Point groups in four dimensions

[7] For cross-referencing, also given here are quaternion based notations by Patrick du Val (1964)[8] and John Conway (2003).

Each group defined by a Goursat tetrahedron fundamental domain bounded by mirror planes.

The term polychoron (plural polychora, adjective polychoric), from the Greek roots poly ("many") and choros ("room" or "space") and was advocated[10] by Norman Johnson and George Olshevsky in the context of uniform polychora (4-polytopes), and their related 4-dimensional symmetry groups.

Lower rank Coxeter groups can only bound hosohedron or hosotope fundamental domains on the 3-sphere.

[12] Extended symmetries exist in uniform polychora with symmetric ring-patterns within the Coxeter diagram construct.

The omnitruncated dual polychora have cells that match the fundamental domains of the symmetry group.

A hierarchy of 4D point groups and some subgroups. Vertical positioning is grouped by order. Blue, green, and pink colors show reflectional, hybrid, and rotational groups.
Some 4D point groups in Conway's notation
The 16-cell edges projected onto a 3-sphere represent 6 great circles of B4 symmetry. 3 circles meet at each vertex. Each circle represents axes of 4-fold symmetry.
The 24-cell edges projected onto a 3-sphere represent the 16 great circles of F4 symmetry. Four circles meet at each vertex. Each circle represents axes of 3-fold symmetry.
The 600-cell edges projected onto a 3-sphere represent 72 great circles of H4 symmetry. Six circles meet at each vertex. Each circle represent axes of 5-fold symmetry.
[4,3], , octahedral pyramidal group is isomorphic to 3d octahedral symmetry
[3,3], , tetrahedral pyramidal group is isomorphic to 3d tetrahedral symmetry