geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, a Goursat tetrahedron is a

tetrahedral In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ...

fundamental domain Given a topological space and a group acting on it, the images of a single point under the group action form an orbit of the action. A fundamental domain or fundamental region is a subset of the space which contains exactly one point from each o ...

of a

Wythoff construction In geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling. It is often referred to as Wythoff's kaleidoscopic construction. Construction process ...

. Each tetrahedral face represents a reflection hyperplane on 3-dimensional surfaces: the 3-sphere, Euclidean 3-space, and hyperbolic 3-space.

Coxeter Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington to ...

named them after

Édouard Goursat Édouard Jean-Baptiste Goursat (21 May 1858 – 25 November 1936) was a French mathematician, now remembered principally as an expositor for his ''Cours d'analyse mathématique'', which appeared in the first decade of the twentieth century. It se ...

who first looked into these domains. It is an extension of the theory of

Schwarz triangle In geometry, a Schwarz triangle, named after Hermann Schwarz, is a spherical triangle that can be used to tile a sphere (spherical tiling), possibly overlapping, through reflections in its edges. They were classified in . These can be defined mor ...

s for Wythoff constructions on the sphere.

Graphical representation

A Goursat tetrahedron can be represented graphically by a tetrahedral graph, which is in a dual configuration of the fundamental domain tetrahedron. In the graph, each node represents a face (mirror) of the Goursat tetrahedron. Each edge is labeled by a rational value corresponding to the reflection order, being π/

dihedral angle A dihedral angle is the angle between two intersecting planes or half-planes. In chemistry, it is the clockwise angle between half-planes through two sets of three atoms, having two atoms in common. In solid geometry, it is defined as the uni ...

. :

A 4-node Coxeter-Dynkin diagram represents this tetrahedral graph with order-2 edges hidden. If many edges are order 2, the

Coxeter group In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean refl ...

can be represented by a bracket notation. Existence requires each of the 3-node subgraphs of this graph, (p q r), (p u s), (q t u), and (r s t), must correspond to a

Extended symmetry

An extended symmetry of the Goursat tetrahedron is a

semidirect product In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product: * an ''inner'' semidirect product is a particular way in w ...

of the

symmetry and the

symmetry (the Goursat tetrahedron in these cases).

Coxeter notation In geometry, Coxeter notation (also Coxeter symbol) is a system of classifying symmetry groups, describing the angles between fundamental reflections of a Coxeter group in a bracketed notation expressing the structure of a Coxeter-Dynkin diagram ...

supports this symmetry as double-brackets like [X_means_full_Coxeter_group_symmetry_[X.html" ;"title=".html" ;"title="[X">[X means full Coxeter group symmetry [X">.html" ;"title="[X">[X means full Coxeter group symmetry [X with Y as a symmetry of the Goursat tetrahedron. If Y is a pure reflective symmetry, the group will represent another Coxeter group of mirrors. If there is only one simple doubling symmetry, Y can be implicit like X with either reflectional or rotational symmetry depending on the context. The extended symmetry of each Goursat tetrahedron is also given below. The highest possible symmetry is that of the regular

tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...

as ,3 and this occurs in the prismatic point group ,2,2or ^[3,3/sup>.html"_;"title=",3.html"_;"title="^{[3,3">^{[3,3/sup>">,3.html"_;"title="^{[3,3">^{[3,3/sup>and_the_paracompact_hyperbolic_group_[3^{[3,3.html" ;"title=",3">^{[3,3/sup>.html" ;"title=",3.html" ;"title="^{[3,3">^{[3,3/sup>">,3.html" ;"title="^{[3,3">^{[3,3/sup>and the paracompact hyperbolic group [3^{[3,3">,3">^{[3,3/sup>.html" ;"title=",3.html" ;"title="^{[3,3">^{[3,3/sup>">,3.html" ;"title="^{[3,3">^{[3,3/sup>and the paracompact hyperbolic group [3^{[3,3/sup>].

See Tetrahedron#Isometries of irregular tetrahedra for 7 lower symmetry isometries of the tetrahedron.

Whole number solutions

The following sections show all of the whole number Goursat tetrahedral solutions on the 3-sphere, Euclidean 3-space, and Hyperbolic 3-space. The extended symmetry of each tetrahedron is also given.

The colored tetrahedal diagrams below are vertex figure

In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off.
Definitions

Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connect ...
s for omnitruncated polytopes and honeycombs from each symmetry family. The edge labels represent polygonal face orders, which is double the Coxeter graph branch order. The dihedral angle

A dihedral angle is the angle between two intersecting planes or half-planes. In chemistry, it is the clockwise angle between half-planes through two sets of three atoms, having two atoms in common. In solid geometry, it is defined as the uni ...
of an edge labeled ''2n'' is π/''n''. Yellow edges labeled 4 come from right angle (unconnected) mirror nodes in the Coxeter diagram.

3-sphere (finite) solutions

The solutions for the 3-sphere with density 1 solutions are: ( Uniform polychora)

Euclidean (affine) 3-space solutions

Density 1 solutions: Convex uniform honeycomb

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

Twenty-eight such honey ...
s:

Compact hyperbolic 3-space solutions

Density 1 solutions: (Convex uniform honeycombs in hyperbolic space

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 Wyt ...
) ( Coxeter diagram#Compact (Lannér simplex groups))

Paracompact hyperbolic 3-space solutions

Density 1 solutions: (See Coxeter diagram#Paracompact (Koszul simplex groups))

Rational solutions

There are hundreds of rational solutions for the 3-sphere, including these 6 linear graphs which generate the Schläfli-Hess polychora, and 11 nonlinear ones from Coxeter:

In all, there are 59 sporadic tetrahedra with rational angles, and 2 infinite families.https://arxiv.org/abs/2011.14232 Space vectors forming rational angles, Kiran S. Kedlaya, Alexander Kolpakov, Bjorn Poonen, Michael Rubinstein, 2020

See also

* Point group

In geometry, a point group is a mathematical group of symmetry operations (isometries in a Euclidean space) that have a fixed point in common. The coordinate origin of the Euclidean space is conventionally taken to be a fixed point, and every p ...
for ''n''-simplex solutions on (''n''-1)-sphere.

References

* ''Regular Polytopes

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 -faces (for all , where is the dimension of the polytope) — cells, ...
'', (3rd edition, 1973), Dover edition, (page 280, Goursat's tetrahedra

* Norman Johnson (mathematician), Norman Johnson ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. (1966) He proved the enumeration of the Goursat tetrahedra by Coxeter is complete
* Goursat, Edouard, ''Sur les substitutions orthogonales et les divisions régulières de l'espace'', Annales Scientifiques de l'École Normale Supérieure, Sér. 3, 6 (1889), (pp. 9–102, pp. 80–81 tetrahedra)
* {{KlitzingPolytopes, ../explain/goursat.htm, Dynkin Diagrams, Goursat tetrahedra
* Norman Johnson (mathematician), Norman Johnson, ''Geometries and Transformations'' (2018), Chapters 11,12,13
* N. W. Johnson, R. Kellerhals, J. G. Ratcliffe, S. T. Tschantz, ''The size of a hyperbolic Coxeter simplex'', Transformation Groups 1999, Volume 4, Issue 4, pp 329–35

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