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# Contents

## Idea

Various kinds of (mathematical) structures have classifications by which the possible examples/cases of these structures are elements either

1. of some infinite sequence of such structures, following some systematic construction rule,

2. or else of a (typically finite) set of examples which follow no such rule.

In the second case, these “unsystematic” examples of the given kind of structure are then often called exceptional or sporadic.

Exceptional structures are often related to one another. For example, the exceptional Lie group G2 is the symmetry group of the octonions, and the other exceptional Lie groups are related to the octonions in the Freudenthal magic square, while the Albert algebra is the exceptional Jordan algebra of 3-by-3 hermitian matrices over the octonions. The Leech lattice is related to the Monster simple group, and can also be connected with the octonions (Wilson 09). Moreover, all these structures tend to appear as aspects of M-theory.