One may believe with more dimensions, there are more regular polytopes. However, the fourth dimension is as complex as it gets. This is because each dimension's increasing "freedom" nullifies its complexity. In fact, all higher dimensions each only have three regular polytopes. 

A Geometric Approach

Differentiating between figures in the first few dimensions can be quite a task. One of the simplest ways to view higher dimensions is by slicing. A simple three-dimensional figure, a cube, can be sliced parallel to its sides to give a square, a two-dimensional figure. A hypercube, the cubic equivalent in four dimensions, therefore can be sliced to give cubes.

When you take a sheet of paper and look at it from the top, you see a rectangle. If you turn in on its side, you see a line. Between the rectangle and line, you see rhombuses, parallelograms, and other quadrilaterals. 

When you try to draw a cube, notice what shapes are used to create it. If you drew a cube from a corner view, you would need parallelograms and other quadrilaterals. From that view, you won't see any squares, although squares are what make up a cube. This strange phenomenon occurs anytime a polytope is rotated. So when you look at a hypercube rotating, don't expect to see just perfect cubes, rather look for distorted ones.

Stability in Higher Dimensions

The square has a perimeter and an area. However, the cube is defined geometrically by a "intersection" of six squares, also perpendicular and parallel to each other. Notice the cube has surface area and volume, but not perimeter. 

The same notion can be applied toward the hypercube, the cubic equivalent in the fourth dimension. The hypercube does not have surface area because it is again an "intersection", however of multiple cubes. Its "hyperfaces" are usually distorted when viewed, this is another reason the fourth dimension is difficult to visualize. This distortion occurs because we are trying to view a complex figure on a flat computer screen, or paper, etc. The hypercube has a "hypervolume", which is also difficult to interpret because it is defined as the side length to the fourth power. We cannot apply the hypervolume because of obvious reasons: hypercubes do not exist literally and they have no modern application.

We can see that each new dimension loses one geometric property from the previous and gains an entirely new one in the process. The table below illustrates this fact. This pattern is a probable reason why higher dimensions may not be as complicated as they seem. 

Use the following links to navigate through information about the six regular polychora.

• simplex
• hypercube
• cross polytope

• 24 cell
• 120 cell
• 600 cell