Thorold Gosset

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John Herbert de Paz Thorold Gosset (16 October 1869 [1] December 1962) was an English lawyer and an amateur mathematician. In mathematics, he is noted for discovering and classifying the semiregular polytopes in dimensions four and higher.

England Country in north-west Europe, part of the United Kingdom

England is a country that is part of the United Kingdom. It shares land borders with Wales to the west and Scotland to the north-northwest. The Irish Sea lies west of England and the Celtic Sea lies to the southwest. England is separated from continental Europe by the North Sea to the east and the English Channel to the south. The country covers five-eighths of the island of Great Britain, which lies in the North Atlantic, and includes over 100 smaller islands, such as the Isles of Scilly and the Isle of Wight.

Lawyer legal professional who helps clients and represents them in a court of law

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Mathematician person with an extensive knowledge of mathematics

A mathematician is someone who uses an extensive knowledge of mathematics in his or her work, typically to solve mathematical problems.

Contents

Biography

Thorold Gosset was born in Thames Ditton, the son of John Jackson Gosset, a civil servant and statistical officer for HM Customs, [2] and his wife Eleanor Gosset (formerly Thorold). [3] He was admitted to Pembroke College, Cambridge as a pensioner on 1 October 1888, graduated BA in 1891, was called to the bar of the Inner Temple in June 1895, and graduated LLM in 1896. [1] In 1900 he married Emily Florence Wood, [4] and they subsequently had two children, named Kathleen and John. [5]

Thames Ditton village in United Kingdom

Thames Ditton is a suburban village by and on the River Thames, in the Elmbridge borough of Surrey, England. Apart from a large inhabited island in the river, it lies on the southern bank, centred 12.2 miles (19.6 km) southwest of Charing Cross in central London. Thames Ditton is just outside Greater London but within the Greater London Urban Area as defined by the Office for National Statistics. Its clustered village centre and shopping area on a winding High Street is surrounded by housing, schools and sports areas. Its riverside faces the Thames Path and Hampton Court Palace Gardens and golf course in the London Borough of Richmond upon Thames. Its most commercial area is spread throughout its conservation area and contains restaurants, cafés, shops and businesses.

HM Customs

HM Customs was the national Customs service of England until a merger with the Department of Excise in 1909. The phrase 'HM Customs', in use since the Middle Ages, referred both to the customs dues themselves and to the office of state established for their collection, assessment and administration.

Pembroke College, Cambridge college of the University of Cambridge

Pembroke College is a constituent college of the University of Cambridge, England. The college is the third-oldest college of the university and has over seven hundred students and fellows. Physically, it is one of the university's larger colleges, with buildings from almost every century since its founding, as well as extensive gardens. Its members are termed "Valencians".

Mathematics

According to H. S. M. Coxeter, [6] after obtaining his law degree in 1896 and having no clients, Gosset amused himself by attempting to classify the regular polytopes in higher-dimensional (greater than three) Euclidean space. After rediscovering all of them, he attempted to classify the "semi-regular polytopes", which he defined as polytopes having regular facets and which are vertex-uniform, as well as the analogous honeycombs, which he regarded as degenerate polytopes. In 1897 he submitted his results to James W. Glaisher, then editor of the journal Messenger of Mathematics . Glaisher was favourably impressed and passed the results on to William Burnside and Alfred Whitehead. Burnside, however, stated in a letter to Glaisher in 1899 that "the author's method, a sort of geometric intuition" did not appeal to him. He admitted that he never found the time to read more than the first half of Gosset's paper. In the end Glaisher published only a brief abstract of Gosset's results. [7]

Regular polytope polytope whose symmetry group acts transitively on its flags

In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or j-faces — cells, faces and so on — are also transitive on the symmetries of the polytope, and are regular polytopes of dimension ≤ n.

Euclidean space Generalization of Euclidean geometry to higher dimensions

In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and similar spaces of higher dimension. It is named after the Ancient Greek mathematician Euclid of Alexandria. The term "Euclidean" distinguishes these spaces from other types of spaces considered in modern geometry. Euclidean spaces also generalize to higher dimensions.

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

Gosset's results went largely unnoticed for many years. His semiregular polytopes were rediscovered by Elte in 1912 [8] and later by H.S.M. Coxeter who gave both Gosset and Elte due credit.

Coxeter introduced the term Gosset polytopes for three semiregular polytopes in 6, 7, and 8 dimensions discovered by Gosset: the 221, 321, and 421 polytopes. The vertices of these polytopes were later seen to arise as the roots of the exceptional Lie algebras E6, E7 and E8.

2<sub> 21</sub> polytope uniform 6-polytope

In 6-dimensional geometry, the 221 polytope is a uniform 6-polytope, constructed within the symmetry of the E6 group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an 6-ic semi-regular figure. It is also called the Schläfli polytope.

3<sub> 21</sub> polytope uniform 7-polytope

In 7-dimensional geometry, the 321 polytope is a uniform 7-polytope, constructed within the symmetry of the E7 group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an 7-ic semi-regular figure.

4<sub> 21</sub> polytope semiregular uniform 8-polytope

In 8-dimensional geometry, the 421 is a semiregular uniform 8-polytope, constructed within the symmetry of the E8 group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an 8-ic semi-regular figure.

A new and more precise definition of the Gosset Series of polytopes has been given by Conway in 2008 [9]

See also

Related Research Articles

The term semiregular polyhedron is used variously by different authors.

Rectified 600-cell

In geometry, the rectified 600-cell or rectified hexacosichoron is a convex uniform 4-polytope composed of 600 regular octahedra and 120 icosahedra cells. Each edge has two octahedra and one icosahedron. Each vertex has five octahedra and two icosahedra. In total it has 3600 triangle faces, 3600 edges, and 720 vertices.

Rectified 5-cell uniform polychoron

In four-dimensional geometry, the rectified 5-cell is a uniform 4-polytope composed of 5 regular tetrahedral and 5 regular octahedral cells. Each edge has one tetrahedron and two octahedra. Each vertex has two tetrahedra and three octahedra. In total it has 30 triangle faces, 30 edges, and 10 vertices. Each vertex is surrounded by 3 octahedra and 2 tetrahedra; the vertex figure is a triangular prism.

Snub 24-cell

In geometry, the snub 24-cell or snub disicositetrachoron is a convex uniform 4-polytope composed of 120 regular tetrahedral and 24 icosahedral cells. Five tetrahedra and three icosahedra meet at each vertex. In total it has 480 triangular faces, 432 edges, and 96 vertices.

Uniform 6-polytope vertex-transitive 6-polytope bounded by uniform facets

In six-dimensional geometry, a uniform polypeton is a six-dimensional uniform polytope. A uniform polypeton is vertex-transitive, and all facets are uniform 5-polytopes.

Demihypercube polytope constructed from alternation of an hypercube

In geometry, demihypercubes are a class of n-polytopes constructed from alternation of an n-hypercube, labeled as n for being half of the hypercube family, γn. Half of the vertices are deleted and new facets are formed. The 2n facets become 2n(n-1)-demicubes, and 2n(n-1)-simplex facets are formed in place of the deleted vertices.

5-demicube semiregular 5-polytope

In five-dimensional geometry, a demipenteract or 5-demicube is a semiregular 5-polytope, constructed from a 5-hypercube (penteract) with alternated vertices removed.

Gosset–Elte figures

In geometry, the Gosset–Elte figures, named by Coxeter after Thorold Gosset and E. L. Elte, are a group of uniform polytopes which are not regular, generated by a Wythoff construction with mirrors all related by order-2 and order-3 dihedral angles. They can be seen as one-end-ringed Coxeter–Dynkin diagrams.

Semiregular polytope polytope that is vertex-uniform and has all its facets being regular polytopes

In geometry, by Thorold Gosset's definition a semiregular polytope is usually taken to be a polytope that is vertex-uniform and has all its facets being regular polytopes. E.L. Elte compiled a longer list in 1912 as The Semiregular Polytopes of the Hyperspaces which included a wider definition.

In geometry, a uniform k21 polytope is a polytope in k + 4 dimensions constructed from the En Coxeter group, and having only regular polytope facets. The family was named by their Coxeter symbol k21 by its bifurcating Coxeter–Dynkin diagram, with a single ring on the end of the k-node sequence.

Uniform 5-polytope vertex-transitive 5-polytope bounded by uniform facets

In geometry, a uniform 5-polytope is a five-dimensional uniform polytope. By definition, a uniform 5-polytope is vertex-transitive and constructed from uniform 4-polytope facets.

Petrie polygon

In geometry, a Petrie polygon for a regular polytope of n dimensions is a skew polygon in which every (n – 1) consecutive sides belongs to one of the facets. The Petrie polygon of a regular polygon is the regular polygon itself; that of a regular polyhedron is a skew polygon such that every two consecutive side belongs to one of the faces.

1<sub> 22</sub> polytope uniform 6-polytope

In 6-dimensional geometry, the 122 polytope is a uniform polytope, constructed from the E6 group. It was first published in E. L. Elte's 1912 listing of semiregular polytopes, named as V72 (for its 72 vertices).

1<sub> 32</sub> polytope uniform 7-polytope

In 7-dimensional geometry, 132 is a uniform polytope, constructed from the E7 group.

In geometry, the 521 honeycomb is a uniform tessellation of 8-dimensional Euclidean space. The symbol 521 is from Coxeter, named for the length of the 3 branches of its Coxeter-Dynkin diagram.

Emanuel Lodewijk Elte was a Dutch mathematician. He is noted for discovering and classifying semiregular polytopes in dimensions four and higher.

In four-dimensional Euclidean geometry, the snub 24-cell honeycomb, or snub icositetrachoric honeycomb is a uniform space-filling tessellation by snub 24-cells, 16-cells, and 5-cells. It was discovered by Thorold Gosset with his 1900 paper of semiregular polytopes. It is not semiregular by Gosset's definition of regular facets, but all of its cells (ridges) are regular, either tetrahedra or icosahedra.

References

  1. 1 2 "Gosset, John Herbert de Paz Thorold (GST888JH)". A Cambridge Alumni Database. University of Cambridge.
  2. UK Census 1871, RG10-863-89-23
  3. "Register of Marriages". St George Hanover Square 1a. General Register Office for England and Wales. Jan–Mar 1868: 429.
  4. "Register of Marriages". St George Hanover Square 1a. General Register Office for England and Wales. Jun–Sep 1900: 1014.
  5. UK Census 1911, RG14-181-9123-19
  6. Coxeter, H. S. M. (1973). Regular Polytopes (3rd ed.). New York: Dover Publications. ISBN   0-486-61480-8. A brief account of Gosset and his contribution to mathematics is given on page 164.
  7. Gosset, Thorold (1900). "On the regular and semi-regular figures in space of n dimensions". Messenger of Mathematics . 29: 4348.
  8. Elte, E. L. (1912). The Semiregular Polytopes of the Hyperspaces. Groningen: University of Groningen. ISBN   1-4181-7968-X.
  9. Conway,, John H. (2008). The symmetries of Things (1st ed.). Wellesley, Massachusetts: A.K. Peters Ltd. ISBN   978-1-56881-220-5. A new account of Gosset Series is given on pages 411-413.