coil shape

I have always laid my windings(turns)side by side very neatly so that I get the number of turns require per inch, even when done by hand.

Thank you for your reply - no, I mean the actual shape of the winding - the shape of the winding form.

I have seen them in a wide variety of forms -triangular, circular. What detrmines this shape - or is it arbitrary

When I saw your question originally, I thought you were referring to the section through the wire. It's clearer now, it seems to me that the form would be determined by the physical location of the winding. So the shape of the coil in plan could be designed (rounded) trapezoidal to sit together on a circle at a radius. The flux density would be harder to calculate than for a circular coil, as you have an effective diameter, but empirically people have probably worked out what is best.

Form follows function, I suspect.