# Projection of a Torus

Projection of a Torus
A four-dimensional torus projected into three-dimensional space.

# Basic Description

It is impossible to visualize a complete four-dimensional object, since we have only ever lived in three-dimensional space. However, there are ways to capture parts of the four-dimensional object in three-dimensional space. A useful analogy is a world map. We can capture the essence of the three-dimensional globe on a two-dimensional map, but only by using a projection, which distorts the three-dimensional object in some way to fit on a two-dimensional surface.

A similar process is carried out to create this page's main image. A four-dimensional object, described further below, is projected into three-dimensions using two different projections.

# A More Mathematical Explanation

The four-dimensional object is defined parametrically by $(x_1,x_2,x_3,x_4)=(cos(u),sin(u),cos(v),sin(v))$

The four-dimensional object is defined parametrically by $(x_1,x_2,x_3,x_4)=(cos(u),sin(u),cos(v),sin(v))$

# About the Creator of this Image

Thomas F. Banchoff is a geometer, and a professor at Brown University since 1967.