Exceptional object

Many branches of mathematics study objects of a given type and prove a classification theorem.

In many cases, these exceptional objects play a further and important role in the subject.

[1][2][3] A related phenomenon is exceptional isomorphism, when two series are in general different, but agree for some small values.

In three dimensions, one finds two more regular polyhedra — the dodecahedron (12-hedron) and the icosahedron (20-hedron) — making five Platonic solids.

[5] Moreover, the pattern is similar if non-convex polytopes are included: in two dimensions, there is a regular star polygon for every rational number

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.

The codewords of the extended binary Golay code have a length of 24 bits and have weights 0, 8, 12, 16, or 24.

The bits of a 24-bit word can be thought of as specifying the possible subsets of a 24 element set.

has one outer automorphism (corresponding to conjugation by an odd element of

This exceptional outer automorphism is realized inside of the Mathieu group

[13][14] These five or six framed cobordism classes of manifolds having Kervaire invariant 1 are exceptional objects related to exotic spheres.

The first three cases are related to the complex numbers, quaternions and octonions respectively: a manifold of Kervaire invariant 1 can be constructed as the product of two spheres, with its exotic framing determined by the normed division algebra.

[15] Due to similarities of dimensions, it is conjectured that the remaining cases (dimensions 30, 62 and 126) are related to the Rosenfeld projective planes, which are defined over algebras constructed from the octonions.

Specifically, it has been conjectured that there is a construction that takes these projective planes and produces a manifold with nonzero Kervaire invariant in two dimensions lower, but this remains unconfirmed.

[16] In quantum information theory, there exist structures known as SIC-POVMs or SICs, which correspond to maximal sets of complex equiangular lines.

Some of the known SICs—those in vector spaces of 2 and 3 dimensions, as well as certain solutions in 8 dimensions—are considered exceptional objects and called "sporadic SICs".

They differ from the other known SICs in ways that involve their symmetry groups, the Galois theory of the numerical values of their vector components, and so forth.

[17] The sporadic SICs in dimension 8 are related to the integral octonions.

Likewise, exceptional objects related to the number 24 include the following.

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, a development that related modular functions to the Monster group.

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.

[21] Many of the exceptional objects in mathematics and physics have been found to be connected to each other.

Developments 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 algebra E8 is connected to the Monster group.

(In fact, well before the proof of the Monstrous moonshine conjecture, the elliptic j-function was discovered to encode the representations of E8.

The Jordan superalgebras are a parallel set of exceptional objects with supersymmetry.

For example, the golden ratio φ has the simplest continued fraction approximation, and accordingly is most difficult to approximate by rationals; however, it is but one of infinitely many such quadratic numbers (continued fractions).