Square (geometry) in the context of Cubes

In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordinate is needed to specify a point on it – for example, the point at 5 on a number line. A surface, such as the boundary of a cylinder or sphere, has a dimension of two (2D) because two coordinates are needed to specify a point on it – for example, both a latitude and longitude are required to locate a point on the surface of a sphere. A two-dimensional Euclidean space is a two-dimensional space on the plane. The inside of a cube, a cylinder or a sphere is three-dimensional (3D) because three coordinates are needed to locate a point within these spaces.

In classical mechanics, space and time are different categories and refer to absolute space and time. That conception of the world is a four-dimensional space but not the one that was found necessary to describe electromagnetism. The four dimensions (4D) of spacetime consist of events that are not absolutely defined spatially and temporally, but rather are known relative to the motion of an observer. Minkowski space first approximates the universe without gravity; the pseudo-Riemannian manifolds of general relativity describe spacetime with matter and gravity. 10 dimensions are used to describe superstring theory (6D hyperspace + 4D), 11 dimensions can describe supergravity and M-theory (7D hyperspace + 4D), and the state-space of quantum mechanics is an infinite-dimensional function space.

The square mile (abbreviated as sq mi and sometimes as mi) is an imperial and US unit of measure for area. One square mile is equal to the area of a square with each side measuring a length of one mile.

Squaring the circle is a problem in geometry first proposed in Greek mathematics. It is the challenge of constructing a square with the area of a given circle by using only a finite number of steps with a compass and straightedge. The difficulty of the problem raised the question of whether specified axioms of Euclidean geometry concerning the existence of lines and circles implied the existence of such a square.

In 1882, the task was proven to be impossible, as a consequence of the Lindemann–Weierstrass theorem, which proves that pi (${\displaystyle \pi }$) is a transcendental number.That is, ${\displaystyle \pi }$ is not the root of any polynomial with rational coefficients. It had been known for decades that the construction would be impossible if ${\displaystyle \pi }$ were transcendental, but that fact was not proven until 1882. Approximate constructions with any given non-perfect accuracy exist, and many such constructions have been found.

In geometry, the square tiling, square tessellation or square grid is a regular tiling of the Euclidean plane consisting of four squares around every vertex. John Horton Conway called it a quadrille.

A quincunx (/ˈkwɪn.kʌŋks/ KWIN-kunks) is a geometric pattern consisting of five points arranged in a cross, with four of them forming a square or rectangle and a fifth at its center. The same pattern has other names, including "in saltire" or "in cross" in heraldry (depending on the orientation of the outer square), the five-point stencil in numerical analysis, and the five dots tattoo. It forms the arrangement of five units in the pattern corresponding to the five-spot on six-sided dice, playing cards, and dominoes. It is represented in Unicode as U+2059 ⁙ FIVE DOT PUNCTUATION or (for the die pattern) U+2684 ⚄ DIE FACE-5.

A shawl (from Persian: شال shāl) is a simple item of clothing, loosely worn over the shoulders, upper body and arms, and sometimes also over the head. It is usually a rectangular piece of cloth, but can also be square or triangular in shape. Other shapes include oblong shawls.

A cube is a three-dimensional solid object in geometry. A cube has eight vertices and twelve straight edges of the same length, so that these edges form six square faces of the same size. It is an example of a polyhedron.

The cube is found in many popular cultures, including toys and games, the arts, optical illusions, and architectural buildings. Cubes can be found in crystal structures, science, and technological devices. It is also found in ancient texts, such as Plato's work Timaeus, which described a set of solids now called Platonic solids, associating a cube with the classical element of earth. A cube with unit length is the canonical unit of volume in three-dimensional space, relative to which other solid objects are measured.

In geometry, a tesseract or 4-cube is a four-dimensional hypercube, analogous to a two-dimensional square and a three-dimensional cube. Just as the perimeter of the square consists of four edges and the surface of the cube consists of six square faces, the hypersurface of the tesseract consists of eight cubical cells, meeting at right angles. The tesseract is one of the six convex regular 4-polytopes.

The tesseract is also called an 8-cell, C₈, (regular) octachoron, or cubic prism. It is the four-dimensional measure polytope, taken as a unit for hypervolume. Coxeter labels it the γ₄ polytope. The term hypercube without a dimension reference is frequently treated as a synonym for this specific polytope.

In mathematics, a unit square is a square whose sides have length 1. Often, the unit square refers specifically to the square in the Cartesian plane with corners at the four points (0, 0), (1, 0), (0, 1), and (1, 1).

In geometry, a hypercube is an n-dimensional analogue of a square (n = 2) and a cube (n = 3); the special case for n = 4 is known as a tesseract. It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, perpendicular to each other and of the same length. A unit hypercube's longest diagonal in n dimensions is equal to ${\displaystyle {\sqrt {n}}}$.

An n-dimensional hypercube is more commonly referred to as an n-cube or sometimes as an n-dimensional cube. The term measure polytope (originally from Elte, 1912) is also used, notably in the work of H. S. M. Coxeter who also labels the hypercubes the γn polytopes.

A square inch (plural: square inches) is a unit of area, equal to the area of a square with sides of one inch. The square inch is a common unit of measurement in the United States and the United Kingdom. Pounds per square inch (psi), a unit of pressure, are derived from this unit of area.