Hypercube displays projections of hypercubes into three dimensions. By clicking on the applet, you can switch between 4– and 5-dimensional hypercubes.
A hypercube in 4 dimensions has 16 vertices, 32 edges, 24 square faces, and 8 cubic hyperfaces which form its boundary. When projected down to three dimensions, its projection is a rhombic dodecahedron, with 12 of the hypercube’s 24 faces projected into the interior, and 12 forming the surface.
A hypercube in 5 dimensions has 32 vertices, 80 edges, 80 square faces, 40 cubic faces, and 10 (4-dimensional) hypercubic faces which form its boundary. When projected down to three dimensions, its projection is a rhombic icosahedron, with 60 of the hypercube’s 80 faces projected into the interior, and 20 forming the surface.
These notes describe in detail the components and symmetries of hypercubes of arbitrary dimension.