conceptually and/or visually. In fact, the idea of dimensions can even be applied to
abstract objects that can’t be directly visualized. Mathematicians most often display
such dimensions on graphs using a single point (for example, x) to represent one
dimension; two points (usually xand y,or an ordered pair) to represent two dimen-
sions; and three points (usually x, y, z) for three dimensions. The four- (and higher)
dimensional analogs of three-dimensional objects often retain the prefix “hyper-” such
as hypercube and hyperplane. The basic geometric structures of higher-dimensional
geometry—the line, plane, space, and hyperspace—all consist of an infinite number of
points arranged in specific ways.What is Euclidean space?
Euclidean (also seen as Euclidian) space is often called Cartesian space, or more simply,
n-space. It is made up of ndimensions and is a set of points, with each point represented
by a coordinate of ncomponents. Space with two to three dimensions—and that does not
use Einstein’s and others’ ideas of relativistic physics—are considered Euclidean space.In Euclidean space, distance is defined by certain “rules”: The distance between
170 two points is positive, unless they are the same points; the distance from points ato b
The ups and downs of stock markets, such as this one in Tokyo, Japan, are often illustrated using two-dimension-
al line graphs. Sometimes, though, economists employ even more complex, four-dimensional models.
Stone/Getty Images.