Three-dimensional space
Three values. That is all it takes to pin down where anything sits in the space around you. In geometry, three-dimensional space is the mathematical setting in which exactly three numbers, called coordinates, are needed to fix the position of a single point. People shorten the name to 3D space, to 3-space, and very rarely to tri-dimensional space. The version most of us mean is three-dimensional Euclidean space, the model that stands in for physical space itself. When three perpendicular directions are chosen, those values often go by familiar everyday words: width or breadth, height or depth, and length. Yet this comfortable picture is only one example of something broader. More general three-dimensional spaces carry the name 3-manifolds. So why are there exactly three dimensions and not four, or two? Who first argued that three was the limit? And how did a region that feels so obvious become a stage for cones, quaternions, knots, and curved spacetime? The answers run from an ancient philosopher to the algebra of rotations.
Aristotle set down the case for three dimensions in plain terms. A magnitude divisible one way is a line, he wrote; two ways, a surface; three ways, a body. Beyond these there is no other magnitude, he insisted, because the three dimensions are all that there are, and that which is divisible in three directions is divisible in all. That claim about a ceiling on dimension shaped how the subject was understood for centuries. Books XI to XIII of Euclid's Elements then turned the idea into worked geometry. Book XI builds notions of perpendicularity, parallelism, and orthogonality of lines and planes, along with angles and parallelepiped solids. Book XII reaches toward infinitesimals through the method of exhaustion, finding the area of a circle and the volume of a pyramid, cone, cylinder, or sphere. Book XIII closes by constructing the five regular Platonic solids inside a sphere, covering the cube, octahedra, icosahedra, and dodecahedra. That count of five regular solids would later sit inside a larger family of nine.
René Descartes gave three-dimensional space its coordinate skeleton in the 17th century. His work La Géométrie launched analytic geometry, describing every point by three numbers. Three coordinate axes are drawn, each perpendicular to the other two, meeting at a single crossing point called the origin. The position of any point becomes an ordered triple of real numbers, each one the distance from the origin measured along an axis. Pierre de Fermat reached similar ground on his own, in a manuscript titled Ad locos planos et solidos isagoge, meaning Introduction to Plane and Solid Loci. It went unpublished during Fermat's lifetime, yet his hunt for the extrema of a curve laid groundwork for differential calculus. Isaac Newton offered a different route entirely, introducing the polar coordinate system, a non-Cartesian alternative suited to certain geometries. Cartesian coordinates are not the only way to locate a point; cylindrical coordinates and spherical coordinates also do the job, and the number of possible methods is in fact infinite.
Alexis Clairaut, in the 18th century, studied algebraic curves in space, the concept of tangent space, and curvature, bringing calculus to bear on all three. Leonhard Euler pushed further into the fabric of surfaces. He studied the geodesic on a surface and derived the first analytical geodesic equation. He then introduced the first intrinsic coordinate systems on a surface, opening the theory of intrinsic geometry that underpins modern geometric ideas. In 1760, Euler proved a theorem expressing the curvature of a space curve in terms of the principal curvatures, now called Euler's theorem. Gaspard Monge, later in the same century, added important contributions to the study of curves and surfaces in space. Together the work of Euler and Monge laid the foundations of differential geometry. A surface of revolution shows how simply such shapes can arise: revolve a plane curve, the generatrix, about a fixed axis line. If the generatrix line meets the axis, the result is a right circular cone with its vertex at the meeting point. If the generatrix runs parallel to the axis instead, the surface becomes a circular cylinder.
William Rowan Hamilton, in the 19th century, reached three-dimensional space through the quaternions, a hypercomplex number system. To work with them, Hamilton coined the very terms scalar and vector, first defined in their three-dimensional sense inside his framework. Three-dimensional space could then be described by quaternions whose scalar component vanished. Hamilton did not study them explicitly, yet his work quietly introduced the notions of a basis, given by the quaternion elements, along with the dot product and the cross product. Those two products correspond to the scalar part and the vector part of multiplying two vector quaternions. Josiah Willard Gibbs was the one who finally identified the two products in their own right. The modern notation for the dot and cross product first appeared in his classroom teaching notes, and again in the 1901 textbook Vector Analysis, written by Edwin Bidwell Wilson from Gibbs' lectures. The abstract idea of vector spaces grew from Hermann Grassmann and Giuseppe Peano, the latter giving the first modern definition of a vector space as an algebraic structure. Arthur Cayley carried matrix mathematics into n-dimensional geometry.
Two distinct points always determine a single straight line. Add a third distinct point and the picture branches: three points are either collinear or fix a unique plane. Four distinct points open a further possibility, since they can be collinear, coplanar, or span the entire space. Two distinct lines have three fates of their own. They can intersect, run parallel, or be skew, meaning they neither meet nor share a common plane. Two parallel lines, or two intersecting lines, always lie together in one plane. Two distinct planes either meet in a common line or stay parallel and never touch. Three distinct planes, with no two parallel, can meet in a common line, meet in a single common point, or share no point at all, in which case their three lines of intersection run mutually parallel. A hyperplane is a subspace one dimension below the whole space, so in three-dimensional space the hyperplanes are exactly the planes. Each plane satisfies a single linear equation in Cartesian coordinates, while a line needs a pair of independent linear equations. Varignon's theorem adds a tidy result: the midpoints of any quadrilateral form a parallelogram, and so they are coplanar.
A sphere in 3-space is the set of all points at a fixed distance from a central point P. It is also called a 2-sphere, because like every surface it is intrinsically two-dimensional, and the solid it encloses is called a ball, or 3-ball. A stranger relative arises from a 4-ball, whose three-dimensional surface is the 3-sphere, made of points equidistant from the origin of Euclidean space. That 3-sphere is itself an example of a 3-manifold, a space that looks locally like 3-D space: each of its points has a neighborhood homeomorphic to an open subset of three-dimensional space. Regular shapes come in a fixed count here. In three dimensions there are nine regular polytopes, the five convex Platonic solids and the four nonconvex Kepler-Poinsot polyhedra. Curved surfaces defined by a second-degree equation are called quadric surfaces, and there are six non-degenerate types: the ellipsoid, the hyperboloid of one sheet, the hyperboloid of two sheets, the elliptic cone, the elliptic paraboloid, and the hyperbolic paraboloid. The hyperboloid of one sheet and the hyperbolic paraboloid are ruled surfaces, built from straight lines. Each carries two families of generating lines, and every such family is called a regulus.
Independence is what makes three the right number. Space has three dimensions because the length of a box is independent of its width or breadth, so every point can be written as a linear combination of three independent vectors. A vector can be pictured as an arrow whose length is its magnitude and whose heading is its direction. The dot product of two non-zero vectors involves the angle between them, and it can compute the work done by a constant force pulling a block down an inclined plane at an angle to its motion. The cross product, written with the symbol times, takes two vectors and returns a vector perpendicular to both, normal to the plane that holds them. It can find the torque on a bolt turned by a wrench, or the Lorentz force on an electron moving through a magnetic field. Its components can be written using the Einstein summation convention with the Levi-Civita symbol. Together, space and this product form an algebra that is neither commutative nor associative, yet is a Lie algebra with the cross product as the Lie bracket. It is isomorphic to the Lie algebra of three-dimensional rotations. In place of associativity, the cross product obeys the Jacobi identity. A binary product yielding a perpendicular vector exists only in three and seven dimensions. Three-dimensional space also stands apart in topology, since at least three dimensions are required to tie a knot in a piece of string, and the curved spacetime of General Relativity shows how the same 3-manifold can bend in many ways while staying continuous.
Common questions
What is three-dimensional space in geometry?
Three-dimensional space is a mathematical space in which three values, called coordinates, are required to determine the position of a point. It is also known as 3D space, 3-space, or rarely tri-dimensional space, and most commonly refers to three-dimensional Euclidean space, which models physical space.
Why are there exactly three dimensions according to Aristotle?
Aristotle argued that a magnitude divisible one way is a line, two ways a surface, and three ways a body. He held that beyond these there is no other magnitude, because the three dimensions are all that there are, and that which is divisible in three directions is divisible in all.
Who developed Cartesian coordinates for three-dimensional space?
René Descartes introduced Cartesian coordinates for three-dimensional space in the 17th century in his work La Géométrie, launching analytic geometry. Pierre de Fermat independently developed similar ideas in his unpublished manuscript Ad locos planos et solidos isagoge.
What are the regular polytopes in three-dimensional space?
There are nine regular polytopes in three dimensions: the five convex Platonic solids and the four nonconvex Kepler-Poinsot polyhedra. The five Platonic solids include the cube, octahedra, icosahedra, and dodecahedra.
What is the cross product in three-dimensional space?
The cross product is a binary operation on two vectors in three-dimensional space, denoted by the symbol times, that produces a vector perpendicular to both and normal to the plane containing them. It can compute torque on a bolt turned by a wrench or the Lorentz force on an electron moving through a magnetic field, and such a binary product exists only in three and seven dimensions.
How is three-dimensional space related to quaternions?
William Rowan Hamilton described three-dimensional space using quaternions with a vanishing scalar component, and coined the terms scalar and vector within that framework. His work indirectly introduced the dot product and cross product, which Josiah Willard Gibbs later identified in their own right.
What are the six non-degenerate quadric surfaces in three-dimensional space?
The six non-degenerate quadric surfaces are the ellipsoid, the hyperboloid of one sheet, the hyperboloid of two sheets, the elliptic cone, the elliptic paraboloid, and the hyperbolic paraboloid. The hyperboloid of one sheet and the hyperbolic paraboloid are ruled surfaces made up of straight lines.
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