The famous Gieseking hyperbolic ideal simplex manifold (1912) and the other 3 double-simplex manifolds, found also by I. PROK with his computer program, can be described in Poincaré half-space model of H3 and by the complex projective line C∞. The Dehn type surgery leads to compact manifolds and cone manifolds (especially orbifolds) through complex 2×2 matrix algebra and attractive pictures. Hyperbolic space groups (above) as prospective crystal structures might have future applications (fullerenes, nanotubes)!?!