A *representation theorem* in general means a canonical way of expressing a class of objects using another class of objects, usually more fundamental or easier to grasp. Beside the well-known *group representation theory*, which concerns about viewing (finite or Lie) groups as matrices, here are three lesser known (to me!) representation theorems.

**Birkhoff’s Representation Theorem.** Every distributive lattice is isomorphic to a sublattice of the power set lattice of some set.

**Riesz–Markov–Kakutani Representation Theorem.** Any positive linear functional on the space of compactly supported continuous functions on a locally compact Hausdorff space can be viewed as integration against a measure.

**Kapovich-Millson Universality Theorem.** Any compact smooth manifold is diffeomorphic to a component of the configuration space of some planar linkage.

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Ph.D. student in the Department of Computer Science, University of Illinois at Urbana-Champaign.