In elliptic n-space, the flat pole of an 
-flat is a point located an arc
length of 
radians distant from each point of the 
-flat. For an 
-dimensional spherical simplex,
there are 
such poles, one for each of its 
facets. Passing an 
-flat through each subset of 
of these poles then divides the space into 
simplices. The polar simplex is the simplex having edges
that are supplements of the dihedral angles of
the original simplex.
There are twice as many simplexes in spherical n-space, with diametrically opposite simplexes being congruent, so the chosen simplex is the one located in the same hemisphere as the original simplex.
The polar simplex of a polar simplex is the original simplex. The principal circumcenter of a simplex is the incenter of its polar simplex, and the principal circumradius of a simplex is the complement of the inradius of its polar simplex. The altitudes of a simplex and its polar simplex lie on the lines connecting corresponding vertices.
