Equi-isoclinic planes of Euclidean spaces

In a Euclidean space, a p-set of equi-isoclinic planes is a set of p isoclinic planes of which each pair has the same non-zero angle.The Euclidean 4-space E⁴ contains a unique congruence class of quadruples of equi-isoclinic planes, whereas quintuples of equi-isoclinic planes do not exist in E⁴.In the following a method is given to derive sets of equi-isoclinic planes in Euclidean spaces. We find again the well-known sets of equi-isoclinic planes of E⁴. The quadruples of equi-isoclinic planes in E⁵ are derived. It turns out that E⁵ contains one congruence class of such quadruples which are not flat quadruples and one congruence class of quintuples of equi-isoclinic planes, whereas sextuples of equi-isoclinic planes do not exist in E⁵.It appears that the symmetry group of that quintuple is isomorphic to the symmetric group S₅.