Robert Williams (geometer)
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Robert Edward Williams (born 1942) 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 In geometry, a space-filling polyhedron is a polyhedron that can be used to fill all of three-dimensional space via translations, rotations and/or reflections, where ''filling'' means that; taken together, all the instances of the polyhedron c ...
, 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 The Mojave Desert (; ; ) is a desert in the rain shadow of the southern Sierra Nevada mountains and Transverse Ranges in the Southwestern United States. Named for the Indigenous peoples of the Americas, indigenous Mohave people, it is located pr ...
in California, USA with desert tortoises.


Biography—life, theories, and work

Robert Williams was born in
Cincinnati, Ohio Cincinnati ( ; colloquially nicknamed Cincy) is a city in Hamilton County, Ohio, United States, and its county seat. Settled in 1788, the city is located on the northern side of the confluence of the Licking River (Kentucky), Licking and Ohio Ri ...
, 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 Southern Illinois University is a system of public universities in the southern region of the U.S. state of Illinois. Its headquarters is in Carbondale, Illinois. Board of trustees The university is governed by the nine member SIU Board of T ...
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 In geometry, a geodesic () is a curve representing in some sense the locally shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connec ...
, 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 In geometry, a space-filling polyhedron is a polyhedron that can be used to fill all of three-dimensional space via translations, rotations and/or reflections, where ''filling'' means that; taken together, all the instances of the polyhedron c ...
, 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 The RAND Corporation, doing business as RAND, is an American nonprofit global policy think tank, research institute, and public sector consulting firm. RAND engages in research and development (R&D) in several fields and industries. Since the ...
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 Science is a systematic discipline that builds and organises knowledge in the form of testable hypotheses and predictions about the universe. Modern science is typically divided into twoor threemajor branches: the natural sciences, which stu ...
'' 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 A hierarchy (from Greek: , from , 'president of sacred rites') is an arrangement of items (objects, names, values, categories, etc.) that are represented as being "above", "below", or "at the same level as" one another. Hierarchy is an importan ...
to be a legitimate area of scientific research. In the spring of 1970, Williams became a visiting lecturer in Design at
Southern Illinois University Southern Illinois University is a system of public universities in the southern region of the U.S. state of Illinois. Its headquarters is in Carbondale, Illinois. Board of trustees The university is governed by the nine member SIU Board of T ...
. 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
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and
Mathworld ''MathWorld'' is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science ...
. 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''.


Environmental design work

Williams uses the geometry of Natural Structure, Catenatic Geometry principles, and Symbolic analysis as fundamental components of his architectural, environmental design, and cosmology work. In 1967, he became a charter member of
Experiments in Art and Technology Experiments in Art and Technology (E.A.T.), a non-profit and tax-exempt organization, was established in 1967 to develop collaborations between artists and engineers. The group operated by facilitating person-to-person contacts between artists and ...
(E.A.T.) founded by engineers Billy Klüver and Fred Waldhauer and artists
Robert Rauschenberg Milton Ernest "Robert" or "Bob" Rauschenberg (October 22, 1925 – May 12, 2008) was an American painter and graphic artist whose early works anticipated the Pop art movement. Rauschenberg is well known for his Combine painting, Combines (1954â ...
and
Robert Whitman Robert Whitman (May 23, 1935 – January 19, 2024) was an American artist best known for his seminal theater pieces of the early 1960s combining visual and sound images, actors, film, slides, and evocative props in environments of his own makin ...
. In addition to theoretical work, Williams was awarded U.S. Patent No. 6,532,701] in 2003 for a shelter system of clustered modular enclosures. He designed and constructed of these modular, moveable, expandable-contractible enclosures to raise the endangered
desert tortoise The desert tortoise (''Gopherus agassizii'') is a species of tortoise in the Family (biology), family Testudinidae. The species is native to the Mojave Desert, Mojave and Sonoran Deserts of the southwestern United States and northwestern Mexico ...
(Gopherus agassizii) at the Fort Irwin Military Reservation and
Edwards Air Force Base Edwards Air Force Base (AFB) is a United States Air Force installation in California. Most of the base sits in Kern County, California, Kern County, but its eastern end is in San Bernardino County, California, San Bernardino County and a souther ...
in California, USA. Of all of Williams' design and architectural work, he considers his association with biologists, David Morafka and Kenneth Nagy to repopulate the Western Mojave Desert with the desert tortoise as his most rewarding environmental design work.


Catenatic Geometry and Sacred Geometry

In both his books and lectures, Williams is a keen popularizer of the geometries in natural structures and how they can be used in environmental design. His current work focuses on two concepts first introduced in ''Natural Structure: Toward a Form Language.''


Catenatic Geometry

Following the lead of mathematicians L. Fejes Tóth and C. A. Rogers, Williams formalized the concepts underlying Catenatic Geometry. In ''The Kiss Catenatic'' he expanded the concept of small circles covering a sphere to include interconnected platen circuits that model multi-level linked units of the 3-dimensional matrix chain. He presented examples of the use of Catenatic Geometry in discussions of
dark matter In astronomy, dark matter is an invisible and hypothetical form of matter that does not interact with light or other electromagnetic radiation. Dark matter is implied by gravity, gravitational effects that cannot be explained by general relat ...
and
dark energy In physical cosmology and astronomy, dark energy is a proposed form of energy that affects the universe on the largest scales. Its primary effect is to drive the accelerating expansion of the universe. It also slows the rate of structure format ...
,
red-shift In physics, a redshift is an increase in the wavelength, and corresponding decrease in the frequency and photon energy, of electromagnetic radiation (such as light). The opposite change, a decrease in wavelength and increase in frequency and e ...
,
fundamental forces In physics, the fundamental interactions or fundamental forces are interactions in nature that appear not to be reducible to more basic interactions. There are four fundamental interactions known to exist: * gravity * electromagnetism * weak int ...
, discrete units of space, and the
expansion of the universe The expansion of the universe is the increase in proper length, distance between Gravitational binding energy, gravitationally unbound parts of the observable universe with time. It is an intrinsic and extrinsic properties (philosophy), intrins ...
.


Sacred Geometry

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 Myth is a genre of folklore consisting primarily of narratives that play a fundamental role in a society. For scholars, this is very different from the vernacular usage of the term "myth" that refers to a belief that is not true. Instead, the ...
Robert Lawlor.Lawlor, R. Sacred Geometry: Philosophy and practice, London: Thames & Hudson, 1989 (1st edition 1979, 1980, or 1982), In ''The Integration of Universal Constants'' Williams presented relationships among numerous diverse subjects: geometric form,
color spectrum The visible spectrum is the band of the electromagnetic spectrum that is visible to the human eye. Electromagnetic radiation in this range of wavelengths is called ''visible light'' (or simply light). The optical spectrum is sometimes consider ...
, the music
octave In music, an octave (: eighth) or perfect octave (sometimes called the diapason) is an interval between two notes, one having twice the frequency of vibration of the other. The octave relationship is a natural phenomenon that has been referr ...
, the
periodic table The periodic table, also known as the periodic table of the elements, is an ordered arrangement of the chemical elements into rows (" periods") and columns (" groups"). It is an icon of chemistry and is widely used in physics and other s ...
,
astronomy Astronomy is a natural science that studies celestial objects and the phenomena that occur in the cosmos. It uses mathematics, physics, and chemistry in order to explain their origin and their overall evolution. Objects of interest includ ...
,
astrology Astrology is a range of Divination, divinatory practices, recognized as pseudoscientific since the 18th century, that propose that information about human affairs and terrestrial events may be discerned by studying the apparent positions ...
,
psychology Psychology is the scientific study of mind and behavior. Its subject matter includes the behavior of humans and nonhumans, both consciousness, conscious and Unconscious mind, unconscious phenomena, and mental processes such as thoughts, feel ...
,
tarot Tarot (, first known as ''trionfi (cards), trionfi'' and later as ''tarocchi'' or ''tarocks'') is a set of playing cards used in tarot games and in fortune-telling or divination. From at least the mid-15th century, the tarot was used to play t ...
,
chakras A chakra (; ; ) is one of the various focal points used in a variety of ancient meditation practices, collectively denominated as Tantra, part of the inner traditions of Hinduism and Buddhism. The concept of the chakra arose in Hinduism. B ...
,
gender Gender is the range of social, psychological, cultural, and behavioral aspects of being a man (or boy), woman (or girl), or third gender. Although gender often corresponds to sex, a transgender person may identify with a gender other tha ...
,
seasons A season is a division of the year based on changes in weather, ecology, and the number of daylight hours in a given region. On Earth, seasons are the result of the axial parallelism of Earth's axial tilt, tilted orbit around the Sun. In temperat ...
of the year, among others. The relationships are depicted in six integrated cosmology charts.


Publications

* 1967. ''Geometry, Structure, Environment''. Masters Thesis. Carbondale: Southern Illinois University. * 1968. ''Handbook of Structure''. McDonnell-Douglas Advanced Research Laboratories. Research Communication 75. * 1972. ''Natural Structure: Toward a Form Language''. Moorpark, California: Eudaemon Press. * 1978. ''The Geometrical Foundation of Natural Structure''. New York: Dover Publications, Inc. . * 2009. ''The Geometry of Natural Structure (40th Anniversary Edition)''. San Francisco: Eudaemon Press. * 2009. ''The Kiss Catenatic: The Introduction of Catenatic Geometry and its Environs''. San Francisco: Eudaemon Press. * 2009. ''The Integration of Universal Constants''. San Francisco: Eudaemon Press. . * 2020. ''Concealed Symmetry: Modulators Within Universal Process.'' Santa Paula: Eudaemon Press. .


References


U. S. Patent Office publications

* "Shelter system of clustered modular enclosures". U.S. Patent 6,532,701 (March 18, 2003). * "A Unified Method and System for Multi-Dimensional Mapping of Spatial-Energy Relationships Among Micro- and Macro-Events in the Universe". U.S. Patent Application No. 10/714,142, Publication No. US-2005-0143919-A1 (June 30, 2005).


External links

* Robert Williams: ''Catenatic Geometry & Natural Structure'' YouTube video series 2009:
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{{DEFAULTSORT:Williams, Robert American geometers 20th-century American mathematicians 1942 births Living people 21st-century American mathematicians People from Cincinnati Mathematicians from Ohio California State University, Northridge alumni Southern Illinois University alumni