# Platonic Solids (2008)

The Platonic Solids project explores how a purely operations-based geometric process can generate complex form.

Rather than studying the possibilities in combining numerous primitives, this project examines the potential inherent in a single primitive given an appropriate process. It takes the most primitive forms, the platonic solids, and repeatedly employs one single operation – the division of a form’s faces into smaller faces – until a new form is produced.

All of the forms shown are generated using the same single process, Only the variables that control the process' division operation are allowed to change. This single process affects both the form's topography and topology. It influences attributes such as the degree of branching, porosity, and fractalization - just to name a few. The process also works at multiple scales: it affects not only the overall shape, but it determines the surface development as well as the generation of minuscule textures.

The resulting forms display a novel aesthetic and an astounding complexity that largely defies attempts at reductionism.

The processes at the core of this project are the Catmull-Clark and Doo-Sabin algorithms. They were both conceived of in the late 1970's with the aim of generating smooth surfaces from coarses polygonal meshes.

The processes can be understood by considering their two parts: topological rules and weighting rules. The topological rules specify how to obtain the combinatorics of the refined mesh from the combinatorics of the input mesh by generating new vertices, edges and faces. The weighting rules specify how to calculate the positions of these new vertices based on interpolation between vertices of the input mesh.

By introducing parameters to allow for variations in these weighting rules, non-rounded forms with highly diverse attributes can be produced. Whereas the traditional weighting rules specify the positions of new vertices strictly as interpolations of previous-generation vertices, these rules are amended to allow for extrusion along face, edge and vertex normals. It is primarily through these two changes to the established schemes the complex geometries in this project become possible.