Feature

Fractal Geology

Mathematics can paint a picture of how metamorphic rock forms.

By Simon Cox

Spectacular computer-generated pictures, and their use in the movie industry, are the only aspect of fractals with which most people are familiar. Fractals and spatial geometry are relatively new concepts, and have only become popular since the advent of computers and the mathematical formulation by Benoit Mandelbrot in 1967.

At the Otago Geology Department, I have been using the concepts of fractals and spatial geometry to measure the amount and style of quartz veins in the schist rocks of central Otago.

At school we were taught the dimensions:

• zero for a point
• one for a line (along its length)
• two for a plane (length x width)
• three for a volume (length x width x height)

However, many natural objects such as clouds, rivers, and coastlines, have a roughness or complexity that does not lend itself to description in terms of lines and planes. The roughness is better described using "fractional", or fractal, dimensions.

On a map, for example, the tortuous curving trace of a coastline or a river has a dimension somewhere between the one dimension of a straight line and the two dimensions of a plane. A very tortuous river may have a dimension of 1.8 for example, compared with a gently curving river of dimension 1.3. Curving surfaces can also be described using fractal geometry, having a fractal dimension between that of a two-dimensional plane and a three-dimensional volume.

Analysis

In order to measure the fractal dimension of veins in the schist, a video camera was used to film the rock. Rock along the shores of Lake Hawea is particularly photogenic, as the outcrops of rock there are clean and not covered in dirt or lichen. Back at the university, frames are grabbed from the video as images of 512 x 512 pixels (picture elements), and analysed on a Macintosh computer.

The images can be analysed in a variety of ways to provide information about the "spatial geometry" of the quartz veins. In the images, pixels which represent veins are white with greyscale values between 0-30. Rock on the other hand is a darker grey colour, with greyscale between 30-256.

A "threshold" value is chosen to distinguish between what is rock and what is vein -- in this case around 30. Then, simply by counting how many pixels in the image have values less than 30, and comparing this with the number having greyscale values higher than 30, the proportion of veins can be determined. More complicated analysis of the images provides information on the number of veins of a particular thickness, and the fractal dimension of the veins.

But why all the interest in the quartz veins of Otago anyway?

Quartz veins form in metamorphic rocks when they are buried deep in the earth. Intense heat and pressure dehydrates the rock, driving off hot fluids, which percolate through fractures in the rock, sometimes even causing the rock to fracture. As these fluids move through the rock, they become enriched in many elements, including silica and gold. As the fluids are cooled by passing through rocks of lower temperature, or as a result of differences in stress and strain in the rock, the silica (and sometimes gold) is deposited in the fractures as veins.

The quartz veins we see in the schists of Otago have now been uplifted to the surface of the earth. Erosion has taken away the rocks that originally lay on top of the schists and buried them.

In the South Island, we have a unique natural laboratory in which to study the effects of fluids on deformation of the earth's crust. This is currently a "hot topic" in geology with important implications for the generation of earthquakes and mineral deposits. At Lake Hawea, and through the mountains to the West Coast, there are a whole range of rocks produced under different conditions. Some were originally buried to depths of around 10 kilometres and at temperatures of 300^oC; others were buried to 18 kilometres and 550^oC.

By studying differences in the spatial geometry of veins, together with the differences in chemistry due to vein development, I've been deriving a picture of fluid flow at different levels in the earth. Although previous theories have postulated that circulation of large amounts of fluid are necessary for a rock to deform and veins to develop, this new approach is suggesting the crust may be drier than recently thought.

Simon Cox is a Research Fellow at the University of Otago.