In 8-dimensional geometry, there are 255 uniform polytopes with E<sub>8</sub> symmetry. The three simplest forms are the 4<sub>21</sub>, 2<sub>41</sub>, and 1<sub>42</sub> polytopes, composed of 240, 2160 and 17280 vertices respectively.
These polytopes can be visualized as symmetric orthographic projections in Coxeter planes of the E<sub>8</sub> Coxeter group, and other subgroups.
Symmetric orthographic projections of these 255 polytopes can be made in the E<sub>8</sub>, E<sub>7</sub>, E<sub>6</sub>, D<sub>7</sub>, D<sub>6</sub>, D<sub>5</sub>, D<sub>4</sub>, D<sub>3</sub>, A<sub>7</sub>, A<sub>5</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>, E<sub>7</sub>, E<sub>8</sub> have [12], [18], [30] symmetry respectively. In addition there are two other degrees of fundamental invariants, order [20] and [24] for the E<sub>8</sub> group that represent Coxeter planes.
11 of these 255 polytopes are each shown in 14 symmetry planes, with vertices and edges drawn, and vertices colored by the number of overlapping vertices in each projective position.