Mathematics in Snowflake's Fractals

The fractals drawn by Snowflake have all kinds of interesting mathematics associated with them. (If you already know how the pictures are drawn, the following will make a lot more sense...)

The quintessential fractal

Based on (and named after) Koch's famous "snowflake curve", fractals like the ones drawn by Snowflake are a classic example of fractals in general. The concept of iterating a simple rule, and considering the infinite limit of that iterative process, is at the core of most, if not all, fractals in mathematics. Several of the concepts that characterize fractals are easy to see examples of and discuss with Snowflake.

Self Similarity

Self similarity is loosely considered the unifying quality of all things fractal. The curves explored by Snowflake are exactly self similar (that is, Snowflake draws curves that are approximations to exactly self similar curves). Essentially, that means that if you magnify any particular piece of the curve, it would look exactly like the original. (Some fractals are self similar in less exact ways--for example, the Mandelbrot set exhibits "quasi-self similarity". Magnified bits are like the orginal set, but with more intricacies. Some coastlines exhibit self similarity in that they show the same (non-Euclidean) dimension at different scales.)

Non-Euclidean dimension

One of the mind-boggling things about fractals is that they challenge our traditional notion of dimension. Fractal structures often fit best between the integer dimensions that we are used to from Euclidean geometry. For example, a common measure of dimension for the classical Koch snowflake curve is (log 4)/(log 3), approximately 1.26. With Snowflake, students can create their own fractals and learn how to find the fractal dimensions of these curves.

Sensitivity to initial conditions

Often the behind-the-scenes reason for "chaotic" behavior, exponential sensitivity to initial conditions (which basically means that little changes in the beginning of a process can lead to major differences in the outcome) can be seen by making small changes in the initial curve and seeing dramatic changes in the result upon iteration.

Mathematics in General

Besides the mathematics that are more or less peculiar to fractals, there is a wealth of traditional mathematical topics that arise naturally in the context of playing with a program like Snowflake.

Exponentiation

If students are not familiar with exponents and exponential notation, activities with Snowflake can stimulate discussion and motivate the introduction of these concepts. The total number of segments drawn, the lengths of these segements, and the total length of the nth iteration are all simple exponential functions of the number of iterations.

Students that have already learned about exponents but need an opportunity to discover exponential patterns themselves can do so in an interesting and engaging way with Snowflake-based activities.

Operations with fractions

Activities similar to those used for exploring exponentiation could be used to give students practice with addition and multiplication of fractions.

Symmetry

Curves generated by Snowflake can be extremely rich in symmetries, with a single shape having rotation, translation, and reflection symmetries, plus the additional symmetry aspects related to self similarity.

Infinity and Limits

The whole idea of Snowflake is to draw various levels of approximations to curves which are actually limits of an infinite iterative sequence. Having the opportunity to see the approximations at increasingly accurate levels can help students imagine the infinite continuation of the process.

Converging and diverging sequences also arise naturally in considering the length of the curve after n iterations and the area bounded by the nth iteration and the zeroth (or actually any other) iteration.


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