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1. 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.

2. 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).

3. Connelly's flexible polyhedron is an example of a polyhedral sphere embedded in ${\mathbb R}^3$ which admits nontrivial deformations (so that each boundary face stays rigid). Cauchy proved (with some gaps fixed over 100 years later) that there are no flexible convex polyhedra, but general rigidity problem was open for over 150 years. People tended to believe that such polyhedra do not exist (for instance, "generic" polyhedral spheres are rigid). Connelly started by trying to prove non-existence and ended up constructing a counter-example, again, simple and ingenious.

2 fixed link to wikipedia (%27)
1. 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.

2. Mnev's Universality Theorem: http://en.wikipedia.org/wiki/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).