Introduction to Hyperspace and Warped Space
Some of the concepts in this chapter are among the most esoteric in all of mathematics, with farreaching applications. Hyperspace (space of more than three dimensions) and warped space can be envisioned even by young children. Some people who think they are not mathematically inclined find nonEuclidean geometry interesting, perhaps because some of it resembles science fiction.
As we have seen, the rectangular (or Cartesian) coordinate plane is defined by two number lines that intersect perpendicularly at their zero points. The lines form axes, often called the x axis and the y axis. Points in such a system are identified by ordered pairs of the form ( x , y ). The point defined by (0,0) is called the origin. Cartesian threespace is defined by three number lines that intersect at a single point, corresponding to the zero point of each line, and such that each line is perpendicular to the plane determined by the other two lines. The lines form axes, representing variables such as x , y , and z . Points are defined by ordered triples of the form ( x , y , z ). The origin is the point defined by (0,0,0). What about Cartesian fourspace ? Or fivespace ? Or infinityspace ?
Imagine That!
A system of rectangular coordinates in four dimensions—Cartesian fourspace or 4D space—is defined by four number lines that intersect at a single point, corresponding to the zero point of each line, and such that each of the lines is perpendicular to the other three. The lines form axes, representing variables such as w , x , y , and z . Alternatively, the axes can be labeled x _{1} , x _{2} , x _{3} , and x _{4} . Points are identified by ordered quadruples of the form ( w , x , y , z ) or ( x _{1} , x _{2} , x _{3} , x _{4} ). The origin is defined by (0,0,0,0). As with the variables or numbers in ordered pairs and triples, there are no spaces after the commas. Everything is all scrunched together.
At first you might think, “Cartesian fourspace isn’t difficult to imagine,” and draw an illustration such as Fig. 111 to illustrate it. But when you start trying to plot points in this system, you’ll find out there is a problem. You can’t define points in such a rendition of fourspace without ambiguity. There are too many possible values of the ordered quadruple (w,x,y,z) , and not enough points in 3D space to accommodate them all. In 3D space as we know it, four number lines such as those shown in Fig. 111 cannot be oriented so they intersect at a single point with all four lines perpendicular to the other three.
Imagine the point in a room where the walls meet the floor. Unless the building has an unusual architecture or is sagging because of earth movement, there are three straight line segments defined by this intersection. One of the line segments runs up and down between the two walls, and the other two run horizontally between the two walls and the floor. The line segments are all mutually perpendicular at the point where they intersect. They are like the x , y , and z axes in Cartesian threespace. Now think of a fourth line segment that has one end at the intersection point of the existing three line segments, and that is perpendicular to them all. Such a line segment can’t exist in ordinary space! But in four dimensions, or hyperspace, it can. If you were a 4D creature, you would not be able to understand how 3D creatures could possibly have trouble envisioning four line segments all coming together at mutual right angles.
Mathematically, we can work with Cartesian fourspace, even though it cannot be directly visualized. This makes 4D geometry a powerful mathematical tool. As it turns out, the universe we live in requires four or more dimensions in order to be fully described. Albert Einstein was one of the first scientists to put forth the idea that the “fourth dimension” exists.

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