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In mathematics, the idea of irreducibility plays a fundamental role. It concerns objects which cannot be broken down into simpler objects of the same kind. Mathematicians then focus their research on these "simplest" objects, the irreducibles. For example irreducible numbers are prime, the building blocks of all other numbers.

Each of the following objects, sampled from across mathematical disciplines, has an irreducible version: Numbers, polynomials, algebraic varieties, Cantor sets, algebraic curves, lattices, non-deterministic sets, topological spaces, hyperbolic geometries, simplices (simplexes), polyhedra, contour integrals, and harmonic series.