In 6-dimensional geometry, there are 39 uniform polytopes with E<sub>6</sub> symmetry. The two simplest forms are the 2<sub>21</sub> and 1<sub>22</sub> polytopes, composed of 27 and 72 vertices respectively.
They can be visualized as symmetric orthographic projections in Coxeter planes of the E<sub>6</sub> Coxeter group, and other subgroups.
Symmetric orthographic projections of these 39 polytopes can be made in the E<sub>6</sub>, D<sub>5</sub>, D<sub>4</sub>, D<sub>2</sub>, A<sub>5</sub>, A<sub>4</sub>, A<sub>3</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> has 12 symmetry.
Six symmetry planes graphs are shown for 9 of the 39 polytopes in the E<sub>6</sub> symmetry. The vertices and edges drawn with 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.