The images below display three-dimensional and two-dimensional projections of the six four-dimensional regular polytopes, analogous to the five Platonic solids in three-dimensions.

The two-dimensional projections (white background) are orthographic (i.e. as if the shape was observed from infinity, with a flat perspective. The edge width is proportional to the average distance of the vertices to the projection plane. The projection planes are chosen randomly for each image.

The three-dimensional projections are in perspective. Some basic shading gives an idea of the three-dimensional configuration.

The 5-cell is the four-dimensional simplex, analogous to the tetrahedron in three dimensions.

Stereographic image: look at the left picture with your right eye and vice versa to see a three-dimensional projection of the 5-cell

The 8-cell is the hypercube, the four-dimensional analogue of the cube. 

Stereographic image: look at the left picture with your right eye and vice versa to see a three-dimensional projection of the 8-cell

The 16-cell is the dual to the 8-cell. It is a four-dimensional analogue of the octahedron.

Stereographic image: look at the left picture with your right eye and vice versa to see a three-dimensional projection of the 16-cell

The 24-cell has no direct analogue in three-dimensions. It is self-dual.

Stereographic image: look at the left picture with your right eye and vice versa to see a three-dimensional projection of the 24-cell

The 120-cell is the four-dimensional analog of the dodecahedron.

Stereographic image: look at the left picture with your right eye and vice versa to see a three-dimensional projection of the 120-cell

The 600-cell is the four-dimensional analogue of the icosahedron. It is dual to the 120-cell.

Stereographic image: look at the left picture with your right eye and vice versa to see a three-dimensional projection of the 600-cell

Dall-e's interpretation of four-dimensional polytopes and their projections.