CAN PACHYCEPHALOSAUR (DINOSAURIA: ORNITHISCHIA) CRANIAL MORPHOLOGY BE QUANTIFIED BY ESTIMATING FRACTAL DIMENSION?

Perfect fractals exhibit self-similarity at infinite scales and can be described by their fractal dimension, a discrete value between 1 (a line) and 2 (a plane). Fractal geometry occurs in nature (corals, plants, ammonite sutures, etc.), and studying fractal development produces unique insights regarding the natural system involved. This study addresses the evolutionary development of fractal geometry in pachycephalosaur skulls. Can cranial morphology be described by fractal dimension? If so, is there meaningful evolutionary variation in Db?

Fractal dimension can be estimated using box-counting, where multiple grids are placed onto an image to generate per-box presence/absence data. Differences in grids and box-counts are compared with a linear regression that produces an estimate of the fractal dimension (Db). Db was estimated for 5 taxa and 2 ontogimorphs of pachycephalosaurs using images from published literature. Db for each taxon is closer to 2 than 1, with Stegoceras, Prenocephale, and the Stygimoloch ontogimorph producing Db values closest to 2. The Dracorex ontogimorph and Pachycephalosaurus each produce Db values which are closer to 1 than any other taxa. These results identify some useful relationships and open questions. Fractals are present in each taxon and each ontogimorph. Fractals are most explicit, however, in derived taxa and ontogenetic end-members, suggesting that fractals may be a medium through which ontogeny recapitulates phylogeny in Pachycephalosauridae.

These results have powerful implications. As fractals continue to be used, it will be necessary to include more data, additional tools, and comparison to other methods. Quantifying morphology with a discrete value (Db) represents a fundamentally different approach to generating shape data, and the limitations of this methodology are not presently identified.