an example of a 0/1-polytope, with all coordinates as 0 or 1. It can also be seen one facet of a regular (n+1)-orthoplex. There is a canonical map from the standard n-simplex to an arbitrary n-simplex with vertices (v0, ..., vn) given by The coefficients ti are called the barycentric coordinates of a point in the n-simplex. Such a general simplex is often called an affine n-simplex, to emphasize that the canonical map is an affine transformation. It is also sometimes called an oriented affine n-simplex to emphasize that the canonical map may be orientation preserving or reversing. More generally, there is a canonical map from the standard -simplex (with n vertices) onto any polytope with n vertices, given by the same equation (modifying indexing): These are known as generalized barycentric coordinates, and express every polytope as the image of a simplex: A commonly used function from Rn to the interior of the standard -simplex is the softmax function, or normalized exponential function; this generalizes the standard logistic function. Examples Δ0 is the point . Δ1 is the line segment joining (1, 0) and (0, 1) in R2. Δ2 is the equilateral triangle with vertices (1, 0, 0), (0, 1, 0) and (0, 0, 1) in R3. Δ3 is the regular tetrahedron with vertices (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0) and (0, 0, 0, 1) in R4. Δ4 is the regular 5-cell with vertices (1, 0, 0, 0, 0), (0, 1, 0, 0, 0), (0, 0, 1, 0, 0), (0, 0, 0, 1, 0) and (0, 0, 0, 0, 1) in R5. Increasing coordinates An alternative coordinate system is given by taking the indefinite sum: This yields the alternative presentation by order, namely as nondecreasing n-tuples between 0 and 1: Geometrically, this is an n-dimensional subset of (maximal dimension, codimension 0) rather than of (codimension 1). The facets, which on the standard simplex correspond to one coordinate vanishing, here correspond to successive coordinates being equal, while the interior corresponds to the inequalities becoming strict (increasing sequences). A key distinction between these presentations is the behavior under permuting coordinates – the standard simplex is stabilized by permuting coordinates, while permuting elements of the "ordered simplex" do not leave it invariant, as permuting an ordered sequence generally makes it unordered. Indeed, the ordered simplex is a (closed) fundamental domain for the action of the symmetric group