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, or postulates, and proved many properties of space as theorems.

Today, mathematicians study Euclidean spaces of any dimension, 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 dimension, and mathematicians often represent these spaces using Cartesian coordinates.

Definition

Euclidean space is a way to describe the space we live in, based on ideas from ancient Greek mathematicians. They started by looking at the world around them and created simple rules, called postulates or axioms, to explain how shapes and spaces work. This way of thinking about space 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. In the 1800s, mathematicians expanded these ideas to spaces with more than three dimensions, using both old and new methods.

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. This lets us work with space without needing to pick a special starting point or direction, making the math more flexible and useful.

Prototypical examples

In math, a special kind of space called a Euclidean vector space can be thought of as a Euclidean space. One common example is Rⁿ, which means a space with n dimensions, where we use a special way to measure distances called the dot product.

This example is important because every Euclidean space with n dimensions can be matched exactly to Rⁿ by picking a starting point and a special set of 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 very 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 it follows a simple rule: 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, and it plays a big role in many geometric proofs.

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.

When we talk about Euclidean space, we can think of it 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 and how shapes behave in the space. In simple terms, this topology is built using small balls around each point, which helps define what areas are "open" or accessible.

Euclidean spaces are also very neat and organized. Small, closed shapes in these spaces stay well-behaved and fit inside larger shapes, making the space easy to study and work with.

Axiomatic definitions

The way we think about Euclidean space today is different from how ancient Greek mathematicians like Euclid thought about it. Back then, people believed Euclidean space was just a description of the real world around us, not something that needed a strict definition.

Later, in the late 1800s, mathematicians realized they needed better ways to define space, especially after discovering non-Euclidean geometries. Two main ideas emerged. Felix Klein suggested describing geometries by looking at their symmetries, following his Erlangen program. Another approach came from David Hilbert, who used a set of rules inspired by Euclid's postulates. These rules are part of synthetic geometry and don’t need to use real numbers.

Later, mathematicians like G. D. Birkhoff and Alfred Tarski created even simpler rule sets that sometimes use real numbers. In Geometric Algebra, Emil Artin showed that all these different ways to define Euclidean space actually mean the same thing.

Usage

Since the time of the ancient Greeks, Euclidean space has been used to help 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. For example, they help us study shapes and patterns in more complex ways.

Other geometric spaces

Since the late 1800s, many types of spaces have been studied 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 where triangle angles add up to more than 180°, and hyperbolic geometry where they add up to less. 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.