Euclidean space

Also known as Euclidean vector space, finite-dimensional real vector space, Euclidean n-space

generalization of Euclidean geometry to higher-dimensional vector spaces

AI overview

Euclidean space is a mathematical framework that extends the familiar geometry of flat surfaces and three-dimensional environments we experience into any number of dimensions using vectors. It matters because it provides a foundational tool for physics, engineering, computer science, and many other fields to model and solve problems involving spatial relationships and distances in abstract high-dimensional settings.

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Euler characteristic: 1

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A point in three-dimensional Euclidean space can be located by three coordinates.

Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's Elements, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer dimension n, which are called Euclidean n-spaces when one wants to specify their dimension. For n equal to one or two, they are commonly called respectively Euclidean lines and Euclidean planes. The qualifier "Euclidean" is used to distinguish Euclidean spaces from other spaces that were later considered in physics and modern mathematics.