Thorold Gosset
Encyclopedia
Thorold Gosset 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.
According to H. S. M. Coxeter, Gosset, after attaining his law degree in 1895 and having no clients, amused himself by attempting to classify the regular polytope
s 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 polytope
s having regular
facets and which are vertex-uniform, as well as the analogous honeycomb
s, 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 Glashier published only a brief abstract of Gosset results.
Gosset's results went largely unnoticed for many years. His semiregular polytopes were rediscovered by Elte
in 1912 and later by H.S.M. Coxeter who gave both Gosset and Elte due credit.
Semiregular figures Gosset enumerated: (his names in parentheses)
England
England is a country that is part of the United Kingdom. It shares land borders with Scotland to the north and Wales to the west; the Irish Sea is to the north west, the Celtic Sea to the south west, with the North Sea to the east and the English Channel to the south separating it from continental...
lawyer
Lawyer
A lawyer, according to Black's Law Dictionary, is "a person learned in the law; as an attorney, counsel or solicitor; a person who is practicing law." Law is the system of rules of conduct established by the sovereign government of a society to correct wrongs, maintain the stability of political...
and an amateur mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
. In mathematics, he is noted for discovering and classifying the semiregular polytopes in dimensions four and higher.
According to H. S. M. Coxeter, Gosset, after attaining his law degree in 1895 and having no clients, amused himself by attempting to classify the regular polytope
Regular polytope
In mathematics, a regular polytope is a polytope whose symmetry is transitive 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...
s in higher dimensional (greater than three) Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
. After rediscovering all of them, he attempted to classify the "semi-regular polytopes", which he defined as polytope
Polytope
In elementary geometry, a polytope is a geometric object with flat sides, which exists in any general number of dimensions. A polygon is a polytope in two dimensions, a polyhedron in three dimensions, and so on in higher dimensions...
s having regular
Regular polytope
In mathematics, a regular polytope is a polytope whose symmetry is transitive 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...
facets and which are vertex-uniform, as well as the analogous honeycomb
Honeycomb (geometry)
In geometry, a honeycomb is a space filling or close packing of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions....
s, which he regarded as degenerate polytopes. In 1897 he submitted his results to James W. Glaisher
James Whitbread Lee Glaisher
James Whitbread Lee Glaisher son of James Glaisher, the meteorologist, was a prolific English mathematician.He was educated at St Paul's School and Trinity College, Cambridge, where he was second wrangler in 1871...
, then editor of the journal Messenger of Mathematics
Messenger of Mathematics
The Messenger of Mathematics is a defunct mathematics journal.The editor-in-chief was William Allen Whitworth with Charles Taylor and volumes 1–58 were published between 1872 and 1929...
. Glaisher was favourably impressed and passed the results on to William Burnside
William Burnside
William Burnside was an English mathematician. He is known mostly as an early contributor to the theory of finite groups....
and Alfred Whitehead
Alfred North Whitehead
Alfred North Whitehead, OM FRS was an English mathematician who became a philosopher. He wrote on algebra, logic, foundations of mathematics, philosophy of science, physics, metaphysics, and education...
. 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 Glashier published only a brief abstract of Gosset results.
Gosset's results went largely unnoticed for many years. His semiregular polytopes were rediscovered by Elte
E. L. Elte
Emanuel Lodewijk Elte was a Dutch mathematician. He is noted for discovering and classifying semiregular polytopes in dimensions four and higher....
in 1912 and later by H.S.M. Coxeter who gave both Gosset and Elte due credit.
See also
- Gosset graphGosset graphThe Gosset graph, named after Thorold Gosset, is a specific regular graph with 56 vertices and valency 27.-Construction:...
Semiregular figures Gosset enumerated: (his names in parentheses)
- Convex uniform honeycombConvex uniform honeycombIn geometry, a convex uniform honeycomb is a uniform tessellation which fills three-dimensional Euclidean space with non-overlapping convex uniform polyhedral cells.Twenty-eight such honeycombs exist:* the familiar cubic honeycomb and 7 truncations thereof;...
s, two 3D honeycombs:- Tetrahedral-octahedral honeycombTetrahedral-octahedral honeycombThe tetrahedral-octahedral honeycomb or alternated cubic honeycomb is a space-filling tessellation in Euclidean 3-space. It is composed of alternating octahedra and tetrahedra in a ratio of 1:2....
(Simple tetroctahedric check) - Gyrated alternated cubic honeycomb (Complex tetroctahedric check)
- Tetrahedral-octahedral honeycomb
- Uniform polychoraUniform polychoronIn geometry, a uniform polychoron is a polychoron or 4-polytope which is vertex-transitive and whose cells are uniform polyhedra....
, three 4-polytopes:- Rectified 5-cellRectified 5-cellIn four dimensional geometry, the rectified 5-cell is a uniform polychoron 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...
(Tetroctahedric) - Snub 24-cellSnub 24-cellIn geometry, the snub 24-cell is a convex uniform polychoron 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....
(Tetricosahedric)- Snub 24-cell honeycombSnub 24-cell honeycombIn 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...
- Snub 24-cell honeycomb
- Rectified 600-cellRectified 600-cellIn geometry, a rectified 600-cell is a uniform polychoron formed as the rectification of the regular 600-cell.There are four rectifications of the 600-cell, including the zeroth, the 600-cell itself...
(Octicosahedric)
- Rectified 5-cell
- Semiregular E-polytopeSemiregular E-polytopeIn geometry, a uniform k21 polytope is a polytope in k + 4 dimensions constructed from the En Coxeter group, and having only regular polytope facets...
s, four polytopes, and one honeycomb:- 5-demicube (5-ic semi-regular), a 5-polytope5-polytopeIn five-dimensional geometry, a 5-polytope is a 5-dimensional polytope, bounded by facets. Each polyhedral cell being shared by exactly two polychoron facets. A proposed name for 5-polytopes is polyteron.-Definition:...
- 221 polytope (6-ic semi-regular), a 6-polytope6-polytopeIn six-dimensional geometry, a uniform polypeton is a six-dimensional uniform polytope. A uniform polypeton is vertex-transitive, and all facets are uniform polytera....
- 321 polytope (7-ic semi-regular), a 7-polytope7-polytopeIn seven-dimensional geometry, a 7-polytope is a polytope contained by 6-polytope facets. Each 5-polytope ridge being shared by exactly two 6-polytope facets....
- 421 polytope (8-ic semi-regular), an 8-polytope8-polytopeIn eight-dimensional geometry, a polyzetton is a polytope contained by 7-polytope facets. Each 6-polytope ridge being shared by exactly two 7-polytope facets....
- 521 honeycomb5 21 honeycombIn geometry, the 521 honeycomb is a uniform tessellation of 8-dimensional Euclidean space.This honeycomb was first studied by Gosset who called it a 9-ic semi-regular figure...
(9-ic check) (8D honeycomb)
- 5-demicube (5-ic semi-regular), a 5-polytope