  Consider the KS diagram  Γ3 shown in Figure 3 which obviously is a part of KS’s  Γ1,  but which has additional concrete assignments of eight points satisfying the orthogonality relations (and thus proving directly that  Γ3  is constructible). From our previous colouring constraints (joined points are not both white and a triangle has exactly one white point) we see immediately that  Γ3  is colourable only if the outermost points are not both white (which would require, as shown in Fig. 3, that two joined points are white — contrary to the constraints). Moreover, we easily calculate the angle between the two outermost points to be  cos−1(1/3).[8]  So we conclude that if one wants to colour all eight points and wants to colour white one of the outer ones, then the other must be black. Taking into account that we can insert a diagram between any two points in R3 which are separated by exactly the angle cos−1(1/3) and translating our problem back from a colouring problem into KS’s example (constraint VC2), we end with a constraint  VC2′: 