|Died||July 13, 2017 86)(aged|
|Alma mater||University of Toronto|
|Known for||Johnson solid (1966)|
|Institutions||Wheaton College, Norton, Massachusetts|
|Doctoral advisor||H. S. M. Coxeter|
Norman Woodason Johnson (November 12, 1930 – July 13, 2017) was a mathematician at Wheaton College, Norton, Massachusetts.
Norman Johnson was born on November 12, 1930 in Chicago. His father had a bookstore and published a local newspaper.
Johnson earned his undergraduate mathematics degree in 1953 at Carleton College in Northfield, Minnesotafollowed by a master's degree from the University of Pittsburgh. After graduating in 1953, Johnson did alternative civilian service as a conscientious objector. He earned his PhD from the University of Toronto in 1966 with a dissertation title of The Theory of Uniform Polytopes and Honeycombs under the supervision of H. S. M. Coxeter. From there, he accepted a position in the Mathematics Department of Wheaton College in Massachusetts and taught until his retirement in 1998.
In 1966, he enumerated 92 convex non-uniform polyhedra with regular faces. Victor Zalgaller later proved (1969) that Johnson's list was complete, and the set is now known as the Johnson solids.
Johnson is also credited with naming all the uniform star polyhedra and their duals, as published in Magnus Wenninger's model building books: Polyhedron models (1971) and Dual models (1983).
He completed final edits for his book Geometries and Transformations just before his death on July 13, 2017, and nearly completed his manuscript on uniform polytopes.
In geometry, every polyhedron is associated with a second dual figure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. Such dual figures remain combinatorial or abstract polyhedra, but not all are also geometric polyhedra. Starting with any given polyhedron, the dual of its dual is the original polyhedron.
Harold Scott MacDonald "Donald" Coxeter, was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century.
In geometry, the gyroelongated square bipyramid, heccaidecadeltahedron, or tetrakis square antiprism is one of the Johnson solids (J17). As the name suggests, it can be constructed by gyroelongating an octahedron by inserting a square antiprism between its congruent halves. It is one of the eight strictly-convex deltahedra.
In geometry, the gyroelongated square pyramid is one of the Johnson solids (J10). As its name suggests, it can be constructed by taking a square pyramid and "gyroelongating" it, which in this case involves joining a square antiprism to its base.
In geometry, the tridiminished icosahedron is one of the Johnson solids (J63).
In geometry, the elongated square gyrobicupola or pseudo-rhombicuboctahedron is one of the Johnson solids (J37). It is not usually considered to be an Archimedean solid, even though its faces consist of regular polygons that meet in the same pattern at each of its vertices, because unlike the 13 Archimedean solids, it lacks a set of global symmetries that take every vertex to every other vertex. It strongly resembles, but should not be mistaken for, the small rhombicuboctahedron, which is an Archimedean solid. It is also a canonical polyhedron.
In geometry, the gyroelongated square cupola is one of the Johnson solids (J23). As the name suggests, it can be constructed by gyroelongating a square cupola (J4) by attaching an octagonal antiprism to its base. It can also be seen as a gyroelongated square bicupola (J45) with one square bicupola removed.
In geometry, the gyroelongated pentagonal birotunda is one of the Johnson solids (J48). As the name suggests, it can be constructed by gyroelongating a pentagonal birotunda by inserting a decagonal antiprism between its two halves.
In geometry, the metabidiminished rhombicosidodecahedron is one of the Johnson solids (J81).
In geometry, the augmented dodecahedron is one of the Johnson solids (J58), consisting of a dodecahedron with a pentagonal pyramid (J2) attached to one of the faces. When two or three such pyramids are attached, the result may be a parabiaugmented dodecahedron, a metabiaugmented dodecahedron or a triaugmented dodecahedron.
In geometry, the augmented truncated tetrahedron is one of the Johnson solids (J65). It is created by attaching a triangular cupola (J3) to one hexagonal face of a truncated tetrahedron.
In geometry, the parabiaugmented truncated dodecahedron is one of the Johnson solids (J69). As its name suggests, it is created by attaching two pentagonal cupolas (J5) onto two parallel decagonal faces of a truncated dodecahedron.
In geometry, the triaugmented truncated dodecahedron is one of the Johnson solids (J71); of them, it has the greatest volume in proportion to the cube of the side length. As its name suggests, it is created by attaching three pentagonal cupolas (J5) onto three nonadjacent decagonal faces of a truncated dodecahedron.
A uniform polyhedron has regular polygons as faces and is vertex-transitive. It follows that all vertices are congruent.
The quarter cubic honeycomb, quarter cubic cellulation or bitruncated alternated cubic honeycomb is a space-filling tessellation in Euclidean 3-space. It is composed of tetrahedra and truncated tetrahedra in a ratio of 1:1. It is called "quarter-cubic" because its symmetry unit – the minimal block from which the pattern is developed by reflections – is four times that of the cubic honeycomb.
In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors. It describes a kaleidoscopic construction: each graph "node" represents a mirror and the label attached to a branch encodes the dihedral angle order between two mirrors, that is, the amount by which the angle between the reflective planes can be multiplied to get 180 degrees. An unlabeled branch implicitly represents order-3, and each pair of nodes that is not connected by a branch at all represents a pair of mirrors at order-2.
In hyperbolic geometry, a uniform honeycomb in hyperbolic space is a uniform tessellation of uniform polyhedral cells. In 3-dimensional hyperbolic space there are nine Coxeter group families of compact convex uniform honeycombs, generated as Wythoff constructions, and represented by permutations of rings of the Coxeter diagrams for each family.
In geometry, John Horton Conway defines architectonic and catoptric tessellations as the uniform tessellations of Euclidean 3-space and their duals, as three-dimensional analogue of the Platonic, Archimedean, and Catalan tiling of the plane. The singular vertex figure of an architectonic tessellation is the dual of the cell of catoptric tessellation. The cubille is the only Platonic (regular) tessellation of 3-space, and is self-dual. There are other uniform honeycombs constructed as prismatic stacks which are excluded from these categories.
Ruth Kellerhals is a Swiss mathematician at the University of Fribourg, whose field of study is hyperbolic geometry, geometric group theory and polylogarithm identities.
In geometry, uniform honeycombs in hyperbolic space are tessellations of convex uniform polyhedron cells. In 3-dimensional hyperbolic space there are 23 Coxeter group families of paracompact uniform honeycombs, generated as Wythoff constructions, and represented by ring permutations of the Coxeter diagrams for each family. These families can produce uniform honeycombs with infinite or unbounded facets or vertex figure, including ideal vertices at infinity, similar to the hyperbolic uniform tilings in 2-dimensions.