# Multidimensional geometry

## Multidimensional geometry Topics

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### Hypersphere

The -hypersphere (often simply called the -sphere) is a generalization of the circle (called by geometers the 2-sphere) and usual sphere (called by geometers the 3-sphere) to dimensions . The -sphere is therefore defined (again, to a geometer; see below) as the set of -tuples of points (, , ..., ) such that(1)where is the radius of the hypersphere.Unfortunately, geometers and topologists adopt incompatible conventions for the meaning of "-sphere," with geometers referring to the number of coordinates in the underlying space ("thus a two-dimensional sphere is a circle," Coxeter 1973, p. 125) and topologists referring to the dimension of the surface itself ("the -dimensional sphere is defined to be the set of all points in satisfying ," Hocking and Young 1988, p. 17; "the -sphere is ," Maunder 1997, p. 21). A geometer would therefore regard the object described by(2)as a 2-sphere,..

### Tesseract

The tesseract is the hypercube in , also called the 8-cell or octachoron. It has the Schläfli symbol , and vertices . The figure above shows a projection of the tesseract in three-space (Gardner 1977). The tesseract is composed of 8 cubes with 3 to an edge, and therefore has 16 vertices, 32 edges, 24 squares, and 8 cubes. It is one of the six regular polychora.The tesseract has 261 distinct nets (Gardner 1966, Turney 1984-85, Tougne 1986, Buekenhout and Parker 1998).In Madeleine L'Engle's novel A Wrinkle in Time, the characters in the story travel through time and space using tesseracts. The book actually uses the idea of a tesseract to represent a fifth dimension rather than a four-dimensional object (and also uses the word "tesser" to refer to movement from one three dimensional space/world to another).In the science fiction novel Factoring Humanity by Robert J. Sawyer, a tesseract is used by humans on Earth to enter the fourth..

### Parallelotope

Move a point along a line from an initial point to a final point. It traces out a line segment . When is translated from an initial position to a final position, it traces out a parallelogram . When is translated, it traces out a parallelepiped . The generalization of to dimensions is then called a parallelotope. has vertices ands, where is a binomial coefficient and , 1, ..., (Coxeter 1973). These are also the coefficients of .

### Flag

A collection of faces of an -dimensional polytope or simplicial complex, one of each dimension 0, 1, ..., , which all have a common nonempty intersection. In normal three dimensions, the flag consists of a half-plane, its bounding ray, and the ray's endpoint.

### Cross polytope

The cross polytope is the regular polytope in dimensions corresponding to the convex hull of the points formed by permuting the coordinates (, 0, 0, ..., 0). A cross-polytope (also called an orthoplex) is denoted and has vertices and Schläfli symbol . The cross polytope is named because its vertices are located equidistant from the origin along the Cartesian axes in Euclidean space, which each such axis perpendicular to all others. A cross polytope is bounded by -simplexes, and is a dipyramid erected (in both directions) into the th dimension, with an -dimensional cross polytope as its base.In one dimension, the cross polytope is the line segment . In two dimensions, the cross polytope is the filled square with vertices , , , . In three dimensions, the cross polytope is the convex hull of the octahedron with vertices , , , , , . In four dimensions, the cross polytope is the 16-cell, depicted in the above figure by projecting onto one of the four mutually..

### Simplex

A simplex, sometimes called a hypertetrahedron (Buekenhout and Parker 1998), is the generalization of a tetrahedral region of space to dimensions. The boundary of a -simplex has 0-faces (polytope vertices), 1-faces (polytope edges), and -faces, where is a binomial coefficient. An -dimensional simplex can be denoted using the Schläfli symbol . The simplex is so-named because it represents the simplest possible polytope in any given space.The content (i.e., hypervolume) of a simplex can be computedusing the Cayley-Menger determinant.In one dimension, the simplex is the line segment . In two dimensions, the simplex is the convex hull of the equilateral triangle. In three dimensions, the simplex is the convex hull of the tetrahedron. The simplex in four dimensions (the pentatope) is a regular tetrahedron in which a point along the fourth dimension through the center of is chosen so that . The regular simplex in dimensions with is denoted..

### Associahedron

The associahedron is the -dimensional generalization of the pentagon. It was discovered by Stasheff in 1963 and it is also known as the Stasheff polytope. The number of nodes in the -associahedron is equivalent to the number of binary trees with nodes, which is the Catalan number .The associahedron is the basic tool in the study of homotopy associative Hopf spaces.Loday (2004) provides the following method for associahedron construction. Take , the set of planar binary trees with leaves. Define as the number of leaves to the left of the th vertex and as the number of leaves to the right of the th vertex. For in , defineThe -associahedron is then defined as the convex hull of .The associahedron can be obtained by removing facets from the permutohedron,and is related to the cyclohedron and permutohedron...

### Polytope

The word polytope is used to mean a number of related, but slightly different mathematical objects. A convex polytope may be defined as the convex hull of a finite set of points (which are always bounded), or as a bounded intersection of a finite set of half-spaces. Coxeter (1973, p. 118) defines polytope as the general term of the sequence "point, line segment, polygon, polyhedron, ...," or more specifically as a finite region of -dimensional space enclosed by a finite number of hyperplanes. The special name polychoron is sometimes given to a four-dimensional polytope. However, in algebraic topology, the underlying space of a simplicial complex is sometimes called a polytope (Munkres 1991, p. 8). The word "polytope" was introduced by Alicia Boole Stott, the somewhat colorful daughter of logician George Boole (MacHale 1985).The part of the polytope that lies in one of the bounding hyperplanes is called..

### Permutohedron

The permutohedron is the -dimensional generalization of the hexagon. The -permutohedron is the convex hull of all permutations of the vector in . The number of vertices is .

### Howe's theorem

Let be a primitive polytope with eight vertices. Then there is a unimodular map that maps to the polyhedron whose vertices are (0, 0, 0), (1, 0, 0), (0, 1, 0), (0, 0, 1), (0, 1, 1), (1, , ), (1, , ), and (1, , ) with , , and . Furthermore, any primitive polyhedron with fewer than eight vertices can be embedded in one with eight vertices.

### Zonotope

A zonotope is a set of points in -dimensional space constructed from vectors by taking the sum of , where each is a scalar between 0 and 1. Different choices of scalars give different points, and the zonotope is the set of all such points. Alternately it can be viewed as a Minkowski sum of line segments connecting the origin to the endpoint of each vector. It is called a zonotope because the faces parallel to each vector form a so-called zone wrapping around the polytope (Eppstein 1996).A three-dimensional zonotope is called a zonohedron.There is some confusion in the definition of zonotopes (Eppstein 1996). Wells (1991, pp. 274-275) requires the generating vectors to be in general position (all -tuples of vectors must span the whole space), so that all the faces of the zonotope are parallelotopes. Others (Bern et al. 1995; Ziegler 1995, pp. 198-208; Eppstein 1996) do not make this restriction. Coxeter (1973) starts with one definition..

### Hypersine

The hypersine (-dimensional sine function) is a function of a vertex angle of an -dimensional parallelotope or simplex. If the content of the parallelotope is and the contents of the facets of the parallelotope that meet at vertex are , then the value of the -dimensional sine of that vertex is(1)Changing the length of an edge of the parallelotope by a factor changes the content by the same factor and the contents of all but one of the facets by the same factor. Thus, a change in edge length does not affect the value of the right-hand side, and the sine function is dependent solely on the angles between the edges of the parallelotope, not their lengths. In addition, the sines of all of the vertex angles of the parallelotope are the same, since the opposite facets have the same content, and one of each pair of opposite facets meets at each vertex. If we extend the facets at a vertex, all of the vertex angles thus formed have the same sine, as they are simply translations..

### Tetracyclic plane

The set of all points that can be put into one-to-one correspondence with sets of essentially distinct values of four homogeneous coordinates , not all simultaneously zero, which are connected by the relation

### Sphere embedding

A 4-sphere has positive curvature,with(1)(2)Since(3)(4)(5)To stay on the surface of the sphere,(6)(7)(8)(9)(10)(11)With the addition of the so-called expansion parameter, this is the Robertson-Walker line element.

### Hypercube

The hypercube is a generalization of a 3-cube to dimensions, also called an -cube or measure polytope. It is a regular polytope with mutually perpendicular sides, and is therefore an orthotope. It is denoted and has Schläfli symbol .The following table summarizes the names of -dimensional hypercubes.object1line segment2square3cube4tesseractThe number of -cubes contained in an -cube can be found from the coefficients of , namely , where is a binomial coefficient. The number of nodes in the -hypercube is therefore (OEIS A000079), the number of edges is (OEIS A001787), the number of squares is (OEIS A001788), the number of cubes is (OEIS A001789), etc.The numbers of distinct nets for the -hypercube for , 2, ... are 1, 11, 261, ... (OEIS A091159; Turney 1984-85).The above figure shows a projection of the tesseract in three-space. A tesseract has 16 polytope vertices, 32 polytope edges, 24 squares, and eight cubes.The dual of the tesseract..

### Hyperbolic polar sine

The hyperbolic polar sine is a function of an -dimensional simplex in hyperbolic space. It is analogous to the polar sine of an -dimensional simplex in elliptic or spherical space. If the edges between vertices and have length , the value of the hyperbolic polar sine of the -dimensional hyperbolic simplex in space with Gaussian curvature is given byThe hyperbolic polar sine is used in the generalizedlaw of sines for a hyperbolic simplex.The limit of the hyperbolic polar sine of an -dimensional hyperbolic simplex as the curvature of the space approaches zero is , where is the content of the Euclidean simplex with the same edge lengths.

### Polar sine

The polar sine is a function of a vertex angle of an -dimensional parallelotope or simplex. If the content of the parallelotope is and the lengths of the edges of the parallelotope that meet at vertex are , , ..., , then the value of the polar sine of that vertex isChanging the length of an edge of the parallelotope by a factor changes the content by the same factor. Thus, a change in edge length does not affect the value of the right-hand side, and the polar sine function is dependent solely on the angles between the edges of the parallelotope, not their lengths. Also the polar sines of all of the vertex angles of the parallelotope are the same, since the right-hand side of the definition does not depend on the vertex chosen. If we extend the facets at a vertex, all of the vertex angles thus formed have the same polar sine, as they are simply translations of the vertex angles of the parallelotope.If a sphere is centered at the vertex of an -dimensional angle, the rays..

### Generalized law of sines

The generalized law of sines applies to a simplex in space of any dimension with constant Gaussian curvature. Let us work up to that. Initially in two-dimensional space, we define a generalized sine function for a one-dimensional simplex (line segment) with content (length) in space of constant Gaussian curvature as(1)For particular values of , we have(2)giving(3)Thus in elliptic space (), the function is the sine function; in Euclidean space (), the function is simply the content itself; and in hyperbolic space (), the function is the hyperbolic sine function. Thus for a two-dimensional simplex with edges of length , , and , we can express the law of sines for space with any constant Gaussian curvature as(4)For Euclidean space (), equation (4) specializes to(5)For the elliptic plane or the unit sphere (), equation (4) specializes to(6)For the hyperbolic plane (), equation (4) specializes to(7)Our generalization for the two-dimensional..

### Regular polychoron

The necessary condition for the polychoron to be regular (with Schläfli symbol ) and finite isSufficiency can be established by consideration ofthe six figures satisfying this condition.There are sixteen regular polychora, six of which are convex (Wells 1986, p. 68) and ten of which are stellated (Wells 1991, p. 209). The regular convex polychora have four principal types of symmetry axes, and the projections into three-spaces orthogonal to these may be called the "canonical" projections.Of the six regular convex polychora, five are typically regarded as being analogous to the Platonic solids: the 4-simplex (a hyper-tetrahedron), the 4-cross polytope (a hyper-octahedron), the 4-cube (a hyper-cube), the 600-cell (a hyper-icosahedron), and the 120-cell (a hyper-dodecahedron). The 24-cell, however, has no perfect analogy in higher or lower spaces. The pentatope and 24-cell are self-dual, the 16-cell..

### Pentatope

The pentatope is the simplest regular figure in four dimensions, representing the four-dimensional analog of the solid tetrahedron. It is also called the 5-cell, since it consists of five vertices, or pentachoron. The pentatope is the four-dimensional simplex, and can be viewed as a regular tetrahedron in which a point along the fourth dimension through the center of is chosen so that . The pentatope has Schläfli symbol .It is one of the six regular polychora.The skeleton of the pentatope is isomorphic to the complete graph , known as the pentatope graph.The pentatope is self-dual, has five three-dimensional facets (each the shape of a tetrahedron), 10 ridges (faces), 10 edges, and five vertices. In the above figure, the pentatope is shown projected onto one of the four mutually perpendicular three-spaces within the four-space obtained by dropping one of the four vertex components (R. Towle)...

### Hyperplane

Let , , ..., be scalars not all equal to 0. Then the set consisting of all vectorsin such thatfor a constant is a subspace of called a hyperplane.More generally, a hyperplane is any codimension-1 vector subspace of a vector space. Equivalently, a hyperplane in a vector space is any subspace such that is one-dimensional. Equivalently, a hyperplane is the linear transformation kernel of any nonzero linear map from the vector space to the underlying field.

### Regge calculus

Regge calculus is a finite element method utilized in numerical relativity in attempts of describing spacetimes with few or no symmetries by way of producing numerical solutions to the Einstein field equations (Khavari 2009). It was developed initially by Italian mathematician Tullio Regge in the 1960s (Regge 1961).Modern forays into Regge's method center on the triangulation of manifolds, particularly on the discrete approximation of 4-dimensional Riemannian and Lorentzian manifolds by way of cellular complexes whose 4-dimensional triangular simplices share their boundary tetrahedra (i.e., 3-dimensional simplices) to enclose a flat piece of spacetime (Marinelli 2013). Worth noting is that Regge himself devised the framework in more generality, though noted that no such generality is lost by assuming a triangular approximation (Regge 1961).The benefit of this technique is that the structures involved are rigid and hence are..

### Pentaspherical space

The set of all points that can be put into one-to-one correspondence with sets of essentially distinct values of five homogeneous coordinates , not all simultaneously zero, which are connected by the relation

### Minkowski's conjecture

Minkowski's conjecture states that every lattice tiling of by unit hypercubes contains two hypercubes that meet in an -dimensional face. Minkowski first considered the problem in 1896, when he stated it as a theorem whose proof would be provided later. However, it subsequently appeared as on open problem in Minkowski's 1907 book, suggesting the purported proof was erroneous. The conjecture was subsequently proved in eight and fewer dimensions by Peron (1940) and in general by Hajós (1942).Keller's conjecture is a generalization of Minkowski's conjecture (which however is known to be true only in dimensions six and less and to be false in at least dimensions 8, 10, and 12).

### Kissing number

The number of equivalent hyperspheres in dimensions which can touch an equivalent hypersphere without any intersections, also sometimes called the Newton number, contact number, coordination number, or ligancy. Newton correctly believed that the kissing number in three dimensions was 12, but the first proofs were not produced until the 19th century (Conway and Sloane 1993, p. 21) by Bender (1874), Hoppe (1874), and Günther (1875). More concise proofs were published by Schütte and van der Waerden (1953) and Leech (1956). After packing 12 spheres around the central one (which can be done, for example, by arranging the spheres so that their points of tangency with the central sphere correspond to the vertices of an icosahedron), there is a significant amount of free space left (above figure), although not enough to fit a 13th sphere.Exact values for lattice packings are known for to 9 and (Conway and Sloane 1993, Sloane and..

### Keller's conjecture

Keller conjectured that tiling an -dimensional space with -dimensional hypercubes of equal size yields an arrangement in which at least two hypercubes have an entire -dimensional "side" in common. This conjecture generalizes Minkowski's conjecture.Corrádi and Szabó (1990) reformulated the conjecture by showing that if there exists a clique of size (the largest possible) in the class of graphs which have now become known as Keller graphs, then Keller's conjecture is false in that dimension. However, note that the nonexistence of such a clique does not necessarily imply the truth of the conjecture, only that no counterexample exists with hypercubes whose coordinates are integers or half-integers (Debroni et al. 2011).Perron (1940) proved Keller's conjecture to be true in dimensions six and less, and it has been shown to be false in dimensions 8, 10, and 12 by Lagarias and Shor (1992), who found a cliques of size..

### Hypersphere packing

In two dimensions, there are two periodic circle packings for identical circles: square lattice and hexagonal lattice. In 1940, Fejes Tóth proved that the hexagonal lattice is the densest of all possible plane packings (Conway and Sloane 1993, pp. 8-9).The analog of face-centered cubic packing is the densest lattice packing in four and five dimensions. In eight dimensions, the densest lattice packing is made up of two copies of face-centered cubic. In six and seven dimensions, the densest lattice packings are cross sections of the eight-dimensional case. In 24 dimensions, the densest packing appears to be the Leech lattice. For high dimensions (-D), the densest known packings are nonlattice.The densest lattice packings of hyperspheres in dimensions are known rigorously for , 2, ..., 8, and have packing densities summarized in the following table, which also gives the corresponding Hermite constants (Gruber and Lekkerkerker..