on the n-cube, meaning that the orbit of the ordered simplex under the n! elements of the symmetric group divides the n-cube into mostly disjoint simplices (disjoint except for boundaries), showing that this simplex has volume Alternatively, the volume can be computed by an iterated integral, whose successive integrands are A further property of this presentation is that it uses the order but not addition, and thus can be defined in any dimension over any ordered set, and for example can be used to define an infinite-dimensional simplex without issues of convergence of sums. Projection onto the standard simplex Especially in numerical applications of probability theory a projection onto the standard simplex is of interest. Given with possibly negative entries, the closest point on the simplex has coordinates where is chosen such that can be easily calculated from sorting . The sorting approach takes complexity, which can be improved to complexity via median-finding algorithms. Projecting onto the simplex is computationally similar to projecting onto the ball. Corner of cube Finally, a simple variant is to replace "summing to 1" with "summing to at most 1"; this raises the dimension by 1, so to simplify notation, the indexing changes: This yields an n-simplex as a corner of the n-cube, and is a standard orthogonal simplex. This is the simplex used in the simplex method, which is based at the origin, and locally models a vertex on a polytope with n facets. Cartesian coordinates for a regular n-dimensional simplex in Rn One way to write down a regular n-simplex in Rn is to choose two points to be the first two vertices, choose a third point to make an equilateral triangle, choose a fourth point to make a regular tetrahedron, and so on. Each step requires satisfying equations that ensure that each newly chosen vertex, together with the previously chosen vertices, forms a regular simplex. There are several sets of equations that can be written down and used for this purpose. These include the equality of all the distances between vertices; the equality of all the distances from vertices to the center of the simplex; the fact that the angle subtended through the new vertex by any two previously chosen vertices is ; and the fact that the angle subtended through the center of the simplex by any two vertices is . It is also possible to directly write down a particular