The surface of New Zealand is dissected by a complex network of fault lines. Understanding the mathematics behind this pattern could lead to better prediction of earthquakes, both large and small, that shake the country on an average of 14,000 times a year. The object of this thesis is to investigate whether fault lines are fractal and, if so, what their fractal dimension is. A spatial pattern of fault line orientation is studied, using a permutation test, specifically adapted for this study. It is found that the estimated fractal dimension is very dependent on the method chosen to analyse spatial distribution. The characteristic of the estimated fractal pattern for the South Island of New Zealand as described in this study and appears to agree with findings for the seismic area of Japan. The orientation of the fault lines depends on spatial location. This result supports leading theories of plate tectonics. This study helps build an overall picture of the fault system in New Zealand and brings us one step further towards a more complete knowledge which may one day help us to predict and prepare for earthquakes in the future.