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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