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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 that don't fit into any series. These are known as '''exceptional objects'''.

Frequently these exceptional objects play a further and important role in the subject. Surprisingly, the exceptional objects in one branch of mathematics are often related 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.

== Regular polytopes ==
{{Main|Regular polytope}}
The prototypical examples of exceptional objects arise when we classify the [[regular polytope]]s. In 2 dimensions we have a series of [[regular polygon|regular ''n''-gons]] for ''n'' ≥ 3. In every dimension above 2 we find analogues of the cube, tetrahedron and octahedron. In 3 dimensions we find two more regular polyhedra – the [[dodecahedron]] (12-cell) and the [[icosahedron]] (20-cell) – making 5 [[Platonic solids]]. In 4 dimensions we have a total of 6 [[Convex regular 4-polytope|regular polytopes]] including the [[120-cell]], the [[600-cell]] and the [[24-cell]]. There are no other regular polytopes; in higher dimensions the only regular polytopes are of the [[hypercube]], [[simplex]], [[orthoplex]] series. So we have three series and 5 exceptional polytopes.

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. In three dimensions there are four [[Kepler–Poinsot polyhedron|Kepler–Poinsot polyhedra]], and in four dimensions ten [[Schläfli–Hess polychoron|Schläfli–Hess polychora]]; in higher dimensions there are no non-convex regular figures.

These can be generalized to [[tessellation]]s of other spaces, especially [[uniform tessellation]]s, notably tilings of Euclidean space ([[honeycomb (geometry)|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 ===
{{see|Schwarz triangle}}
{| class="wikitable" align=right
|[[Image:Sphere symmetry group td.png|120px]] (3 3 2)
|[[Image:Sphere symmetry group oh.png|120px]] (4 3 2)
|[[Image:Sphere symmetry group ih.png|120px]] (5 3 2)
|-
|[[Image:Tile 3,6.svg|120px]] (3 3 3)
|[[Image:Tile V488 bicolor.svg|120px]] (4 4 2)
|[[Image:Tile V46b.svg|120px]] (6 3 2)
|}

Related to tilings and the regular polyhedra, there are exceptional [[Schwarz triangle]]s (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 triangle]]s. 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 triangle|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 ==
{{Main|Sporadic group}}
[[File:Finitesubgroups.svg|thumb|The relations between the sporadic groups, most being related to the monster.]]

The finite simple groups have been [[Classification of finite simple groups|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 "[[Pariah group|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 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 by 3 self-adjoint matrices over the octonions.

== Simple Lie groups ==

The [[simple Lie group]]s form a number of series ([[classical Lie group]]s) labelled A, B, C and D. In addition we have the exceptional groups [[G2 (mathematics)|G2]] (the automorphism group of the octonions), [[F4 (mathematics)|F4]], [[E6 (mathematics)|E6]], [[E7 (mathematics)|E7]], [[E8 (mathematics)|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 so the exceptional Lie groups correspond to exceptional root systems and exceptional [[Dynkin diagram]]s.

== Supersymmetric algebras ==
There are a few exceptional objects with [[supersymmetry]]. The [[Lie superalgebra]]s '''G(3)''' in 31 dimensions and '''F(4)''' in 40 dimensions and the Jordan superalgebras '''K3''' and '''K10''' being the main finite dimensional examples.

== Unimodular lattices ==

Up to isometry there is only one even [[unimodular lattice]] in 15 dimensions or less — the [[E8 lattice|E8 lattice]]. Up to dimension [[24 (number)|24]] there is only one even unimodular lattice with no [[root system|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 [[Conway group|Co1]] is the automorphism group itself modulo ±1. The groups [[Conway group|Co2]] and [[Conway group|Co3]], as well as a number of other sporadic groups, arise as stabilisers of various subsets of the Leech lattice.

== Codes ==

Some [[Error-correcting code|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 <math>M_{24}</math>, one of the sporadic simple groups, is the group of automorphisms of the [[binary Golay code|extended binary Golay code]] and four more of the sporadic simple groups arise as various types of stabilizer subgroup of <math>M_{24}</math>.

== Block designs ==

An exceptional [[block design]] is the [[Steiner system]] S(5,8,24) whose automorphism group is the sporadic simple [[Mathieu group]] <math>M_{24}</math>.

== Snarks ==

In [[graph theory]], we can have [[snark (graph theory)|snark]]s.

== Outer automorphisms ==
Certain families of groups generically have a certain [[outer automorphism group]], but in particular cases they have other, exceptional outer automorphisms.

Among families of finite simple groups, the only example<ref>ATLAS p. xvi</ref> is in the [[automorphisms of the symmetric and alternating groups]]: for <math>n \geq 3, n \neq 6</math> the [[alternating group]] <math>A_n</math> has one outer automorphism (corresponding to conjugation by an odd element of <math>S_n</math>) and the [[symmetric group]] <math>S_n</math> has no outer automorphisms. However, for <math>n=6,</math> there is an [[Automorphisms of the symmetric and alternating groups#exceptional outer automorphism|exceptional outer automorphism]] of <math>S_6</math> (of order 2), and correspondingly, the outer automorphism group of <math>A_6</math> is not <math>C_2</math> (the group of order 2) but rather <math>C_2 \times C_2</math> (the [[Klein four-group]]).

If one instead considers A6 as the (isomorphic) [[projective special linear group]] PSL(2,9), then the outer automorphism is not exceptional; thus the exceptionalness can be seen as due to the [[exceptional isomorphism]] <math>A_6 \cong \operatorname{PSL}(2,9).</math> This exceptional outer automorphism is realized inside of the Mathieu group M12 and similarly, M12 acts on a set of 12 elements in 2 different ways.

[[File:Dynkin diagram D4.png|thumb|The symmetries of the Dynkin diagram D4 correspond to the [[outer automorphism]]s of Spin(8) in triality]]
Among [[Lie groups]], the [[spin group]] Spin(8) has an exceptionally large outer automorphism group (namely <math>S_3</math>), which corresponds to the exceptional symmetries of the [[Dynkin diagram]] D4. This phenomenon is referred to as ''[[triality]].''

The exceptional symmetry of the D4 diagram also gives rise to the [[Steinberg group (Lie theory)|Steinberg groups]].

== Algebraic topology ==
{{Expand section|date=January 2010}}
The five or six (depending on the status of ''n''=126) (framed [[cobordism class]]es of) manifolds of [[Kervaire invariant]] one, which exist in dimension <math>n=2^k-2</math> for <math>n=2, 6, 14, 30, 62,</math> and possibly 126, but no higher, are exceptional objects related to [[exotic sphere]]s. It is conjectured that these are related to the [[Rosenfeld projective plane]]s (over octonions and related) and exceptional Lie algebras, due to similarities of dimensions, but no connection has been established.

== Connections ==
Numerous connections have been observed between some, though not all, of these exceptional objects. Most common are objects related to [[8 (number)|8]] and [[24 (number)|24]] dimensions, noting that 24 = 8 · 3. By contrast, the [[pariah group]]s stand apart, as the name suggests.

=== 8 dimensions ===
* The octonions are 8-dimensional.
* The [[E8 lattice|E8 lattice]] can be realized as the integral octonions (up to a scale factor).
* The exceptional Lie groups can be seen as symmetries of the octonions and structures derived from the octonions; further, the E8 algebra is related to the E8 lattice, as the notation implies (the lattice is generated by the root system of the algebra).
* Triality occurs for Spin(8), which also connects to 8 · 3  = 24.

=== 24 dimensions ===
* The Leech lattice is 24-dimensional.
* Most sporadic simple groups can be related to the Leech lattice, or more broadly the monster.
* The exceptional [[Jordan algebra]] has a representation in terms of 24×24 real matrices together with the Jordan product rule.

=== Other phenomena ===
These objects are connected to various other phenomena in math which may be considered surprising but not themselves "exceptional". For example, in [[algebraic topology]], 8-fold real [[Bott periodicity]] can be seen as coming from the octonions. In the theory of [[modular forms]], the 24-dimensional nature of the Leech lattice underlies the presence of 24 in the formulas for the [[Dedekind eta function]] and the [[modular discriminant]], which connection is deepened by [[monstrous moonshine]], which related modular functions to the monster group.

=== Physics ===
In [[string theory]] and superstring theory we often find that particular dimensions are singled out as a result of exceptional algebraic phenomena. For example, [[bosonic string theory]] requires a spacetime of dimension 26 which is directly related to the presence of 24 in the [[Dedekind eta function]]. Similarly, the possible dimensions of [[supergravity]] are related to the dimensions of the [[division algebras]].

== Unexceptional objects ==
=== Pathologies ===
{{See also|Pathological (mathematics)}}
"Exceptional" object is reserved for objects that are unusual, meaning rare, the exception, not for ''unexpected'' or ''non-standard'' objects. These unexpected-but-typical (or common) phenomena are generally referred to as [[Pathological (mathematics)|pathological]], such as [[nowhere differentiable function]]s, or "exotic", as in [[exotic sphere]]s – there are exotic spheres in arbitrarily high dimension (not only a finite set of exceptions), and in many dimensions most (differential structures on) spheres are exotic.

=== Extremal objects ===
Exceptional objects must be distinguished from ''extremal'' objects: those that fall in a family and are the most extreme example by some measure are of interest, but not unusual in the way exceptional objects are. For example, the [[golden ratio]] ''φ'' has the simplest [[continued fraction]] approximation, and accordingly is most difficult to [[Diophantine approximation|approximate by rationals]]; however, it is but one of infinitely many such quadratic numbers (continued fractions).

Similarly, the (2,3,7) [[Schwarz triangle]] is the smallest hyperbolic Schwarz triangle, and the associated [[(2,3,7) triangle group]] is of particular interest, being the universal [[Hurwitz group]], and thus being associated with the [[Hurwitz curve]]s, the maximally symmetric algebraic curves. However, it falls in a family of such triangles ((2,4,7), (2,3,8), (3,3,7), etc.), and while the smallest, is not exceptional or unlike the others.

== Connections ==
Many of the exceptional objects in mathematics and physics have been found to be connected to each other. Conjectures such as the [[Monstrous moonshine]] conjectures show how, for example, the [[Monster group]] is connected to [[String Theory]]. The theory of [[modular forms]] shows how the E8 is connected to the Monster group. Other interesting connections include how the [[Leech lattice]] is connected via the [[Golay code]] to the adjacency matrix of the [[dodecahedron]] (another exceptional object). Below is a [[mind map]] showing how some of the exceptional objects in mathematical physics are related.

[[File:Exceptionalmindmap.png|600px]]

The connections can partly be explained by thinking of the algebras as a tower of lattice [[vertex operator algebra]]s. It just so happens that the vertex algebras at the bottom are so simple that they are isomorphic to familiar non-vertex algebras. Thus the connections can be seen simply as the consequence of some lattices being sub-lattices of others.

=== Supersymmetries ===
The [[Jordan superalgebra]]s are a parallel set of exceptional objects with [[supersymmetry]]. These are the [[Lie superalgebra]]s which are related to Lorentzian lattices. This subject is less developed so the connections between the objects is less developed. There are new conjectures parallel to the [[Monstrous moonshine]] conjectures for these super-objects involving different sporadic groups.

[[File:supermindmap.png|550px]]

== See also ==
* [[Exceptional isomorphism]]

== References ==
{{reflist}}
{{refbegin}}
* {{citation
|last=Stillwell
|first=John
|jstor=2589218
|title=Exceptional Objects
|journal=[[American Mathematical Monthly]]
|date=November 1998
|pages=850–858
}}
* [[John Baez]] and [[John Huerta]], ''Division Algebras and Supersymmetry II''. {{arxiv|1003.3436}}.
* [http://math.ucr.edu/home/baez/week106.html This Week's Finds in Mathematical Physics, Week 106], [[John Baez]]
* [http://math.ucr.edu/home/baez/platonic.html Platonic Solids in all Dimensions], [[John Baez]]
{{refend}}

[[Category:Mathematical terminology]]

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