Note on self-propagation of turbulent spots
numbers of reflections are 9, 15 and n + 1. This completes the proof for the case of three mirrors. For more than three mirrors all the considerations preceding equation 2 are valid, but the corresponding inequality has no integer solutions at all. To prove only Synge's result, which is included above, a much simpler method suffices. If a point arbitrarily near a vertex is reflected successively once in each of the sides meeting there the image remains arbitrarily close to the vertex. The third reflection must bring it to the antipode, an angular distance arbitrarily close to t radius. Thus the angular distance from the vertex to the third side is arbitrarily near x/2, and this applies to each side. Thus the triangle must be an octant of a sphere, which completes the necessity proof. The sufficiency is proved by reflecting an octant three times in its sides. It is of interest to compare our results with those described on pps. 81-83 of Coxeter's "Regular Polytopes" (Methuen, London, 1948). There the discrete groups of reflections generated by three or more concurrent planes are deduced. It is first found that all the dihedral angles must be of the form ir/n, then equation 2 is applied, and the solutions are given by equation 3, with none for more than three planes. In the problem of discrete groups all sequences of reflections are permitted, while in the present problem only "admissible" sequences of reflections are permitted, which is a weaker condition. On the other hand we have the condition that some admissible sequence must carry each point into its antipode, and this condition is strong enough to limit the solutions to those given in equation 4—a subset of the configurations generating discrete groups. The new parallel reflecting configurations given herein may have practical value in some applications because they reflect back only those rays which are incident from a certain angular region, and the angular region may be made much smaller than that of the mutually orthogonal mirrors. I wish to express my indebtedness to my colleague, Prof. Wilhelm Magnus, who supplied a crucial part of the necessity proof given above.