Consider the following explanatory article describing the solution of the particle-in-a-box, and the particle-in-a-circular box problems . The first part of the work consists of the description of 1 particle in a 'box' of infinite potential in 1 dimension. The particle in a 'rectangular box' is then discussed (as a simple extension to 2 dimensions). The final part of the work begins, "Squares are nice, but we learn more from circles." And proceeds to describe 1 particle in a circular box of infinite potential in 2 dimensions. More succinctly, using the above syntax: The geometric 'endpoints' defined by the square and the circle have quite natural characteristics to them, representing the orthogonal and the infinitely smooth respectively. In addition there exists an exact solution to 1 particle in an equilateral triangle , however the general polygonal case requires an approximate solution derived via a perturbation from the circle .