6.4 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Exceptional object | 1/3 | https://en.wikipedia.org/wiki/Exceptional_object | reference | science, encyclopedia | 2026-05-05T07:23:52.521982+00:00 | kb-cron |
In mathematics, an exceptional object is one of finitely many exceptions to some classification of objects. Many branches of mathematics study objects of a given type and prove a classification theorem. A common theme is that the classification results in a number of series of objects and a finite number of exceptions — often with desirable properties — that do not fit into any series; these are known as exceptional objects. In many cases, these exceptional objects play a further and important role in the subject. Furthermore, the exceptional objects in one branch of mathematics often relate to the exceptional objects in others. A related phenomenon is exceptional isomorphism, when two series are in general different, but agree for some small values. For example, spin groups in low dimensions are isomorphic to other classical Lie groups.
== Regular polytopes ==
The prototypical examples of exceptional objects arise in the classification of regular polytopes: in two dimensions, there is a series of regular n-gons for n ≥ 3. In every dimension above 2, one can find analogues of the cube, tetrahedron and octahedron. In three dimensions, one finds two more regular polyhedra — the dodecahedron (12-hedron) and the icosahedron (20-hedron) — making five Platonic solids. In four dimensions, a total of six regular polytopes exist, including the 120-cell, the 600-cell and the 24-cell. There are no other regular polytopes, as the only regular polytopes in higher dimensions are of the hypercube, simplex, orthoplex series. In all dimensions combined, there are therefore three series and five exceptional polytopes. Moreover, the pattern is similar if non-convex polytopes are included: in two dimensions, there is a regular star polygon for every rational number
p
/
q
>
2
{\displaystyle \textstyle p/q>2}
. In three dimensions, there are four Kepler–Poinsot polyhedra, and in four dimensions, ten Schläfli–Hess polychora; in higher dimensions, there are no non-convex regular figures. These can be generalized to tessellations of other spaces, especially uniform tessellations, notably tilings of Euclidean space (honeycombs), which have exceptional objects, and tilings of hyperbolic space. There are various exceptional objects in dimension below 6, but in dimension 6 and above, the only regular polyhedra/tilings/hyperbolic tilings are the simplex, hypercube, cross-polytope, and hypercube lattice.
=== Schwarz triangles ===
Related to tilings and the regular polyhedra, there are exceptional Schwarz triangles (triangles that tile the sphere, or more generally Euclidean plane or hyperbolic plane via their triangle group of reflections in their edges), particularly the Möbius triangles. In the sphere, there are 3 Möbius triangles (and 1 1-parameter family), corresponding to the 3 exceptional Platonic solid groups, while in the Euclidean plane, there are 3 Möbius triangles, corresponding to the 3 special triangles: 60-60-60 (equilateral), 45-45-90 (isosceles right), and 30-60-90. There are additional exceptional Schwarz triangles in the sphere and Euclidean plane. By contrast, in the hyperbolic plane, there is a 3-parameter family of Möbius triangles, and none exceptional.
== Finite simple groups ==
The finite simple groups have been classified into a number of series as well as 26 sporadic groups. Of these, 20 are subgroups or subquotients of the monster group, referred to as the "Happy Family", while 6 are not, and are referred to as "pariahs". Several of the sporadic groups are related to the Leech lattice, most notably the Conway group Co1, which is the automorphism group of the Leech lattice, quotiented out by its center.
== Division algebras == There are only three finite-dimensional associative division algebras over the reals — the real numbers, the complex numbers and the quaternions. The only non-associative division algebra is the algebra of octonions. The octonions are connected to a wide variety of exceptional objects. For example, the exceptional formally real Jordan algebra is the Albert algebra of 3×3 self-adjoint matrices over the octonions.
== Simple Lie groups == The simple Lie groups form a number of series (classical Lie groups) labelled A, B, C and D. In addition, there are the exceptional groups G2 (the automorphism group of the octonions), F4, E6, E7, E8. These last four groups can be viewed as the symmetry groups of projective planes over O, C⊗O, H⊗O and O⊗O, respectively, where O is the octonions and the tensor products are over the reals. The classification of Lie groups corresponds to the classification of root systems, and thus the exceptional Lie groups correspond to exceptional root systems and exceptional Dynkin diagrams.
== Supersymmetric algebras == There are a few exceptional objects with supersymmetry. The classification of superalgebras by Kac and Tierry-Mieg indicates that the Lie superalgebras G(3) in 31 dimensions and F(4) in 40 dimensions, and the Jordan superalgebras K3 and K10, are examples of exceptional objects.
== Unimodular lattices == Up to isometry, there is only one even unimodular lattice in 15 dimensions or less — the E8 lattice. Up to dimension 24, there is only one even unimodular lattice without roots, the Leech lattice. Three of the sporadic simple groups were discovered by Conway while investigating the automorphism group of the Leech lattice. For example, Co1 is the automorphism group itself modulo ±1. The groups Co2 and Co3, as well as a number of other sporadic groups, arise as stabilisers of various subsets of the Leech lattice.
== Codes == Some codes also stand out as exceptional objects, in particular the perfect binary Golay code, which is closely related to the Leech lattice. The Mathieu group
M
24
{\displaystyle M_{24}}
, one of the sporadic simple groups, is the group of automorphisms of the extended binary Golay code, and four more of the sporadic simple groups arise as various types of stabilizer subgroup of
M
24
{\displaystyle M_{24}}
.