This research explores the principles, implementation, and optimization of a competitive volume compression system based on fractal image compression. The extension of fractal image compression to volumetric data is trivial in theory. However, the simple addition of a dimension to existing fractal image compression algorithms results in infeasible compression times and noncompetitive volume compression results. This paper extends several fractal image compression enhancements to perform properly and efficiently on volumetric data, and introduces a new 3D edge classification scheme based on principal component analysis. Numerous experiments over the many parameters of fractal volume compression suggest aggressive settings of its system parameters. At this peak efficiency, fractal volume compression surpasses vector quantization and approaches within 1 dB PSNR of the discrete cosine transform. When compared to the DCT, fractal volume compression represents surfaces in volumes exceptionally well at high compression rates, and the artifacts of its compression error appear as noise instead of deceptive smoothing or distracting ringing.