•Steam machine (1722, Newcomen)

Watt's parallelogram, approximate solution


1864 Peaullicier

1871 Lipkin, 1873 Peaullicier

Hart, Kempe

Lubricants, British Parlament 1870-80

(The Penguin Dictionary of Curious and Interesting Geometry, 1991).

• Kapovich, Millson: “Universality theorems for planar linkages.”


Theorem (Kempe)
Let S be a real algebraic plane curve, p a point in S. There is an abstract, complex and closed mechanism L, a Zariski closed set Z, which is a union of irreducible components of M(L) (realizations of L) and a closed (in the usual topology) neighborhood U of p in S, such that the restriction of the input to Z is onto U.

But Z is not open in M(L), U≠S, even if S is compact and the application Z--->U is not a trivial covering (injective).

Theorem (Thurston)
Let M be a compact smooth manifold, then there is a mechanism L such that M is diffeomorphic to a union of components of M(L).

Computing degrees of freedom for planar linkages