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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** 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.

A **lawyer** or **attorney** is a person who practices law, as an advocate, attorney, attorney at law, barrister, barrister-at-law, bar-at-law, civil law notary, counsel, counselor, counsellor, counselor at law, solicitor, chartered legal executive, or public servant preparing, interpreting and applying law, but not as a paralegal or charter executive secretary. Working as a lawyer involves the practical application of abstract legal theories and knowledge to solve specific individualized problems, or to advance the interests of those who hire lawyers to perform legal services.

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

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** 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** 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** 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".

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] }

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*.

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 (

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 2_{21}, 3_{21}, and 4_{21} polytopes. The vertices of these polytopes were later seen to arise as the roots of the exceptional Lie algebras E_{6}, E_{7} and E_{8}.

In 6-dimensional geometry, the **2 _{21}** polytope is a uniform 6-polytope, constructed within the symmetry of the E

In 7-dimensional geometry, the **3 _{21}** polytope is a uniform 7-polytope, constructed within the symmetry of the E

In 8-dimensional geometry, the **4 _{21}** is a semiregular uniform 8-polytope, constructed within the symmetry of the E

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

The term **semiregular polyhedron** is used variously by different authors.

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.

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.

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.

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.

In geometry, **demihypercubes** are a class of n-polytopes constructed from alternation of an n-hypercube, labeled as *hγ _{n}* for being

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

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.

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 k_{21} polytope** is a polytope in

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.

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.

In 6-dimensional geometry, the **1 _{22}** polytope is a uniform polytope, constructed from the E

In 7-dimensional geometry, **1 _{32}** is a uniform polytope, constructed from the E7 group.

In geometry, the **5 _{21} honeycomb** is a uniform tessellation of 8-dimensional Euclidean space. The symbol 5

**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.

- 1 2 "Gosset, John Herbert de Paz Thorold (GST888JH)".
*A Cambridge Alumni Database*. University of Cambridge. - ↑ UK Census 1871, RG10-863-89-23
- ↑ "Register of Marriages". St George Hanover Square 1a. General Register Office for England and Wales. Jan–Mar 1868: 429.
- ↑ "Register of Marriages". St George Hanover Square 1a. General Register Office for England and Wales. Jun–Sep 1900: 1014.
- ↑ UK Census 1911, RG14-181-9123-19
- ↑ 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. - ↑ Gosset, Thorold (1900). "On the regular and semi-regular figures in space of
*n*dimensions".*Messenger of Mathematics*.**29**: 43–48. - ↑ Elte, E. L. (1912).
*The Semiregular Polytopes of the Hyperspaces*. Groningen: University of Groningen. ISBN 1-4181-7968-X. - ↑ 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.

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