Up to this time, we have not made us of trigonometric expressions in describing geometric objects. In the spirit of coordinate geometry, these expressions are extremely useful for specifying points on circles, spheres, and hyperspheres. In the plane, a unit circle is the collection of points (x, y) at distance 1 from the origin. By the Pythagorean theorem, this condition may be expressed as the algebraic statement x² + y² = 1. Analogously, in three-space the unit sphere is defined as the collection of points at unit distance from the origin, so in terms of coordinates, the unit sphere is the collection of points (x, y, u) such that x² + y² + u² = 1. In four-space, the unit hypersphere is the collection of points (x, y, u, v) such that x² + y² + u² + v² = 1.

To determine coordinates for the unit circle, mathematicians invented two functions, cosine and sine. These functions describe the position on a circle of a point reached by starting at (1, 0) and traveling counterclockwise around the circle a certain distance. If that distance is t, then the coordinates of the point on the unit circle are given by (cos(t), sin(t)).

From the coordinates for a circle, we can get geographical coordinates for the points of the unit sphere in three-space. Specifically, if t is the longitude of a point and s is its latitude, then the coordinates of the point are (cos(t)cos(s), sin(t)cos(s), sin(s)). It is straightforward to show that the sum of the squares of these coordinates is 1, the defining property of the unit sphere.

Similarly, we may obtain points on the unit hypersphere by using three angular coordinates, t, s, r, and defining (cos(t)cos(s)cos(r), sin(t)cos(s)cos(r), sin(s)cos(r), sin(r)). Another representation using angular coordinates t, s, and r is even more symmetric, namely (cos(t)cos(s), sin(t)cos(s), cos(r)sin(s), sin(r)sin(s)). In each case we can show that the sum of the squares of the coordinates is 1, thus establishing that the points lie on a unit hypersphere. In the second representation, for a fixed choice of the variable s, we obtain the circle of circles. Choosing s equal to 45 degrees, we obtain the Clifford torus, with the simple formula: 1/2^1/2 (cos(t), sin(t), cos(r), sin(r)). In this way we define a collection of torus surfaces in the hypersphere, and these are precisely the surfaces that we have already seen in our study of stereographic projection and in our analysis of orbits of dynamical systems. The orbits for a pair of synchronized pendulums are given by (cos(t)cos(s), sin(t)cos(s), cos(t + c)sin(s), sin(t + c)sin(s)) for any fixed choice of s and c. It is remarkable that a graphics computer can take such simple equations and produce pictures of such rich quality.

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