PHI & PI

Pi bothered me in 8th grade. Why does Pi exist? Why is it that the ratio of circumference and a diameter produce a non-repeating, apparently infinitely long number?

Now PHI (the Golden Ratio) is starting to bother me. I wrote earlier that I think PHI has something to do with emergence/synergy.

When the ratio of the whole to the greatest, is same as the ratio of the greatest to the smallest, an irrational number arises. It shows the same behaviour as Pi. Infinite, non-repeating.

Circular reference can create paradoxes that can create 'self-awareness'.

Is there a relationship between PHI and PI?