The applet you see above creates several different polyloids in four dimensions which are analogs to polyhedrons in three dimensions. You can choose which shape to see, as well as rotate it and change the perspective with which you are looking at it.
The hypercube is created by joining three cubes along each edge. The hypercube has 16 vertices, 32 edges, 24 faces, and 8 spatial sides. It can be visualized as two parallel cubes with corresponding vertices joined.
The hexdecaloid is created by joining four tetrahedrons along each edge. The hexdecaloid has 8 vertices, 24 edges, 32 faces, and 16 spatial sides, and it is dual to the hypercube.
The pentaloid is created by joining three tetrahedrons along each edge. The pentaloid has 5 vertices, 10 edges, 10 faces, and 5 spatial sides. Its edges form the complete graph of order 5, and it is the simplest polyloid. It is also its own dual.
The cubaloid is created by joining three octahedrons along each edge. The cubaloid has 24 vertices, 96 edges, 96 faces, and 24 spatial sides. Its edges are the same as those for a stellated hypercube where the heights of the stellations are equal to the edge lengths. The cubaloid has all 120 degree diloidal angles and represents the best regular solution to the sphere packing problem in four dimensions.