  We solve a problem of Littlewood: there exist seven infinite circular
cylinders of unit radius which mutually touch each other. In fact, we exhibit
two such sets of cylinders. Our approach is algebraic and uses symbolic and
numerical computational techniques. We consider a system of polynomial
equations describing the position of the axes of the cylinders in the 3
dimensional space. To have the same number of equations (namely 20) as the
number of variables, the angle of the first two cylinders is fixed to 90
degrees, and a small family of direction vectors is left out of consideration.
Homotopy continuation method has been applied to solve the system. The number
of paths is about 121 billion, it is hopeless to follow them all. However,
after checking 80 million paths, two solutions are found. Their validity, i.e.,
the existence of exact real solutions close to the approximate solutions at
hand, was verified with the alphaCertified method as well as by the interval
Krawczyk method.
