Three-dimensional space requires three coordinates to pinpoint any location within it. This mathematical concept models our physical universe and helps us understand spatial relationships.
Historical Development
Euclid first explored 3D geometry in Books XI to XIII of his Elements, defining concepts like orthogonality and describing the five regular Platonic solids. In the 17th century, René Descartes and Pierre de Fermat introduced coordinate systems to map three-dimensional space mathematically.
Coordinate Systems
Three perpendicular axes (x, y, z) intersect at the origin point. Each position in space gets defined by three numbers that show the distance from this origin along each axis. Alternative systems include:
- Cylindrical coordinates
- Spherical coordinates
Geometric Properties
Lines and Planes
Two points create a line. Three non-collinear points define a plane. Four points can be:
- All on one line
- All in one plane
- Spread through 3D space
Spheres
A sphere contains all points equidistant from its center. The volume of a sphere follows the formula:
V=43πr3V=34πr3Its surface area is:
A=4πr2A=4πr2
Mathematical Applications
William Rowan Hamilton enhanced our understanding through quaternions, introducing scalar and vector terms. Josiah Willard Gibbs later developed the modern dot and cross product notations, essential tools for describing three-dimensional relationships. The space serves as both a practical model for physical reality and a foundation for more complex mathematical concepts in physics and engineering.
https://en.wikipedia.org/wiki/Three-dimensional_space
This article includes a list of general references, but it lacks sufficient corresponding inline citations. (April 2016) |
In geometry, a three-dimensional space (3D space, 3-space or, rarely, tri-dimensional space) is a mathematical space in which three values (coordinates) are required to determine the position of a point. Most commonly, it is the three-dimensional Euclidean space, that is, the Euclidean space of dimension three, which models physical space. More general three-dimensional spaces are called 3-manifolds. The term may also refer colloquially to a subset of space, a three-dimensional region (or 3D domain), a solid figure.
Technically, a tuple of n numbers can be understood as the Cartesian coordinates of a location in a n-dimensional Euclidean space. The set of these n-tuples is commonly denoted and can be identified to the pair formed by a n-dimensional Euclidean space and a Cartesian coordinate system. When n = 3, this space is called the three-dimensional Euclidean space (or simply "Euclidean space" when the context is clear). In classical physics, it serves as a model of the physical universe, in which all known matter exists. When relativity theory is considered, it can be considered a local subspace of space-time. While this space remains the most compelling and useful way to model the world as it is experienced, it is only one example of a 3-manifold. In this classical example, when the three values refer to measurements in different directions (coordinates), any three directions can be chosen, provided that these directions do not lie in the same plane. Furthermore, if these directions are pairwise perpendicular, the three values are often labeled by the terms width/breadth, height/depth, and length.
