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71-487: The tesseract is also called an 8-cell , C 8 , (regular) octachoron , or cubic prism . It is the four-dimensional measure polytope , taken as a unit for hypervolume. Coxeter labels it the γ 4 polytope. The term hypercube without a dimension reference is frequently treated as a synonym for this specific polytope . The Oxford English Dictionary traces the word tesseract to Charles Howard Hinton 's 1888 book A New Era of Thought . The term derives from

142-445: A ( t + 1 ) {\displaystyle (t+1)} -dilate of P {\displaystyle {\mathcal {P}}} differs, in terms of integer lattice points, from a t {\displaystyle t} -dilate of P {\displaystyle {\mathcal {P}}} only by lattice points gained on the boundary. Equivalently, P {\displaystyle {\mathcal {P}}}

213-546: A Companion of the Order of Canada . In 1973 he received the Jeffery–Williams Prize . A festschrift in his honour, The Geometric Vein , was published in 1982. It contained 41 essays on geometry, based on a symposium for Coxeter held at Toronto in 1979. A second such volume, The Coxeter Legacy , was published in 2006 based on a Toronto Coxeter symposium held in 2004. Polytope In elementary geometry ,

284-416: A hypersphere circumscribed about a regular polytope is the distance from the polytope's center to one of the vertices, and for the tesseract this radius is equal to its edge length; the diameter of the sphere, the length of the diagonal between opposite vertices of the tesseract, is twice the edge length. Only a few uniform polytopes have this property, including the four-dimensional tesseract and 24-cell ,

355-422: A polytope is a geometric object with flat sides ( faces ). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions n as an n -dimensional polytope or n -polytope . For example, a two-dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope. In this context, "flat sides" means that

426-491: A polytope is called a net . There are 261 distinct nets of the tesseract. The unfoldings of the tesseract can be counted by mapping the nets to paired trees (a tree together with a perfect matching in its complement ). The construction of hypercubes can be imagined the following way: [REDACTED] The 8 cells of the tesseract may be regarded (three different ways) as two interlocked rings of four cubes. The tesseract can be decomposed into smaller 4-polytopes. It

497-458: A bounding surface, ignoring its interior. In this light convex polytopes in p -space are equivalent to tilings of the ( p −1)-sphere , while others may be tilings of other elliptic , flat or toroidal ( p −1)-surfaces – see elliptic tiling and toroidal polyhedron . A polyhedron is understood as a surface whose faces are polygons , a 4-polytope as a hypersurface whose facets ( cells ) are polyhedra, and so forth. The idea of constructing

568-421: A composite Schläfli symbol {4,3} × { }, with symmetry order 96. As a 4-4 duoprism , a Cartesian product of two squares , it can be named by a composite Schläfli symbol {4}×{4}, with symmetry order 64. As an orthotope it can be represented by composite Schläfli symbol { } × { } × { } × { } or { }, with symmetry order 16. Since each vertex of

639-523: A figure a polyschem . The German term polytop was coined by the mathematician Reinhold Hoppe , and was introduced to English mathematicians as polytope by Alicia Boole Stott . Nowadays, the term polytope is a broad term that covers a wide class of objects, and various definitions appear in the mathematical literature. Many of these definitions are not equivalent to each other, resulting in different overlapping sets of objects being called polytopes . They represent different approaches to generalizing

710-400: A higher polytope from those of lower dimension is also sometimes extended downwards in dimension, with an ( edge ) seen as a 1-polytope bounded by a point pair, and a point or vertex as a 0-polytope. This approach is used for example in the theory of abstract polytopes . In certain fields of mathematics, the terms "polytope" and "polyhedron" are used in a different sense: a polyhedron is

781-680: 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

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852-493: A popular theme in art, architecture, and science fiction. Notable examples include: The word tesseract has been adopted for numerous other uses in popular culture, including as a plot device in works of science fiction, often with little or no connection to the four-dimensional hypercube; see Tesseract (disambiguation) . Harold Scott MacDonald Coxeter Harold Scott MacDonald " Donald " Coxeter CC FRS FRSC (9 February 1907 – 31 March 2003)

923-455: A real representation as a tesseract or 4-4 duoprism in 4-dimensional space. 4 {4} 2 has 16 vertices, and 8 4-edges. Its symmetry is 4 [4] 2 , order 32. It also has a lower symmetry construction, [REDACTED] [REDACTED] [REDACTED] , or 4 {}× 4 {}, with symmetry 4 [2] 4 , order 16. This is the symmetry if the red and blue 4-edges are considered distinct. Since their discovery, four-dimensional hypercubes have been

994-443: A rhombic dodecahedron into four congruent rhombohedra , giving a total of eight possible rhombohedra, each a projected cube of the tesseract. This projection is also the one with maximal volume. One set of projection vectors are u = (1,1,−1,−1) , v = (−1,1,−1,1) , w = (1,−1,−1,1) . The tetrahedron forms the convex hull of the tesseract's vertex-centered central projection. Four of 8 cubic cells are shown. The 16th vertex

1065-415: A ridge, while H. S. M. Coxeter uses cell to denote an ( n  − 1)-dimensional element. The terms adopted in this article are given in the table below: An n -dimensional polytope is bounded by a number of ( n  − 1)-dimensional facets . These facets are themselves polytopes, whose facets are ( n  − 2)-dimensional ridges of the original polytope. Every ridge arises as

1136-597: A set of half-spaces . This definition allows a polytope to be neither bounded nor finite. Polytopes are defined in this way, e.g., in linear programming . A polytope is bounded if there is a ball of finite radius that contains it. A polytope is said to be pointed if it contains at least one vertex. Every bounded nonempty polytope is pointed. An example of a non-pointed polytope is the set { ( x , y ) ∈ R 2 ∣ x ≥ 0 } {\displaystyle \{(x,y)\in \mathbb {R} ^{2}\mid x\geq 0\}} . A polytope

1207-455: A tesseract is adjacent to four edges, the vertex figure of the tesseract is a regular tetrahedron . The dual polytope of the tesseract is the 16-cell with Schläfli symbol {3,3,4}, with which it can be combined to form the compound of tesseract and 16-cell. Each edge of a regular tesseract is of the same length. This is of interest when using tesseracts as the basis for a network topology to link multiple processors in parallel computing :

1278-402: A tesseract with side length s : This configuration matrix represents the tesseract. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole tesseract. The diagonal reduces to the f-vector (16,32,24,8). The nondiagonal numbers say how many of the column's element occur in or at the row's element. For example,

1349-640: A uniform duoprism , the tesseract exists in a sequence of uniform duoprisms : { p }×{4}. The regular tesseract, along with the 16-cell , exists in a set of 15 uniform 4-polytopes with the same symmetry . The tesseract {4,3,3} exists in a sequence of regular 4-polytopes and honeycombs , { p ,3,3} with tetrahedral vertex figures , {3,3}. The tesseract is also in a sequence of regular 4-polytope and honeycombs , {4,3, p } with cubic cells . The regular complex polytope 4 {4} 2 , [REDACTED] [REDACTED] [REDACTED] , in C 2 {\displaystyle \mathbb {C} ^{2}} has

1420-427: A unit tesseract is centered at the origin, so that its coordinates are the more symmetrical ( ± 1 2 , ± 1 2 , ± 1 2 , ± 1 2 ) . {\displaystyle {\bigl (}{\pm {\tfrac {1}{2}}},\pm {\tfrac {1}{2}},\pm {\tfrac {1}{2}},\pm {\tfrac {1}{2}}{\bigr )}.} This

1491-613: A vector of all ones, and the inequality is component-wise. It follows from this definition that P {\displaystyle {\mathcal {P}}} is reflexive if and only if ( t + 1 ) P ∘ ∩ Z d = t P ∩ Z d {\displaystyle (t+1){\mathcal {P}}^{\circ }\cap \mathbb {Z} ^{d}=t{\mathcal {P}}\cap \mathbb {Z} ^{d}} for all t ∈ Z ≥ 0 {\displaystyle t\in \mathbb {Z} _{\geq 0}} . In other words,

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1562-577: 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

1633-398: Is finite if it is defined in terms of a finite number of objects, e.g., as an intersection of a finite number of half-planes. It is an integral polytope if all of its vertices have integer coordinates. A certain class of convex polytopes are reflexive polytopes. An integral d {\displaystyle d} -polytope P {\displaystyle {\mathcal {P}}}

1704-416: Is called an edge , and consists of a line segment. A 2-dimensional face consists of a polygon , and a 3-dimensional face, sometimes called a cell , consists of a polyhedron . A polytope may be convex . The convex polytopes are the simplest kind of polytopes, and form the basis for several different generalizations of the concept of polytopes. A convex polytope is sometimes defined as the intersection of

1775-529: Is defined by its vertices. Polytopes in lower numbers of dimensions have standard names: A polytope comprises elements of different dimensionality such as vertices, edges, faces, cells and so on. Terminology for these is not fully consistent across different authors. For example, some authors use face to refer to an ( n  − 1)-dimensional element while others use face to denote a 2-face specifically. Authors may use j -face or j -facet to indicate an element of j dimensions. Some use edge to refer to

1846-428: Is necessary, see for example the rules described for dual polyhedra . Depending on circumstance, the dual figure may or may not be another geometric polytope. If the dual is reversed, then the original polytope is recovered. Thus, polytopes exist in dual pairs. If a polytope has the same number of vertices as facets, of edges as ridges, and so forth, and the same connectivities, then the dual figure will be similar to

1917-428: Is possible to project tesseracts into three- and two-dimensional spaces, similarly to projecting a cube into two-dimensional space. The cell-first parallel projection of the tesseract into three-dimensional space has a cubical envelope. The nearest and farthest cells are projected onto the cube, and the remaining six cells are projected onto the six square faces of the cube. The face-first parallel projection of

1988-423: Is projected to infinity and the four edges to it are not shown. (Edges are projected onto the 3-sphere ) The tesseract, like all hypercubes , tessellates Euclidean space . The self-dual tesseractic honeycomb consisting of 4 tesseracts around each face has Schläfli symbol {4,3,3,4} . Hence, the tesseract has a dihedral angle of 90°. The tesseract's radial equilateral symmetry makes its tessellation

2059-813: Is reflexive if and only if its dual polytope P ∗ {\displaystyle {\mathcal {P}}^{*}} is an integral polytope. Regular polytopes have the highest degree of symmetry of all polytopes. The symmetry group of a regular polytope acts transitively on its flags ; hence, the dual polytope of a regular polytope is also regular. There are three main classes of regular polytope which occur in any number of dimensions: Dimensions two, three and four include regular figures which have fivefold symmetries and some of which are non-convex stars, and in two dimensions there are infinitely many regular polygons of n -fold symmetry, both convex and (for n ≥ 5) star. But in higher dimensions there are no other regular polytopes. In three dimensions

2130-415: Is reflexive if for some integral matrix A {\displaystyle \mathbf {A} } , P = { x ∈ R d : A x ≤ 1 } {\displaystyle {\mathcal {P}}=\{\mathbf {x} \in \mathbb {R} ^{d}:\mathbf {Ax} \leq \mathbf {1} \}} , where 1 {\displaystyle \mathbf {1} } denotes

2201-546: Is the Cartesian product of the closed interval [ − 1 2 , 1 2 ] {\displaystyle {\bigl [}{-{\tfrac {1}{2}}},{\tfrac {1}{2}}{\bigr ]}} in each axis. Another commonly convenient tesseract is the Cartesian product of the closed interval [−1, 1] in each axis, with vertices at coordinates (±1, ±1, ±1, ±1) . This tesseract has side length 2 and hypervolume 2 = 16 . An unfolding of

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2272-575: Is the convex hull of the compound of two demitesseracts ( 16-cells ). It can also be triangulated into 4-dimensional simplices ( irregular 5-cells ) that share their vertices with the tesseract. It is known that there are 92 487 256 such triangulations and that the fewest 4-dimensional simplices in any of them is 16. The dissection of the tesseract into instances of its characteristic simplex (a particular orthoscheme with Coxeter diagram [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] )

2343-424: Is the most basic direct construction of the tesseract possible. The characteristic 5-cell of the 4-cube is a fundamental region of the tesseract's defining symmetry group , the group which generates the B 4 polytopes . The tesseract's characteristic simplex directly generates the tesseract through the actions of the group, by reflecting itself in its own bounding facets (its mirror walls ). The radius of

2414-468: Is typically taken as the basic unit for hypervolume in 4-dimensional space. The unit tesseract in a Cartesian coordinate system for 4-dimensional space has two opposite vertices at coordinates [0, 0, 0, 0] and [1, 1, 1, 1] , and other vertices with coordinates at all possible combinations of 0 s and 1 s. It is the Cartesian product of the closed unit interval [0, 1] in each axis. Sometimes

2485-655: 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

2556-484: The Greek téssara ( τέσσαρα 'four') and aktís ( ἀκτίς 'ray'), referring to the four edges from each vertex to other vertices. Hinton originally spelled the word as tessaract . As a regular polytope with three cubes folded together around every edge, it has Schläfli symbol {4,3,3} with hyperoctahedral symmetry of order 384. Constructed as a 4D hyperprism made of two parallel cubes, it can be named as

2627-401: The 11-cell . An abstract polytope is a partially ordered set of elements or members, which obeys certain rules. It is a purely algebraic structure, and the theory was developed in order to avoid some of the issues which make it difficult to reconcile the various geometric classes within a consistent mathematical framework. A geometric polytope is said to be a realization in some real space of

2698-590: 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

2769-599: 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

2840-437: The amplituhedron was discovered as a simplifying construct in certain calculations of theoretical physics. In the field of optimization , linear programming studies the maxima and minima of linear functions; these maxima and minima occur on the boundary of an n -dimensional polytope. In linear programming, polytopes occur in the use of generalized barycentric coordinates and slack variables . In twistor theory ,

2911-468: The convex polytopes to include other objects with similar properties. The original approach broadly followed by Ludwig Schläfli , Thorold Gosset and others begins with the extension by analogy into four or more dimensions, of the idea of a polygon and polyhedron respectively in two and three dimensions. Attempts to generalise the Euler characteristic of polyhedra to higher-dimensional polytopes led to

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2982-443: The regular skew polyhedra and the infinite series of tilings represented by the regular apeirogon , square tiling, cubic honeycomb, and so on. The theory of abstract polytopes attempts to detach polytopes from the space containing them, considering their purely combinatorial properties. This allows the definition of the term to be extended to include objects for which it is difficult to define an intuitive underlying space, such as

3053-424: The star polytopes . Some regular polytopes are stars. Since a (filled) convex polytope P in d {\displaystyle d} dimensions is contractible to a point, the Euler characteristic χ {\displaystyle \chi } of its boundary ∂P is given by the alternating sum: This generalizes Euler's formula for polyhedra . The Gram–Euler theorem similarly generalizes

3124-568: The topological idea of a polytope as the piecewise decomposition (e.g. CW-complex ) of a manifold . Branko Grünbaum published his influential work on Convex Polytopes in 1967. In 1952 Geoffrey Colin Shephard generalised the idea as complex polytopes in complex space, where each real dimension has an imaginary one associated with it. Coxeter developed the theory further. The conceptual issues raised by complex polytopes, non-convexity, duality and other phenomena led Grünbaum and others to

3195-628: The uniform polytopes , convex and nonconvex, in four or more dimensions remains an outstanding problem. The convex uniform 4-polytopes were fully enumerated by John Conway and Michael Guy using a computer in 1965; in higher dimensions this problem was still open as of 1997. The full enumeration for nonconvex uniform polytopes is not known in dimensions four and higher as of 2008. In modern times, polytopes and related concepts have found many important applications in fields as diverse as computer graphics , optimization , search engines , cosmology , quantum mechanics and numerous other fields. In 2013

3266-453: The unique regular body-centered cubic lattice of equal-sized spheres, in any number of dimensions. The tesseract is 4th in a series of hypercube : The tesseract (8-cell) is the third in the sequence of 6 convex regular 4-polytopes (in order of size and complexity). Hyper- tetrahedron 5-point Hyper- octahedron 8-point Hyper- cube 16-point 24-point Hyper- icosahedron 120-point Hyper- dodecahedron 600-point As

3337-500: The vertex figure , here tetrahedra, (4,6,4). The next row is vertex figure ridge, here a triangle, (3,3). [ 16 4 6 4 2 32 3 3 4 4 24 2 8 12 6 8 ] {\displaystyle {\begin{bmatrix}{\begin{matrix}16&4&6&4\\2&32&3&3\\4&4&24&2\\8&12&6&8\end{matrix}}\end{bmatrix}}} It

3408-404: The 2 in the first column of the second row indicates that there are 2 vertices in (i.e., at the extremes of) each edge; the 4 in the second column of the first row indicates that 4 edges meet at each vertex. The bottom row defines they facets, here cubes, have f-vector (8,12,6). The next row left of diagonal is ridge elements (facet of cube), here a square, (4,4). The upper row is the f-vector of

3479-403: The additional property that, for any two simplices that have a nonempty intersection, their intersection is a vertex, edge, or higher dimensional face of the two. However this definition does not allow star polytopes with interior structures, and so is restricted to certain areas of mathematics. The discovery of star polyhedra and other unusual constructions led to the idea of a polyhedron as

3550-557: The alternating sum of internal angles ∑ φ {\textstyle \sum \varphi } for convex polyhedra to higher-dimensional polytopes: Not all manifolds are finite. Where a polytope is understood as a tiling or decomposition of a manifold, this idea may be extended to infinite manifolds. plane tilings , space-filling ( honeycombs ) and hyperbolic tilings are in this sense polytopes, and are sometimes called apeirotopes because they have infinitely many cells. Among these, there are regular forms including

3621-630: The associated abstract polytope. Structures analogous to polytopes exist in complex Hilbert spaces C n {\displaystyle \mathbb {C} ^{n}} where n real dimensions are accompanied by n imaginary ones. Regular complex polytopes are more appropriately treated as configurations . Every n -polytope has a dual structure, obtained by interchanging its vertices for facets, edges for ridges, and so on generally interchanging its ( j  − 1)-dimensional elements for ( n  −  j )-dimensional elements (for j  = 1 to n  − 1), while retaining

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3692-468: The connectivity or incidence between elements. For an abstract polytope, this simply reverses the ordering of the set. This reversal is seen in the Schläfli symbols for regular polytopes, where the symbol for the dual polytope is simply the reverse of the original. For example, {4, 3, 3} is dual to {3, 3, 4}. In the case of a geometric polytope, some geometric rule for dualising

3763-558: The convex Platonic solids include the fivefold-symmetric dodecahedron and icosahedron , and there are also four star Kepler-Poinsot polyhedra with fivefold symmetry, bringing the total to nine regular polyhedra. In four dimensions the regular 4-polytopes include one additional convex solid with fourfold symmetry and two with fivefold symmetry. There are ten star Schläfli-Hess 4-polytopes , all with fivefold symmetry, giving in all sixteen regular 4-polytopes. A non-convex polytope may be self-intersecting; this class of polytopes include

3834-406: The development of topology and the treatment of a decomposition or CW-complex as analogous to a polytope. In this approach, a polytope may be regarded as a tessellation or decomposition of some given manifold . An example of this approach defines a polytope as a set of points that admits a simplicial decomposition . In this definition, a polytope is the union of finitely many simplices , with

3905-505: The distance between two nodes is at most 4 and there are many different paths to allow weight balancing. A tesseract is bounded by eight three-dimensional hyperplanes . Each pair of non-parallel hyperplanes intersects to form 24 square faces. Three cubes and three squares intersect at each edge. There are four cubes, six squares, and four edges meeting at every vertex. All in all, a tesseract consists of 8 cubes, 24 squares, 32 edges, and 16 vertices. A unit tesseract has side length 1 , and

3976-414: 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

4047-536: The following decades, even during his lifetime. In 1882 Reinhold Hoppe , writing in German, coined the word polytop to refer to this more general concept of polygons and polyhedra. In due course Alicia Boole Stott , daughter of logician George Boole , introduced the anglicised polytope into the English language. In 1895, Thorold Gosset not only rediscovered Schläfli's regular polytopes but also investigated

4118-414: The generic object in any dimension (referred to as polytope in this article) and polytope means a bounded polyhedron. This terminology is typically confined to polytopes and polyhedra that are convex . With this terminology, a convex polyhedron is the intersection of a finite number of halfspaces and is defined by its sides while a convex polytope is the convex hull of a finite number of points and

4189-430: The hexagonal prism in a way analogous to how the faces of the 3D cube project onto six rhombs in a hexagonal envelope under vertex-first projection. The two remaining cells project onto the prism bases. The vertex-first parallel projection of the tesseract into three-dimensional space has a rhombic dodecahedral envelope. Two vertices of the tesseract are projected to the origin. There are exactly two ways of dissecting

4260-459: The ideas of semiregular polytopes and space-filling tessellations in higher dimensions. Polytopes also began to be studied in non-Euclidean spaces such as hyperbolic space. An important milestone was reached in 1948 with H. S. M. Coxeter 's book Regular Polytopes , summarizing work to date and adding new findings of his own. Meanwhile, the French mathematician Henri Poincaré had developed

4331-434: The intersection of two facets (but the intersection of two facets need not be a ridge). Ridges are once again polytopes whose facets give rise to ( n  − 3)-dimensional boundaries of the original polytope, and so on. These bounding sub-polytopes may be referred to as faces , or specifically j -dimensional faces or j -faces. A 0-dimensional face is called a vertex , and consists of a single point. A 1-dimensional face

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4402-469: The more general study of abstract combinatorial properties relating vertices, edges, faces and so on. A related idea was that of incidence complexes, which studied the incidence or connection of the various elements with one another. These developments led eventually to the theory of abstract polytopes as partially ordered sets, or posets, of such elements. Peter McMullen and Egon Schulte published their book Abstract Regular Polytopes in 2002. Enumerating

4473-508: The original and the polytope is self-dual. Some common self-dual polytopes include: Polygons and polyhedra have been known since ancient times. An early hint of higher dimensions came in 1827 when August Ferdinand Möbius discovered that two mirror-image solids can be superimposed by rotating one of them through a fourth mathematical dimension. By the 1850s, a handful of other mathematicians such as Arthur Cayley and Hermann Grassmann had also considered higher dimensions. Ludwig Schläfli

4544-458: The sides of a ( k + 1) -polytope consist of k -polytopes that may have ( k – 1) -polytopes in common. Some theories further generalize the idea to include such objects as unbounded apeirotopes and tessellations , decompositions or tilings of curved manifolds including spherical polyhedra , and set-theoretic abstract polytopes . Polytopes of more than three dimensions were first discovered by Ludwig Schläfli before 1853, who called such

4615-719: The square is 2 , {\displaystyle {\sqrt {2}},} for the cube is 3 , {\displaystyle {\sqrt {3}},} and only for the tesseract is 4 = 2 {\displaystyle {\sqrt {4}}=2} edge lengths. An axis-aligned tesseract inscribed in a unit-radius 3-sphere has vertices with coordinates ( ± 1 2 , ± 1 2 , ± 1 2 , ± 1 2 ) . {\displaystyle {\bigl (}{\pm {\tfrac {1}{2}}},\pm {\tfrac {1}{2}},\pm {\tfrac {1}{2}},\pm {\tfrac {1}{2}}{\bigr )}.} For

4686-399: The tesseract into three-dimensional space has a cuboidal envelope. Two pairs of cells project to the upper and lower halves of this envelope, and the four remaining cells project to the side faces. The edge-first parallel projection of the tesseract into three-dimensional space has an envelope in the shape of a hexagonal prism . Six cells project onto rhombic prisms, which are laid out in

4757-445: The three-dimensional cuboctahedron , and the two-dimensional hexagon . In particular, the tesseract is the only hypercube (other than a zero-dimensional point) that is radially equilateral . The longest vertex-to-vertex diagonal of an n {\displaystyle n} -dimensional hypercube of unit edge length is n t , {\displaystyle {\sqrt {n{\vphantom {t}}}},} which for

4828-495: 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

4899-540: 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

4970-497: 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

5041-450: Was the first to consider analogues of polygons and polyhedra in these higher spaces. He described the six convex regular 4-polytopes in 1852 but his work was not published until 1901, six years after his death. By 1854, Bernhard Riemann 's Habilitationsschrift had firmly established the geometry of higher dimensions, and thus the concept of n -dimensional polytopes was made acceptable. Schläfli's polytopes were rediscovered many times in

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