A Computational Study of Sofas and Cars
Philip Gibbs · 2014
Abstract: There is a class of geometric problem that seeks to find the shape of largest area that can pass down a corridor of given form or turn round inside a given shape. A popular example is the moving sofa problem for a shape that can be moved round an L-shaped corner in a corridor of width one. This problem has a conjectured solution proposed by Gerver in 1992. We investigate some of these problems numerically giving strong empirical evidence that Gerver was right and that a similar solution can be constructed for the related Conway car problem. The Sofa Problem In 1966 Moser posed his moving sofa problem [1]. What is the shape of largest area that can be manoeuvred round a right angle turn in a corridor of width one? A shape that obviously works is a semi-circle of radius one, but Hammersley [2] extended the shape using a rectangular mid-section with a semi-circle cut out of one side. By Thales ’ theorem the semi-circle can slide round the corner touching the inner side in three places while the remaining arcs of the original semi-circle each touch the outer sides of the corridor in two further places. The maximum area of such a shape is given when the radius of the removed semi-circle is equal to