enhedskubus

The enhedskubus, or unit cube, is a cube in three-dimensional Euclidean space with side length 1. It can be described as the set of all points (x, y, z) in R^3 with 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, and 0 ≤ z ≤ 1. Equivalently, it is the Cartesian product [0,1] × [0,1] × [0,1]. It is the 3-dimensional analogue of the square and serves as the standard unit volume in three dimensions.

Its volume is 1 cubic unit and its surface area is 6 square units. It has 12 edges of length 1, 6 square faces, and 8 vertices at all combinations of 0 and 1: (0,0,0), (0,0,1), …, (1,1,1). The space diagonal from (0,0,0) to (1,1,1) has length sqrt(3). Face diagonals have length sqrt(2). The center is at (0.5, 0.5, 0.5). The inscribed sphere has radius 1/2, tangent to all faces; the circumscribed sphere has radius sqrt(3)/2, passing through all vertices.

The unit cube is the regular 3-cube, a member of the family of hypercubes. It has 24 rotational symmetries (and 48 total symmetries including reflections). It tiles R^3 by translations along the coordinate axes and serves as a fundamental domain for the integer lattice Z^3.

In applications, the unit cube functions as a standard measure space for sampling, integration, and probability in three dimensions, and as a basic building block in computer graphics and numerical methods. The term enhedskubus is Danish for unit cube.