variety of forms, but each snowflake is hexagonal, the shape of snowflakes, involving water in the atmosphere of the crystallization process. When water molecules in the atmosphere are cooled below freezing, they begin to sublimation and form water crystals, or ice crystals. Ice crystals, like all other crystals, have the most basic properties of a geometric shape with their own rules. Ice crystals belong to six square crystal system, six-square crystal system has four crystal axes, three of which are on one plane, intersecting with each other at 60 degrees, and the other is perpendicular to the plane of the three auxiliary axes. The most typical shape of the six-cubic system is six prism. However, when the spindle direction crystals develop slowly during the crystallization process, and the direction of the auxiliary axis develops rapidly, the crystals appear as hexagonal slices.

Atmospheric water vapor in the crystallization process, is often the crystal in the direction of the main axis growth rate is slow, and three auxiliary axis direction is much faster, ice crystals are more than six side flake. When the atmosphere is very rich in moisture, the surrounding water molecules constantly to the original formation of the wafer, wherein, the snow of the six corners of the first, so that the top point will appear some protruding objects and twigs. These forks grow to a certain degree and diverge. Secondary division and the mother branch are maintained at a 60 degree angle. In this way, a hexagonal star-shaped snowflake is formed. Each snowflake is a hexagonal star in its entirety, but there are many differences in subtle forms. The formation and appearance of various snowflakes are closely related to different meteorological conditions, especially the air temperature and humidity.

The clouds are the place where snowflakes are born, and snowflakes are produced in these tiny nuclei in the clouds. Ice crystals turn into snowflakes the journey is tortuous and complex, it with the air flow up and down, left and right to move repeatedly, the surrounding temperature and water vapor conditions are constantly changing, so that the growth of ice crystals are different parts. Because the ice crystals in the production and growth process encountered in the temperature, humidity and airflow conditions are very diverse, so the hexagonal shape of the snowflake has become a variety of shapes. There are three trends in the shape of the nucleation: a long, thin hexagonal crystal, two sharp pins and a thin hexagonal wafer. If the concentration of water vapor around them is low, the ice crystals grow slowly, and the edges grow evenly, and if the surrounding water vapor concentration is larger, the growth process will not only increase in volume, but also change shape, the most common is the falling hexagonal snowflakes in the sky. Why do they tend to be hexagonal? The original ice crystal growth to consume the nearby water vapor, so, the closer to the ice crystals, the more thin water vapor, a little far away from the natural water to add, they first encountered is the sharp angle is being stretched forward, so, each sharp angle quickly lengthen, gradually become dendritic. The same reason, these "branches" on the growing new small branch, the cycle of the formation of the hexagonal snowflakes we see. The ice crystals before the formation of snowflakes are affected by the surrounding environment, the crystal growth rate of the positive hexagon and the side box on the bottom surface is different, and the shape changes accordingly, for example, the temperature will bring subtle changes to the crystalline surface, the bottom level expands into hexagonal when near 0°c, and the needle shape is reduced to -5~-10 °c on the side began to generate six prism and the side of the prism, -15°c formed dendritic, in the fall to -10~-21°c, the positive hexagon began to expand, and then regenerated into six prism.

Animated images:

Finally, Xu Zhimo's poem "The Joy of Snowflakes"

If I were a snowflake,
Dashing in the air,
I must recognize my direction
Flying, flying, flying,
There's my direction on the ground.

Not to the valley of the cold Lonesome,
Not to the foothills of the Sorrow,
Nor go to the desolate streets to be disconsolate
Flying, flying, flying,
You see, I have my direction!

In the mid-air, Juan Juan's flying,
To recognize the quiet place,
Waiting for her to visit in the garden
Flying, flying, flying,
Ah, she has the fragrance of cinnabar Plum!

When I was light in my body,
Yingying, touched her skirt,
Close to her soft heart
Dissolving, dissolving, dissolving
Dissolved into her soft-hearted heart!

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Snowflakes six out of---several algorithms generated by the snowflake image, and wish everyone Christmas Eve and Happy Xmas