Dimension

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Dimension

One of the most misunderstood concepts in mathematics, dimension is a central concept in modern geometry, topology, algebra, and the theory of fractals. To make matters even more complicated, each of these branches of mathematics has its own version of dimension (or even several versions).

In simplest terms, the dimension of a mathematical object is the number of independent parameters required to describe that object. It is well known that the dimension of a line is 1, the dimension of a plane is 2, and the dimension of space is 3. For example, any point in space can be described by three coordinates; a box in space is described by three parameters, its height, width and depth.

However, most people have difficulty imagining dimensions larger than 3; it is sometimes asserted that "The fourth dimension is time." As a statement about our physical universe, this may be correct; as a mathematical statement, it is false. Mathematical structures can have any number of dimensions, and none of these need to have any connection with time.

Here are four different ways that mathematicians approach the concept of dimension.

• The vector space approach. An n-dimensional Euclidean vector space is described as the set of all points with coordinates (x₁, x₂,..., x_n), where the numbers x₁, x₂,..., x_n can take any real values. More generally, vector spaces can be defined with coordinates in any, such as the, and the dimension equals the number of coordinates.
• The differential geometric approach. Differential geometry deals primarily with the properties of smooth, which are generalizations of curves and surfaces. On a small scale, any manifold "looks like" an n-dimensional Euclidean vector space. For example, any small piece of a sphere looks like a flat plane; therefore the sphere is 2-dimensional.
• The topological approach. Topologists use the "Lebesgue covering dimension" or "topological dimension," an ingenious method that makes no mention of real numbers or Euclidean space, and hence might be considered more fundamental. Roughly speaking, a topological space has dimension n if it can be covered by open sets that overlap no more than (n+1) at a time, and if these coverings can be made arbitrarily fine. For example, a plane can be covered by disks that intersect no more than 3 at a time, and the disks may be made as small as desired. Hence the plane is two-dimensional.
• The fractal approach. The various approaches to fractal dimension all take into account the scaling of an object. They take the viewpoint that the dimension of a box is 3 not because it resides in Euclidean 3-dimensional space, but because its volume scales as the third power of its linear size. Remarkably, for fractals the "volume"--measured, for example, by the number of disks of a given (fixed) size needed to cover the fractal--need not scale as an integer power. For example, the has a fractal dimension of 1.26..., because when its size is tripled the number of disks required to cover it increases by a factor of four, or 3^(1.26...). Yet the topological dimension of the snowflake curve is 1. The fractal dimension, in fact, is the only concept of dimension that allows fractional values; this motivated the name "fractal."