pendulums are identical (La = Lb), θ is 45°. The angle θ is analogous to the Cabibbo angle (though that angle applies to quarks rather than neutrinos). When the number of oscillators (particles) is increased to three, the orthogonal matrix can no longer be described by a single angle; instead, three are required (Euler angles). Furthermore, in the quantum case, the matrices may be complex. This requires the introduction of complex phases in addition to the rotation angles, which are associated with CP violation but do not influence the observable effects of neutrino oscillation. Theory, graphically Two neutrino probabilities in vacuum In the approximation where only two neutrinos participate in the oscillation, the probability of oscillation follows a simple pattern: The blue curve shows the probability of the original neutrino retaining its identity. The red curve shows the probability of conversion to the other neutrino. The maximum probability of conversion is equal to sin22θ. The frequency of the oscillation is controlled by Δm2. Three neutrino probabilities If three neutrinos are considered, the probability for each neutrino to appear is somewhat complex. The graphs below show the probabilities for each flavor, with the plots in the left column showing a long range to display the slow "solar" oscillation, and the plots in the right column zoomed in, to display the fast "atmospheric" oscillation. The parameters used to create these graphs (see below) are consistent with current measurements, but since some parameters are still quite uncertain, some aspects of these plots are only qualitatively correct. The illustrations were created using the following parameter values: sin2(2θ13) = 0.10 (Determines the size of the small wiggles.) sin2(2θ23) = 0.97 sin2(2θ12) = 0.861 δ = 0 (If the actual value of this phase is large, the probabilities will be somewhat distorted, and will be different for neutrinos and antineutrinos.) Normal mass hierarchy: m1 ≤ m2 ≤ m3 Δm = Δm ≈ Δm = Observed values of oscillation parameters . PDG combination of Daya Bay, RENO, and Double Chooz results. . This corresponds to θsol (solar), obtained from KamLand, solar, reactor and accelerator data. at 90% confidence level, corresponding to (atmospheric) (normal mass hierarchy) and the sign of are currently unknown. Solar neutrino experiments combined with KamLAND have measured the so-called solar parameters and Atmospheric neutrino experiments such as Super-Kamiokande together with the K2K and MINOS long baseline accelerator neutrino experiment have determined the so-called atmospheric