4-dimensional 4-dimensional 4-dimensional
Image source: Wikipedia's Tesseract

Four-dimensional hypercube (a 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.


Square, cube and tesseract

Square, cube and tesseract.

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.

Square, cube and tesseract

A drawing of the first four dimensions.

On the left is zero dimensions (a point) and on the right is four dimensions (A tesseract). There is an axis and labels on the right and which level of dimensions it is on the bottom. The arrows alongside the shapes indicate the direction of extrusion.



quare, cube and tesseract

Single rotated tesseract.

Single rotated tesseract

A 3D projection of an 8-cell performing a simple rotation about a plane which bisects the figure from front-left to back-right and top to bottom

Double rotated tesseract

Double rotated tesseract.

A 3D projection of an 8-cell performing a double rotation about two orthogonal planes.