page_title: Robert Williams (geometer)
section_title: Biography—life, theories, and work
hierarchical_section_title: Robert Williams (geometer) / Biography—life, theories, and work
caption_attribution_description: English: Topological transformation from Lord Kelvin's alpha-tetrakaidecahedron to The Williams Beta-tetrakaidecahedron.
context_page_description: Robert Edward Williams 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.
context_section_description: Robert Williams was born in Cincinnati, Ohio, the son of Robert Finley Williams and Edna Rita Brotherton. His father was the oldest member of the Williams Brothers, a quartet of musical entertainers, who appeared on recordings, radio, and television, from the late 1930s to the present.
  Williams's work was originally inspired by the design principles in natural structure systems promoted by R. Buckminster Fuller. He was introduced to the work of Fuller by designer Peter Pearce in 1963.  He finished graduate studies in structural design at Southern Illinois University in 1967, where Fuller was University Professor. While at SIU, he invented a system of clustering dome structures by using small circle Catenatic Geometry principles rather than great circles, or geodesics, as Fuller had designed into geodesic dome structures. From his research with naturally packed cell systems (biological cells, soap bubble packings, and metal crystallites) he also discovered a new space-filling polyhedron, the β-tetrakaidecahedron, the faces of which closely approximate the actual distribution of the kinds of faces found in experimental samples of cell geometry in natural systems.
  Williams met astronomer, Albert George Wilson at the Rand Corporation in 1966. Wilson invited him to conduct research at the McDonnell-Douglas Corporation Advanced Research Laboratories (DARL) in Huntington Beach, California, USA. After graduate studies, he joined Dr. Wilson in September 1967 and continued his research into general structure principles in natural systems. He was the geometry and structure consultant to NASA engineer, Charles A. Willits, on the initiatory work in the development of large scale structure systems for space stations. The first of four editions of his structural geometry research was published by DARL in 1969, with the title:Handbook of Structure. His paper in the journal Science proposed that his discovery of the β-tetrakaidecahedron is the most reasonable alternative  to Lord Kelvin's α-tetrakaidecahedron.
  As an organizer and presenter at the First International Conference on Hierarchical Structures sponsored by DARL in 1968, Williams was an early proponent advocating the discipline of Hierarchical Structure to be a legitimate area of scientific research.
  In the spring of 1970, Williams became a visiting lecturer in Design at Southern Illinois University. A year later he returned to California, and started the design company Mandala Design Associates. In 1972, Eudaemon Press published Natural Structure: Toward a Form Language, an expanded edition of the original Handbook of Structure. In 1979, Dover Publications published the third edition titled, The Geometrical Foundation of Natural Structure, in its series of classical explanations of science. These works are cited in many books on geometry, science, and design. Numerous references to these works are found in geometry articles in Wikipedia and Mathworld.
  On the fortieth anniversary of the initial DARL publication, Eudaemon Press published a commemorative 40th anniversary edition: The Geometry of Natural Structure: A Language of Form Source Book for Scientists and Designers.As a companion volume, Eudaemon Press also published Williams's recent work: The Kiss Catenatic: The Introduction of Catenatic Geometry and its Environs.
caption: Topological transformation from the Lord Kelvin α-tetrakaidecahedron (a) to the Williams β-tetrakaidecahedron (c), having 8 pentagons, 4 hexagons, and 2 square faces.
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