Tesseract

A tesseract, also known as a hypercube, is a four-dimensional cube, or, alternately, it is the extension of the idea of a square to a four-dimensional space in the same way that a cube is the extension of the idea of a square to a three-dimensional space.

Above, we can see a projection of a rotating hypercube into a three-dimensional space.

In the image below, we see one attempt to represent a diagram of a hypercube, although the image is deceiving in the sense that relative sizes and angles in this image have been distorted. Every quadrilateral in the image below that is formed by four vertices of the hypercube is a square (although they do not look like squares in the diagram!) Similarly, every pair of lines that meet at a vertex are at right angles to each other (as we would expect in a square or a cube). It is impossible to perfectly represent a hypercube in two-dimensions, or even to construct one in three-dimensions.

A square is a two-dimensional closed figure with lines of equal length that meet each other at right angles. A cube is a three-dimensional figure with lines of equal length that meet each other at right angles. For the square, two lines meet at each vertex (corner). For the cube, because we've added another dimension, we have three lines meeting at each vertex.

A tesseract is a four-dimensional closed figure with lines of equal length that meet each other at right angles. Since we've added another dimension, four lines meet at each vertex at right angles. Just as with a cube, each 2D face of the tesseract is a square. In fact, a tesseract has 3D "faces", each of which is a cube.

You can work out the properties of a tesseract by extrapolating from the idea of a square and cube (see the problems below). However, they are also listed behind the button here:

A hypercube has the following properties:

• 16 vertices (0D: points)
• 32 edges (1D: lines)
• 24 faces (2D: squares)
• 8 cells (3D: cubes)

It is difficult to visualize objects in higher dimensions. We've seen above a couple of different representations above. Here's another, in which the property that all the lines in a tesseract are the same length is more clearly shown. While this picture helps us see that all the 2D faces of a tesseract are squares, it's harder to see in this picture that the 3D cells are all cubes:

Each different representation helps us develop some intuition about the relationships in the shape. Many other representations are possible, each with its own compromises.

* Square
* Hyperplane