Three-dimensional space
Books XI to XIII of Euclid's Elements dealt with three-dimensional geometry. Book XI develops notions of perpendicularity, parallelism, and orthogonality of lines and planes. The construction and properties of angles appear throughout these texts. Parallelepiped solids form a key subject in the early chapters. Book XII discusses infinitesimals and the method of exhaustion for finding volume. A pyramid or cone requires this method to calculate its space. A cylinder or sphere also falls under this mathematical scope. Book XIII describes the construction of five regular Platonic solids inside a sphere. These shapes include the cube, octahedra, icosahedra and dodecahedra. Aristotle recognized the existence of three dimensions before Euclid wrote his works.
In the 17th century, three-dimensional space was described with Cartesian coordinates. René Descartes developed analytic geometry in his work La Géométrie. Pierre de Fermat independently developed similar ideas in the manuscript Ad locos planos et solidos isagoge. This manuscript remained unpublished during Fermat's lifetime. Three coordinate axes are given, each perpendicular to the other two at the origin. They are usually labeled x, y, and z. Relative to these axes, the position of any point in three-dimensional space is given by an ordered triple of real numbers. Each number gives the distance from the origin measured along the given axis. Other popular methods include cylindrical coordinates and spherical coordinates. Isaac Newton introduced the polar coordinate system as an alternative non-Cartesian system.
In three dimensions, there are nine regular polytopes. Five convex Platonic solids exist alongside four nonconvex Kepler-Poinsot polyhedra. The symmetry group for the tetrahedron is Td with order 24. The cube has symmetry Oh with order 48. The dodecahedron carries symmetry Ih with order 120. A sphere in 3-space consists of all points at a fixed distance r from a central point c. The solid enclosed by the sphere is called a ball or 3-ball. The volume of the ball equals four-thirds pi times r cubed. The surface area of the sphere equals four pi times r squared. Another type of sphere arises from a 4-ball whose three-dimensional surface is the 3-sphere. Points equidistant to the origin characterize those on the unit 3-sphere centered at the origin.
William Rowan Hamilton developed quaternions in the 19th century. He coined the terms scalar and vector within his geometric framework. Josiah Willard Gibbs identified dot products and cross products as distinct operations. Edwin Bidwell Wilson wrote Vector Analysis based on Gibbs' lectures in 1901. The cross product A × B produces a vector perpendicular to both inputs. It is normal to the plane containing them. This operation computes torque on a bolt turned by a wrench. It also calculates Lorentz force on an electron traveling through a magnetic field. The magnitude relates to the angle between vectors via sine. The space forms a Lie algebra with the Jacobi identity holding true. Binary products with vector results exist only in three and seven dimensions.
Vector calculus concerns infinitesimal changes to vector fields in three-dimensional Euclidean space. The del operator or nabla symbol guides differentiation. Gradient indicates direction of greatest increase for a function. Divergence shows net flux around a point like particle density change. Curl indicates rotational circulation of a vector field. Stokes theorem relates surface integral of curl to line integral over boundary. Divergence theorem connects volume integral to surface integral over boundary. A line integral sums function values along every infinitesimal increment of curve C. Surface integrals generalize multiple integrals to integration over surfaces. Volume integrals measure regions within three-dimensional domains. These tools model fluid flow and electromagnetism effectively.
Three-dimensional space has topological properties distinguishing it from other dimensions. At least three dimensions are required to tie a knot in string. Generic three-dimensional spaces are 3-manifolds locally resembling R3. Globally, these manifolds can curve in various manners while remaining continuous. Curved spacetime found in General Relativity serves as an example. In finite geometry, PG(3,2) contains Fano planes as its subspaces. Any three skew lines in PG(3,q) lie within exactly one regulus. Three distinct planes meeting pairwise form unique points or common lines. Four distinct points determine the entire space if not collinear or coplanar. Two parallel lines or intersecting lines lie in a unique plane. Skew lines do not meet and do not share a common plane.
Common questions
What books of Euclid's Elements dealt with three-dimensional geometry?
Books XI to XIII of Euclid's Elements dealt with three-dimensional geometry. Book XI develops notions of perpendicularity, parallelism, and orthogonality of lines and planes. Book XII discusses infinitesimals and the method of exhaustion for finding volume.
When was three-dimensional space described with Cartesian coordinates?
Three-dimensional space was described with Cartesian coordinates in the 17th century. René Descartes developed analytic geometry in his work La Géométrie during this period. Pierre de Fermat independently developed similar ideas in the manuscript Ad locos planos et solidos isagoge.
How many regular polytopes exist in three dimensions?
There are nine regular polytopes in three dimensions. Five convex Platonic solids exist alongside four nonconvex Kepler-Poinsot polyhedra. The symmetry group for the tetrahedron is Td with order 24 while the cube has symmetry Oh with order 48.
Who developed quaternions in the 19th century?
William Rowan Hamilton developed quaternions in the 19th century. He coined the terms scalar and vector within his geometric framework. Josiah Willard Gibbs identified dot products and cross products as distinct operations later.
Why does three-dimensional space allow knots to form in string?
At least three dimensions are required to tie a knot in string. Generic three-dimensional spaces are 3-manifolds locally resembling R3. Globally these manifolds can curve in various manners while remaining continuous.