Three-dimensional space

Did You Know?

• At least three dimensions are required to tie a knot in a piece of string.
• In finite geometry, the simplest instance of a three-dimensional projective space is PG(3,2), which has Fano planes as its 2-dimensional subspaces.
• Any three skew lines in PG(3,q) are contained in exactly one regulus.

History

The idea that space has three dimensions was noted by the ancient philosopher Aristotle. Later, in Euclid's Elements, the books numbered eleven through thirteen explored three-dimensional geometry. The eleventh book covered concepts like perpendicular and parallel lines and planes, along with angles and parallelepiped solids. The twelfth book used infinitesimals and the method of exhaustion to find areas and volumes, such as those of a circle, pyramid, cone, cylinder, or sphere. The thirteenth book showed how to construct the five regular Platonic solids—the cube, octahedron, icosahedron, and dodecahedron—inside a sphere.

In the 1600s, three-dimensional space gained a new description through Cartesian coordinates, part of analytic geometry developed by Rene Descartes in his work La Geometrie. Pierre de Fermat independently came up with similar ideas in a manuscript called Ad locos planos et solidos isagoge, though it was not published during his life. Fermat's work on finding the extremes of a curve helped lay the groundwork for differential calculus. Isaac Newton introduced the polar coordinate system as another way to describe space, useful for certain geometric problems.

During the 1700s, Alexis Clairaut studied algebraic curves in space, along with tangent spaces and curvature, applying calculus to these ideas. Leonhard Euler worked on geodesics on surfaces, deriving the first analytical equation for a geodesic, and later created the first intrinsic coordinate systems for surfaces, starting the field of intrinsic geometry. In 1760, Euler proved a theorem relating the curvature of a space curve on a surface to its principal curvatures, now called Euler's theorem. Gaspard Monge also made key contributions to the study of curves and surfaces in space. The work of Euler and Monge formed the basis for differential geometry.

In the 1800s, William Rowan Hamilton developed quaternions, a hypercomplex number system. He coined the terms scalar and vector, defining them in a three-dimensional sense within his geometric framework for quaternions. Three-dimensional space could then be described using quaternions with a zero scalar component.

In Euclidean geometry

In mathematics, analytic geometry, also known as Cartesian geometry, uses three coordinates to locate any point in three-dimensional space. These coordinates are based on three axes—typically labeled x, y, and z—that meet at a single point called the origin, with each axis perpendicular to the other two. A point’s position is given as an ordered triple of real numbers, where each number measures the distance from the origin along its corresponding axis, which is also the distance from the point to the plane formed by the other two axes.

Other coordinate systems, such as cylindrical and spherical coordinates, can also describe points, though infinitely many methods exist. Two distinct points always define a straight line. Three distinct points are either collinear or define a unique plane. Four distinct points may be collinear, coplanar, or span the entire space.

Two distinct lines can intersect, be parallel, or be skew; intersecting or parallel lines lie in a single plane, while skew lines neither meet nor share a plane. Two distinct planes either intersect along a common line or are parallel. Three distinct planes, with no two parallel, may share a common line, meet at a single point, or have no common point; in the last case, the intersection lines of each pair are mutually parallel. A line can lie within a plane, intersect it at one point, or be parallel to it.

A hyperplane is a subspace one dimension lower than the full space; in three dimensions, hyperplanes are planes (two-dimensional subspaces). In Cartesian coordinates, a hyperplane’s points satisfy one linear equation, so planes are described by linear equations. A line can be described by two independent linear equations, each representing a plane that intersects along that line. Varignon’s theorem states that the midpoints of any quadrilateral in three-dimensional space form a parallelogram and lie in the same plane.

In linear algebra

In linear algebra, space has three dimensions because the length of a box is independent of its width or breadth; every point in space can be described by a linear combination of three independent vectors. A vector can be pictured as an arrow, with magnitude its length and direction the direction the arrow points. A vector in three-dimensional space can be represented by an ordered triple of real numbers called its components. The dot product of two vectors A = [A1, A2, A3] and B = [B1, B2, B3] is defined as the sum of the products of their corresponding components.

The magnitude of a vector A is denoted by the square root of the dot product of A with itself, giving the Euclidean length. Without reference to components, the dot product of two non-zero Euclidean vectors A and B is given by the product of their magnitudes and the cosine of the angle θ between them. For a physical example, the dot product can compute the work performed by a constant force vector applied at an angle to the downslope direction of motion.

The cross product is a binary operation on two vectors in three-dimensional space, denoted by the symbol ×; the cross product A × B is a vector perpendicular to both A and B and normal to the plane containing them. It can compute torque on a bolt or the Lorentz force on an electron. The cross product is a function from pairs of vectors to vectors; its components can be written using the Levi-Civita symbol. Its magnitude is related to the angle between the vectors.

The space and cross product form an algebra over a field that is not commutative nor associative but is a Lie algebra with the cross product as the Lie bracket, isomorphic to the Lie algebra of three-dimensional rotations. Instead of associativity, the cross product satisfies the Jacobi identity.

In calculus

Vector calculus is concerned with infinitesimal and cumulative changes to vector fields, primarily in three-dimensional Euclidean space. The del operator, or nabla, is used for differentiation. The gradient indicates the direction of greatest increase of a function and its magnitude; in a rectangular coordinate system, the gradient of a differentiable function f is given by the sum of its partial derivatives with respect to each coordinate multiplied by the respective unit vectors i, j, and k. The divergence indicates the net flux of a vector field around a point, whether the location is a source or sink; for a differentiable vector field F = U i + V j + W k, divergence is the scalar-valued function equal to the sum of the partial derivatives of its components.

The curl is a vector indicating the rotational circulation of a vector field; expanded in Cartesian coordinates, the curl ∇ × F for F composed of [Fx, Fy, Fz] is given by a determinant involving partial derivatives and unit vectors. In index notation with Einstein summation convention, the curl uses the Levi-Civita symbol. A line integral of a scalar field f along a piecewise smooth curve C is defined as the integral of f evaluated along a parametrization r of C, multiplied by the magnitude of the derivative of r. For a vector field F, the line integral along C in the direction of r is defined as the integral of the dot product of F with the derivative of r. A subtype of line integral found in physics is the plane closed loop, which determines the circulation of the function around the loop.