A very esteemed member here taught me this years ago.
The general formula to determine the number of phases in any system is: 360 degrees divided by the angular displacement between the phases equals the number of phases in that system.
360/120=3phases
360/180=2phases
360/360=1phase
360/90=4phases, but since only half of the system was needed & used it became misnamed two phase. An is still causing much confusion to this date.
The beauty of a real 2 phase system is that with an angular displacement of 180 degrees the math stays on the real number line, at least for resistive loads or adding the voltages between coils.
I have a terminology quibble with the above. Based on this discussion I absolutely agree that 180 degrees is a different phase angle than 0 degrees, and thus I agree than in a balanced '2 phase' system you will have _4_ distinct phase angles.
The terminology quibble is in how this is normally used. I will go back to the literature on 'high phase order' motors. In a conventional 3 phase motor you have at least _6_ phase bands in the stator; 3 for each of the supply phases and 3 for the inverses of these. The inverse phase bands are required because you have coils for each of the phase circuits, and thus half of the conductors travel down the stator slots in one direction, and half go the other way. But this is still called a _three_ phase motor. What the HPO people call a six phase motor generally has 12 phase bands, and can be understood as two three phase coil sets displaced by 30 degrees.
As I have seen the terminology used, terminals with phase displacements of 180 degrees are considered separate parts of the same 'phase' for purpose of counting system phases. Thus a phase displacement of 90 degrees (which I agree has 4 separate phase angles available) is _called_ '2 phase', and a phase displacement of 180 degrees is _called_ single phase. A phase displacement of 120 degrees is called 3 phase, but if you look at how 3 phase is used to run motors you will clearly see 6 separate phase angles in use.
An exception to this seems to be in transformers for rectifiers; 'hexaphase' transformers have 6 terminals with a phase angle of 60 degrees, and a transformer for a 12 pulse rectifier has phase angle differences of 30 degrees.
-Jon