Why 3 phase? Why not 4 phase or 5 phase?

BECOZ3 PHASE SYSTEM IS MORE EFFICIENT THAN4 OR FIVE PHASE AND THE SECOND REASON IS THAT3 PHASE IS CHEAPER THAN4 OR5

3 phase power generating and delivering is more efficient than2 phase which in turn is more efficient than1 phase. Similarly, going up in the number of phases result in increased efficiency as well, i.e.4 phase is more efficient than3 phase and5 phase is more efficient than4 phase,...etc. The increase of efficiency as the number of phases increase is attributed to that power delivery becomes more continuous as the number of phases increases. A single phase (with resistive load) power delivery will have zero instantaneous power twice a cycle but no zero instantaneous power in the case of3 phase power and higher phases number with a more smooth power delivery as phases go up. It can be easily calculated to show that3 phase delivery is about150% more efficient than single phase. This is the optimum choice as going higher than that the increased efficiency does not justify the increased complexity of using more phases. I am not sure for the case of Aluminum melting industry which you say it uses48 phase for melting, I can only make an intelligent guess and let others judge it. Aluminum melting uses induction heating which involves high currents(powers). Hence it may be justified to increase the complexity and use a high number of phases in order to increase the efficiency of the melting process as possible. The complexity can be justified as control is probably using high power diodes and thyristors. It is not like in the case of power generation where having a3 phase generator/ motor is optimum and more than that means more brushes, 

For3 Phase system the phase angel between transmission line is360/3=120 Degree. As the number of phase increases the phase difference will decrease and more number of times transposition is required. It affects the Installation cost of transmission of Transmission towers.

If we use3 phase the power output will be even n reaching its sine maximum peak.. whereas in4 phase the flux will not be even and it may cancel out.. in5 phase sine does not reach its maximum peak.. so4 and5 phase can not be used efficient while3 phase can be used more efficiently..

Nicely answered, Concur Fully !! 

1 to produce rotational magnatic feild we need at lease tree phase    its like shaire we can not  make stable shair with one or two stand we need at least3 , sow in electrical enginiring we use three becouse its   its

the  best between choises  less phase  mean less complication and moore economic

Dwar Arun,

I will mathematically demonstrate to you why is power generated and distributed in three phase for your better understanding. You should know first the most important reason for distributing power in3 phase system:

early scientists have found that the maximum power output and maximum power delivery rate can be achieved in a3 phase system ! 

when they came to mathematically calculate power delivered by a three phase system, they have found that only in the case of3 phases (and you can found this calculation in below link) the oscillating part of the power signal (the result of multiplication of I and V sinusoids) cancels out and a constant power component remains.  thus power is deliver at a constant rate in three phases system and it doesnt bounce back and forth between load and source therefore its the most efficient poly-phase system to generate and distribute electricity.

In the page below, scroll down almost to the end were written three phase power equation and start reading from there: you can come back and read my text again for better understanding.

http://electrical4u.com/electric-power-single-and-three-phase/ 

All answers are correct in one way or other