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The Catalan numbers, named for the Belgian mathematician Eugène Charles Catalan, are a sequence of natural numbers that are important in combinatorial mathematics. The sequence begins:

1, 1, 2, 5, 14, 42, 132, ...

The Catalan numbers are solutions to numerous counting problems which often have a recursive flavour. In fact, one author lists over 60 different possible interpretations of these numbers. For example, the n^th Catalan number is the number of full binary trees with n internal nodes, or n+1 leaves. It is also the number of ways of associating n applications of a binary operator as well as the number of ways of a convex polygon with n + 2 sides can be cut into triangles by connecting vertices with straight lines.

The Lorenz attractor is a non-linear dynamical system derived from the simplified equations of convection rolls in certain atmospheric equations. For a certain set of parameters the system exhibits chaotic behavior and forms what is called a strange attractor.

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• it is unknown whether π and e are algebraically independent?
• there are precisely six convex regular polytopes in 4-dimensions? These are analogs of the five Platonic solids known to the ancient Greeks.
• that the set of rational numbers is equal in size to the subset of integers; that is, they can be put in one-to-one correspondence?
• the Riemann hypothesis is one of the most important unsolved problems in mathematics?
• Chaitin's constant Ω is a real number which is definable but not computable?

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