Peaucellier–Lipkin inversor: http://en.wikipedia.org/wiki/Peaucellier-Lipkin_linkage By mid-19th century it was widely believed that one cannot transform circular motion to linear motion. For instance, Chebyshev tried quite hard but gave up and invented his polynomials instead, to deal with the issue approximately. The construction of inversor is simple and ingenious.
Mnev's Universality Theorem dealing with configuration spaces of linear arrangements and convex polytopes. The idea is that one can encode elementary algebraic operations into elementary geometric objects (actually, this goes back to Von Staudt in 19th century).
Misha
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