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Around 300 BC, the Greek mathematician Euclid undertook a study of relationships among distances and angles, first in a plane (an idealized flat surface) and then in space. An example of such a relationship is that the sum of the angles in a triangle is always 180 degrees. Today these relationships are known as two- and three-dimensional Euclidean geometry.

In modern mathematical language, distance and angle can be generalized easily to 4-dimensional, 5-dimensional, and even higher-dimensional spaces. An n-dimensional space with notions of distance and angle that obey the Euclidean relationships is called an n-dimensional Euclidean space. Most of this article is devoted to developing the modern language necessary for the conceptual leap to higher dimensions.

An essential property of a Euclidean space is its flatness. Other spaces exist in geometry that are not Euclidean. For example, the surface of a sphere is not; a triangle on a sphere (suitably defined) will have angles that sum to something greater than 180 degrees. In fact, there is essentially only one Euclidean space of each dimension, while there are many non-Euclidean spaces of each dimension. Often these other spaces are constructed by systematically deforming Euclidean space.

One way to think of the Euclidean plane is as a set of points satisfying certain relationships, expressible in terms of distance and angle. For example, there are two fundamental operations on the plane. One is translation, which means a shifting of the plane so that every point is shifted in the same direction and by the same distance. The other is rotation about a fixed point in the plane, in which every point in the plane turns about that fixed point through the same angle. One of the basic tenets of Euclidean geometry is that two figures (that is, subsets) of the plane should be considered equivalent (congruent) if one can be transformed into the other by some sequence of translations, rotations and reflections. (See Euclidean group.)

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