In 6-dimensional geometry, there are 64 uniform polytopes with B<sub>6</sub> symmetry. There are two regular forms, the 6-orthoplex, and 6-cube with 12 and 64 vertices respectively. The 6-demicube is added with half the symmetry.
They can be visualized as symmetric orthographic projections in Coxeter planes of the B<sub>6</sub> Coxeter group, and other subgroups.
Symmetric orthographic projections of these 64 polytopes can be made in the B<sub>6</sub>, B<sub>5</sub>, B<sub>4</sub>, B<sub>3</sub>, B<sub>2</sub>, A<sub>5</sub>, A<sub>3</sub>, Coxeter planes. A<sub>k</sub> has [k+1] symmetry, and B<sub>k</sub> has [2k] symmetry.
These 64 polytopes are each shown in these 8 symmetry planes, with vertices and edges drawn, and vertices colored by the number of overlapping vertices in each projective position.