Symmetric quadrature rules on a triangle

We present a class of quadrature rules on triangles in 2 which, somewhat similar to Gaussian rules on intervals in 1, have rapid convergence, positive weights, and symmetry. By a scheme combining simple group theory and numerical optimization, we obtain quadrature rules of this kind up to the order 30 on triangles. This scheme, essentially a formalization and generalization of the approach used by Lyness and Jespersen over 25 years ago, can be easily extended to other regions in 2 and surfaces in higher dimensions, such as squares, spheres. We present example formulae and relevant numerical results.