Unit cube
A unit cube, also known as a unit hypercube, is a cube whose sides are all of length 1. More formally, in n-dimensional space, a unit cube is a cube with side length 1.
In 3-dimensional space (the familiar everyday space), a unit cube typically refers to a cube with vertices at the eight points (x, y, z) where x, y, and z are each either 0 or 1. This specific configuration places one vertex at the origin (0, 0, 0) and extends along the positive x, y, and z axes.
The term can also be generalized to n-dimensional space, where the unit cube is the set of all points (x1, x2, ..., xn) such that 0 ≤ xi ≤ 1 for all i from 1 to n. The "standard" unit n-cube has one vertex at the origin and extends along the positive coordinate axes.
The volume of a unit cube in n-dimensional space is always 1, since the volume is calculated as the product of the side lengths (1 * 1 * ... * 1 = 1).
Unit cubes are fundamental objects in various areas of mathematics, including geometry, topology, and computer graphics. They are often used as building blocks for more complex structures and as a convenient way to represent and manipulate data. They serve as a fundamental example in subjects such as measure theory and integration, specifically when dealing with Lebesgue measure.