A Flurry Of Snowflakes

 

Introduction

A fresh snowfall is one of nature's miracles, brightening the drabbest landscape, adding excitement to the dullest day and forcing us to acknowledge its transforming power and timeless beauty. A snowfall is of course an accumulation of inumerable snowflakes, which are themselves aggregates of tiny snow crystals. Under a microscope snow crystals turn out to be fascinating objects. Peering through the lens you enter a realm of pristine beauty and infinite variety. Before long, you also notice that snow crystals show an interesting six-fold symmetry. No matter how dissimilar two crystals might be, they will both depend on the number six for their basic structure:

 

 

Underlying the natural complexity on display in these photographs is the geometric form of the hexagon—of which the central snow crystal is a near approximation. The hexagonal symmetry of the snowflake is also seen in a purely mathematical object, known as the Koch Snowflake.

 

The Koch Snowflake

The intricate structure of natural snow crystals is described in mathematical terms by the language of fractal geometry, because snow crystals approximate to a class of geometric figures known as fractals. These objects have mind-blowing properties such as fractional dimension (hence ‘fractal’) and self-similarity (they display similarities of form at different scales). The prototypical fractal snowflake is the Koch snowflake, created by the following iterative process [1]:

Step 1: Subdivide each side of an equilateral triangle into three equal portions. With each central portion do the following: draw an equilateral triangle which has the central portion as its base, then subtract that base.

Steps 2, 3, 4 etc: Repeat step 1 with each side of the new figure.

The first three steps of this process are illustrated below:


 

The process can go on for ever, but the figure created from step 3 is practically identical to the true fractal, which can of course never be completed. [2]

Note that the first step creates a hexagram, the outline of the Star of David. This is a clue to a more intuitive way of creating the Koch Snowflake, by merging the triangle, and each triangle thereafter created, with an inverted copy of itself, as shown here:


 

This method also has the advantage of creating internal structure within the fractal. So now after step 1 we see the Star of David proper, with an internal hexagon clearly delineated. Steps 2 and 3 create closer approximations of the Koch snowflake, nested within which are snowflakes of a different design. Standalone versions of these internal snowflakes can be created from a hexagon by a variation of the process leading to the Koch Snowflake, where we draw triangles that points inwards, rather than outwards. [3]

 

 

The Koch Snowflake Emulated

The creation of the Koch Snowflake can be emulated for one or more steps with every third triangle in the triangular number series (those with a single unit at their geometric centre). This is accomplished, as shown above, by the merging of the triangle with an inverted copy of itself. For instance, triangle 28 (T(7)) can give rise to a hexagram/hexagon pair, as shown here:

\

 

Triangle 28 can only be taken one step towards the Koch snowflake. A good example of a triangle that can be taken two steps towards the Koch Snowflake is triangle 253 (T(22)). This process is illustrated below:

 

 

 

 

 

 

The Internal Snowflake

The internal snowflake, with 151 units, can also be created ‘standalone’ by the variation of iterative process shown above, where we draw triangles pointing into the figure. This is illustrated below.

 

 

 

Building the Snowflake from Crystals

Real snowflakes are accretions of smaller crystals. Similarly, snowflake 151 can be built from two simple geometric snow crystals, as illustrated below.

 

Here we see six hexagons surrounding a central hexagram. This 6-plus-1 pattern is familiar to us in the Creation story, in which God manifested the heavens and the earth in six days before resting on the seventh day. The process of accretion can be continued indefinitely to create a series of fractal snowflakes that are a unique and important feature of the New Bible Code.

 

Bill Downie 17/5/04

Updated 24/9/12

 

Notes

1. In mathematics an iterative process is one in which, at each step of the process, the same operation is repeated on the outcome of the previous step.

2. For the mathematically-inclined reader, the dimension of the outline of the Koch snowflake is log4/log3, or 1.261859... Amazing but true.

3. Technically, these are the starting figure and first two iterations of a Koch anti-snowflake.