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The complete set of 6 platonic solids, with a $1 coin for size comparison (the $1 coin is about 2.5cm or 1 inch in diameter).polytopes

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The pentachoron, or 4-simplex. This is the 4D equivalent of the tetrahedron, and is made up of 5 tetrahedra: the big one on the outside, and then four other tetrahedron, each having one of the outside triangles as a face, and having the apex at the central point. In 4D space, all of these tetrahedra would be regular tetrahedra (all equal length edges, all equal angles), but you lose this when you represent the shape in 3D.

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The tesseract or 4D hypercube. It is analogous to the 3D cube, and is probably the best known of any 4D shape. It is made up of 8 cubes. In this representation, two of the cubes actually look like cubes: the small one in the centre, and the big one that makes up the "outside". The remaining 6 cubes are distorted: they look like pyramids that have had their tops chopped off, where one of the outside faces of the big cube forms the base for this truncated pyramid, and the corresponding face on the small cube forms the top of the truncated pyramid. The sloping sides of the pyramid look like non-regular trapezia (you can see one to the immediate NE of the coin), but in the real 4D tesseract these would actually be squares.

I used to have a tesseract made of straws — and in a different 3D projection — hanging outside my office door. I must build one outside my new office.

The 16 cell, which is analogous to the octahedron in 3D. It is made up of 16 tetrahedra (analogously, the octahedron is made up of 8 equilateral triangles).

The 24 cell. This one does not have a 3D counterpart. It is made up of 24 octahedra.

Just as identical cubes pack together and fill 3D space (and the tesseract does the same in 4D space), the 24 cell "tiles" 4D space, the only regular convex polytope that does so in any of the dimensions above 2 apart from hypercubes.

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The 120 cell, which is analogous to the dodecahedron. I really love this one.

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A couple of close-ups of the 120 cell, so you can see how detailed it is. The quality of the 3D printing used to produce these shapes is stunning.

And the 600 cell to finish up with, which is the 4D analog of the icosahedron.

Given that this is made up of 600 tetrahedra it is an extremely complex shape.

A close up of the 600 cell. The sheer magnitude of having 600 tetrahedra here makes it difficult to fully appreciate just how amazing this is. (My head started hurting a shape or two ago as I tried to envisage the actual 4D shapes, and when I got to this one I just whimpered a bit and gave up!)