Robert Williams (geometer)
Encyclopedia
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, 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.
, 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 Wikepedia 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.
and Fred Waldhauer
and artists Robert Rauschenberg
and Robert Whitman
. 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 18000 square feet (1,672.3 m²) of these modular, moveable, expandable-contractible enclosures to raise the endangered desert tortoise
(Gopherus agassizii) at the Fort Irwin Military Reservation
and Edwards Air Force Base
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.
and dark energy
, red-shift, fundamental forces, discrete units of space, and the expansion of the universe.
With respect to the symbolic Level, he followed the lead of symbologist and mythographer Robert Lawlor
. 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.
Biography—life, theories, and work
Robert Williams was born in Cincinnati, OhioCincinnati, Ohio
Cincinnati is a city in the U.S. state of Ohio. Cincinnati is the county seat of Hamilton County. Settled in 1788, the city is located to north of the Ohio River at the Ohio-Kentucky border, near Indiana. The population within city limits is 296,943 according to the 2010 census, making it Ohio's...
, the son of Robert Finley Williams and Edna Rita Brotherton. His father was the oldest member of the Williams Brothers
Williams Brothers
The Williams Brothers were a singing quartet that performed extensively on radio, movies, nightclubs, and television from 1938 through the 1990s.-History:...
, 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
Southern Illinois University is a state university system based in Carbondale, Illinois, in the Southern Illinois region of the state, with multiple campuses...
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
Albert George Wilson
Albert George Wilson is an American astronomer.He was born in Houston, Texas. He received his Ph.D. in mathematics from Caltech in 1947; his thesis title was Axially Symmetric Thermal Stresses in a Semi-Infinite Solid....
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
Science (journal)
Science is the academic journal of the American Association for the Advancement of Science and is one of the world's top scientific journals....
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
Southern Illinois University
Southern Illinois University is a state university system based in Carbondale, Illinois, in the Southern Illinois region of the state, with multiple campuses...
. 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 Wikepedia and Mathworld
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 Digital Library grant to the University of Illinois at...
.
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(E.A.T.) founded by engineers Billy KlüverBilly Klüver
Billy Klüver Johan Wilhelm Klüver was an electrical engineer at Bell Telephone Laboratories who founded Experiments in Art and Technology. Klüver lectured extensively on art and technology and social issues to be addressed by the technical community. He published numerous articles on these subjects...
and Fred Waldhauer
Fred Waldhauer
Frederick Donald Waldhauer was an American electrical engineer known for his work in hearing aids and combining art and technology.Waldhauer was born on December 6, 1927, and grew up in Brooklyn, New York, USA...
and artists Robert Rauschenberg
Robert Rauschenberg
Robert Rauschenberg was an American artist who came to prominence in the 1950s transition from Abstract Expressionism to Pop Art. Rauschenberg is well-known for his "Combines" of the 1950s, in which non-traditional materials and objects were employed in innovative combinations...
and Robert Whitman
Robert Whitman
Robert Whitman is 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 making...
. 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 18000 square feet (1,672.3 m²) of these modular, moveable, expandable-contractible enclosures to raise the endangered desert tortoise
Desert Tortoise
The desert tortoise is a species of tortoise native to the Mojave desert and Sonoran desert of the southwestern United States and northwestern Mexico. They can be located in western Arizona, southeastern California, southern Nevada, and southwestern Utah. The species name agassizii is in honor of...
(Gopherus agassizii) at the Fort Irwin Military Reservation
Fort Irwin Military Reservation
Fort Irwin & the National Training Center is a major training area for the United States Military and is a census-designated place located in the Mojave Desert in northern San Bernardino County, California. Fort Irwin sits at an elevation of . The 2010 United States census reported Fort Irwin's...
and Edwards Air Force Base
Edwards Air Force Base
Edwards Air Force Base is a United States Air Force base located on the border of Kern County, Los Angeles County, and San Bernardino County, California, in the Antelope Valley. It is southwest of the central business district of North Edwards, California and due east of Rosamond.It is named in...
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 Toth 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 matterDark matter
In astronomy and cosmology, dark matter is matter that neither emits nor scatters light or other electromagnetic radiation, and so cannot be directly detected via optical or radio astronomy...
and dark energy
Dark energy
In physical cosmology, astronomy and celestial mechanics, dark energy is a hypothetical form of energy that permeates all of space and tends to accelerate the expansion of the universe. Dark energy is the most accepted theory to explain recent observations that the universe appears to be expanding...
, red-shift, fundamental forces, discrete units of space, and the expansion of the universe.
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 Robert Lawlor
Robert Lawlor
Robert Lawlor is a mythographer, symbologist and New Age author of several books.After training as a painter and a sculptor, he became a yoga student of Sri Aurobindo and lived for many years in Pondicherry, India, where he was a founding member of Auroville. In India, he discovered the works of...
. In The Integration of Universal Constants Williams presented relationships among numerous diverse subjects: geometric form, color spectrum, the music octave
Octave
In music, an octave is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems"...
, the periodic table
Periodic table
The periodic table of the chemical elements is a tabular display of the 118 known chemical elements organized by selected properties of their atomic structures. Elements are presented by increasing atomic number, the number of protons in an atom's atomic nucleus...
, astronomy
Astronomy
Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...
, astrology
Astrology
Astrology consists of a number of belief systems which hold that there is a relationship between astronomical phenomena and events in the human world...
, psychology
Psychology
Psychology is the study of the mind and behavior. Its immediate goal is to understand individuals and groups by both establishing general principles and researching specific cases. For many, the ultimate goal of psychology is to benefit society...
, tarot
Tarot
The tarot |trionfi]] and later as tarocchi, tarock, and others) is a pack of cards , used from the mid-15th century in various parts of Europe to play a group of card games such as Italian tarocchini and French tarot...
, chakras, gender
Gender
Gender is a range of characteristics used to distinguish between males and females, particularly in the cases of men and women and the masculine and feminine attributes assigned to them. Depending on the context, the discriminating characteristics vary from sex to social role to gender identity...
, seasons 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. ISBN 0-486-23729-X.
- 2009. The Geometry of Natural Structure (40th Anniversary Edition). San Francisco: Eudaemon Press. ISBN 978-0-9823465-1-8
- 2009. The Kiss Catenatic: The Introduction of Catenatic Geometry and its Environs. San Francisco: Eudaemon Press. ISBN 978-0-9823465-2-5
- 2009. The Integration of Universal Constants. San Francisco: Eudaemon Press. (2009). ISBN 978-0-9823465-0-1.
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).