I don't have a favourite, but regard the main regular polygons as of equal merit and value, each with its distinct properties: the Triangle, the Square and Rectangle, the Hexagon and the Octagon.
Of the regular curves, one of my Maths teachers reckoned the "most beautiful" is the Sine Curve (the trace of y = sinx from 0º to 360º to obtain a full cycle); and yes it is graceful but more vitally, without it we would have no sound, no sea-waves, no electricity to speak of so no way to chat about it like this....
Though without the Circle we would not have the electricity either because we would not have rotating machinery to generate it. Nor wheeled transport.
And to square the circle so to speak, put the Right-angled Triangle polygon and the Circle non-polygon together and process them in the right way, as you can do on graph-paper, and what do they produce? The Sine Curve!
Benedict Arnold 