  Consider now (for a reductio ad absurdum) that a0 and a9 have different colours. We arbitrarily colour a0 white and a9 black. The colouring constraints then force us to colour the rest of the diagram as is done in Fig. 1, but this requires that a5 and a6 are orthogonal and both white — which is forbidden. Hence, two points closer than sin−1(1/3) cannot have different colours. Contrapositively, two points of different colour cannot be closer than sin−1(1/3).