Fractal snowflake

The image I have used as the logo for this website is a fractal that I generated by accident during my school career, and have not seen elsewhere.

I have had an interest in fractals since approximately 1997, especially in writing computer programs to generate them. One of the earliest fractals I generated was the Sierpinski gasket using the Chaos game algorithm. However, while writing a program to do so (in BBC BASIC on an Acorn RISC OS machine), I made an error that led to this fractal. The conventional Chaos Game algorithm involves iteratively generating points the collection of which becomes (an approximation to) the fractal: after initialisation, each subsequent point in the iteration $\mathbf{x}_{n+1}$ is generated by randomly selecting one of the (typically 3) “attractor” points $\mathbf{a}_i$ (at the corners of the triangle) and choosing the point midway between the current point $\mathbf{x}_n$ and that attractor:

$\displaystyle \mathbf{x}_{n+1} = \frac{\mathbf{x}_n + \mathbf{a}_i}{2}$

However, in the program I wrote, I mistakenly had the algorithm as:

$\displaystyle \mathbf{x}_{n+1} = \frac{\mathbf{x}_n - \mathbf{a}_i}{2}$ (note the minus sign in place of the plus above)

Giving rise to the following shape:

Properties

The fractal has similar properties to the Sierpinski Triangle:

• It has infinite length but zero area
• If it is enlarged by a linear scale factor of 2, the resulting shape consists of 3 copies of the original fractal, so it has a Hausdorff dimension of $\frac{\log{3}}{\log{2}} \approx 1.585$

Other constructions

I have also generated this fractal geometrically, starting with an equilateral triangle with extended sides and applying a geometrical rewrite rule.