We consider the space V_{n} of quadratic rational maps with one named critical point of period n, quotiented by M\"obius conjugacy preserving named critical points. Thus, the space V_{1} is the space of all quadratic polynomials up to affine conjugacy. The parameter space V_{1} must be one of the most studied in all dynamics. But V_{1} differs from most other parameter spaces in an important respect: there is a natural ``base map'' and, within the Mandelbrot set, natural paths, up to a natural homotopy, to any other map in the Mandelbrot set. The simplest parameter space for which this fails to be true is V_{3}.

One can say (truthfully) that there is no canonical choice of fundamental domain for V_{3,m}, which is obtained from V_{3} by removing a natural dynamically defined puncture set. I shall exhibit a fundamental domain, using the dynamical planes of three maps within V_{3} and a theory known as the ``Resident's View''. This enables one to at least formulate an analogue of MLC for this family.

Three parts of the fundamental domain are straightforward (although the proof in one of these cases is probably new). The structure of the fourth part is much more interesting, involving a spiral in the dynamical plane of the so-called ``aeroplane'' quadratic polynomial z\mapsto z^{2}+c for the (unique) real parameter c for which the critical point 0 has period 3.