He is an American designer, mathematician, and architect. He is noted for books on the geometry of natural structure, the discovery of a new space-filling polyhedron, the development of theoretical principles of Catenatic Geometry, and the invention of the Ars-Vivant Wild-life Protector System for repopulating the Western Mojave Desert in California, USA with desert tortoises.

Williams uses the geometry of Natural Structure, Catenatic Geometry principles, and Symbolic analysis as fundamental components of his architectural, environmental design, and cosmology work.

Three areas fascinate me: Geometry-Cosmology- Architecture. My professional interest-the communication of: (1) Catenatic Geometry: A unifying concept that demonstrates the interconnectedness of every event, energy and form-entity, in the Universe. Catenatic Geometry is a major descriptive tool to show these connections, whether they be visible or invisible, short-lived or enduring. (2) Cosmology: Based upon my cosmology work from the viewpoint of Catenatic Geometry, it is apparent that a Grand Unifying Theory in physics and Cosmology is impossible to achieve using what is termed the Standard Model. Catenatic Geometry can demonstrate the single source of both Dark Matter and Dark energy, among other seemingly unrelated physical concepts. (3) Architecture: With fundamental research showing increased energy efficiencies of alternative geometries of design, the AIA actively ignores this work. Schools of Architecture neither teach nor encourage alternative geometries for the field.

From the beginning of his geometry research, Williams considered polyhedral geometry as the basis of a Form Language comprising three levels: the Formative (geometry), the Purportive (psychology), and the Symbolic. With respect to the symbolic Level, he followed the lead of symbologist and mythographer Robert Lawlor.[21] In The Integration of Universal Constants Williams presented relationships among numerous diverse subjects: geometric form, color spectrum, the music octave, the periodic table, astronomy, astrology, psychology, tarot, chakras, gender, seasons of the year, among others. The relationships are depicted in six integrated cosmology charts.