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 n {\displaystyle {\sqrt {n}}} .
31-398: 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 a hyperrectangle (also called an n-orthotope ). A unit hypercube is
62-733: 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
93-399: 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;
124-516: 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
155-399: 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
186-680: A nightly cocktail made from Kahlúa (a coffee liqueur), peach schnapps , and soy milk . Since 1978, the Canadian Mathematical Society have awarded the Coxeter–James Prize in his honor. He was made a Fellow of the Royal Society in 1950 and in 1997 he was awarded their Sylvester Medal . In 1990, he became a Foreign Member of the American Academy of Arts and Sciences and in 1997 was made
217-574: 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. Harold Scott MacDonald Coxeter Harold Scott MacDonald " Donald " Coxeter CC FRS FRSC (9 February 1907 – 31 March 2003)
248-477: 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
279-577: A year as a Rockefeller Fellow , where he worked with Hermann Weyl , Oswald Veblen , and Solomon Lefschetz . Returning to Trinity for a year, he attended Ludwig Wittgenstein 's seminars on the philosophy of mathematics . In 1934 he spent a further year at Princeton as a Procter Fellow. In 1936 Coxeter moved to the University of Toronto. In 1938 he and P. Du Val , H. T. Flather, and John Flinders Petrie published The Fifty-Nine Icosahedra with University of Toronto Press . In 1940 Coxeter edited
310-515: 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} :
341-407: 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
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#1732794171077372-412: 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
403-458: 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
434-489: 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
465-580: 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
496-402: 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
527-879: 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
558-655: The Circle Limit series based on hyperbolic tessellations . He also inspired some of the innovations of Buckminster Fuller . Coxeter, M. S. Longuet-Higgins and J. C. P. Miller were the first to publish the full list of uniform polyhedra (1954). He worked for 60 years at the University of Toronto and published twelve books. Coxeter was a vegetarian . He attributed his longevity to his vegetarian diet, daily exercise such as fifty press-ups and standing on his head for fifteen minutes each morning, and consuming
589-590: The Royal Academy of Arts . A maternal cousin was the architect Sir Giles Gilbert Scott . In his youth, Coxeter composed music and was an accomplished pianist at the age of 10. He felt that mathematics and music were intimately related, outlining his ideas in a 1962 article on "Music and Mathematics" in the Canadian Music Journal . He was educated at King Alfred School, London , and St George's School, Harpenden , where his best friend
620-650: The Royal Society of Canada , the Royal Society , and the Order of Canada . He was an author of 12 books, including The Fifty-Nine Icosahedra (1938) and Regular Polytopes (1947). Many concepts in geometry and group theory are named after him, including the Coxeter graph , Coxeter groups , Coxeter's loxodromic sequence of tangent circles , Coxeter–Dynkin diagrams , and the Todd–Coxeter algorithm . Coxeter
651-533: 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
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#1732794171077682-430: 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
713-465: 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
744-497: 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
775-414: The eleventh edition of Mathematical Recreations and Essays , originally published by W. W. Rouse Ball in 1892. He was elevated to professor in 1948. He was elected a Fellow of the Royal Society of Canada in 1948 and a Fellow of the Royal Society in 1950. He met M. C. Escher in 1954 and the two became lifelong friends; his work on geometric figures helped inspire some of Escher's works, particularly
806-502: 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
837-492: 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}
868-1044: 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
899-495: Was John Flinders Petrie, later a mathematician for whom Petrie polygons were named. He was accepted at King's College, Cambridge , in 1925, but decided to spend a year studying in hopes of gaining admittance to Trinity College , where the standard of mathematics was higher. Coxeter won an entrance scholarship and went to Trinity in 1926 to read mathematics. There he earned his BA (as Senior Wrangler ) in 1928, and his doctorate in 1931. In 1932 he went to Princeton University for
930-540: Was a British-Canadian geometer and mathematician. He is regarded as one of the greatest geometers of the 20th century. Coxeter was born in England and educated at the University of Cambridge , with student visits to Princeton University . He worked for 60 years at the University of Toronto in Canada, from 1936 until his retirement in 1996, becoming a full professor there in 1948. His many honours included membership in
961-497: Was born in Kensington , England, to Harold Samuel Coxeter and Lucy ( née Gee ). His father had taken over the family business of Coxeter & Son, manufacturers of surgical instruments and compressed gases (including a mechanism for anaesthetising surgical patients with nitrous oxide ), but was able to retire early and focus on sculpting and baritone singing; Lucy Coxeter was a portrait and landscape painter who had attended