Hypercube

As used in geometry, a hypercube is an extrapolation of the cube or square to n dimensions. When n is not specified, it's generally assumed to be 4. For example, a 4th dimensional hypercube is called a tesseract. Therefore, an n-dimensional hypercube is also known as an n-cube. It is best drawn and represented in non-Euclidean geometry.

Tesseract

A tesseract is the 4th dimensional hypercube. It is made by combining two cubes. The net of a tesseract is composed of 8 cubes. It has the Schlaefli symbol ${4,3,3}$. One simple coordinate system for its vertices are $(\pm1, \pm1, \pm1, \pm1)$. The alternated tesseract is a 4D cross-polytope, which coincidentally, is also it's dual.

Extra Notes

The alternated hypercube is known as a demicube. The dual of the hypercube is known as the cross-polytope. For dimensions n≥3, the only n-dimensional regular honeycomb is made of the hypercube.

Links

To see an $\mathfrak{e}$xample of a 4D cube, click here: [1]