Norman Johnson (mathematician)

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Norman Johnson
Norman Johnson (mathematician).jpg
Born(1930-11-12)November 12, 1930
DiedJuly 13, 2017(2017-07-13) (aged 86)
Citizenship United States
Alma mater University of Toronto
Known for Johnson solid (1966)
Scientific career
Fields Mathematics
Institutions Wheaton College, Norton, Massachusetts
Doctoral advisor H. S. M. Coxeter

Norman Woodason Johnson (November 12, 1930July 13, 2017) was a mathematician at Wheaton College, Norton, Massachusetts. [1]


Early life and education

Norman Johnson was born on November 12, 1930 in Chicago. His father had a bookstore and published a local newspaper. [1]

Johnson earned his undergraduate mathematics degree in 1953 at Carleton College in Northfield, Minnesota [2] followed by a master's degree from the University of Pittsburgh. [1] After graduating in 1953, Johnson did alternative civilian service as a conscientious objector. [1] 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. [1]


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. [3] [4]

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). [5]

Death and final works

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. [1]


Related Research Articles

<span class="mw-page-title-main">Convex uniform honeycomb</span> Spatial tiling of convex uniform polyhedra

In geometry, a convex uniform honeycomb is a uniform tessellation which fills three-dimensional Euclidean space with non-overlapping convex uniform polyhedral cells.

<span class="mw-page-title-main">Polyhedron</span> 3D shape with flat faces, straight edges and sharp corners

In geometry, a polyhedron is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices.

In elementary geometry, a polytope is a geometric object with flat sides (faces). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions n as an n-dimensional polytope or n-polytope. For example, a two-dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope. In this context, "flat sides" means that the sides of a (k + 1)-polytope consist of k-polytopes that may have (k – 1)-polytopes in common.

<span class="mw-page-title-main">4-polytope</span> Four-dimensional geometric object with flat sides

In geometry, a 4-polytope is a four-dimensional polytope. It is a connected and closed figure, composed of lower-dimensional polytopal elements: vertices, edges, faces (polygons), and cells (polyhedra). Each face is shared by exactly two cells. The 4-polytopes were discovered by the Swiss mathematician Ludwig Schläfli before 1853.

<span class="mw-page-title-main">Tridiminished icosahedron</span> 63rd Johnson solid

In geometry, the tridiminished icosahedron is one of the Johnson solids. The name refers to one way of constructing it, by removing three pentagonal pyramids from a regular icosahedron, which replaces three sets of five triangular faces from the icosahedron with three mutually adjacent pentagonal faces.

<span class="mw-page-title-main">Augmented tridiminished icosahedron</span> 64th Johnson solid

In geometry, the augmented tridiminished icosahedron is one of the Johnson solids. It can be obtained by joining a tetrahedron to another Johnson solid, the tridiminished icosahedron.

<span class="mw-page-title-main">Gyroelongated pentagonal birotunda</span> 48th Johnson solid

In geometry, the gyroelongated pentagonal birotunda is one of the Johnson solids. As the name suggests, it can be constructed by gyroelongating a pentagonal birotunda by inserting a decagonal antiprism between its two halves.

<span class="mw-page-title-main">Augmented pentagonal prism</span> 52nd Johnson solid

In geometry, the augmented pentagonal prism is one of the Johnson solids. As the name suggests, it can be constructed by augmenting a pentagonal prism by attaching a square pyramid to one of its equatorial faces.

<span class="mw-page-title-main">Augmented dodecahedron</span> 58th Johnson solid

In geometry, the augmented dodecahedron is one of the Johnson solids, consisting of a dodecahedron with a pentagonal pyramid 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.

<span class="mw-page-title-main">Metagyrate diminished rhombicosidodecahedron</span> 78th Johnson solid

In geometry, the metagyrate diminished rhombicosidodecahedron is one of the Johnson solids. It can be constructed as a rhombicosidodecahedron with one pentagonal cupola rotated through 36 degrees, and a non-opposing pentagonal cupola removed.

<span class="mw-page-title-main">Augmented truncated cube</span> 66th Johnson solid

In geometry, the augmented truncated cube is one of the Johnson solids. As its name suggests, it is created by attaching a square cupola onto one octagonal face of a truncated cube.

<span class="mw-page-title-main">Parabiaugmented truncated dodecahedron</span> 69th Johnson solid

In geometry, the parabiaugmented truncated dodecahedron is one of the Johnson solids. As its name suggests, it is created by attaching two pentagonal cupolas onto two parallel decagonal faces of a truncated dodecahedron.

<span class="mw-page-title-main">Uniform polyhedron</span> Isogonal polyhedron with regular faces

In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive. It follows that all vertices are congruent.

<span class="mw-page-title-main">Triangular prism</span> Prism with a 3-sided base

In geometry, a triangular prism or trigonal prism is a prism with 2 triangular bases. If the edges pair with each triangle's vertex and if they are perpendicular to the base, it is a right triangular prism. A right triangular prism may be both semiregular and uniform.

<i>Regular Polytopes</i> (book) Geometry book

Regular Polytopes is a geometry book on regular polytopes written by Harold Scott MacDonald Coxeter. It was originally published by Methuen in 1947 and by Pitman Publishing in 1948, with a second edition published by Macmillan in 1963 and a third edition by Dover Publications in 1973. The Basic Library List Committee of the Mathematical Association of America has recommended that it be included in undergraduate mathematics libraries.

<span class="mw-page-title-main">Architectonic and catoptric tessellation</span> Uniform Euclidean 3D tessellations and their duals

In geometry, John Horton Conway defines architectonic and catoptric tessellations as the uniform tessellations of Euclidean 3-space with prime space groups 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 the corresponding catoptric tessellation, and vice versa. The cubille is the only Platonic (regular) tessellation of 3-space, and is self-dual. There are other uniform honeycombs constructed as gyrations or 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.

In three-dimensional hyperbolic geometry, the alternated hexagonal tiling honeycomb, h{6,3,3}, or , is a semiregular tessellation with tetrahedron and triangular tiling cells arranged in an octahedron vertex figure. It is named after its construction, as an alteration of a hexagonal tiling honeycomb.

<span class="mw-page-title-main">Regular skew apeirohedron</span> Infinite regular skew polyhedron

In geometry, a regular skew apeirohedron is an infinite regular skew polyhedron. They have either skew regular faces or skew regular vertex figures.


  1. 1 2 3 4 5 6 Weiss, Asia Ivić; Stehle, Eva Marie (2017). "Norman W. Johnson (12 November 1930 to 13 July 2017)". The Art of Discrete and Applied Mathematics. 1: #N1.01. doi: 10.26493/2590-9770.1231.403 . ISSN   2590-9770. Archived from the original on 2022-05-19. Retrieved 2022-05-19.
  2. "Norman Johnson '53". Carleton College . 2017-07-18. Archived from the original on 2022-05-19. Retrieved 2022-05-19.
  3. Hart, George W. "Johnson solids". George W. Hart . Archived from the original on 2021-08-30. Retrieved 2016-06-10.
  4. Weisstein, Eric W. "Johnson Solid". MathWorld . Retrieved 2016-06-10.
  5. Wenninger, Magnus (1983). Dual Models. Cambridge University Press. p. xii. doi:10.1017/CBO9780511569371. ISBN   978-0-521-54325-5. MR   0730208.