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60-422: The cube is the three-dimensional hypercube , a family of polytopes also including the two-dimensional square and four-dimensional tesseract . A cube with unit side length is the canonical unit of volume in three-dimensional space, relative to which other solid objects are measured. The cube can be represented in many ways, one of which is the graph known as the cubical graph . It can be constructed by using

120-450: A {\displaystyle 2a} is the locus of all points ( x , y , z ) {\displaystyle (x,y,z)} such that max { | x − x 0 | , | y − y 0 | , | z − z 0 | } = a . {\displaystyle \max\{|x-x_{0}|,|y-y_{0}|,|z-z_{0}|\}=a.} The cube

180-733: A Minkowski sum : the d -dimensional hypercube is the Minkowski sum of d mutually perpendicular unit-length line segments, and is therefore an example of a zonotope . The 1- skeleton of a hypercube is a hypercube graph . A unit hypercube of dimension n {\displaystyle n} is the convex hull of all the 2 n {\displaystyle 2^{n}} points whose n {\displaystyle n} Cartesian coordinates are each equal to either 0 {\displaystyle 0} or 1 {\displaystyle 1} . These points are its vertices . The hypercube with these coordinates

240-441: A compass and straightedge solely. Ancient mathematicians could not solve this old problem until French mathematician Pierre Wantzel in 1837 proved it was impossible. With edge length a {\displaystyle a} , the inscribed sphere of a cube is the sphere tangent to the faces of a cube at their centroids, with radius 1 2 a {\textstyle {\frac {1}{2}}a} . The midsphere of

300-440: 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

360-476: A hyperrectangle (also called an n-orthotope ). A unit hypercube is a hypercube whose side has length one unit . Often, the hypercube whose corners (or vertices ) are the 2 points in R with each coordinate equal to 0 or 1 is called the unit hypercube. A hypercube can be defined by increasing the numbers of dimensions of a shape: This can be generalized to any number of dimensions. This process of sweeping out volumes can be formalized mathematically as

420-399: A ( 1 {\displaystyle 1} -dimensional) line segment has 2 {\displaystyle 2} endpoints; a ( 2 {\displaystyle 2} -dimensional) square has 4 {\displaystyle 4} sides or edges; a 3 {\displaystyle 3} -dimensional cube has 6 {\displaystyle 6} square faces;

480-517: A ( 4 {\displaystyle 4} -dimensional) tesseract has 8 {\displaystyle 8} three-dimensional cubes as its facets. The number of vertices of a hypercube of dimension n {\displaystyle n} is 2 n {\displaystyle 2^{n}} (a usual, 3 {\displaystyle 3} -dimensional cube has 2 3 = 8 {\displaystyle 2^{3}=8} vertices, for instance). The number of

540-444: A cube is the sphere tangent to the edges of a cube, with radius 1 2 a {\textstyle {\frac {1}{\sqrt {2}}}a} . The circumscribed sphere of a cube is the sphere tangent to the vertices of a cube, with radius 3 2 a {\textstyle {\frac {\sqrt {3}}{2}}a} . For a cube whose circumscribed sphere has radius R {\displaystyle R} , and for

600-404: A decimal-based system of measurement devised by Edmund Gunter in 1620. The base unit is Gunter's chain of 66 feet (20 m) which is subdivided into 4 rods, each of 16.5 ft or 100 links of 0.66 feet. A link is abbreviated "lk", and links "lks", in old deeds and land surveys done for the government. Astronomical measure uses: In atomic physics, sub-atomic physics, and cosmology,

660-485: A gap. The cube can be represented as the cell , and examples of a honeycomb are cubic honeycomb , order-5 cubic honeycomb , order-6 cubic honeycomb , and order-7 cubic honeycomb . The cube can be constructed with six square pyramids , tiling space by attaching their apices. Polycube is a polyhedron in which the faces of many cubes are attached. Analogously, it can be interpreted as the polyominoes in three-dimensional space. When four cubes are stacked vertically, and

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720-782: A given point in its three-dimensional space with distances d i {\displaystyle d_{i}} from the cube's eight vertices, it is: 1 8 ∑ i = 1 8 d i 4 + 16 R 4 9 = ( 1 8 ∑ i = 1 8 d i 2 + 2 R 2 3 ) 2 . {\displaystyle {\frac {1}{8}}\sum _{i=1}^{8}d_{i}^{4}+{\frac {16R^{4}}{9}}=\left({\frac {1}{8}}\sum _{i=1}^{8}d_{i}^{2}+{\frac {2R^{2}}{3}}\right)^{2}.} The cube has octahedral symmetry O h {\displaystyle \mathrm {O} _{\mathrm {h} }} . It

780-415: A part of the hypercube graph, it is also an example of a unit distance graph . Like other graphs of cuboids, the cubical graph is also classified as a prism graph . An object illuminated by parallel rays of light casts a shadow on a plane perpendicular to those rays, called an orthogonal projection . A polyhedron is considered equiprojective if, for some position of the light, its orthogonal projection

840-453: A plane by the process known as polar reciprocation . One property of dual polyhedrons generally is that the polyhedron and its dual share their three-dimensional symmetry point group . In this case, the dual polyhedron of a cube is the regular octahedron , and both of these polyhedron has the same symmetry, the octahedral symmetry. The cube is face-transitive , meaning its two squares are alike and can be mapped by rotation and reflection. It

900-1067: A polyhedron by connecting along the edges of those polygons. Eleven nets for the cube are shown here. In analytic geometry , a cube may be constructed using the Cartesian coordinate systems . For a cube centered at the origin, with edges parallel to the axes and with an edge length of 2, the Cartesian coordinates of the vertices are ( ± 1 , ± 1 , ± 1 ) {\displaystyle (\pm 1,\pm 1,\pm 1)} . Its interior consists of all points ( x 0 , x 1 , x 2 ) {\displaystyle (x_{0},x_{1},x_{2})} with − 1 < x i < 1 {\displaystyle -1<x_{i}<1} for all i {\displaystyle i} . A cube's surface with center ( x 0 , y 0 , z 0 ) {\displaystyle (x_{0},y_{0},z_{0})} and edge length of 2

960-589: A side length corresponding to that of the base. Similarly, the exponent 3 will yield a perfect cube , an integer which can be arranged into a cube shape with a side length of the base. As a result, the act of raising a number to 2 or 3 is more commonly referred to as " squaring " and "cubing", respectively. However, the names of higher-order hypercubes do not appear to be in common use for higher powers. Unit of length A unit of length refers to any arbitrarily chosen and accepted reference standard for measurement of length. The most common units in modern use are

1020-477: A simple combinatorial argument: for each of the 2 n {\displaystyle 2^{n}} vertices of the hypercube, there are ( n m ) {\displaystyle {\tbinom {n}{m}}} ways to choose a collection of m {\displaystyle m} edges incident to that vertex. Each of these collections defines one of the m {\displaystyle m} -dimensional faces incident to

1080-443: A space—called honeycomb —in which each face of any of its copies is attached to a like face of another copy. There are five kinds of parallelohedra, one of which is the cuboid. Every three-dimensional parallelohedron is zonohedron , a centrally symmetric polyhedron whose faces are centrally symmetric polygons , An elementary way to construct a cube is using its net , an arrangement of edge-joining polygons constructing

1140-405: A time interval of 1 ⁄ 299792458 seconds." It is approximately equal to 1.0936 yd . Other SI units are derived from the meter by adding prefixes , as in millimeter or kilometer, thus producing systematic decimal multiples and submultiples of the base unit that span many orders of magnitude. For example, a kilometer is 1000 m . In the centimeter–gram–second system of units ,

1200-515: Is 2 {\displaystyle 2} , and its n {\displaystyle n} -dimensional volume is 2 n {\displaystyle 2^{n}} . Every hypercube admits, as its faces, hypercubes of a lower dimension contained in its boundary. A hypercube of dimension n {\displaystyle n} admits 2 n {\displaystyle 2n} facets, or faces of dimension n − 1 {\displaystyle n-1} :

1260-411: Is Hanner polytope , because it can be constructed by using Cartesian product of three line segments. Its dual polyhedron, the regular octahedron, is constructed by direct sum of three line segments. According to Steinitz's theorem , the graph can be represented as the skeleton of a polyhedron; roughly speaking, a framework of a polyhedron. Such a graph has two properties. It is planar , meaning

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1320-409: Is p vertices and pn facets. Any positive integer raised to another positive integer power will yield a third integer, with this third integer being a specific type of figurate number corresponding to an n -cube with a number of dimensions corresponding to the exponential. For example, the exponent 2 will yield a square number or "perfect square", which can be arranged into a square shape with

1380-450: Is vertex-transitive , meaning all of its vertices are equivalent and can be mapped isometrically under its symmetry. It is also edge-transitive , meaning the same kind of faces surround each of its vertices in the same or reverse order, all two adjacent faces have the same dihedral angle . Therefore, the cube is regular polyhedron because it requires those properties. The cube is a special case among every cuboids . As mentioned above,

1440-406: Is a regular polygon. The cube is equiprojective because, if the light is parallel to one of the four lines joining a vertex to the opposite vertex, its projection is a regular hexagon . Conventionally, the cube is 6-equiprojective. The cube can be represented as configuration matrix . A configuration matrix is a matrix in which the rows and columns correspond to the elements of a polyhedron as in

1500-478: Is a set of polyhedrons known since antiquity. It was named after Plato in his Timaeus dialogue, who attributed these solids with nature. One of them, the cube, represented the classical element of earth because of its stability. Euclid 's Elements defined the Platonic solids, including the cube, and using these solids with the problem involving to find the ratio of the circumscribed sphere's diameter to

1560-412: Is also the cartesian product [ 0 , 1 ] n {\displaystyle [0,1]^{n}} of n {\displaystyle n} copies of the unit interval [ 0 , 1 ] {\displaystyle [0,1]} . Another unit hypercube, centered at the origin of the ambient space, can be obtained from this one by a translation . It is the convex hull of

1620-458: Is also the cartesian product [ − 1 / 2 , 1 / 2 ] n {\displaystyle [-1/2,1/2]^{n}} . Any unit hypercube has an edge length of 1 {\displaystyle 1} and an n {\displaystyle n} -dimensional volume of 1 {\displaystyle 1} . The n {\displaystyle n} -dimensional hypercube obtained as

1680-474: Is composed of reflection symmetry , a symmetry by cutting into two halves by a plane. There are nine reflection symmetries: the five are cut the cube from the midpoints of its edges, and the four are cut diagonally. It is also composed of rotational symmetry , a symmetry by rotating it around the axis, from which the appearance is interchangeable. It has octahedral rotation symmetry O {\displaystyle \mathrm {O} } : three axes pass through

1740-406: Is equal to n {\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 γ n polytopes. The hypercube is the special case of

1800-452: Is not in the same face, formulated as a 3 {\displaystyle a{\sqrt {3}}} . Both formulas can be determined by using Pythagorean theorem . The surface area of a cube A {\displaystyle A} is six times the area of a square: A = 6 a 2 . {\displaystyle A=6a^{2}.} The volume of a cuboid is the product of its length, width, and height. Because all

1860-489: Is one of three regular polytope families, labeled by Coxeter as γ n . The other two are the hypercube dual family, the cross-polytopes , labeled as β n, and the simplices , labeled as α n . A fourth family, the infinite tessellations of hypercubes , is labeled as δ n . Another related family of semiregular and uniform polytopes is the demihypercubes , which are constructed from hypercubes with alternate vertices deleted and simplex facets added in

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1920-520: Is the largest cube that can pass through a hole cut into the unit cube, despite having sides approximately 6% longer. A polyhedron that can pass through a copy of itself of the same size or smaller is said to have the Rupert property . A geometric problem of doubling the cube —alternatively known as the Delian problem —requires the construction of a cube with a volume twice the original by using

1980-581: Is the length of the edges of the hypercube. These numbers can also be generated by the linear recurrence relation . For example, extending a square via its 4 vertices adds one extra line segment (edge) per vertex. Adding the opposite square to form a cube provides E 1 , 3 = 12 {\displaystyle E_{1,3}=12} line segments. The extended f-vector for an n -cube can also be computed by expanding ( 2 x + 1 ) n {\displaystyle (2x+1)^{n}} (concisely, (2,1) ), and reading off

2040-402: The 2 n {\displaystyle 2^{n}} points whose vectors of Cartesian coordinates are Here the symbol ± {\displaystyle \pm } means that each coordinate is either equal to 1 / 2 {\displaystyle 1/2} or to − 1 / 2 {\displaystyle -1/2} . This unit hypercube

2100-879: The m {\displaystyle m} -dimensional hypercubes (just referred to as m {\displaystyle m} -cubes from here on) contained in the boundary of an n {\displaystyle n} -cube is For example, the boundary of a 4 {\displaystyle 4} -cube ( n = 4 {\displaystyle n=4} ) contains 8 {\displaystyle 8} cubes ( 3 {\displaystyle 3} -cubes), 24 {\displaystyle 24} squares ( 2 {\displaystyle 2} -cubes), 32 {\displaystyle 32} line segments ( 1 {\displaystyle 1} -cubes) and 16 {\displaystyle 16} vertices ( 0 {\displaystyle 0} -cubes). This identity can be proven by

2160-493: The Cartesian product of graphs . The cube was discovered in antiquity. It was associated with the nature of earth by Plato , the founder of Platonic solid. It was used as the part of the Solar System , proposed by Johannes Kepler . It can be derived differently to create more polyhedrons, and it has applications to construct a new polyhedron by attaching others. A cube is a special case of rectangular cuboid in which

2220-634: The metric units , used in every country globally. In the United States the U.S. customary units are also in use. British Imperial units are still used for some purposes in the United Kingdom and some other countries. The metric system is sub-divided into SI and non-SI units. The base unit in the International System of Units (SI) is the meter , defined as "the length of the path travelled by light in vacuum during

2280-533: The vertex figure are regular simplexes . The regular polygon perimeter seen in these orthogonal projections is called a Petrie polygon . The generalized squares ( n = 2) are shown with edges outlined as red and blue alternating color p -edges, while the higher n -cubes are drawn with black outlined p -edges. The number of m -face elements in a p -generalized n -cube are: p n − m ( n m ) {\displaystyle p^{n-m}{n \choose m}} . This

2340-464: The United States continue to use: The Australian building trades adopted the metric system in 1966 and the units used for measurement of length are meters (m) and millimeters (mm). Centimeters (cm) are avoided as they cause confusion when reading plans . For example, the length two and a half meters is usually recorded as 2500 mm or 2.5 m; it would be considered non-standard to record this length as 250 cm. American surveyors use

2400-601: The basic unit of length is the centimeter , or 1 ⁄ 100 of a meter. Other non-SI units are derived from decimal multiples of the meter. The basic unit of length in the imperial and U.S. customary systems is the yard , defined as exactly 0.9144 m by international treaty in 1959. Common imperial units and U.S. customary units of length include: In addition, the following are used by sailors : Aviators use feet for altitude worldwide (except in Russia and China) and nautical miles for distance. Surveyors in

2460-433: The coefficients of the resulting polynomial . For example, the elements of a tesseract is (2,1) = (4,4,1) = (16,32,24,8,1). An n -cube can be projected inside a regular 2 n -gonal polygon by a skew orthogonal projection , shown here from the line segment to the 16-cube. The hypercubes are one of the few families of regular polytopes that are represented in any number of dimensions. The hypercube (offset) family

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2520-465: The considered vertex. Doing this for all the vertices of the hypercube, each of the m {\displaystyle m} -dimensional faces of the hypercube is counted 2 m {\displaystyle 2^{m}} times since it has that many vertices, and we need to divide 2 n ( n m ) {\displaystyle 2^{n}{\tbinom {n}{m}}} by this number. The number of facets of

2580-497: The convex hull of the points with coordinates ( ± 1 , ± 1 , ⋯ , ± 1 ) {\displaystyle (\pm 1,\pm 1,\cdots ,\pm 1)} or, equivalently as the Cartesian product [ − 1 , 1 ] n {\displaystyle [-1,1]^{n}} is also often considered due to the simpler form of its vertex coordinates. Its edge length

2640-513: The cube can be represented as the rectangular cuboid with edges equal in length and all of its faces are all squares. The cube may be considered as the parallelepiped in which all of its edges are equal edges. The cube is a plesiohedron , a special kind of space-filling polyhedron that can be defined as the Voronoi cell of a symmetric Delone set . The plesiohedra include the parallelohedrons , which can be translated without rotating to fill

2700-414: The cube's opposite faces centroid, six through the cube's opposite edges midpoints, and four through the cube's opposite vertices; each of these axes is respectively four-fold rotational symmetry (0°, 90°, 180°, and 270°), two-fold rotational symmetry (0° and 180°), and three-fold rotational symmetry (0°, 120°, and 240°). The dual polyhedron can be obtained from each of the polyhedron's vertices tangent to

2760-409: The cube, twelve vertices and eight edges. The cubical graph is a special case of hypercube graph or n {\displaystyle n} - cube—denoted as Q n {\displaystyle Q_{n}} —because it can be constructed by using the operation known as the Cartesian product of graphs . To put it in a plain, its construction involves two graphs connecting

2820-440: The edge length. Following its attribution with nature by Plato, Johannes Kepler in his Harmonices Mundi sketched each of the Platonic solids, one of them is a cube in which Kepler decorated a tree on it. In his Mysterium Cosmographicum , Kepler also proposed the Solar System by using the Platonic solids setting into another one and separating them with six spheres resembling the six planets. The ordered solids started from

2880-412: The edges are equal in length. Like other cuboids, every face of a cube has four vertices, each of which connects with three congruent lines. These edges form square faces, making the dihedral angle of a cube between every two adjacent squares being the interior angle of a square, 90°. Hence, the cube has six faces, twelve edges, and eight vertices. Because of such properties, it is categorized as one of

2940-453: The edges of a cube are equal in length, it is: V = a 3 . {\displaystyle V=a^{3}.} One special case is the unit cube , so-named for measuring a single unit of length along each edge. It follows that each face is a unit square and that the entire figure has a volume of 1 cubic unit. Prince Rupert's cube , named after Prince Rupert of the Rhine ,

3000-403: The edges of a graph are connected to every vertex without crossing other edges. It is also a 3-connected graph , meaning that, whenever a graph with more than three vertices, and two of the vertices are removed, the edges remain connected. The skeleton of a cube can be represented as the graph, and it is called the cubical graph , a Platonic graph . It has the same number of vertices and edges as

3060-428: The five Platonic solids , a polyhedron in which all the regular polygons are congruent and the same number of faces meet at each vertex. Given a cube with edge length a {\displaystyle a} . The face diagonal of a cube is the diagonal of a square a 2 {\displaystyle a{\sqrt {2}}} , and the space diagonal of a cube is a line connecting two vertices that

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3120-502: The gaps, labeled as hγ n . n -cubes can be combined with their duals (the cross-polytopes ) to form compound polytopes: The graph of the n -hypercube's edges is isomorphic to the Hasse diagram of the ( n −1)- simplex 's face lattice . This can be seen by orienting the n -hypercube so that two opposite vertices lie vertically, corresponding to the ( n −1)-simplex itself and the null polytope, respectively. Each vertex connected to

3180-492: The hypercube can be used to compute the ( n − 1 ) {\displaystyle (n-1)} -dimensional volume of its boundary: that volume is 2 n {\displaystyle 2n} times the volume of a ( n − 1 ) {\displaystyle (n-1)} -dimensional hypercube; that is, 2 n s n − 1 {\displaystyle 2ns^{n-1}} where s {\displaystyle s}

3240-463: The innermost to the outermost: regular octahedron , regular icosahedron , regular dodecahedron , regular tetrahedron , and cube. The cube can appear in the construction of a polyhedron, and some of its types can be derived differently in the following: The honeycomb is the space-filling or tessellation in three-dimensional space, meaning it is an object in which the construction begins by attaching any polyhedrons onto their faces without leaving

3300-524: The middle row indicates that there are two vertices in (i.e., at the extremes of) each edge, denoted as 2; the middle column of the first row indicates that three edges meet at each vertex, denoted as 3. The following matrix is: [ 8 3 3 2 12 2 4 4 6 ] {\displaystyle {\begin{bmatrix}{\begin{matrix}8&3&3\\2&12&2\\4&4&6\end{matrix}}\end{bmatrix}}} The Platonic solid

3360-415: The other four are attached to the second-from-top cube of the stack, the resulting polycube is Dali cross , after Salvador Dali . The Dali cross is a tile space polyhedron, which can be represented as the net of a tesseract . A tesseract is a cube analogous' four-dimensional space bounded by twenty-four squares, and it is bounded by the eight cubes known as its cells . Hypercube In geometry ,

3420-451: The pair of vertices with an edge to form a new graph. In the case of the cubical graph, it is the product of two Q 2 {\displaystyle Q_{2}} ; roughly speaking, it is a graph resembling a square. In other words, the cubical graph is constructed by connecting each vertex of two squares with an edge. Notationally, the cubical graph can be denoted as Q 3 {\displaystyle Q_{3}} . As

3480-421: The preferred unit of length is often related to a chosen fundamental physical constant, or combination thereof. This is often a characteristic radius or wavelength of a particle. Some common natural units of length are included in this table: Archaic units of distance include: In everyday conversation, and in informal literature, it is common to see lengths measured in units of objects of which everyone knows

3540-1044: The top vertex then uniquely maps to one of the ( n −1)-simplex's facets ( n −2 faces), and each vertex connected to those vertices maps to one of the simplex's n −3 faces, and so forth, and the vertices connected to the bottom vertex map to the simplex's vertices. This relation may be used to generate the face lattice of an ( n −1)-simplex efficiently, since face lattice enumeration algorithms applicable to general polytopes are more computationally expensive. Regular complex polytopes can be defined in complex Hilbert space called generalized hypercubes , γ n = p {4} 2 {3}... 2 {3} 2 , or [REDACTED] [REDACTED] [REDACTED] [REDACTED] .. [REDACTED] [REDACTED] [REDACTED] [REDACTED] . Real solutions exist with p = 2, i.e. γ n = γ n = 2 {4} 2 {3}... 2 {3} 2 = {4,3,..,3}. For p > 2, they exist in C n {\displaystyle \mathbb {C} ^{n}} . The facets are generalized ( n −1)-cube and

3600-405: The vertices, edges, and faces. The diagonal of a matrix denotes the number of each element that appears in a polyhedron, whereas the non-diagonal of a matrix denotes the number of the column's elements that occur in or at the row's element. As mentioned above, the cube has eight vertices, twelve edges, and six faces; each element in a matrix's diagonal is denoted as 8, 12, and 6. The first column of

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