simplex is If P is a general parallelotope, the same assertions hold except that it is no longer true, in dimension > 2, that the simplexes need to be pairwise congruent; yet their volumes remain equal, because the n-parallelotope is the image of the unit n-hypercube by the linear isomorphism that sends the canonical basis of to . As previously, this implies that the volume of a simplex coming from a n-path is: Conversely, given an n-simplex of , it can be supposed that the vectors form a basis of . Considering the parallelotope constructed from and , one sees that the previous formula is valid for every simplex. Finally, the formula at the beginning of this section is obtained by observing that From this formula, it follows immediately that the volume under a standard n-simplex (i.e. between the origin and the simplex in Rn+1) is The volume of a regular n-simplex with unit side length is as can be seen by multiplying the previous formula by xn+1, to get the volume under the n-simplex as a function of its vertex distance x from the origin, differentiating with respect to x, at (where the n-simplex side length is 1), and normalizing by the length of the increment, , along the normal vector. Dihedral angles of the regular n-simplex Any two (n − 1)-dimensional faces of a regular n-dimensional simplex are themselves regular (n − 1)-dimensional simplices, and they have the same dihedral angle of cos−1(1/n). This can be seen by noting that the center of the standard simplex is , and the centers of its faces are coordinate permutations of . Then, by symmetry, the vector pointing from to is perpendicular to the faces. So the vectors normal to the faces are permutations of , from which the dihedral angles are calculated. Simplices with an "orthogonal corner" An "orthogonal corner" means here that there is a vertex at which all adjacent edges are pairwise orthogonal. It immediately follows that all adjacent faces are pairwise orthogonal. Such simplices are generalizations of right triangles and for them there exists an n-dimensional version of the Pythagorean theorem: The sum of the squared (n − 1)-dimensional volumes of the facets adjacent to the orthogonal corner equals the squared (n − 1)-dimensional volume of the facet opposite of the orthogonal corner. where are facets being pairwise orthogonal to each other but not orthogonal to ,