The Koide formula is an unexplained empirical equation discovered by Yoshio Koide in 1981. In its original form, it relates the masses of the three charged leptons; later authors have extended the relation to neutrinos, quarks, and other families of particles. Formula The Koide formula is where the masses of the electron, muon, and tau are measured respectively as = , = , and = ; the digits in parentheses are the uncertainties in the last digits. This gives = . No matter what masses are chosen to stand in place of the electron, muon, and tau, The upper bound follows from the fact that the square roots are necessarily positive, and the lower bound follows from the Cauchy–Bunyakovsky–Schwarz inequality. The experimentally determined value, , lies at the center of the mathematically allowed range. But note that removing the requirement of positive roots it is possible to fit an extra tuple in the quark sector (the one with strange, charm and bottom). The mystery is in the physical value. Not only is the result peculiar, in that three ostensibly arbitrary numbers give a simple fraction, but also in that in the case of electron, muon, and tau, is exactly halfway between the two extremes of all possible combinations: (if the three masses were equal) and 1 (if one mass dominates). Robert Foot also interpreted the Koide formula as a geometrical relation, in which the value is the squared cosine of the angle between the vector and the vector (see dot product). That angle is almost exactly 45 degrees: When the formula is assumed to hold exactly ( = ), it may be used to predict the tau mass from the (more precisely known) electron and muon masses; that prediction is = . Please note that solving the Koide formula can also predict the third particle mass to be around 3.37 MeV/c2. While the original formula arose in the context of preon models, other ways have been found to derive it (both by Sumino and by Koide – see references below). As a whole, however, understanding remains incomplete. Similar matches have been found for triplets of quarks depending on running masses. With alternating quarks, chaining Koide equations for consecutive triplets, it is possible to reach a result of 173.263947(6) GeV for the mass of the top quark. Speculative extension Carl Brannen has proposed the lepton masses are given by the squares