The Snowflakes in Mathematics
By Qien Shensun, 16th Apr 2023
Translated by Yifan Wang, 17th Aug 2023
Snowflakes are often correlated with winter, but when snowflakes are mentioned, I believe not many people would think about mathematics. However, snowflakes do indeed exist in mathematics, in the form of the “Koch snowflake”. A quite well-known mathematics concept, the Koch snowflake is both beautiful and interesting. Today, lets dive into the secrets embedded within the Koch snowflake.
However, before we jump too quickly into the subject, we need to first understand a very important concept - "fractals".
Indeed, fractals exists in many forms, and can be commonly found in nature, such as in tree branches, lighting, clouds and other natural phenomenon. In mathematics, fractals can be generated using mathematical formulas and computers. Apart from looking cool in art, fractals have a myriad of applications in various research fields, including natural and computer sciences.
Now we know where we can find and how we can use fractals, lets move on to its proper mathematical definition:
A fractal is a mathematical concept that describes the structure of a complex geometric shape, and its dimensions are usually non-integer! Which leads it to display a strange characteristic known as “self-similarity”, which describes a process in which as we zoom in on the fractal, the shapes found at different scales are still geometrically similar to one another. This phenomenon is also called extended symmetry or expanded symmetry. Well although this all sounds quite abstract, we can indeed find this magical fractal phenomenon everywhere.
Now that we have a basic understanding of fractals, we can formally introduce our protagonist of the day – the Koch Snowflake!
A Koch snowflake is a fractal pattern consisting of a series of similar and self-similar triangles. It can be built by recursively dividing each line segment into thirds and building a new equilateral triangle in the middle segment with sides the length of the newly divided thirds
Although the shape might look daunting to draw or even visualize, it can actually be done in three relatively easy steps:
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Starting with an equilateral triangle, we first divide each side into thirds and construct a new equilateral triangle on the middle segments of each original side.
2. After this, we should see a shape that resembles a hexagram. We take this new hexagram and repeat exactly what we did before, taking each of the now 12 sides, dividing them into thirds and constructing an equilateral triangle on the middle segments.
3. We will then repeat the same step for every proceeding shape created until we reach a level of detail we want.
By following those steps infinitely, we can create a Koch snowflake with infinite detail and perimeter. Strangely, although its boundary length approaches infinity, its area is still limited...
This raises yet another question: How does a Koch snowflake infinitely increase its perimeter whilst having a limited area?
Assuming that the original perimeter of the triangle at the beginning is L, according to the recursive relationship, we can derive a formula for the perimeter of the Koch snowflake at different depths of recursion, denoted by n:
Using this formula, we can find if there is indeed a limit to the perimeter of the Koch snowflake as n is increased to infinity. Thus, we derive:
Now we have proved that the perimeter of the Koch snowflake is infinite, how do we measure its area?
Let’s start by denoting the area of the Koch snowflake at a nth level of recursion as Sn and begin with a base triangle with the area of 1 unit squared:
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We can know that the area of the small triangle added each time is one ninth that of the triangles added in the previous level of recursion.
2. By following the instructions for constructing a Koch snowflake, we need to add a small triangle on each side of the shape. Thus, we can find the area of the new shape by adding the area of the small triangles multiplied by the number of sides to the area of the current shape. By doing so, we derive:
3. If we keep doing this, we can eventually derive a general formula for the area of a Koch snowflake:
At last, we can use the general formula to see what happens as we increase n to infinity.
We can now see that the area remains at a fixed value 8/5 unit squared as n approaches infinity, so now we have mathematically proved that the Koch snowflake will be a geometric figure with infinite perimeter but finite area!
With that, I believe that everyone has now mastered the knowledge about Koch snowflakes. Of course, in addition to Koch snowflakes, fractals and cauliflower shapes often arouse the interest of mathematicians, as well as biologists! An article from “Science” shown in the picture shows a group of biologists and mathematicians collaborating to explore the mysteries of cauliflower fractals.
That's all for today's sharing! If you are still interested in fractals or Koch snowflakes and want to continue to study the topic, you can explore this interesting topic by yourself——