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**Snow Crystals, Fractals and Self Similarity**

**Introduction**

The pieces of snow falling to the ground in winter have two very special scientific characteristics:

- No two flakes are exactly alike
- If you cut one, the remaining portions will have the same shape as what you cut from

These amazing qualities have been studied in the realm of **Chaos Theory**.

**Uniqueness**

According to a renowned scientist, finding two identical snowflakes is like ”shuffling a deck of cards and getting the exact same shuffle back. You could shuffle every second for the entire life of the universe, and you wouldn’t come close to getting two of the same”.^{1 } In order to explain this, we must look at the way that snow crystals are formed.

As snow is water, the basic blocks are in the majority of molecules two hydrogen atoms and one oxygen atom. While the atoms are all alike, about 1 in 5000 molecules^{2} will be a so called ”heavy water” or ”deuterium oxide” compound, where the hydrogen atoms are the ^{2}H isotope^{3} with two particles (one proton + one neutron) in the nucleus instead of the normal ^{1}H isotope.

These abberrations are placed at random points in the lattice. And since there are some 10^{18} atoms in a normal snowflake, about 10^{14} – 10^{15} deuterium molecules will be placed at some locations in the atomic structure.^{2} The likelihood of two different configurations of such a high number being exactly the same is so low that it can be considered impossible!^{2}

**Self Similarity**

The classical example of a system with self similarity is the Babushka or Matryoshka doll.^{4 } When opening it, a doll with the exact shape and look comes out, only smaller, and this keeps repeating. The mathematical definition of this is a *fractal*, and this has been observed and used in many early cultural art forms.^{5}

Snow is not perfectly self similar, but a mathematical fractal is like snow to a degree of 95-99 %.^{6,7 } It has been shown in a model of how snowflakes are formed that they indeed take a shape given by a fractal pattern.^{8}

**The Koch snowflake**

Swedish researcher Helge von Koch used fractal theory to build figures of snowflakes as early as 1904. The process was to start with a triangle and then continuously split each side and add another triangle at the half point. This yielded the initial sequence of shapes shown in Figure 1.

Figure 1. Four first Koch snowflakes [9].

**References**

[1] Prof. Kenneth Libbrecht, Californa Institute of Technology, quote from PBS Newshour.

[2] http://www.its.caltech.edu/~atomic/snowcrystals/alike/alike.htm - Snowflakes - No Two Alike? Caltech.

[3] http://pubchem.ncbi.nlm.nih.gov/compound/deuterium_oxide - Open Chemistry Database.

[4] https://en.wikipedia.org/wiki/Matryoshka_doll - Wikipedia.

[5] http://polynomial.me.uk/2009/09/20/fractals-of-brain-fractals-of-mind/ - Self Similarity ~ Fractals, Fractals Everywhere, Karl Richard 2009.

[6] *Fractals – A Brief Explanation. *Junior University, available through http://juni.osfc.ac.uk/Aim_Higher/Maths/

[7] Prof. Craig A. Tovey, Georgia Institue of Technology, answered on http://www.researchgate.net/post/Are_snowflakes_based_on_fractal_geometry .

[8] Nittmann, J. and Stanley, H.E.: *Non-deterministic approach to anisotropic growth patterns with continuously tunable morphology: the fractal properties of some real snowflakes*, Journal of Physics A, vol. 20, Number 17, also available at http://iopscience.iop.org/0305-4470/20/17/010

[9]. http://mathworld.wolfram.com/KochSnowflake.html - Wolfram MathWorld.

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