This article concerns the use of hypersphere-close-pack ( HCP) grids and quadratic multivariate Taylor polynomial interpolation functions with several additional higher-order terms for multidimensional finite differencing ( MDFD). MDFD is a general methodology for the numerical solution of partial differential equations ( PDE) defined on irregular domains. MDFD uses domain embedding, which means that expensive body-fitted gridding is not required, but the methodology is flexible enough to allow any grid. The HCP grid allows efficient, stable, and robust implementation of MDFD. In this article, we discuss MDFD and related methods, summarize the HCP grids in two and three dimensions, outline the algorithms used, and present the results of a number of diagnostic examples. The examples concern numerical solution of the unsteady parabolic heat equation defined for a number of domains of different shapes. The examples suggest, among other things, that the algorithms work with domains that have no similarity to the grid geometry and that the solutions appear to have global O(h(2)) convergence.