Norman Johnson | |
---|---|

Born | |

Died | July 13, 2017 86) | (aged

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, 1930 – July 13, 2017) was a mathematician at Wheaton College, Norton, Massachusetts.^{ [1] }

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] }

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] }

- ———— (1960-05-01). "A Geometric Model for the Generalized Symmetric Group".
*Canadian Mathematical Bulletin*.**3**(2): 133–142. doi: 10.4153/CMB-1960-016-7 . S2CID 124822323. - Grünbaum, Branko; ———— (January 1965). "The Faces of a Regular-Faced Polyhedron".
*Journal of the London Mathematical Society*. s1-40 (1): 577–586. doi:10.1112/jlms/s1-40.1.577. - ———— (January 1966). "Convex polyhedra with regular faces".
*Canadian Journal of Mathematics*.**18**: 169–200. doi: 10.4153/cjm-1966-021-8 . ISSN 0008-414X. MR 0185507. S2CID 122006114. Zbl 0132.14603. - ———— (1966).
*The theory of uniform polytopes and honeycombs*(PhD thesis). University of Toronto. OL 14849556M. Archived from the original on 2022-05-20. Retrieved 2022-05-20. - ———— (December 1969). "
*Euclidean Geometry and Convexity*by Russell V. Benson (review)".*The American Mathematical Monthly*.**76**(10): 1165–1160. doi:10.2307/2317227. JSTOR 2317227. - ———— (January 1981). "Absolute Polarities and Central Inversions". In Davis, C.; Grünbaum, B.; Sherk, F. A. (eds.).
*The Geometric Vein*. New York City: Springer Nature. pp. 443–464. doi:10.1007/978-1-4612-5648-9_28. ISBN 978-1-4612-5648-9. - ————; Weiss, Asia Ivić (July 1999). "Quaternionic modular groups".
*Linear Algebra and Its Applications*.**295**(1): 159–189. doi: 10.1016/S0024-3795(99)00107-X . - ————; Weiss, Asia Ivić (December 1999). "Quadratic Integers and Coxeter Groups".
*Canadian Journal of Mathematics*.**51**(6): 1307–1336. doi: 10.4153/CJM-1999-060-6 . S2CID 111383205. - ————; Kellerhals, Ruth; Ratcliffe, John G.; Tschantz, Steven T. (December 1999). "The size of a hyperbolic Coxeter simplex".
*Transformation Groups*.**4**(4): 329–353. doi:10.1007/BF01238563. S2CID 123105209. - ————; Kellerhals, Ruth; Ratcliffe, John G.; Tschantz, Steven T. (2002-04-15). "Commensurability classes of hyperbolic Coxeter groups".
*Linear Algebra and Its Applications*.**345**(1–3): 119–147. doi: 10.1016/S0024-3795(01)00477-3 . - ———— (2012). "Regular Inversive Polytopes". In Deza, Michel; Petitjean, Michel; Markov, Krassimir (eds.).
*Mathematics of Distances and Applications*. Sofia: ITHEA. Archived from the original on 2022-05-20. Retrieved 2022-05-19. - ———— (2018-06-07).
*Geometries and Transformations*. ISBN 978-1-107-10340-5. OCLC 1043026091. OL 27839953M . Retrieved 2022-05-20.

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

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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

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.

* 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.

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.

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

- 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. - ↑ "Norman Johnson '53".
*Carleton College*. 2017-07-18. Archived from the original on 2022-05-19. Retrieved 2022-05-19. - ↑ Hart, George W. "Johnson solids".
*George W. Hart*. Archived from the original on 2021-08-30. Retrieved 2016-06-10. - ↑ Weisstein, Eric W. "Johnson Solid".
*MathWorld*. Retrieved 2016-06-10. - ↑ Wenninger, Magnus (1983).
*Dual Models*. Cambridge University Press. p. xii. doi:10.1017/CBO9780511569371. ISBN 978-0-521-54325-5. MR 0730208.

- Norman W. Johnson at the Mathematics Genealogy Project
- Norman W. Johnson Endowed Fund in Mathematics and Computer Science at Wheaton College

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