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 is generated by randomly selecting one of the (typically 3) “attractor” points (at the corners of the triangle) and choosing the point midway between the current point and that attractor:

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

*(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

## Other constructions

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