Fractal image compression is a relatively recent image compression method which exploits similarities in different parts of the image. The basic idea is to represent an image by fractals and each of which is the fixed point of an Iterated Function System (IFS). Therefore, an input image can be represented by a series of IFS codes rather than pixels. In this way, an impressive compression ratio 10000:1 can be achieved. The application of fractal image compression presented in this paper is based on a novel image structure, Spiral Architecture, which has hexagonal instead of square pixels as the basic element. In the paper evidence would suggest that introducing Spiral Architecture into fractal image compression will improve the compression performance in compression ratio with little suffering in image quality. There are also much research could be done in this area to further improve the results.