Three-dimensional space

In geometry, a three-dimensional space is a special kind of mathematical space. In this space, three numbers, called coordinates, tell us the exact position of a single point. We often call this 3D space, 3-space, or tri-dimensional space. The most common type is three-dimensional Euclidean space. It helps us describe the world around us, just like how we see and feel space every day.

Three-dimensional space is very important in classical physics. It is used as a model for the universe, where all known matter exists. Even when we study advanced ideas like relativity theory, three-dimensional space helps us understand parts of something larger called space-time.

Simply put, when we talk about width, height, and length, we are describing three-dimensional space. These three directions are not all on the same flat surface, called a plane. If they are at right angles to each other, or perpendicular, they help us measure and understand the size and shape of objects in our world.

History

The idea of three-dimensional space has been explored for many years. Ancient Greek thinkers like Aristotle wondered how objects take up space. Euclid wrote about shapes and sizes.

Later, mathematicians found new ways to describe space. In the 1600s, René Descartes created a system using three numbers to show any place. We now call these numbers coordinates. This made solving geometry problems easier. Other thinkers added to these ideas, helping us understand the world better.

In Euclidean geometry

Main article: Coordinate system

In three-dimensional space, we use three numbers to show where a point is. Think of three lines that meet at one spot called the origin. These lines are named x, y, and z. By measuring how far a point is along each line, we can find its place in space.

We can describe a point in different ways. Besides using x, y, and z, we can also use cylindrical or spherical coordinates. Each way has its own rules.

Main article: Sphere

A sphere is all the points that are the same distance from a center point. The space inside a sphere is called a ball. We can find the size and area of a sphere using simple math.

Main article: Polyhedron

In three dimensions, there are special shapes called regular polytopes. These include five simple shapes known as Platonic solids and four more complex shapes.

Main article: Surface of revolution

Some surfaces are made by spinning a line or curve around an axis. For example, spinning a straight line can make a cone or a cylinder.

Main article: Quadric surface

Quadric surfaces are shapes defined by special math rules. There are several types, like ellipsoids, hyperboloids, cones, and paraboloids. These shapes are useful in many parts of math and science.

In linear algebra

In linear algebra, three-dimensional space uses the idea of independence. Space has three dimensions because the length of a box does not depend on its width or breadth. Every point in space can be described using three independent vectors.

Dot product, angle, and length

The dot product helps us understand angles and lengths between vectors. For example, it can calculate the work done by a force pushing an object on an inclined plane.

Cross product

The cross product is a special way to find a vector that is perpendicular to two other vectors. It has many uses in physics and engineering, like calculating the torque on a bolt or the Lorentz force on an electron in a magnetic field.

Main article: Cross product

In calculus

Main article: vector calculus

Vector calculus looks at changes in vector fields in three-dimensional Euclidean space . It uses a special symbol called "del" (∇) to help with calculations.

Gradient, divergence and curl

The gradient shows the direction where a value grows the most. The divergence tells us if a point is adding to or taking away from a field. The curl measures how much a field spins around a point.

Line, surface, and volume integrals

A line integral adds up values along a curve. A surface integral does the same but over a flat area. A volume integral adds up values through a space, like finding the total amount inside a shape.

Main article: Fundamental theorem of line integrals

Main article: Stokes' theorem

Main article: Divergence theorem

In topology

Three-dimensional space has special properties. It is different from spaces with more or fewer dimensions. For example, you need at least three dimensions to tie a knot in a piece of string.

In the study of shapes, three-dimensional spaces are called 3-manifolds. These spaces can curve and bend in many ways while staying connected, like the curved spacetime described in the theory of General Relativity.

In finite geometry

In finite geometry, we can learn about dimensions with special math rules. A simple example is called PG(3,2), which uses Fano planes as its two-dimensional parts. This example is part of Galois geometry, where we study projective geometry with finite fields. For any Galois field GF(q), there is a three-dimensional projective space called PG(3,q). For example, three skew lines in PG(3,q) always fit into one special set called a regulus.