The regular pentagram was the badge of the Pythogoreans.But the Pythagoreans first thought that all the rations were rational,i.e,. where a nd b are rational numbers.
Clearly this fraction must follow the equation for the golden ratio(It's a property of the pentagram),and hence we get
If and are of different parity clearly this leads to a contradiction.Also the assumption that and are odd also leads to a contradiction.
Hence both and must be even,that is and ,where and .
Substituting this into the equation and cancelling we get .
Now the same logic may be applied to this equation also which gives and where and .
So we obtain an infinitely decreasing sequence of positive integers,ie and .
Clearly no such infinitely decreasing sequence exists for positive integers.
For more details on this approach you may choose to read my note on Fermat's Method of Infinite Descent