George Phillips Odom, Jr
Encyclopedia
George Phillips Odom, Jr (born 1941) is an American artist and amateur geometer, who is primarily known for his work on the golden ratio
(
).
in the 1960s. Later his career faltered somewhat and he could not repeat his early success. Odom suffered from depressions which ultimately culminated in a suicide attempt and a subsequential hospitalization at the Hudson River Psychiatric Center
in Poughkeepsie, where he became a permanent resident since the early 1980s.
Odom became interested in geometry after visiting an exhibition by Buckminster Fuller
in the 1960s. In the mid 1970s he contacted the Canadian geometer Coxeter
as he felt his art work was of some mathematical interest as well. This led to longtime correspondence with Coxeter and another mathematician father Wenniger, a monk from Minnesota, that spanned over several decades. The two mathematicians became one of Odom's few remaining regular contacts to outside world, after he had moved to the Hudson River Psychiatric Center, where he led a rather isolated life otherwise. Their correspondence was not only about mathematical topics, but it comprised matters of philosophy, psychology, religion and world affairs as well. In mathematics Odom was particularly interested in various geometric shapes and the golden ratio. He discovered the occurrence of golden ratio in a few elementary geometrical figures, where it had not been noticed before. The two mathematicians communicated Odom's results to other people in their lectures and conversations, and Coxeter incorporated them into some of his publications as well. Best known of these is the construction of the golden ratio with the help of an equilateral triangle and its circumcircle. Coxeter posed Odom's construction in the form of a problem, that was published 1983 in the American Mathematical Monthly
as problem #E2007:
Odom also found another construction of the golden ratio, that is based on an equilateral triangle:
Odom used 3-dimensional geometrical shapes in his artwork, which he examined for occurrences of the golden ratio as well. There he discovered two simple occurrences in platonic solid
s and their circumscribed spheres.
For the first one you connect the midpoints A and B of 2 edges of a tetrahedron
surface and extend them one side so that the extended line intersects the circumscribed sphere in C, then B divives AC according to the golden section. This construction also yields the situation of problem #E2007 from above, if one cuts this 3-dimesional figure along the plane in which the tetrahedron surface is embedded.
The second occurrence is in a cube. If you connect the centers A and B of any two cube surfaces and extend them again, such that the extended line intersects the circumscribed sphere in C, then B will divide AC according to the golden section.
The Princeton
mathematician John Horton Conway
visited Odom in Poughkeepsie in 2007.
Golden ratio
In mathematics and the arts, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The golden ratio is an irrational mathematical constant, approximately 1.61803398874989...
(
).Life and work
Odom garnered some recognition early in his career for his light machines made from fibre optics, that he exhibited at the Knoll International Gallery in ManhattanManhattan
Manhattan is the oldest and the most densely populated of the five boroughs of New York City. Located primarily on the island of Manhattan at the mouth of the Hudson River, the boundaries of the borough are identical to those of New York County, an original county of the state of New York...
in the 1960s. Later his career faltered somewhat and he could not repeat his early success. Odom suffered from depressions which ultimately culminated in a suicide attempt and a subsequential hospitalization at the Hudson River Psychiatric Center
Hudson River Psychiatric Center
The Hudson River Psychiatric Center in Poughkeepsie, New York is a building and a psychiatric facility for adults operated by the New York State Office of Mental Health....
in Poughkeepsie, where he became a permanent resident since the early 1980s.
Odom became interested in geometry after visiting an exhibition by Buckminster Fuller
Buckminster Fuller
Richard Buckminster “Bucky” Fuller was an American systems theorist, author, designer, inventor, futurist and second president of Mensa International, the high IQ society....
in the 1960s. In the mid 1970s he contacted the Canadian geometer Coxeter
Harold Scott MacDonald Coxeter
Harold Scott MacDonald "Donald" Coxeter, was a British-born Canadian geometer. Coxeter is regarded as one of the great geometers of the 20th century. He was born in London but spent most of his life in Canada....
as he felt his art work was of some mathematical interest as well. This led to longtime correspondence with Coxeter and another mathematician father Wenniger, a monk from Minnesota, that spanned over several decades. The two mathematicians became one of Odom's few remaining regular contacts to outside world, after he had moved to the Hudson River Psychiatric Center, where he led a rather isolated life otherwise. Their correspondence was not only about mathematical topics, but it comprised matters of philosophy, psychology, religion and world affairs as well. In mathematics Odom was particularly interested in various geometric shapes and the golden ratio. He discovered the occurrence of golden ratio in a few elementary geometrical figures, where it had not been noticed before. The two mathematicians communicated Odom's results to other people in their lectures and conversations, and Coxeter incorporated them into some of his publications as well. Best known of these is the construction of the golden ratio with the help of an equilateral triangle and its circumcircle. Coxeter posed Odom's construction in the form of a problem, that was published 1983 in the American Mathematical Monthly
American Mathematical Monthly
The American Mathematical Monthly is a mathematical journal founded by Benjamin Finkel in 1894. It is currently published 10 times each year by the Mathematical Association of America....
as problem #E2007:
- Let A and B be midpoints of the sides EF and ED of an equilateral triangle DEF. Extend AB to meet the circumcircle (of DEF) at C. Show that B divides AC according to the golden section
Odom also found another construction of the golden ratio, that is based on an equilateral triangle:
- Consider an equilateral triangle ABC with its altitude from C onto AB. Let D be the pedal point of the altitude on AB. Now extend the altitude CD beyond D by |BD| and denote the endpoint of the extension with E. The ray EA intersects the circle around D with radius |CD| in F and A divides now EF according to the golden section.
Odom used 3-dimensional geometrical shapes in his artwork, which he examined for occurrences of the golden ratio as well. There he discovered two simple occurrences in platonic solid
Platonic solid
In geometry, a Platonic solid is a convex polyhedron that is regular, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruent regular polygons, with the same number of faces meeting at each vertex; thus, all its edges are congruent, as are its vertices and...
s and their circumscribed spheres.
For the first one you connect the midpoints A and B of 2 edges of a tetrahedron
Tetrahedron
In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...
surface and extend them one side so that the extended line intersects the circumscribed sphere in C, then B divives AC according to the golden section. This construction also yields the situation of problem #E2007 from above, if one cuts this 3-dimesional figure along the plane in which the tetrahedron surface is embedded.
The second occurrence is in a cube. If you connect the centers A and B of any two cube surfaces and extend them again, such that the extended line intersects the circumscribed sphere in C, then B will divide AC according to the golden section.
The Princeton
Princeton University
Princeton University is a private research university located in Princeton, New Jersey, United States. The school is one of the eight universities of the Ivy League, and is one of the nine Colonial Colleges founded before the American Revolution....
mathematician John Horton Conway
John Horton Conway
John Horton Conway is a prolific mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory...
visited Odom in Poughkeepsie in 2007.