Chinese name
Euclidean space
Foreign name
Euclidean space
brief introduction
Is a special metric space.
meaning
So that we can understand its topological properties.
discoverer
Euclid
quick
navigate by water/air
Strict definition
Introduction to Euclid
brief introduction
Around 300 BC, the ancient Greek mathematician Euclid established the law of the relationship between angle and distance in space, which is now called Euclidean geometry. Euclid first developed "plane geometry" to deal with two-dimensional objects on the plane, and then he analyzed the "solid geometry" of three-dimensional objects. All Euclid's axioms have been arranged in an abstract mathematical space, which is called two-dimensional or three-dimensional Euclid space. [ 1]
These mathematical spaces can be extended to any finite dimension. This space is called? n？ A dimensional Euclidean space (even N-dimensional space for short) or a finite dimensional real inner product space.
These mathematical spaces can also be extended to any dimension, called real inner product space (not necessarily complete), and Hilbert space is also called Euclidean space in advanced algebra textbooks. [1] In order to develop Euclidean space with higher dimensions, the properties of space must be strictly expressed and extended to any dimension. Although the result of this is that mathematics is abstract, it captures the fundamental essence of Euclidean space that we are familiar with, that is, planarity. There are other kinds of space, such as spherical space is not Euclidean space, and the four-dimensional space-time described by relativity when gravity appears is not Euclidean space.
There is a methodology that regards Euclidean plane as a set of points satisfying a certain connection, which can be expressed according to distance and angle. One is translation, that is, moving this plane makes all points move the same distance in the same direction. The second is about the rotation of a fixed point on this plane, and all points on the plane rotate at the same angle around this fixed point. A basic principle of Euclidean geometry is that if a graph can be transformed into another graph through a series of translation and rotation, then two graphs (subsets) of a plane should be considered equivalent (congruence). (See Euclid Group). [2]
The last problem of Euclidean space is that it is not a vector space technically, but an affine space acted by vector space. Intuitively, the difference lies in where the origin should be located in this space. There is no standard choice because it can move around. This technology is ignored in this paper.
Euclidean space, abbreviated as Euclidean space (also called plane space), is a generalization of two-dimensional and three-dimensional spaces that Euclid studied in mathematics. This generalization transforms Euclid's concepts of distance and related length and angle into a coordinate system of any dimension. This is a "standard" example of finite dimension, real number and inner product space. Euclidean space is a special metric space, which enables us to study its topological properties, such as compactness. Inner product space is a generalization of Euclidean space. Both inner product space and metric space are discussed in functional analysis.
Euclidean space plays a role in the definition of manifolds containing Euclidean geometry and non-Euclidean geometry. The mathematical motivation of defining distance function is to define the tee-off around a point in space. This basic concept proves the difference between Euclidean space and other manifolds. Differential geometry introduces differential into moving skills and local Euclidean space, and discusses many properties of non-Euclidean manifolds.
Euclidean space is a 4-dimensional or n-dimensional theoretically infinite space.