Top 10 Best Four Dimensional Shapes

Circles? Spheres? To hell with those! They're just in two and three dimensions respectively!
The Top Ten
1 Klein Bottle

This is the four-dimensional extension of the infamous Möbius Strip. Whereas the Möbius Strip is a line with one boundary, a Klein Bottle is a bottle with none. Just as walking along a Möbius Strip takes you to the underside of where you started, filling a Klein Bottle causes its contents to erupt from the mouth.

2 Tesseract

Just as the glome is to the sphere what the sphere is to the circle, a tesseract is to a cube what a cube is to a square. Its net consists of eight cubes. The tesseract is one of the regular 4-polytopes, or four-dimensional extensions of the Platonic Solids.

3 Glome

A glome is a four-dimensional sphere. Just as the cross-section of a sphere is always a circle, the cross-section of a glome is a sphere.

4 Grand Antiprism

This is a more complex uniform 4-polytope than the Platonic extensions, consisting of 320 cells. 15 tetrahedra for every pentagonal antiprism. This can be made by deconstructing the hexacosichoron, specifically by removing 20 antiprism pyramids, just as the antiprism itself can be made by disassembling an icosahedron.

5 Hexacosichoron

Also aptly known as the 600-cell, this is the most complex extension of the simplest Platonic Solid, consisting of 600 tetrahedral cells. It is the dual polyhedron of the dodecacontachoron.

6 Dodecacontachoron

This is the extension of the dodecahedron into four dimensions, using 120 dodecahedral cells, and is the dual polyhedron of the hexacosichoron.

Take a look at the 3D conception of one of these things rotating. It's so awesome and could be an awesome screensaver.

7 Octacube

This is the extension of the third Platonic shape: the octahedron, into four dimensions. It consists of 24 octahedra.

8 Hexadecachoron

This is the dual polytope of the tesseract and the extension of the tetrahedron, or the simplest Platonic shape.

9 Tetracontoctachoron

This is the truncated version of the octacube, which is the equivalent of cutting the vertices of the octacube itself. The regular octagon is the truncated square, just as this shape is made by cutting the cubic vertices of the octacube.

10 Pentatope

This is a simpler extension of the tetrahedron than the hexadecachoron, with 5 tetrahedra instead of 16. It is this shape that is closer to the tetrahedron in three dimensions and to the equilateral triangle in two.

The Contenders
11 Great Grand Stellated 120-cell
12 Polychoron
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