Image source: Wikipedia's Tesseract

In geometry, the tesseract is the four-dimensional analog of the cube; the tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of 6 square faces, the hypersurface of the tesseract consists of 8 cubical cells. The tesseract is one of the six convex regular 4-polytopes.

The tesseract is also called an 8-cell, regular octachoron, cubic prism, and tetracube (although this last term can also mean a polycube made of four cubes). It is the four-dimensional hypercube, or 4-cube as a part of the dimensional family of hypercubes or "measure polytopes".

The construction of a hypercube can be imagined the following way:

1-dimensional: Two points A and B can be connected to a line, giving a new line segment AB.

2-dimensional: Two parallel line segments AB and CD can be connected to become a square, with the corners marked as ABCD.

3-dimensional: Two parallel squares ABCD and EFGH can be connected to become a cube, with the corners marked as ABCDEFGH.

4-dimensional: Two parallel cubes ABCDEFGH and IJKLMNOP can be connected to become a hypercube, with the corners marked as ABCDEFGHIJKLMNOP.

(Wikipedia)

From left to right: the square, the cube and the tesseract. The two-dimensional (2d) square is bounded by one-dimensional (1d) lines; the three-dimensional (3d) cube by two-dimensional areas; and the four-dimensional (4d) tesseract by three-dimensional volumes. For display on a two-dimensional surface such as a screen, the 3d cube and 4d tesseract require projection.