A. Bowyer proposes an efficient algorithm for computing Dirichlet tessellations and Delaunay triangulations in k-dimensional Euclidean space (k ≥ 2). The algorithm is designed to be extendable to non-Euclidean metric spaces. It has been implemented in ISO FORTRAN, and execution times and stereoscopic images of the tessellation and triangulation are presented. The algorithm is based on the properties of Delaunay triangulations and Dirichlet tessellations, where each Delaunay triangle corresponds to a vertex in the tessellation, which is the circumcenter of the triangle. In k dimensions, each Delaunay simplex corresponds to a vertex in the tessellation, which is the center of a hypersphere passing through the simplex's vertices. The algorithm starts with a Delaunay simplex formed by the first k+1 points and iteratively adds new points, modifying the structure accordingly. The algorithm identifies a vertex to be deleted by the new point, performs a tree search to find all such vertices, and then computes new vertices and their associated points. The algorithm is efficient, with the time complexity depending on the dimensionality of the space. Degeneracies, such as points lying on a hyperplane or in a line, are handled by ensuring that the algorithm remains robust. The algorithm has been implemented in FORTRAN and tested on various data sets, including highly degenerate point patterns. The algorithm is also applicable to non-Euclidean spaces by modifying the relevant routines. The paper includes figures showing the Dirichlet tessellation and Delaunay triangulation in two and three dimensions, as well as timing results for the algorithm in different dimensions. The algorithm is efficient and has been shown to work well in practice.A. Bowyer proposes an efficient algorithm for computing Dirichlet tessellations and Delaunay triangulations in k-dimensional Euclidean space (k ≥ 2). The algorithm is designed to be extendable to non-Euclidean metric spaces. It has been implemented in ISO FORTRAN, and execution times and stereoscopic images of the tessellation and triangulation are presented. The algorithm is based on the properties of Delaunay triangulations and Dirichlet tessellations, where each Delaunay triangle corresponds to a vertex in the tessellation, which is the circumcenter of the triangle. In k dimensions, each Delaunay simplex corresponds to a vertex in the tessellation, which is the center of a hypersphere passing through the simplex's vertices. The algorithm starts with a Delaunay simplex formed by the first k+1 points and iteratively adds new points, modifying the structure accordingly. The algorithm identifies a vertex to be deleted by the new point, performs a tree search to find all such vertices, and then computes new vertices and their associated points. The algorithm is efficient, with the time complexity depending on the dimensionality of the space. Degeneracies, such as points lying on a hyperplane or in a line, are handled by ensuring that the algorithm remains robust. The algorithm has been implemented in FORTRAN and tested on various data sets, including highly degenerate point patterns. The algorithm is also applicable to non-Euclidean spaces by modifying the relevant routines. The paper includes figures showing the Dirichlet tessellation and Delaunay triangulation in two and three dimensions, as well as timing results for the algorithm in different dimensions. The algorithm is efficient and has been shown to work well in practice.