In 7-dimensional geometry, there are 71 uniform polytopes with A7 symmetry. There is one self-dual regular form, the 7-simplex with 8 vertices.
Each can be visualized as symmetric orthographic projections in Coxeter planes of the A7 Coxeter group, and other subgroups.
Symmetric orthographic projections of these 135 polytopes can be made in the A7, A6, A5, A4, A3, A2 Coxeter planes. Ak has [k+1] symmetry. For even k and symmetrically ringed-diagrams, symmetry doubles to [2(k+1)].
These 63 polytopes are each shown in these 6 symmetry planes, with vertices and edges drawn, and vertices colored by the number of overlapping vertices in each projective position.