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First of all, Thank you for inviting me...
There are two reasons:
(1) using three phases results in the lowest conductor cost/watt to transmit power;
(2) using three phases displaced in space around a cylinder by 120 degrees results in a rotating magnetic field when the three phases are excited by three voltage sources displaced in time by 120 electrical degrees - this is a fundamental energy conversion principle for the operation of AC motors and generators.
in addition to
Three phase power is the simplest form of balanced AC. Balanced meaning the power flow can be constant. A single phase system is really a two phase system, and the power through this system varies with time.
6 phases, and even higher (9, 12, 18, 24, etc) systems do exist, and each has small advantages. For example, in a balanced 6 phase system, the 5th and 7th harmonics do not cause torque oscillation or rotor heating in motors and generators The contributions from each 3-phase parts cancel. AC to DC rectifiers often employ AC transformers with multiple, phase shifted windings. A 3 to 6 phase transformer can be made simply by connecting the two sets of simple secondaries one in delta and the other in wye. 12 phase versions allow the input power quality reflected to the three-phase side to meet most specifications.
The reasons these higher order systems are not used in a commercially wide spread fashion are 1) cost and 2) the systems are indeterminate (they have more states (independent variables) than necessary for a given steady state conviction). For example, to provide constant power in six phase system, either subordinate three phase set can provide any fraction of the power necessary, unless another control is instituted such as the two three phase set must be the same. This is just more control and complication.
Cost is the main reason. Rotating electric machines must has at least one stator slot per phase per pole. A six phase machine has at least twice as many windings as a three phase machine. Breakers need twice as many poles, and cabling must have twice as many insulated conductors.
Just an aside. Non-triplen systems work as well. Some papers on 5 phase machines were published in the 80s and 90s, asserting certain benefits in motors. Some such machines have been built. They work.
What you ask has been considered and is employed in limited subsystems, but for main distribution, 3-phase is simple, cost effective and mathematically elegant.
