Fractal structure in snowflakes When water condenses, specific structures will be formed between water molecules. This is because the crystal formed by water is influenced by the structure of water molecules, so it will grow in a specific direction and form a specific structure. When it grows to a certain extent, it will form branches and continue to grow in the original way, thus forming the snowflake shape we see. We can use a simple fractal to describe the snowflake model.
First, we construct a triangle, divide each variant of the triangle into three equal parts, and create a new equilateral triangle with the middle section, thus forming the following image. Suppose that the side length of the original triangle is 1, then the perimeter is 3.
Compared with the original triangle, the changed figure lacks the middle line segment, but adds two sides with the same length, so the total side length becomes four-thirds, so the perimeter becomes four-thirds. Repeat the above process in the same way, and you can get the following chart.
Similarly, each side after the change has become 4/3 times of the original, so the circumference of the whole figure has become 4/3 times of the original, and the total circumference of this figure has become three-quarters of the square. Through many calculations, we can find the law, and every time it changes, the circumference will become three-quarters of the original. So we can get the following formula.
Because fractal geometry is composed of infinite fractals, and the changing n is infinite, we can get the result that the perimeter of snowflakes is infinite.
Realistic factors restrict the development of snowflake structure
Through theoretical calculation, it seems that there is no dispute about the infinite side length of snowflakes, so is this really the case? First of all, it is no problem to calculate through the above fractal geometry knowledge, but the fact that n here can be infinite is ignored. Snowflakes, after all, are real substances and will be restricted by realistic factors. When n tends to infinity, the side length of this geometric structure also tends to infinity, so can this side really tend to infinity? Obviously, it is impossible. Snowflakes are made up of water molecules, which have a certain size. So the size of this side cannot be smaller than the span of water molecules. Therefore, the side length of a snowflake cannot be infinite.
From the above description, we can know that the infinite perimeter of a snowflake is unrealistic, and fractal geometry can indeed exist in mathematical models. However, there are still many constraints in reality. Therefore, it is unreasonable to say that the circumference of snowflakes is infinite, but it is possible to say that the circumference of snowflakes is greater than the diameter of the earth. If you are interested, this can be calculated accurately when we take this side length as the minimum limit.