Three-dimensional space

In geometry, a three-dimensional space is a special kind of mathematical space where three numbers, called coordinates, are needed to find the exact position of a single point. This is often called 3D space, 3-space, or sometimes tri-dimensional space. The most common type is called three-dimensional Euclidean space, which helps us describe the world around us, just like how we see and feel space in our everyday lives.

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

In simple terms, when we talk about width, height, and length — or breadth, depth, and length — we are describing three-dimensional space. These three directions must not all lie in the same flat surface, called a plane, and if they are at right angles to each other, or perpendicular, they help us measure and understand the size and shape of objects in the world.

History

The idea of three-dimensional space has been explored for many centuries. Ancient Greek philosophers like Aristotle thought about how objects take up space, and Euclid wrote about shapes and sizes in his famous work.

Later, mathematicians developed new ways to describe space. In the 1600s, René Descartes introduced a system using three numbers to pinpoint any location, which we now call coordinates. This helped make solving geometry problems much easier. Other great thinkers built on these ideas, leading to many of the mathematical tools we use today to understand the world around us.

In Euclidean geometry

Main article: Coordinate system

In three-dimensional space, we use three numbers called coordinates to describe where a point is. Imagine three lines that cross at a single point called the origin. These lines are called axes and are usually named x, y, and z. By measuring how far a point is along each axis, we can find its exact location in space.

There are different ways to describe a point’s location in three-dimensional space. Besides using x, y, and z coordinates, we can also use cylindrical or spherical coordinates. Each method has its own rules for finding a point’s position.

Main article: Sphere

A sphere is the set of all points that are the same distance from a center point. The space inside a sphere is called a ball. We can calculate the volume and surface area of a sphere using simple formulas.

Main article: Polyhedron

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

Main article: Surface of revolution

Some surfaces are created by spinning a line or curve around an axis. For example, spinning a straight line around an axis can create a cone or a cylinder, depending on the position of the line.

Main article: Quadric surface

Quadric surfaces are shapes defined by special math equations. There are several types, including ellipsoids, hyperboloids, cones, and paraboloids. These shapes are important in many areas of math and science.

In linear algebra

In linear algebra, three-dimensional space depends on the idea of independence. Space has three dimensions because the length of a box is independent of 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 studies changes in vector fields, especially in three-dimensional Euclidean space . It uses a special operator called "del" (∇) for calculations.

Gradient, divergence and curl

The gradient shows the direction where a value increases most. The divergence tells us if a point is a source or sink in a field. The curl measures how much a field rotates 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 surface. 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 that make it different from spaces with more or fewer dimensions. For instance, you need at least three dimensions to tie a knot in a piece of string.

In the study of shapes and smooth surfaces, 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 explore ideas about dimensions using special math rules. The simplest example is something 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 exists a three-dimensional projective space called PG(3,q). For instance, three skew lines in PG(3,q) always fit into one special set called a regulus.