The Orthocentric Simplex in Space of Three and Higher Dimensions

Extract

By a simplex is meant the figure in general flat space of which the triangle and tetrahedron are examples. In space of n dimensions the simplex has (n + 1) vertices and (n + 1) faces, each face consisting of a prime, i.e. a space of (n - 1) dimensions, containing n of the vertices and hence itself a simplex in (n - 1)-dimensional space. There is a general likeness between the triangle and simplexes of higher order, but this is marred by the fact that, in general, a simplex does not possess an orthocentre.