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 Group*.**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, 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 (*J*_{17}). 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 (*J*_{10}). 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 (*J*_{63}).

In geometry, the **elongated square gyrobicupola** or **pseudo-rhombicuboctahedron** is one of the Johnson solids (*J*_{37}). 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 (*J*_{23}). As the name suggests, it can be constructed by gyroelongating a square cupola (*J*_{4}) by attaching an octagonal antiprism to its base. It can also be seen as a gyroelongated square bicupola (*J*_{45}) with one square bicupola removed.

In geometry, the **gyroelongated pentagonal birotunda** is one of the Johnson solids (*J*_{48}). 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 (*J*_{81}).

In geometry, the **augmented dodecahedron** is one of the Johnson solids (*J*_{58}), consisting of a dodecahedron with a pentagonal pyramid (*J*_{2}) 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 (*J*_{65}). It is created by attaching a triangular cupola (*J*_{3}) to one hexagonal face of a truncated tetrahedron.

In geometry, the **parabiaugmented truncated dodecahedron** is one of the Johnson solids (*J*_{69}). As its name suggests, it is created by attaching two pentagonal cupolas (*J*_{5}) onto two parallel decagonal faces of a truncated dodecahedron.

In geometry, the **triaugmented truncated dodecahedron** is one of the Johnson solids (*J*_{71}); 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 (*J*_{5}) 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.

- 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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