▁Z assen haus , ▁Cry stal log raph ic ▁Gr oups ▁of ▁Four - Dim ensional ▁Space . ▁▁ 1 9 8 2 ▁N . ▁P . ▁Warner , ▁The ▁symmetry ▁groups ▁of ▁the ▁regular ▁t ess ell ations ▁of ▁S 2 ▁and ▁S 3 ▁▁▁ 1 9 8 5 ▁E . ▁J . ▁W . ▁Wh itt aker , ▁An ▁at las ▁of ▁hyper st ere og rams ▁of ▁the ▁four - dimensional ▁cry stal ▁classes ▁▁ 1 9 8 5 ▁H . S . M . ▁C ox eter , ▁Reg ular ▁and ▁Sem i - Reg ular ▁Poly top es ▁II , ▁C ox eter ▁notation ▁for ▁ 4 D ▁point ▁groups ▁▁ 2 0 0 3 ▁John ▁Con way ▁ ▁and ▁Smith , ▁On ▁Qu atern ions ▁and ▁Oct on ions , ▁Comple ted ▁qu atern ion - based ▁ 4 D ▁point ▁groups ▁▁ 2 0 1 8 ▁N . ▁W . ▁Johnson ▁Ge omet ries ▁and ▁Trans form ations , ▁Chapter ▁ 1 1 , 1 2 , 1 3 , ▁Full ▁pol ych or ic ▁groups , ▁p . 2 4 9 , ▁du op r ism atic ▁groups ▁p . 2 6 9 ▁ ▁Is omet ries ▁of ▁ 4 D ▁point ▁symmetry ▁▁ ▁There ▁are ▁four ▁basic ▁is omet ries ▁of ▁ 4 - dimensional ▁point ▁symmetry : ▁reflection ▁symmetry , ▁rot ational ▁symmetry , ▁rot ore f lection , ▁and ▁double ▁rotation . ▁ ▁Not ation ▁for ▁groups ▁Point ▁groups ▁in ▁this ▁article