One of the great virtues of our blogs is the manner in which ideas can be floated, discussed and developed. In this case I refer to a posting by Adderphue which gave his method for creating tricornes - by creating a wide-brimmed hat and folding it to shape. This is a solution to creating a tricorne that seems to be within my limited sculpting abilities. One of my current projects (of which more another time) might be solved if I can make tricornes in this manner, and so I am currently experimenting with it.
The big question is: how does one fold a tricorne? Looking into this one very quickly realises that the question itself is rather inadequate to the problem. The hat's shape evolved over time until it finally became the bicorne. So the initial geometry of the unfolded hat, and placement of folds, changes according to which period we are trying to model. A useful starting point is to look at the tricorne at its simplest. This has folds in the form of an equilateral triangle, and gives us the hat worn by these gentlemen.
The maths for this is simple. The figure below shows that the ratio of the hat's central and outer radii is given by r1/r2 = cos 60 = 0.5. All other shapes can be gotten by slight variations on this. For example, if we want each side to have a concave curve to it then we reduce the inner radius slightly. To mimic the tricornes on the Stadden figures, we need to also to raise the front of the hat and pinch the rear corners of the tricorn.