Euclidean space

Euclidean space is the basic space used in geometry to represent physical space around us. It started with the ancient Greek mathematician Euclid, who wrote a book called Elements. In this book, Euclid described three-dimensional space using simple rules.

Today, mathematicians study Euclidean spaces of any size, not just three. These spaces are called Euclidean n-spaces, where n is the number of dimensions. For example, a one-dimensional Euclidean space is a line, and a two-dimensional one is a plane.

Euclidean spaces are different from other types of spaces studied in modern mathematics and physics, called non-Euclidean geometries. There is essentially only one Euclidean space for each size.

Definition

Euclidean space is a way to describe the space we live in, based on ideas from ancient Greek mathematicians. They looked at the world around them and made simple rules to explain how shapes and spaces work. This way of thinking is still used today and is called synthetic geometry.

Later, a mathematician named René Descartes introduced a new method using numbers to describe points in space, called Cartesian coordinates. This helped turn geometry into algebra, making it easier to solve problems with calculations. Today, Euclidean space is often defined using algebra and vectors. This means we can describe space using numbers and rules for how points and directions relate to each other. This helps us understand distances and angles in a clear mathematical way.

The most common way to think about Euclidean space today is as a set of points where we can measure distances and angles using vectors.

Prototypical examples

In math, a special kind of space called a Euclidean vector space is a type of Euclidean space. One common example is Rⁿ, a space with n dimensions. We measure distances in this space using a special method called the dot product.

This example is important because every Euclidean space with n dimensions can match exactly to Rⁿ by choosing a starting point and special directions. This matching is called an isomorphism. So, Rⁿ is often called the standard Euclidean space for dimension n.

Affine structure

Main article: Affine space

Euclidean spaces have special properties called affine properties. These include ideas like lines, subspaces, and parallelism.

Subspaces

Main article: Flat (geometry)

In Euclidean space, a flat or Euclidean subspace is a part of the space that acts like a smaller space on its own. These subspaces have directions connected to them.

Lines and segments

Main article: Line (geometry)

In Euclidean space, a line is a thin path that goes on forever in both directions. There is exactly one line that can pass through two different points. A line segment is a part of a line between two points.

Parallelism

Main article: Parallel (geometry)

Two subspaces are parallel if they have the same direction. In a flat space, two lines either meet at one point or they are parallel and never meet.

Metric structure

A Euclidean space is a special kind of space used in geometry. It has a special way to measure distances and angles, which makes it useful for describing the world around us.

In a Euclidean space, we can measure the distance between any two points. This distance is always positive, and the shortest distance between two points is a straight line. We can also measure the length of lines and the size of angles between them.

One important idea in Euclidean space is orthogonality, which means two lines or directions are at right angles to each other. This helps us understand shapes like squares and rectangles.

Isometries

An isometry is a special kind of mapping between spaces that keeps distances the same.

In Euclidean space, which is the space we use to describe shapes and distances, isometries are very important.

We can think of Euclidean space as the space we live in, with three dimensions: up-down, left-right, and forward-backward. But in math, we can also imagine Euclidean spaces with more or fewer dimensions.

Isometries help us understand how shapes and spaces can move without changing their size or shape. They include simple movements like sliding a shape to a new place (called a translation) or turning it around a point (called a rotation).

Topology

Main article: Real n-space § Topological properties

Euclidean space has a special way of organizing points, called its topology. This helps us understand how points are close to each other.

In these spaces, small shapes stay well-behaved and fit inside larger shapes, making the space easy to study.

Axiomatic definitions

Our idea of Euclidean space is different from what ancient Greek mathematicians like Euclid thought. Long ago, people thought Euclidean space described the real world, but not in a strict way.

In the late 1800s, mathematicians wanted better ways to define space, especially after finding non-Euclidean geometries. Two important ideas came up. Felix Klein said we could describe geometries by looking at their symmetries, using his Erlangen program. David Hilbert used rules based on Euclid's postulates. These rules are part of synthetic geometry and do not need real numbers.

Later, mathematicians like G. D. Birkhoff and Alfred Tarski made even simpler rules, sometimes using real numbers. In Geometric Algebra, Emil Artin showed that all these ways to define Euclidean space really mean the same thing.

Usage

Since ancient times, Euclidean space has helped us understand shapes in the real world. It is important in many sciences like physics, mechanics, and astronomy. We also use it in areas that deal with shapes and positions, such as architecture, geodesy, topography, navigation, industrial design, and technical drawing.

In modern physics, we sometimes think about spaces with more than three dimensions. Euclidean spaces are also used in many parts of mathematics. They help us study shapes and patterns in more complex ways.

Other geometric spaces

Since the late 1800s, people have studied many types of spaces that work like Euclidean spaces but have some unusual properties. These spaces can sometimes be built using Euclidean geometry or fit inside larger Euclidean spaces.

One important type is affine space, which is like Euclidean space but without distances. Affine spaces are used in many areas of math, especially when studying shapes using equations.

Projective space adds special points called "points at infinity" to Euclidean space so that any two lines always meet at exactly one point.

There are also non-Euclidean geometries, like elliptic geometry and hyperbolic geometry. These geometries helped change how we think about math.

Curved spaces are spaces that look like Euclidean space up close but can be bent. For example, the surface of a sphere is a curved space.

A pseudo-Euclidean space is used in Einstein's theory of space-time, where time and space are treated together in special ways.