Introduction

Snowflakes are aggregates of tiny, sculpted pieces of ice known as snow crystals. If you are ever fortunate enough to look at snow crystals under a microscope you will have entered a magical realm of pristine beauty and—because no two snow crystals are identical—of infinite variety. Before long, you will also notice that snow crystals show an interesting sixfold 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 look the same at different scales). The prototypical fractal snowflake is the Koch snowflake, created by the following iterative process (note 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...: Repeat step 1 with each side of the new figure.

The first three steps of this process are illustrated in figure 1:

Figure 1

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. (note 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 in figure 2:

Figure 2

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.

Figure 3

These shapes (note 3) are identical to the internal structure emerging in figure 2.


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 Inner Snowflake

The internal snowflake, with 151 units, can also be created ‘standalone’ by the iterative process shown in figure 3, which starts with a hexagon. 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
Latest update: 10/4/09

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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.