A Counting Problem of Perspective Simplexes In 𝐏𝐆(𝐧,𝐪)

Abstract We start with a tetrahedron (A) = A1A2A3A4 in PG(3,q), q is not a power of 2, and a point P not on the faces of (A) and found that the number of non-degenerate tetrahedrons that are perspective with ( A)from the centre P and exclusive of the points P and Ais is equal to ( q- 1) 4-5. If these tetrahedrons can have one of their vertices as P then the number of such non-degenerate tetrahedrons that are perspective with (A) from the centre P is equal to 4(q- 1)3. These propositions are proved analytically using the homogeneous coordinates of the vertices of the tetrahedrons and other counting results in finite geometries. Further, the propositions are generalised to PG (n,q) with cases n≤q and n>q.