Every uniform polytope is the orbit of a single point bounced between mirrors. In 3D, three mirrors angled just right generate a cube or an icosahedron from one seed point. Here there are four mirrors in 4D, and the checkboxes choose which mirrors the seed sits off of — that's the whole recipe (Wythoff's construction).
Left: the shape's projected wireframe. Right: its live slice, colored by cell. The f-vector counts its vertices, edges, faces, and cells — compare with published tables: they match exactly, because the construction is exact group theory, not approximation.
Pick H4 and check all four rings: the omnitruncated 120-cell, 14,400 vertices, built in an eyeblink. Then try the two exceptional entries — shapes that no mirror recipe produces, made instead by carving pieces off the 600-cell.