The plane can be tiled without gaps by congruent two-dimensional regular simplices (i.e., equilateral triangles). The three-dimensional Euclidean space cannot be tiled by congruent three-dimensional simplices (i.e., equilateral tetrahedra). Can this be done for higher dimensions? I have a vague recollection that I saw a proof that for some dimensions the tiling exists and that the existence is somehow related to the divisibility of the dimension by four, but this may be false. Can anybody please formulate the correct statement and provide either a proof or a reference?
Thanks!