In 7-dimensional geometry, there are 71 uniform polytopes with A<sub>7</sub> 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 A<sub>7</sub> Coxeter group, and other subgroups.
Symmetric orthographic projections of these 71 polytopes can be made in the A<sub>7</sub>, A<sub>6</sub>, A<sub>5</sub>, A<sub>4</sub>, A<sub>3</sub>, A<sub>2</sub> Coxeter planes. A<sub>k</sub> has [k+1] symmetry. For even k and symmetrically ringed-diagrams, symmetry doubles to [2(k+1)].
These 71 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 in progressive order: red, orange, yellow, green, cyan, blue, purple, magenta, red-violet.