inventor Nicola Cabibbo. = 3 For the Standard Model case ( = 3), there are three mixing angles and one CP-violating complex phase. Observations and predictions Cabibbo's idea originated from a need to explain two observed phenomena: the transitions and had similar amplitudes. the transitions with change in strangeness had amplitudes equal to of those with Cabibbo's solution consisted of postulating weak universality (see below) to resolve the first issue, along with a mixing angle , now called the Cabibbo angle, between the and quarks to resolve the second. For two generations of quarks, there are no CP violating phases, as shown by the counting of the previous section. Since CP violations had already been seen in 1964, in neutral kaon decays, the Standard Model that emerged soon after clearly indicated the existence of a third generation of quarks, as Kobayashi and Maskawa pointed out in 1973. The discovery of the bottom quark at Fermilab (by Leon Lederman's group) in 1976 therefore immediately started off the search for the top quark, the missing third-generation quark. Note, however, that the specific values that the angles take on are not a prediction of the standard model: They are free parameters. At present, there is no generally-accepted theory that explains why the angles should have the values that are measured in experiments. Weak universality The constraints of unitarity of the CKM-matrix on the diagonal terms can be written as for all generations . This implies that the sum of all couplings of any one of the up-type quarks to all the down-type quarks is the same for all generations. This relation is called weak universality and was first pointed out by Nicola Cabibbo in 1967. Theoretically it is a consequence of the fact that all SU(2) doublets couple with the same strength to the vector bosons of weak interactions. It has been subjected to continuing experimental tests. The unitarity triangles The remaining constraints of unitarity of the CKM-matrix can be written in the form For any fixed and different and , this is a constraint on three complex numbers, one for each , which says that these numbers form the sides of a triangle in the complex plane. There are six choices of and (three independent), and hence six such triangles, each of which is called a unitary triangle. Their shapes can be very different, but they all have the same area, which