In 7-dimensional geometry, there are 127 uniform polytopes with E<sub>7</sub> symmetry. The three simplest forms are the 3<sub>21</sub>, 2<sub>31</sub>, and 1<sub>32</sub> polytopes, composed of 56, 126, and 576 vertices respectively.
They can be visualized as symmetric orthographic projections in Coxeter planes of the E<sub>7</sub> Coxeter group, and other subgroups.
Symmetric orthographic projections of these 127 polytopes can be made in the E<sub>7</sub>, E<sub>6</sub>, D<sub>6</sub>, D<sub>5</sub>, D<sub>4</sub>, D<sub>3</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, D<sub>k</sub> has 2(k-1) symmetry, and E<sub>6</sub> and E<sub>7</sub> have 12, 18 symmetry respectively.
For 10 of 127 polytopes (7 single rings, and 3 truncations), they are shown in these 8 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.