1.
Plane (geometry)
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In mathematics, a plane is a flat, two-dimensional surface that extends infinitely far. A plane is the analogue of a point, a line. When working exclusively in two-dimensional Euclidean space, the article is used, so. Many fundamental tasks in mathematics, geometry, trigonometry, graph theory and graphing are performed in a space, or in other words. Euclid set forth the first great landmark of mathematical thought, a treatment of geometry. He selected a small core of undefined terms and postulates which he used to prove various geometrical statements. Although the plane in its sense is not directly given a definition anywhere in the Elements. In his work Euclid never makes use of numbers to measure length, angle, in this way the Euclidean plane is not quite the same as the Cartesian plane. This section is concerned with planes embedded in three dimensions, specifically, in R3. In a Euclidean space of any number of dimensions, a plane is determined by any of the following. A line and a point not on that line, a line is either parallel to a plane, intersects it at a single point, or is contained in the plane. Two distinct lines perpendicular to the plane must be parallel to each other. Two distinct planes perpendicular to the line must be parallel to each other. Specifically, let r0 be the vector of some point P0 =. The plane determined by the point P0 and the vector n consists of those points P, with position vector r, such that the vector drawn from P0 to P is perpendicular to n. Recalling that two vectors are perpendicular if and only if their dot product is zero, it follows that the plane can be described as the set of all points r such that n ⋅ =0. Expanded this becomes a + b + c =0, which is the form of the equation of a plane. This is just a linear equation a x + b y + c z + d =0 and this familiar equation for a plane is called the general form of the equation of the plane
2.
Polyhedron
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In geometry, a polyhedron is a solid in three dimensions with flat polygonal faces, straight edges and sharp corners or vertices. The word polyhedron comes from the Classical Greek πολύεδρον, as poly- + -hedron, a convex polyhedron is the convex hull of finitely many points, not all on the same plane. Cubes and pyramids are examples of convex polyhedra, a polyhedron is a 3-dimensional example of the more general polytope in any number of dimensions. Convex polyhedra are well-defined, with several equivalent standard definitions, however, the formal mathematical definition of polyhedra that are not required to be convex has been problematic. Many definitions of polyhedron have been given within particular contexts, some more rigorous than others, some of these definitions exclude shapes that have often been counted as polyhedra or include shapes that are often not considered as valid polyhedra. As Branko Grünbaum observed, The Original Sin in the theory of polyhedra goes back to Euclid, the writers failed to define what are the polyhedra. Nevertheless, there is agreement that a polyhedron is a solid or surface that can be described by its vertices, edges, faces. Natural refinements of this definition require the solid to be bounded, to have a connected interior, and possibly also to have a connected boundary. However, the polyhedra defined in this way do not include the self-crossing star polyhedra, their faces may not form simple polygons, definitions based on the idea of a bounding surface rather than a solid are also common. If a planar part of such a surface is not itself a convex polygon, ORourke requires it to be subdivided into smaller convex polygons, cromwell gives a similar definition but without the restriction of three edges per vertex. Again, this type of definition does not encompass the self-crossing polyhedra, however, there exist topological polyhedra that cannot be realized as acoptic polyhedra. One modern approach is based on the theory of abstract polyhedra and these can be defined as partially ordered sets whose elements are the vertices, edges, and faces of a polyhedron. A vertex or edge element is less than an edge or face element when the vertex or edge is part of the edge or face, additionally, one may include a special bottom element of this partial order and a top element representing the whole polyhedron. However, these requirements are relaxed, to instead require only that the sections between elements two levels apart from line segments. Geometric polyhedra, defined in other ways, can be described abstractly in this way, a realization of an abstract polyhedron is generally taken to be a mapping from the vertices of the abstract polyhedron to geometric points, such that the points of each face are coplanar. A geometric polyhedron can then be defined as a realization of an abstract polyhedron, realizations that forgo the requirement of planarity, that impose additional requirements of symmetry, or that map the vertices to higher dimensional spaces have also been considered. Unlike the solid-based and surface-based definitions, this perfectly well for star polyhedra. However, without restrictions, this definition allows degenerate or unfaithful polyhedra
3.
Polytope
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In elementary geometry, a polytope is a geometric object with flat sides, and may exist in any general number of dimensions n as an n-dimensional polytope or n-polytope. For example, a polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope. Polytopes in more than three dimensions were first discovered by Ludwig Schläfli, the German term polytop was coined by the mathematician Reinhold Hoppe, and was introduced to English mathematicians as polytope by Alicia Boole Stott. The term polytope is nowadays a broad term that covers a class of objects. Many of these definitions are not equivalent, resulting in different sets of objects being called polytopes and they represent different approaches to generalizing the convex polytopes to include other objects with similar properties. 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. However this definition does not allow star polytopes with interior structures, the discovery of star polyhedra and other unusual constructions led to the idea of a polyhedron as a bounding surface, ignoring its interior. A polyhedron is understood as a surface whose faces are polygons, a 4-polytope as a hypersurface whose facets are polyhedra and 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 and this terminology is typically confined to polytopes and polyhedra that are convex. A polytope comprises elements of different dimensionality such as vertices, edges, faces, cells, terminology for these is not fully consistent across different authors. For example, some authors use face to refer to an -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 a ridge, while H. S. M. Coxeter uses cell to denote an -dimensional element, the terms adopted in this article are given in the table below, An n-dimensional polytope is bounded by a number of -dimensional facets. These facets are themselves polytopes, whose facets are -dimensional ridges of the original polytope, Every ridge arises as the intersection of two facets. Ridges are once again polytopes whose facets give rise to -dimensional boundaries of the original polytope and 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 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, 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 defined as the intersection of 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
4.
Polygon
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In elementary geometry, a polygon /ˈpɒlɪɡɒn/ is a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed polygonal chain or circuit. These segments are called its edges or sides, and the points where two edges meet are the vertices or corners. The interior of the polygon is called its body. An n-gon is a polygon with n sides, for example, a polygon is a 2-dimensional example of the more general polytope in any number of dimensions. The basic geometrical notion of a polygon has been adapted in various ways to suit particular purposes, mathematicians are often concerned only with the bounding closed polygonal chain and with simple polygons which do not self-intersect, and they often define a polygon accordingly. A polygonal boundary may be allowed to intersect itself, creating star polygons and these and other generalizations of polygons are described below. The word polygon derives from the Greek adjective πολύς much, many and it has been suggested that γόνυ knee may be the origin of “gon”. Polygons are primarily classified by the number of sides, Polygons may be characterized by their convexity or type of non-convexity, Convex, any line drawn through the polygon meets its boundary exactly twice. As a consequence, all its interior angles are less than 180°, equivalently, any line segment with endpoints on the boundary passes through only interior points between its endpoints. Non-convex, a line may be found which meets its boundary more than twice, equivalently, there exists a line segment between two boundary points that passes outside the polygon. Simple, the boundary of the polygon does not cross itself, there is at least one interior angle greater than 180°. Star-shaped, the interior is visible from at least one point. The polygon must be simple, and may be convex or concave, self-intersecting, the boundary of the polygon crosses itself. Branko Grünbaum calls these coptic, though this term does not seem to be widely used, star polygon, a polygon which self-intersects in a regular way. A polygon cannot be both a star and star-shaped, equiangular, all corner angles are equal. Cyclic, all lie on a single circle, called the circumcircle. Isogonal or vertex-transitive, all lie within the same symmetry orbit. The polygon is cyclic and equiangular
5.
Tessellation
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A tessellation of a flat surface is the tiling of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps. In mathematics, tessellations can be generalized to higher dimensions and a variety of geometries, a periodic tiling has a repeating pattern. The patterns formed by periodic tilings can be categorized into 17 wallpaper groups, a tiling that lacks a repeating pattern is called non-periodic. An aperiodic tiling uses a set of tile shapes that cannot form a repeating pattern. In the geometry of higher dimensions, a space-filling or honeycomb is called a tessellation of space. A real physical tessellation is a made of materials such as cemented ceramic squares or hexagons. Such tilings may be decorative patterns, or may have such as providing durable and water-resistant pavement. Historically, tessellations were used in Ancient Rome and in Islamic art such as in the decorative geometric tiling of the Alhambra palace, in the twentieth century, the work of M. C. Escher often made use of tessellations, both in ordinary Euclidean geometry and in geometry, for artistic effect. Tessellations are sometimes employed for decorative effect in quilting, Tessellations form a class of patterns in nature, for example in the arrays of hexagonal cells found in honeycombs. Tessellations were used by the Sumerians in building wall decorations formed by patterns of clay tiles, decorative mosaic tilings made of small squared blocks called tesserae were widely employed in classical antiquity, sometimes displaying geometric patterns. In 1619 Johannes Kepler made a documented study of tessellations. He wrote about regular and semiregular tessellations in his Harmonices Mundi, he was possibly the first to explore and to explain the structures of honeycomb. Some two hundred years later in 1891, the Russian crystallographer Yevgraf Fyodorov proved that every periodic tiling of the features one of seventeen different groups of isometries. Fyodorovs work marked the beginning of the mathematical study of tessellations. Other prominent contributors include Shubnikov and Belov, and Heinrich Heesch, in Latin, tessella is a small cubical piece of clay, stone or glass used to make mosaics. The word tessella means small square and it corresponds to the everyday term tiling, which refers to applications of tessellations, often made of glazed clay. Tessellation or tiling in two dimensions is a topic in geometry that studies how shapes, known as tiles, can be arranged to fill a plane without any gaps, according to a given set of rules
6.
Square
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In geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles. It can also be defined as a rectangle in which two adjacent sides have equal length, a square with vertices ABCD would be denoted ◻ ABCD. e. A rhombus with equal diagonals a convex quadrilateral with sides a, b, c, d whose area is A =12 =12. Opposite sides of a square are both parallel and equal in length, all four angles of a square are equal. All four sides of a square are equal, the diagonals of a square are equal. The square is the n=2 case of the families of n-hypercubes and n-orthoplexes, a truncated square, t, is an octagon. An alternated square, h, is a digon, the perimeter of a square whose four sides have length ℓ is P =4 ℓ and the area A is A = ℓ2. In classical times, the power was described in terms of the area of a square. This led to the use of the square to mean raising to the second power. The area can also be calculated using the diagonal d according to A = d 22. In terms of the circumradius R, the area of a square is A =2 R2, since the area of the circle is π R2, in terms of the inradius r, the area of the square is A =4 r 2. Because it is a polygon, a square is the quadrilateral of least perimeter enclosing a given area. Dually, a square is the quadrilateral containing the largest area within a given perimeter. Indeed, if A and P are the area and perimeter enclosed by a quadrilateral, then the isoperimetric inequality holds,16 A ≤ P2 with equality if. The diagonals of a square are 2 times the length of a side of the square and this value, known as the square root of 2 or Pythagoras constant, was the first number proven to be irrational. A square can also be defined as a parallelogram with equal diagonals that bisect the angles, if a figure is both a rectangle and a rhombus, then it is a square. If a circle is circumscribed around a square, the area of the circle is π /2 times the area of the square, if a circle is inscribed in the square, the area of the circle is π /4 times the area of the square. A square has an area than any other quadrilateral with the same perimeter
7.
Cube
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In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube is the only regular hexahedron and is one of the five Platonic solids and it has 6 faces,12 edges, and 8 vertices. The cube is also a square parallelepiped, an equilateral cuboid and it is a regular square prism in three orientations, and a trigonal trapezohedron in four orientations. The cube is dual to the octahedron and it has cubical or octahedral symmetry. The cube has four special orthogonal projections, centered, on a vertex, edges, face, the first and third correspond to the A2 and B2 Coxeter planes. The cube can also be represented as a tiling. This projection is conformal, preserving angles but not areas or lengths, straight lines on the sphere are projected as circular arcs on the plane. In analytic geometry, a surface with center and edge length of 2a is the locus of all points such that max = a. For a cube of length a, As the volume of a cube is the third power of its sides a × a × a, third powers are called cubes, by analogy with squares. A cube has the largest volume among cuboids with a surface area. Also, a cube has the largest volume among cuboids with the same linear size. They were unable to solve this problem, and in 1837 Pierre Wantzel proved it to be impossible because the root of 2 is not a constructible number. The cube has three uniform colorings, named by the colors of the faces around each vertex,111,112,123. The cube has three classes of symmetry, which can be represented by coloring the faces. The highest octahedral symmetry Oh has all the faces the same color, the dihedral symmetry D4h comes from the cube being a prism, with all four sides being the same color. The lowest symmetry D2h is also a symmetry, with sides alternating colors. Each symmetry form has a different Wythoff symbol, a cube has eleven nets, that is, there are eleven ways to flatten a hollow cube by cutting seven edges. To color the cube so that no two adjacent faces have the color, one would need at least three colors
8.
4-polytope
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In geometry, a 4-polytope is a four-dimensional polytope. It is a connected and closed figure, composed of lower-dimensional polytopal elements, vertices, edges, faces, each face is shared by exactly two cells. The two-dimensional analogue of a 4-polytope is a polygon, and the three-dimensional analogue is a polyhedron, topologically 4-polytopes are closely related to the uniform honeycombs, such as the cubic honeycomb, which tessellate 3-space, similarly the 3D cube is related to the infinite 2D square tiling. Convex 4-polytopes can be cut and unfolded as nets in 3-space, a 4-polytope is a closed four-dimensional figure. It comprises vertices, edges, faces and cells, a cell is the three-dimensional analogue of a face, and is therefore a polyhedron. Each face must join exactly two cells, analogous to the way in each edge of a polyhedron joins just two faces. Like any polytope, the elements of a 4-polytope cannot be subdivided into two or more sets which are also 4-polytopes, i. e. it is not a compound, the most familiar 4-polytope is the tesseract or hypercube, the 4D analogue of the cube. 4-polytopes cannot be seen in space due to their extra dimension. Several techniques are used to help visualise them, Orthogonal projection Orthogonal projections can be used to show various symmetry orientations of a 4-polytope. They can be drawn in 2D as vertex-edge graphs, and can be shown in 3D with solid faces as visible projective envelopes. Perspective projection Just as a 3D shape can be projected onto a flat sheet, sectioning Just as a slice through a polyhedron reveals a cut surface, so a slice through a 4-polytope reveals a cut hypersurface in three dimensions. A sequence of sections can be used to build up an understanding of the overall shape. The extra dimension can be equated with time to produce an animation of these cross sections. The topology of any given 4-polytope is defined by its Betti numbers, the value of the Euler characteristic used to characterise polyhedra does not generalize usefully to higher dimensions, and is zero for all 4-polytopes, whatever their underlying topology. This inadequacy of the Euler characteristic to distinguish between different topologies in higher dimensions led to the discovery of the more sophisticated Betti numbers. Similarly, the notion of orientability of a polyhedron is insufficient to characterise the surface twistings of toroidal 4-polytopes, like all polytopes, 4-polytopes may be classified based on properties like convexity and symmetry. Self-intersecting 4-polytopes are also known as star 4-polytopes, from analogy with the shapes of the non-convex star polygons. A 4-polytope is regular if it is transitive on its flags and this means that its cells are all congruent regular polyhedra, and similarly its vertex figures are congruent and of another kind of regular polyhedron
9.
Tesseract
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In geometry, the tesseract is the four-dimensional analog of the cube, the tesseract is to the cube as the cube is to the square. Just as the surface of the consists of six square faces. The tesseract is one of the six convex regular 4-polytopes, the tesseract is also called an 8-cell, C8, octachoron, octahedroid, cubic prism, and tetracube. It is the four-dimensional hypercube, or 4-cube as a part of the family of hypercubes or measure polytopes. In this publication, as well as some of Hintons later work, the tesseract can be constructed in a number of ways. As a regular polytope with three cubes folded together around every edge, it has Schläfli symbol with hyperoctahedral symmetry of order 384, constructed as a 4D hyperprism made of two parallel cubes, it can be named as a composite Schläfli symbol ×, with symmetry order 96. As a 4-4 duoprism, a Cartesian product of two squares, it can be named by a composite Schläfli symbol ×, with symmetry order 64, as an orthotope it can be represented by composite Schläfli symbol × × × or 4, with symmetry order 16. Since each vertex of a tesseract is adjacent to four edges, the dual polytope of the tesseract is called the hexadecachoron, or 16-cell, with Schläfli symbol. The standard tesseract in Euclidean 4-space is given as the hull of the points. That is, it consists of the points, A tesseract is bounded by eight hyperplanes, each pair of non-parallel hyperplanes intersects to form 24 square faces in a tesseract. 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, it consists of 8 cubes,24 squares,32 edges, the construction of a hypercube can be imagined the following way, 1-dimensional, Two points A and B can be connected to a line, giving a new line segment AB. 2-dimensional, Two parallel line segments AB and CD can be connected to become a square, 3-dimensional, Two parallel squares ABCD and EFGH can be connected to become a cube, with the corners marked as ABCDEFGH. 4-dimensional, Two parallel cubes ABCDEFGH and IJKLMNOP can be connected to become a hypercube and it is possible to project tesseracts into three- or two-dimensional spaces, as projecting a cube is possible on a two-dimensional space. Projections on the 2D-plane become more instructive by rearranging the positions of the projected vertices, the scheme is similar to the construction of a cube from two squares, juxtapose two copies of the lower-dimensional cube and connect the corresponding vertices. Each edge of a tesseract is of the same length, the regular complex polytope 42, in C2 has a real representation as a tesseract or 4-4 duoprism in 4-dimensional space. 42 has 16 vertices, and 8 4-edges and its symmetry is 42, order 32. It also has a lower construction, or 4×4, with symmetry 44
10.
Platonic solid
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In three-dimensional space, a Platonic solid is a regular, convex polyhedron. It is constructed by congruent regular polygonal faces with the number of faces meeting at each vertex. Five solids meet those criteria, Geometers have studied the mathematical beauty and they are named for the ancient Greek philosopher Plato who theorized in his dialogue, the Timaeus, that the classical elements were made of these regular solids. The Platonic solids have been known since antiquity, dice go back to the dawn of civilization with shapes that predated formal charting of Platonic solids. The ancient Greeks studied the Platonic solids extensively, some sources credit Pythagoras with their discovery. In any case, Theaetetus gave a description of all five. The Platonic solids are prominent in the philosophy of Plato, their namesake, Plato wrote about them in the dialogue Timaeus c.360 B. C. in which he associated each of the four classical elements with a regular solid. Earth was associated with the cube, air with the octahedron, water with the icosahedron, there was intuitive justification for these associations, the heat of fire feels sharp and stabbing. Air is made of the octahedron, its components are so smooth that one can barely feel it. Water, the icosahedron, flows out of hand when picked up. By contrast, a highly nonspherical solid, the hexahedron represents earth and these clumsy little solids cause dirt to crumble and break when picked up in stark difference to the smooth flow of water. Moreover, the cubes being the regular solid that tessellates Euclidean space was believed to cause the solidity of the Earth. Of the fifth Platonic solid, the dodecahedron, Plato obscurely remarks. the god used for arranging the constellations on the whole heaven. Aristotle added an element, aithēr and postulated that the heavens were made of this element. Euclid completely mathematically described the Platonic solids in the Elements, the last book of which is devoted to their properties, propositions 13–17 in Book XIII describe the construction of the tetrahedron, octahedron, cube, icosahedron, and dodecahedron in that order. For each solid Euclid finds the ratio of the diameter of the sphere to the edge length. In Proposition 18 he argues there are no further convex regular polyhedra. Andreas Speiser has advocated the view that the construction of the 5 regular solids is the goal of the deductive system canonized in the Elements
11.
Euclidean tilings by convex regular polygons
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Euclidean plane tilings by convex regular polygons have been widely used since antiquity. The first systematic mathematical treatment was that of Kepler in his Harmonices Mundi and this means that, for every pair of flags, there is a symmetry operation mapping the first flag to the second. This is equivalent to the tiling being an edge-to-edge tiling by congruent regular polygons, There must be six equilateral triangles, four squares or three regular hexagons at a vertex, yielding the three regular tessellations. Vertex-transitivity means that for pair of vertices there is a symmetry operation mapping the first vertex to the second. Note that there are two mirror forms of 34.6 tiling, only one of which is shown in the following table. All other regular and semiregular tilings are achiral, though these yield the same set of tilings in the plane, in other spaces there are Archimedean tilings which are not uniform. Such periodic tilings may be classified by the number of orbits of vertices, edges and tiles. If there are k orbits of vertices, a tiling is known as k -uniform or k -isogonal, if there are t orbits of tiles, as t -isohedral, if there are e orbits of edges, as e -isotoxal. K-uniform tilings with the vertex figures can be further identified by their wallpaper group symmetry. 1-uniform tilings include 3 regular tilings, and 8 semiregular ones, There are 20 2-uniform tilings,61 3-uniform tilings,151 4-uniform tilings,332 5-uniform tilings and 673 6-uniform tilings. Each can be grouped by the m of distinct vertex figures. For edge-to-edge Euclidean tilings, the angles of the polygons meeting at a vertex must add to 360 degrees. A regular n -gon has internal angle 180 degrees, only eleven of these can occur in a uniform tiling of regular polygons, given in previous sections. In particular, if three polygons meet at a vertex and one has an odd number of sides, the two polygons must be the same. If they are not, they would have to alternate around the first polygon, vertex types are listed for each. If two tilings share the two vertex types, they are given subscripts 1,2. There are 61 3-uniform tilings of the Euclidean plane,39 are 3-Archimedean with 3 distinct vertex types, while 22 have 2 identical vertex types in different symmetry orbits. Chavey There are 151 4-uniform tilings of the Euclidean plane, Brian Galebachs search reproduced Krotenheerdts list of 33 4-uniform tilings with 4 distinct vertex types, as well as finding 85 of them with 3 vertex types, and 33 with 2 vertex types
12.
Regular 4-polytope
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In mathematics, a regular 4-polytope is a regular four-dimensional polytope. They are the analogs of the regular polyhedra in three dimensions and the regular polygons in two dimensions. Regular 4-polytopes were first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century, There are six convex and ten star regular 4-polytopes, giving a total of sixteen. The convex regular 4-polytopes were first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century, Schläfli discovered that there are precisely six such figures. Schläfli also found four of the regular star 4-polytopes and he skipped the remaining six because he would not allow forms that failed the Euler characteristic on cells or vertex figures. That excludes cells and vertex figures as, and, the six convex and ten star polytopes described are the only solutions to these constraints. There are four nonconvex Schläfli symbols that have cells and vertex figures, and pass the dihedral test. The regular convex 4-polytopes are the analogs of the Platonic solids in three dimensions and the convex regular polygons in two dimensions. Five of them may be thought of as close analogs of the Platonic solids, There is one additional figure, the 24-cell, which has no close three-dimensional equivalent. Each convex regular 4-polytope is bounded by a set of 3-dimensional cells which are all Platonic solids of the same type and these are fitted together along their respective faces in a regular fashion. The following tables lists some properties of the six convex regular 4-polytopes, the symmetry groups of these 4-polytopes are all Coxeter groups and given in the notation described in that article. The number following the name of the group is the order of the group, John Conway advocates the names simplex, orthoplex, tesseract, octaplex or polyoctahedron, dodecaplex or polydodecahedron, and tetraplex or polytetrahedron. The Euler characteristic for all 4-polytopes is zero, we have the 4-dimensional analog of Eulers polyhedral formula, the topology of any given 4-polytope is defined by its Betti numbers and torsion coefficients. The following table shows some 2-dimensional projections of these 4-polytopes, various other visualizations can be found in the external links below. The Coxeter-Dynkin diagram graphs are given below the Schläfli symbol. The Schläfli–Hess 4-polytopes are the set of 10 regular self-intersecting star polychora. They are named in honor of their discoverers, Ludwig Schläfli, each is represented by a Schläfli symbol in which one of the numbers is 5/2. They are thus analogous to the regular nonconvex Kepler–Poinsot polyhedra and their names given here were given by John Conway, extending Cayleys names for the Kepler–Poinsot polyhedra, along with stellated and great, he adds a grand modifier
13.
Small stellated dodecahedron
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In geometry, the small stellated dodecahedron is a Kepler-Poinsot polyhedron, named by Arthur Cayley, and with Schläfli symbol. It is one of four regular polyhedra. It is composed of 12 pentagrammic faces, with five meeting at each vertex. It shares the vertex arrangement as the convex regular icosahedron. It also shares the same edge arrangement with the great icosahedron and it is the second of four stellations of the dodecahedron. It is central to two lithographs by M. C and its convex hull is the regular convex icosahedron. It also shares its edges with the great icosahedron, compound of small stellated dodecahedron and great dodecahedron Small stellated dodecahedron programing Wenninger, Magnus. Weber, Matthias, Keplers small stellated dodecahedron as a Riemann surface,220, 167–182 Eric W. Weisstein, Small stellated dodecahedron at MathWorld
14.
Square tiling
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In geometry, the square tiling, square tessellation or square grid is a regular tiling of the Euclidean plane. It has Schläfli symbol of, meaning it has 4 squares around every vertex, the internal angle of the square is 90 degrees so four squares at a point make a full 360 degrees. It is one of three regular tilings of the plane, the other two are the triangular tiling and the hexagonal tiling. There are 9 distinct uniform colorings of a square tiling, naming the colors by indices on the 4 squares around a vertex,1111,1112,1112,1122,1123,1123,1212,1213,1234. Cases have simple reflection symmetry, and glide reflection symmetry, three can be seen in the same symmetry domain as reduced colorings, 1112i from 1213, 1123i from 1234, and 1112ii reduced from 1123ii. This tiling is related as a part of sequence of regular polyhedra and tilings, extending into the hyperbolic plane. Like the uniform there are eight uniform tilings that can be based from the regular square tiling. Drawing the tiles colored as red on the faces, yellow at the original vertices. However treating faces identically, there are only three distinct forms, square tiling, truncated square tiling, snub square tiling. Other quadrilateral tilings can be made with topologically equivalent to the square tiling, isohedral tilings have identical faces and vertex-transitivity, there are 17 variations, with 6 identified as triangles that do not connect edge-to-edge, or as quadrilateral with two colinear edges. Symmetry given assumes all faces are the same color, the square tiling can be used as a circle packing, placing equal diameter circles at the center of every point. Every circle is in contact with 4 other circles in the packing, the packing density is π/4=78. 54% coverage. There are 4 uniform colorings of the circle packings, there are 3 regular complex apeirogons, sharing the vertices of the square tiling. Regular complex apeirogons have vertices and edges, where edges can contain 2 or more vertices, Regular apeirogons pr are contrained by, 1/p + 2/q + 1/r =1. Edges have p vertices, and vertex figures are r-gonal, checkerboard List of regular polytopes List of uniform tilings Square lattice Tilings of regular polygons Coxeter, H. S. M. Regular Polytopes, Dover edition, ISBN 0-486-61480-8 p.296, Table II, Regular honeycombs Klitzing, 2D Euclidean tilings o4o4x - squat - O1. The Geometrical Foundation of Natural Structure, A Source Book of Design, p36 Grünbaum, Branko, and Shephard, G. C. CS1 maint, Multiple names, authors list John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 Weisstein, Eric W. Square Grid
15.
Order-5 square tiling
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In geometry, the order-5 square tiling is a regular tiling of the hyperbolic plane. This tiling is related as a part of sequence of regular polyhedra. This hyperbolic tiling is related to an infinite skew polyhedron with the same vertex figure in Euclidean 3-space. John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 Chapter 10, the Beauty of Geometry, Twelve Essays. Square tiling Uniform tilings in hyperbolic plane List of regular polytopes Medial rhombic triacontahedron Weisstein, Weisstein, Eric W. Poincaré hyperbolic disk. Hyperbolic and Spherical Tiling Gallery KaleidoTile 3, Educational software to create spherical, planar and hyperbolic tilings Hyperbolic Planar Tessellations, Don Hatch
16.
Pentagram
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A pentagram is the shape of a five-pointed star drawn with five straight strokes. The word pentagram comes from the Greek word πεντάγραμμον, from πέντε, five + γραμμή, the word pentacle is sometimes used synonymously with pentagram The word pentalpha is a learned modern revival of a post-classical Greek name of the shape. The pentagram is the simplest regular star polygon, the pentagram contains ten points and fifteen line segments. It is represented by the Schläfli symbol, like a regular pentagon, and a regular pentagon with a pentagram constructed inside it, the regular pentagram has as its symmetry group the dihedral group of order 10. The pentagram can be constructed by connecting alternate vertices of a pentagon and it can also be constructed as a stellation of a pentagon, by extending the edges of a pentagon until the lines intersect. Each intersection of edges sections the edges in the golden ratio, also, the ratio of the length of the shorter segment to the segment bounded by the two intersecting edges is φ. As the four-color illustration shows, r e d g r e e n = g r e e n b l u e = b l u e m a g e n t a = φ. The pentagram includes ten isosceles triangles, five acute and five obtuse isosceles triangles, in all of them, the ratio of the longer side to the shorter side is φ. The acute triangles are golden triangles, the obtuse isosceles triangle highlighted via the colored lines in the illustration is a golden gnomon. The pentagram of Venus is the apparent path of the planet Venus as observed from Earth, the tips of the five loops at the center of the figure have the same geometric relationship to one another as the five vertices, or points, of a pentagram. Groups of five intersections of curves, equidistant from the center, have the same geometric relationship. In early monumental Sumerian script, or cuneiform, a pentagram glyph served as a logogram for the word ub, meaning corner, angle, nook, the word Pentemychos was the title of the cosmogony of Pherecydes of Syros. Here, the five corners are where the seeds of Chronos are placed within the Earth in order for the cosmos to appear. The pentangle plays an important symbolic role in the 14th-century English poem Sir Gawain, heinrich Cornelius Agrippa and others perpetuated the popularity of the pentagram as a magic symbol, attributing the five neoplatonic elements to the five points, in typical Renaissance fashion. By the mid-19th century a distinction had developed amongst occultists regarding the pentagrams orientation. With a single point upwards it depicted spirit presiding over the four elements of matter, however, the influential writer Eliphas Levi called it evil whenever the symbol appeared the other way up. It is the goat of lust attacking the heavens with its horns and it is the sign of antagonism and fatality. It is the goat of lust attacking the heavens with its horns, faust, The pentagram thy peace doth mar
17.
Petrie polygon
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In geometry, a Petrie polygon for a regular polytope of n dimensions is a skew polygon such that every consecutive sides belong to one of the facets. The Petrie polygon of a polygon is the regular polygon itself. For every regular polytope there exists an orthogonal projection onto a plane such that one Petrie polygon becomes a regular polygon with the remainder of the interior to it. The plane in question is the Coxeter plane of the group of the polygon. These polygons and projected graphs are useful in visualizing symmetric structure of the regular polytopes. John Flinders Petrie was the son of Egyptologist Flinders Petrie. He was born in 1907 and as a schoolboy showed remarkable promise of mathematical ability, in periods of intense concentration he could answer questions about complicated four-dimensional objects by visualizing them. He first noted the importance of the skew polygons which appear on the surface of regular polyhedra. When my incredulity had begun to subside, he described them to me, one consisting of squares, six at each vertex, in 1938 Petrie collaborated with Coxeter, Patrick du Val, and H. T. Flather to produce The Fifty-Nine Icosahedra for publication, realizing the geometric facility of the skew polygons used by Petrie, Coxeter named them after his friend when he wrote Regular Polytopes. In 1972, a few months after his retirement, Petrie was killed by a car attempting to cross a motorway near his home in Surrey. The idea of Petrie polygons was later extended to semiregular polytopes, the Petrie polygon of the regular polyhedron has h sides, where h+2=24/. The regular duals, and, are contained within the same projected Petrie polygon, three of the Kepler–Poinsot polyhedra have hexagonal, and decagrammic, petrie polygons. The Petrie polygon projections are most useful for visualization of polytopes of dimension four and this table represents Petrie polygon projections of 3 regular families, and the exceptional Lie group En which generate semiregular and uniform polytopes for dimensions 4 to 8. Coxeter, H. S. M. Regular Polytopes, 3rd ed, Section 4.3 Flags and Orthoschemes, Section 11.3 Petrie polygons Ball, W. W. R. and H. S. M. Coxeter Mathematical Recreations and Essays, 13th ed. The Beauty of Geometry, Twelve Essays, Dover Publications LCCN 99-35678 Peter McMullen, Egon Schulte Abstract Regular Polytopes, ISBN 0-521-81496-0 Steinberg, Robert, ON THE NUMBER OF SIDES OF A PETRIE POLYGON Weisstein, Eric W. Petrie polygon. Weisstein, Eric W. Cross polytope graphs, Weisstein, Eric W. Gosset graph 3_21
18.
Vertex figure
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In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. Take some vertex of a polyhedron, mark a point somewhere along each connected edge. Draw lines across the faces, joining adjacent points. When done, these form a complete circuit, i. e. a polygon. This polygon is the vertex figure, more precise formal definitions can vary quite widely, according to circumstance. For example Coxeter varies his definition as convenient for the current area of discussion, most of the following definitions of a vertex figure apply equally well to infinite tilings, or space-filling tessellation with polytope cells. Make a slice through the corner of the polyhedron, cutting all the edges connected to the vertex. The cut surface is the vertex figure and this is perhaps the most common approach, and the most easily understood. Different authors make the slice in different places, Wenninger cuts each edge a unit distance from the vertex, as does Coxeter. For uniform polyhedra the Dorman Luke construction cuts each connected edge at its midpoint, other authors make the cut through the vertex at the other end of each edge. For irregular polyhedra, these approaches may produce a figure that does not lie in a plane. A more general approach, valid for convex polyhedra, is to make the cut along any plane which separates the given vertex from all the other vertices. Cromwell makes a cut or scoop, centered on the vertex. The cut surface or vertex figure is thus a spherical polygon marked on this sphere, many combinatorial and computational approaches treat a vertex figure as the ordered set of points of all the neighboring vertices to the given vertex. In the theory of polytopes, the vertex figure at a given vertex V comprises all the elements which are incident on the vertex, edges, faces. More formally it is the -section Fn/V, where Fn is the greatest face and this set of elements is elsewhere known as a vertex star. A vertex figure for an n-polytope is an -polytope, for example, a vertex figure for a polyhedron is a polygon figure, and the vertex figure for a 4-polytope is a polyhedron. Each edge of the vertex figure exists on or inside of a face of the original polytope connecting two vertices from an original face
19.
Faceting
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In geometry, faceting is the process of removing parts of a polygon, polyhedron or polytope, without creating any new vertices. New edges of a polyhedron may be created along face diagonals or internal space diagonals. A faceted polyhedron will have two faces on each edge and creates new polyhedra or compounds of polyhedra, faceting is the reciprocal or dual process to stellation. For every stellation of some convex polytope, there exists a dual faceting of the dual polytope. For example, a regular pentagon has one symmetry faceting, the pentagram, the regular icosahedron can be faceted into three regular Kepler–Poinsot polyhedra, small stellated dodecahedron, great dodecahedron, and great icosahedron. The regular dodecahedron can be faceted into one regular Kepler–Poinsot polyhedron, the uniform stars and compound of five cubes are constructed by face diagonals. The excavated dodecahedron is a facetting with star hexagon faces, faceting has not been studied as extensively as stellation. In 1619, Kepler described a regular compound of two tetrahedra which fits inside a cube, and which he called the Stella octangula and this seems to be the first known example of faceting. In 1858, Bertrand derived the regular star polyhedra by faceting the regular icosahedron and dodecahedron. In 1974, Bridge enumerated the more straightforward facetings of the regular polyhedra, in 2006, Inchbald described the basic theory of faceting diagrams for polyhedra. For a given vertex, the shows all the possible edges. It is dual to the dual polyhedrons stellation diagram, which all the possible edges and vertices for some face plane of the original core. Note sur la théorie des polyèdres réguliers, Comptes rendus des séances de lAcadémie des Sciences,46, Facetting the dodecahedron, Acta crystallographica A30, pp. 548–552. Inchbald, G. Facetting diagrams, The mathematical gazette,90, alan Holden, Shapes, Space, and Symmetry. Archived from the original on 4 February 2007
20.
Convex polyhedron
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A convex polytope is a special case of a polytope, having the additional property that it is also a convex set of points in the n-dimensional space Rn. Some authors use the terms polytope and convex polyhedron interchangeably. In addition, some require a polytope to be a bounded set. The terms bounded/unbounded convex polytope will be used whenever the boundedness is critical to the discussed issue. Yet other texts treat a convex n-polytope as a surface or -manifold, Convex polytopes play an important role both in various branches of mathematics and in applied areas, most notably in linear programming. A comprehensive and influential book in the subject, called Convex Polytopes, was published in 1967 by Branko Grünbaum, in 2003 the 2nd edition of the book was published, with significant additional material contributed by new writers. In Grünbaums book, and in other texts in discrete geometry. Grünbaum points out that this is solely to avoid the repetition of the word convex. A polytope is called if it is an n-dimensional object in Rn. Many examples of bounded convex polytopes can be found in the article polyhedron, a convex polytope may be defined in a number of ways, depending on what is more suitable for the problem at hand. Grünbaums definition is in terms of a set of points in space. Other important definitions are, as the intersection of half-spaces and as the hull of a set of points. This is equivalent to defining a bounded convex polytope as the hull of a finite set of points. Such a definition is called a vertex representation, for a compact convex polytope, the minimal V-description is unique and it is given by the set of the vertices of the polytope. A convex polytope may be defined as an intersection of a number of half-spaces. Such definition is called a half-space representation, there exist infinitely many H-descriptions of a convex polytope. However, for a convex polytope, the minimal H-description is in fact unique and is given by the set of the facet-defining halfspaces. A closed half-space can be written as an inequality, a 1 x 1 + a 2 x 2 + ⋯ + a n x n ≤ b where n is the dimension of the space containing the polytope under consideration
21.
Euler characteristic
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It is commonly denoted by χ. The Euler characteristic was originally defined for polyhedra and used to prove theorems about them. Leonhard Euler, for whom the concept is named, was responsible for much of early work. In modern mathematics, the Euler characteristic arises from homology and, more abstractly, any convex polyhedrons surface has Euler characteristic V − E + F =2. This equation is known as Eulers polyhedron formula and it corresponds to the Euler characteristic of the sphere, and applies identically to spherical polyhedra. An illustration of the formula on some polyhedra is given below and this version holds both for convex polyhedra and the non-convex Kepler-Poinsot polyhedra. Projective polyhedra all have Euler characteristic 1, like the real plane, while the surfaces of toroidal polyhedra all have Euler characteristic 0. The Euler characteristic can be defined for connected plane graphs by the same V − E + F formula as for polyhedral surfaces, the Euler characteristic of any plane connected graph G is 2. This is easily proved by induction on the number of determined by G. For trees, E = V −1 and F =1, if G has C components, the same argument by induction on F shows that V − E + F − C =1. One of the few graph theory papers of Cauchy also proves this result, via stereographic projection the plane maps to the two-dimensional sphere, such that a connected graph maps to a polygonal decomposition of the sphere, which has Euler characteristic 2. This viewpoint is implicit in Cauchys proof of Eulers formula given below, there are many proofs of Eulers formula. One was given by Cauchy in 1811, as follows and it applies to any convex polyhedron, and more generally to any polyhedron whose boundary is topologically equivalent to a sphere and whose faces are topologically equivalent to disks. Remove one face of the polyhedral surface, after this deformation, the regular faces are generally not regular anymore. The number of vertices and edges has remained the same, therefore, proving Eulers formula for the polyhedron reduces to proving V − E + F =1 for this deformed, planar object. If there is a face more than three sides, draw a diagonal—that is, a curve through the face connecting two vertices that arent connected yet. This adds one edge and one face and does not change the number of vertices, continue adding edges in this manner until all of the faces are triangular. This decreases the number of edges and faces by one each and does not change the number of vertices, remove a triangle with two edges shared by the exterior of the network, as illustrated by the third graph
22.
Vertex (geometry)
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In geometry, a vertex is a point where two or more curves, lines, or edges meet. As a consequence of this definition, the point where two lines meet to form an angle and the corners of polygons and polyhedra are vertices. A vertex is a point of a polygon, polyhedron, or other higher-dimensional polytope. However, in theory, vertices may have fewer than two incident edges, which is usually not allowed for geometric vertices. However, a smooth approximation to a polygon will also have additional vertices. A polygon vertex xi of a simple polygon P is a principal polygon vertex if the diagonal intersects the boundary of P only at x and x, there are two types of principal vertices, ears and mouths. A principal vertex xi of a simple polygon P is called an ear if the diagonal that bridges xi lies entirely in P, according to the two ears theorem, every simple polygon has at least two ears. A principal vertex xi of a simple polygon P is called a mouth if the diagonal lies outside the boundary of P. Any convex polyhedrons surface has Euler characteristic V − E + F =2, where V is the number of vertices, E is the number of edges and this equation is known as Eulers polyhedron formula. Thus the number of vertices is 2 more than the excess of the number of edges over the number of faces, for example, a cube has 12 edges and 6 faces, and hence 8 vertices
23.
Edge (geometry)
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For edge in graph theory, see Edge In geometry, an edge is a particular type of line segment joining two vertices in a polygon, polyhedron, or higher-dimensional polytope. In a polygon, an edge is a segment on the boundary. In a polyhedron or more generally a polytope, an edge is a segment where two faces meet. A segment joining two vertices while passing through the interior or exterior is not an edge but instead is called a diagonal. In graph theory, an edge is an abstract object connecting two vertices, unlike polygon and polyhedron edges which have a concrete geometric representation as a line segment. However, any polyhedron can be represented by its skeleton or edge-skeleton, conversely, the graphs that are skeletons of three-dimensional polyhedra can be characterized by Steinitzs theorem as being exactly the 3-vertex-connected planar graphs. Any convex polyhedrons surface has Euler characteristic V − E + F =2, where V is the number of vertices, E is the number of edges and this equation is known as Eulers polyhedron formula. Thus the number of edges is 2 less than the sum of the numbers of vertices and faces, for example, a cube has 8 vertices and 6 faces, and hence 12 edges. In a polygon, two edges meet at each vertex, more generally, by Balinskis theorem, at least d edges meet at every vertex of a convex polytope. Similarly, in a polyhedron, exactly two faces meet at every edge, while in higher dimensional polytopes three or more two-dimensional faces meet at every edge. Thus, the edges of a polygon are its facets, the edges of a 3-dimensional convex polyhedron are its ridges, archived from the original on 4 February 2007
24.
Set theory
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Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics, the language of set theory can be used in the definitions of nearly all mathematical objects. The modern study of set theory was initiated by Georg Cantor, Set theory is commonly employed as a foundational system for mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Beyond its foundational role, set theory is a branch of mathematics in its own right, contemporary research into set theory includes a diverse collection of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals. Mathematical topics typically emerge and evolve through interactions among many researchers, Set theory, however, was founded by a single paper in 1874 by Georg Cantor, On a Property of the Collection of All Real Algebraic Numbers. Since the 5th century BC, beginning with Greek mathematician Zeno of Elea in the West and early Indian mathematicians in the East, especially notable is the work of Bernard Bolzano in the first half of the 19th century. Modern understanding of infinity began in 1867–71, with Cantors work on number theory, an 1872 meeting between Cantor and Richard Dedekind influenced Cantors thinking and culminated in Cantors 1874 paper. Cantors work initially polarized the mathematicians of his day, while Karl Weierstrass and Dedekind supported Cantor, Leopold Kronecker, now seen as a founder of mathematical constructivism, did not. This utility of set theory led to the article Mengenlehre contributed in 1898 by Arthur Schoenflies to Kleins encyclopedia, in 1899 Cantor had himself posed the question What is the cardinal number of the set of all sets. Russell used his paradox as a theme in his 1903 review of continental mathematics in his The Principles of Mathematics, in 1906 English readers gained the book Theory of Sets of Points by William Henry Young and his wife Grace Chisholm Young, published by Cambridge University Press. The momentum of set theory was such that debate on the paradoxes did not lead to its abandonment, the work of Zermelo in 1908 and Abraham Fraenkel in 1922 resulted in the set of axioms ZFC, which became the most commonly used set of axioms for set theory. The work of such as Henri Lebesgue demonstrated the great mathematical utility of set theory. Set theory is used as a foundational system, although in some areas category theory is thought to be a preferred foundation. Set theory begins with a binary relation between an object o and a set A. If o is a member of A, the notation o ∈ A is used, since sets are objects, the membership relation can relate sets as well. A derived binary relation between two sets is the relation, also called set inclusion. If all the members of set A are also members of set B, then A is a subset of B, for example, is a subset of, and so is but is not. As insinuated from this definition, a set is a subset of itself, for cases where this possibility is unsuitable or would make sense to be rejected, the term proper subset is defined
25.
Cell (geometry)
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In solid geometry, a face is a flat surface that forms part of the boundary of a solid object, a three-dimensional solid bounded exclusively by flat faces is a polyhedron. In more technical treatments of the geometry of polyhedra and higher-dimensional polytopes, in elementary geometry, a face is a polygon on the boundary of a polyhedron. Other names for a polygonal face include side of a polyhedron, for example, any of the six squares that bound a cube is a face of the cube. Sometimes face is used to refer to the 2-dimensional features of a 4-polytope. With this meaning, the 4-dimensional tesseract has 24 square faces, some other polygons, which are not faces, are also important for polyhedra and tessellations. These include Petrie polygons, vertex figures and facets, any convex polyhedrons surface has Euler characteristic V − E + F =2, where V is the number of vertices, E is the number of edges, and F is the number of faces. This equation is known as Eulers polyhedron formula, thus the number of faces is 2 more than the excess of the number of edges over the number of vertices. For example, a cube has 12 edges and 8 vertices, in higher-dimensional geometry the faces of a polytope are features of all dimensions. A face of dimension k is called a k-face, for example, the polygonal faces of an ordinary polyhedron are 2-faces. In set theory, the set of faces of a polytope includes the polytope itself, for any n-polytope, −1 ≤ k ≤ n. For example, with meaning, the faces of a cube include the empty set, its vertices, edges and squares. Formally, a face of a polytope P is the intersection of P with any closed halfspace whose boundary is disjoint from the interior of P, from this definition it follows that the set of faces of a polytope includes the polytope itself and the empty set. In other areas of mathematics, such as the theories of abstract polytopes and star polytopes, abstract theory still requires that the set of faces include the polytope itself and the empty set. A cell is an element of a 4-dimensional polytope or 3-dimensional tessellation. Cells are facets for 4-polytopes and 3-honeycombs, examples, In higher-dimensional geometry, the facets of a n-polytope are the -faces of dimension one less than the polytope itself. A polytope is bounded by its facets, for example, The facets of a line segment are its 0-faces or vertices. The facets of a polygon are its 1-faces or edges, the facets of a polyhedron or plane tiling are its 2-faces. The facets of a 4D polytope or 3-honeycomb are its 3-faces, the facets of a 5D polytope or 4-honeycomb are its 4-faces
26.
Polyhedral combinatorics
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Research in polyhedral combinatorics falls into two distinct areas. A face of a convex polytope P may be defined as the intersection of P, the dimension of a face is the dimension of this hull. The 0-dimensional faces are the vertices themselves, and the 1-dimensional faces are line segments connecting pairs of vertices, note that this definition also includes as faces the empty set and the whole polytope P. If P itself has dimension d, the faces of P with dimension d −1 are called facets of P and the faces with dimension d −2 are called ridges. The faces of P may be ordered by inclusion, forming a face lattice that has as its top element P itself. A key tool in polyhedral combinatorics is the ƒ-vector of a polytope, for instance, a cube has eight vertices, twelve edges, and six facets, so its ƒ-vector is. The dual polytope has a ƒ-vector with the numbers in the reverse order, thus, for instance, the regular octahedron. For the cube the extended ƒ-vector is and for the octahedron it is, although the vectors for these example polyhedra are unimodal, there are higher-dimensional polytopes for which this is not true. For simplicial polytopes, it is convenient to transform these vectors. If we interpret the terms of the ƒ-vector as coefficients of a polynomial ƒ = Σfixd − i −1, then the h-vector lists the coefficients of the polynomial h = ƒ. As Ziegler writes, “for various problems about simplicial polytopes, h-vectors are a more convenient and concise way to encode the information about the face numbers than ƒ-vectors. ”The most important relation among the coefficients of the ƒ-vector of a polytope is Eulers formula Σifi =0. In three dimensions, moving the two 1s at the left and right ends of the extended ƒ-vector to the hand side of the equation transforms this identity into the more familiar form v − e + f =2. By duality, e ≤ 3f −6 and v ≤ 2f −4 and it follows from Steinitzs theorem that any 3-dimensional integer vector satisfying these equalities and inequalities is the ƒ-vector of a convex polyhedron. Asymptotically, this implies there are at most O faces of all dimensions. Balinskis theorem states that the obtained in this way from any d-dimensional convex polytope is d-vertex-connected. A theorem of Blind & Mani-Levitska states that one can reconstruct the face structure of a polytope from its graph. That is, if an undirected graph is the skeleton of a simple polytope. This is in sharp contrast with neighborly polytopes whose graph is a complete graph, thus, the diameter provides a lower bound on the number of steps this method requires
27.
Abstract polytope
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An ordinary geometric polytope is said to be a realization in some real N-dimensional space, typically Euclidean, of the corresponding abstract polytope. The abstract definition allows some more general combinatorial structures than traditional definitions of a polytope, the term polytope is a generalisation of polygons and polyhedra into any number of dimensions. In Euclidean geometry, the six quadrilaterals illustrated are all different, yet they have a common structure in the alternating chain of four vertices and four sides which gives them their name. They are said to be isomorphic or “structure preserving”, the measurable properties of traditional polytopes such as angles, edge-lengths, skewness, straightness and convexity have no meaning for an abstract polytope. What is true for traditional polytopes may not be so for abstract ones, for example, a traditional polytope is regular if all its facets and vertex figures are regular, but this is not necessarily so for an abstract polytope. A traditional geometric polytope is said to be a realisation of the abstract polytope. A realisation is a mapping or injection of the object into a real space, typically Euclidean. The six quadrilaterals shown are all distinct realisations of the abstract quadrilateral, some of them do not conform to traditional definitions of a quadrilateral and are said to be unfaithful realisations. A conventional polytope is a faithful realisation, in an abstract polytope, each structural element - vertex, edge, cell, etc. is associated with a corresponding member or element of the set. The term face often refers to any such element e. g. a vertex, edge or a general k-face, the faces are ranked according to their associated real dimension, vertices have rank =0, edges rank =1 and so on. This usage of incidence also occurs in Finite geometry, although it differs from traditional geometry, for example in the square abcd, edges ab and bc are not abstractly incident. A polytope is defined as a set of faces P with an order relation <. Formally, P will be an ordered set, or poset. Just as the zero is necessary in mathematics, so also set theory requires an empty set which, technically. In an abstract polytope this is known as the least or null face and is a subface of all the others, since the least face is one level below the vertices or 0-faces, its rank is −1 and may be denoted as F−1. There is also a face of which all the others are subfaces. This is called the greatest face, in an n-dimensional polytope, the greatest face has rank = n and may be denoted as Fn. It is sometimes realized as the interior of the geometric figure and these least and greatest faces are sometimes called improper faces, with all others being proper faces
28.
Star polyhedron
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In geometry, a star polyhedron is a polyhedron which has some repetitive quality of nonconvexity giving it a star-like visual quality. There are two kinds of star polyhedron, Polyhedra which self-intersect in a repetitive way. Concave polyhedra of a kind which alternate convex and concave or saddle vertices in a repetitive way. Mathematically these figures are examples of star domains, mathematical studies of star polyhedra are usually concerned with regular, uniform polyhedra, or the duals of the uniform polyhedra. All these stars are of the self-intersecting kind, the regular star polyhedra are self-intersecting polyhedra. They may either have self-intersecting faces, or self-intersecting vertex figures, There are four regular star polyhedra, known as the Kepler-Poinsot polyhedra. The Schläfli symbol implies faces with p sides, and vertex figures with q sides, two of them have pentagrammic faces and two have pentagrammic vertex figures. These images show each form with a single face colored yellow to show the visible portion of that face, There are many uniform star polyhedra including two infinite series, of prisms and of antiprisms, and their duals. The uniform and dual uniform polyhedra are also self-intersecting polyhedra. They may either have self-intersecting faces, or self-intersecting vertex figures or both, the uniform star polyhedra have regular faces or regular star polygon faces. The dual uniform polyhedra have regular faces or regular star polygon vertex figures. Beyond the forms above, there are unlimited classes of self-intersecting polyhedra, two important classes are the stellations of convex polyhedra and their duals, the facettings of the dual polyhedra. For example, the complete stellation of the icosahedron can be interpreted as a polyhedron composed of 12 identical faces. Below is an illustration of this polyhedron with one drawn in yellow. A similarly self-intersecting polytopes in any number of dimensions is called a star polytope, a regular polytope is a star polytope if either its facet or its vertex figure is a star polytope. In four dimensions, the 10 regular star polychora are called the Schläfli-Hess polychora, analogous to the regular star polyhedra, these 10 are all composed of facets which are either one of the five regular Platonic solids or one of the four regular star Kepler-Poinsot polyhedra. For example, the grand stellated 120-cell, projected orthogonally into 3-space, looks like this. A polyhedron which does not cross itself, such that all of the interior can be seen from one point, is an example of a star domain
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120-cell
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In geometry, the 120-cell is the convex regular 4-polytope with Schläfli symbol. It is also called a C120, hecatonicosachoron, dodecacontachoron and hecatonicosahedroid, the boundary of the 120-cell is composed of 120 dodecahedral cells with 4 meeting at each vertex. It can be thought of as the 4-dimensional analog of the dodecahedron and has called a dodecaplex. Just as a dodecahedron can be built up as a model with 12 pentagons,3 around each vertex, there are 120 cells,720 pentagonal faces,1200 edges, and 600 vertices. There are 4 dodecahedra,6 pentagons, and 4 edges meeting at every vertex, there are 3 dodecahedra and 3 pentagons meeting every edge. The dual polytope of the 120-cell is the 600-cell, the vertex figure of the 120-cell is a tetrahedron. The dihedral angle of the 120-cell is 144° The 600 vertices of the 120-cell include all permutations of, the 120-cell consists of 120 dodecahedral cells. For visualization purposes, it is convenient that the dodecahedron has opposing parallel faces, one can stack dodecahedrons face to face in a straight line bent in the 4th direction into a great circle with a circumference of 10 cells. Starting from this initial ten cell construct there are two common visualizations one can use, a stereographic projection, and a structure of intertwining rings. The cell locations lend themselves to a hyperspherical description, pick an arbitrary cell and label it the North Pole. Twelve great circle meridians radiate out in 3 dimensions, converging at the 5th South Pole cell and this skeleton accounts for 50 of the 120 cells. Starting at the North Pole, we can build up the 120-cell in 9 latitudinal layers, with the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere. The centroids of the 30 equatorial cells form the vertices of an icosidodecahedron, the cells labeled interstitial in the following table do not fall on meridian great circles. Layers 2,4,6 and 8 cells are located over the cells faces. Layers 3 and 7s cells are located directly over the pole cells vertices, layer 5s cells are located over the pole cells edges. The 120-cell can be partitioned into 12 disjoint 10-cell great circle rings, starting with one 10-cell ring, one can place another ring alongside it that spirals around the original ring one complete revolution in ten cells. Five such 10-cell rings can be placed adjacent to the original 10-cell ring, although the outer rings spiral around the inner ring, they actually have no helical torsion. The spiraling is a result of the 3-sphere curvature, the inner ring and the five outer rings now form a six ring, 60-cell solid torus
30.
Cubic honeycomb
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The cubic honeycomb or cubic cellulation is the only regular space-filling tessellation in Euclidean 3-space, made up of cubic cells. It has 4 cubes around every edge, and 8 cubes around each vertex and its vertex figure is a regular octahedron. It is a tessellation with Schläfli symbol. John Horton Conway calls this honeycomb a cubille, a geometric honeycomb is a space-filling of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the general mathematical tiling or tessellation in any number of dimensions. Honeycombs are usually constructed in ordinary Euclidean space, like the uniform honeycombs. They may also be constructed in non-Euclidean spaces, such as hyperbolic uniform honeycombs, any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space. The Cartesian coordinates of the vertices are, for all values, i, j, k, with edges parallel to the axes. It is part of a family of hypercube honeycombs, with Schläfli symbols of the form, starting with the square tiling. It is one of 28 uniform honeycombs using convex uniform polyhedral cells, simple cubic lattices can be distorted into lower symmetries, represented by lower crystal systems, There is a large number of uniform colorings, derived from different symmetries. These include, It is related to the regular 4-polytope tesseract, Schläfli symbol, which exists in 4-space and its also related to the order-5 cubic honeycomb, Schläfli symbol, of hyperbolic space with 5 cubes around each edge. It is in a sequence of polychora and honeycomb with octahedral vertex figures and it in a sequence of regular polytopes and honeycombs with cubic cells. The, Coxeter group generates 15 permutations of uniform tessellations,9 with distinct geometry including the cubic honeycomb. The expanded cubic honeycomb is geometrically identical to the cubic honeycomb, the, Coxeter group generates 9 permutations of uniform tessellations,4 with distinct geometry including the alternated cubic honeycomb. This honeycomb is one of five distinct uniform honeycombs constructed by the A ~3 Coxeter group and it is composed of octahedra and cuboctahedra in a ratio of 1,1. John Horton Conway calls this honeycomb a cuboctahedrille, and its dual oblate octahedrille, There are four uniform colorings for the cells of this honeycomb with reflective symmetry, listed by their Coxeter group, and Wythoff construction name, and the Coxeter diagram below. This honeycomb can be divided on trihexagonal tiling planes, using the hexagon centers of the cuboctahedra and this scaliform honeycomb is represented by Coxeter diagram, and symbol s3, with coxeter notation symmetry. The truncated cubic honeycomb or truncated cubic cellulation is a uniform space-filling tessellation in Euclidean 3-space and it is composed of truncated cubes and octahedra in a ratio of 1,1
31.
Order-4 dodecahedral honeycomb
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In the geometry of hyperbolic 3-space, the order-4 dodecahedral honeycomb is one of four compact regular space-filling tessellations. With Schläfli symbol, it has four dodecahedra around each edge and its vertices are constructed from 3 orthogonal axes. Its dual is the cubic honeycomb. A geometric honeycomb is a space-filling of polyhedral or higher-dimensional cells and it is an example of the more general mathematical tiling or tessellation in any number of dimensions. Honeycombs are usually constructed in ordinary Euclidean space, like the uniform honeycombs. They may also be constructed in non-Euclidean spaces, such as hyperbolic uniform honeycombs, any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space. The dihedral angle of a dodecahedron is ~116. 6°, so it is impossible to fit 4 of them on an edge in Euclidean 3-space. However in hyperbolic space a properly scaled dodecahedron can be scaled so that its angles are reduced to 90 degrees. It a half symmetry construction, with two types of hexagonal tilings in the Wythoff construction, beltrami-Klein model There are four regular compact honeycombs in 3D hyperbolic space, There are fifteen uniform honeycombs in the Coxeter group family, including this regular form. There are eleven uniform honeycombs in the bifurcating Coxeter group family and this construction can be represented by alternation with two colors of dodecahedral cells. There are four rectified compact regular honeycombs, The truncated order-4 dodecahedral honeycomb, has octahedron and truncated dodecahedron cells, the cantellated order-4 dodecahedral honeycomb, has rhombicosidodecahedron and cuboctahedron, and cube cells, with a triangular prism vertex figure. The cantitruncated order-4 dodecahedral honeycomb, is a uniform honeycomb constructed with a coxeter diagram, the runcititruncated order-4 dodecahedral honeycomb, is a uniform honeycomb constructed with a coxeter diagram, and a quadrilateral pyramid vertex figure. Johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D, dissertation, University of Toronto,1966 N. W. Johnson, Geometries and Transformations, Chapter 13, Hyperbolic Coxeter groups
32.
Dodecahedron
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In geometry, a dodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the dodecahedron, which is a Platonic solid. There are also three regular star dodecahedra, which are constructed as stellations of the convex form, all of these have icosahedral symmetry, order 120. The pyritohedron is a pentagonal dodecahedron, having the same topology as the regular one. The rhombic dodecahedron, seen as a case of the pyritohedron has octahedral symmetry. The elongated dodecahedron and trapezo-rhombic dodecahedron variations, along with the rhombic dodecahedra are space-filling, there are a large number of other dodecahedra. The convex regular dodecahedron is one of the five regular Platonic solids, the dual polyhedron is the regular icosahedron, having five equilateral triangles around each vertex. Like the regular dodecahedron, it has twelve pentagonal faces. However, the pentagons are not constrained to be regular, and its 30 edges are divided into two sets – containing 24 and 6 edges of the same length. The only axes of symmetry are three mutually perpendicular twofold axes and four threefold axes. Note that the regular dodecahedron can occur as a shape for quasicrystals with icosahedral symmetry. Its name comes from one of the two common crystal habits shown by pyrite, the one being the cube. The coordinates of the eight vertices of the cube are, The coordinates of the 12 vertices of the cross-edges are. When h =1, the six cross-edges degenerate to points, when h =0, the cross-edges are absorbed in the facets of the cube, and the pyritohedron reduces to a cube. When h = √5 − 1/2, the inverse of the golden ratio, a reflected pyritohedron is made by swapping the nonzero coordinates above. The two pyritohedra can be superimposed to give the compound of two dodecahedra as seen in the image here, the regular dodecahedron represents a special intermediate case where all edges and angles are equal. A tetartoid is a dodecahedron with chiral tetrahedral symmetry, like the regular dodecahedron, it has twelve identical pentagonal faces, with three meeting in each of the 20 vertices. However, the pentagons are not regular and the figure has no fivefold symmetry axes, although regular dodecahedra do not exist in crystals, the tetartoid form does
33.
Line segment
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In geometry, a line segment is a part of a line that is bounded by two distinct end points, and contains every point on the line between its endpoints. A closed line segment includes both endpoints, while a line segment excludes both endpoints, a half-open line segment includes exactly one of the endpoints. Examples of line include the sides of a triangle or square. More generally, when both of the end points are vertices of a polygon or polyhedron, the line segment is either an edge if they are adjacent vertices. When the end points both lie on a such as a circle, a line segment is called a chord. Sometimes one needs to distinguish between open and closed line segments, thus, the line segment can be expressed as a convex combination of the segments two end points. In geometry, it is defined that a point B is between two other points A and C, if the distance AB added to the distance BC is equal to the distance AC. Thus in R2 the line segment with endpoints A = and C = is the collection of points. A line segment is a connected, non-empty set, if V is a topological vector space, then a closed line segment is a closed set in V. However, an open line segment is an open set in V if and only if V is one-dimensional. More generally than above, the concept of a segment can be defined in an ordered geometry. A pair of segments can be any one of the following, intersecting, parallel, skew. The last possibility is a way that line segments differ from lines, in an axiomatic treatment of geometry, the notion of betweenness is either assumed to satisfy a certain number of axioms, or else be defined in terms of an isometry of a line. Segments play an important role in other theories, for example, a set is convex if the segment that joins any two points of the set is contained in the set. This is important because it transforms some of the analysis of sets to the analysis of a line segment. The Segment Addition Postulate can be used to add congruent segment or segments with equal lengths and consequently substitute other segments into another statement to make segments congruent. A line segment can be viewed as a case of an ellipse in which the semiminor axis goes to zero, the foci go to the endpoints. A complete orbit of this ellipse traverses the line segment twice, as a degenerate orbit this is a radial elliptic trajectory. In addition to appearing as the edges and diagonals of polygons and polyhedra, some very frequently considered segments in a triangle include the three altitudes, the three medians, the perpendicular bisectors of the sides, and the internal angle bisectors
34.
Uniform tiling
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In geometry, a uniform tiling is a tessellation of the plane by regular polygon faces with the restriction of being vertex-transitive. Uniform tilings can exist in both the Euclidean plane and hyperbolic plane, Uniform tilings are related to the finite uniform polyhedra which can be considered uniform tilings of the sphere. Most uniform tilings can be made from a Wythoff construction starting with a symmetry group, a planar symmetry group has a polygonal fundamental domain and can be represented by the group name represented by the order of the mirrors in sequential vertices. A fundamental domain triangle is, and a triangle, where p, q, r are whole numbers greater than 1. The triangle may exist as a triangle, a Euclidean plane triangle, or a hyperbolic plane triangle. There are a number of schemes for naming these figures, from a modified Schläfli symbol for right triangle domains. The Coxeter-Dynkin diagram is a graph with p, q, r labeled on the edges. If r =2, the graph is linear since order-2 domain nodes generate no reflections, the Wythoff symbol takes the 3 integers and separates them by a vertical bar. If the generator point is off the mirror opposite a domain node, finally tilings can be described by their vertex configuration, the sequence of polygons around each vertex. All uniform tilings can be constructed from various operations applied to regular tilings and these operations as named by Norman Johnson are called truncation, rectification, and Cantellation. Omnitruncation is an operation that combines truncation and cantellation, snubbing is an operation of Alternate truncation of the omnitruncated form. Each is represented by a set of lines of reflection that divide the plane into fundamental triangles and these symmetry groups create 3 regular tilings, and 7 semiregular ones. A number of the tilings are repeated from different symmetry constructors. A prismatic symmetry group represented by represents by two sets of mirrors, which in general can have a rectangular fundamental domain. A further prismatic symmetry group represented by which has a fundamental domain. It constructs two uniform tilings, the prism and apeirogonal antiprism. The stacking of the faces of these two prismatic tilings constructs one non-Wythoffian uniform tiling of the plane. It is called the triangular tiling, composed of alternating layers of squares and triangles
35.
2-face
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In solid geometry, a face is a flat surface that forms part of the boundary of a solid object, a three-dimensional solid bounded exclusively by flat faces is a polyhedron. In more technical treatments of the geometry of polyhedra and higher-dimensional polytopes, in elementary geometry, a face is a polygon on the boundary of a polyhedron. Other names for a polygonal face include side of a polyhedron, for example, any of the six squares that bound a cube is a face of the cube. Sometimes face is used to refer to the 2-dimensional features of a 4-polytope. With this meaning, the 4-dimensional tesseract has 24 square faces, some other polygons, which are not faces, are also important for polyhedra and tessellations. These include Petrie polygons, vertex figures and facets, any convex polyhedrons surface has Euler characteristic V − E + F =2, where V is the number of vertices, E is the number of edges, and F is the number of faces. This equation is known as Eulers polyhedron formula, thus the number of faces is 2 more than the excess of the number of edges over the number of vertices. For example, a cube has 12 edges and 8 vertices, in higher-dimensional geometry the faces of a polytope are features of all dimensions. A face of dimension k is called a k-face, for example, the polygonal faces of an ordinary polyhedron are 2-faces. In set theory, the set of faces of a polytope includes the polytope itself, for any n-polytope, −1 ≤ k ≤ n. For example, with meaning, the faces of a cube include the empty set, its vertices, edges and squares. Formally, a face of a polytope P is the intersection of P with any closed halfspace whose boundary is disjoint from the interior of P, from this definition it follows that the set of faces of a polytope includes the polytope itself and the empty set. In other areas of mathematics, such as the theories of abstract polytopes and star polytopes, abstract theory still requires that the set of faces include the polytope itself and the empty set. A cell is an element of a 4-dimensional polytope or 3-dimensional tessellation. Cells are facets for 4-polytopes and 3-honeycombs, examples, In higher-dimensional geometry, the facets of a n-polytope are the -faces of dimension one less than the polytope itself. A polytope is bounded by its facets, for example, The facets of a line segment are its 0-faces or vertices. The facets of a polygon are its 1-faces or edges, the facets of a polyhedron or plane tiling are its 2-faces. The facets of a 4D polytope or 3-honeycomb are its 3-faces, the facets of a 5D polytope or 4-honeycomb are its 4-faces
36.
Convex uniform honeycomb
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In geometry, a convex uniform honeycomb is a uniform tessellation which fills three-dimensional Euclidean space with non-overlapping convex uniform polyhedral cells. They can be considered the three-dimensional analogue to the uniform tilings of the plane, the Voronoi diagram of any lattice forms a convex uniform honeycomb in which the cells are zonohedra. 1905, Alfredo Andreini enumerated 25 of these tessellations,1991, Norman Johnsons manuscript Uniform Polytopes identified the complete list of 28. 1994, Branko Grünbaum, in his paper Uniform tilings of 3-space, also independently enumerated all 28 and he found the 1905 paper, which listed 25, had 1 wrong, and 4 being missing. Grünbaum states in this paper that Norman Johnson deserves priority for achieving the same enumeration in 1991, alexeyev of Russia had contacted him regarding a putative enumeration of these forms, but that Grünbaum was unable to verify this at the time. Only 14 of the uniform polyhedra appear in these patterns. This set can be called the regular and semiregular honeycombs and it has been called the Archimedean honeycombs by analogy with the convex uniform polyhedra, commonly called Archimedean solids. Recently Conway has suggested naming the set as the Architectonic tessellations, the individual honeycombs are listed with names given to them by Norman Johnson. For cross-referencing, they are given with list indices from Andreini, Williams, Johnson, and Grünbaum. Coxeter uses δ4 for a honeycomb, hδ4 for an alternated cubic honeycomb, qδ4 for a quarter cubic honeycomb. The fundamental infinite Coxeter groups for 3-space are, The C ~3, cubic, The B ~3, alternated cubic, The A ~3 cyclic group, or, There is a correspondence between all three families. Removing one mirror from C ~3 produces B ~3 and this allows multiple constructions of the same honeycombs. If cells are colored based on positions within each Wythoff construction. In addition there are 5 special honeycombs which dont have pure reflectional symmetry and are constructed from reflectional forms with elongation and gyration operations, the total unique honeycombs above are 18. The total unique honeycombs above are 10. Combining these counts,18 and 10 gives us the total 28 uniform honeycombs, the regular cubic honeycomb, represented by Schläfli symbol, offers seven unique derived uniform honeycombs via truncation operations. The reflectional symmetry is the affine Coxeter group, There are four index 2 subgroups that generate alternations, and +, with the first two generated repeated forms, and the last two are nonuniform. The B ~4, group offers 11 derived forms via truncation operations, There are 3 index 2 subgroups that generate alternations, and +
37.
Face (geometry)
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In solid geometry, a face is a flat surface that forms part of the boundary of a solid object, a three-dimensional solid bounded exclusively by flat faces is a polyhedron. In more technical treatments of the geometry of polyhedra and higher-dimensional polytopes, in elementary geometry, a face is a polygon on the boundary of a polyhedron. Other names for a polygonal face include side of a polyhedron, for example, any of the six squares that bound a cube is a face of the cube. Sometimes face is used to refer to the 2-dimensional features of a 4-polytope. With this meaning, the 4-dimensional tesseract has 24 square faces, some other polygons, which are not faces, are also important for polyhedra and tessellations. These include Petrie polygons, vertex figures and facets, any convex polyhedrons surface has Euler characteristic V − E + F =2, where V is the number of vertices, E is the number of edges, and F is the number of faces. This equation is known as Eulers polyhedron formula, thus the number of faces is 2 more than the excess of the number of edges over the number of vertices. For example, a cube has 12 edges and 8 vertices, in higher-dimensional geometry the faces of a polytope are features of all dimensions. A face of dimension k is called a k-face, for example, the polygonal faces of an ordinary polyhedron are 2-faces. In set theory, the set of faces of a polytope includes the polytope itself, for any n-polytope, −1 ≤ k ≤ n. For example, with meaning, the faces of a cube include the empty set, its vertices, edges and squares. Formally, a face of a polytope P is the intersection of P with any closed halfspace whose boundary is disjoint from the interior of P, from this definition it follows that the set of faces of a polytope includes the polytope itself and the empty set. In other areas of mathematics, such as the theories of abstract polytopes and star polytopes, abstract theory still requires that the set of faces include the polytope itself and the empty set. A cell is an element of a 4-dimensional polytope or 3-dimensional tessellation. Cells are facets for 4-polytopes and 3-honeycombs, examples, In higher-dimensional geometry, the facets of a n-polytope are the -faces of dimension one less than the polytope itself. A polytope is bounded by its facets, for example, The facets of a line segment are its 0-faces or vertices. The facets of a polygon are its 1-faces or edges, the facets of a polyhedron or plane tiling are its 2-faces. The facets of a 4D polytope or 3-honeycomb are its 3-faces, the facets of a 5D polytope or 4-honeycomb are its 4-faces
38.
5-polytope
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In five-dimensional geometry, a five-dimensional polytope or 5-polytope is a 5-dimensional polytope, bounded by facets. Each polyhedral cell being shared by exactly two 4-polytope facets, a 5-polytope is a closed five-dimensional figure with vertices, edges, faces, and cells, and 4-faces. A vertex is a point where five or more edges meet, an edge is a line segment where four or more faces meet, and a face is a polygon where three or more cells meet. A cell is a polyhedron, and a 4-face is a 4-polytope, furthermore, the following requirements must be met, Each cell must join exactly two 4-faces. Adjacent 4-faces are not in the same four-dimensional hyperplane, the figure is not a compound of other figures which meet the requirements. The topology of any given 5-polytope is defined by its Betti numbers, the value of the Euler characteristic used to characterise polyhedra does not generalize usefully to higher dimensions, whatever their underlying topology. This inadequacy of the Euler characteristic to distinguish between different topologies in higher dimensions led to the discovery of the more sophisticated Betti numbers. Similarly, the notion of orientability of a polyhedron is insufficient to characterise the surface twistings of toroidal polytopes, 5-polytopes may be classified based on properties like convexity and symmetry. Self-intersecting 5-polytopes are also known as star polytopes, from analogy with the shapes of the non-convex Kepler-Poinsot polyhedra. A uniform 5-polytope has a group under which all vertices are equivalent. The faces of a uniform polytope must be regular, a semi-regular 5-polytope contains two or more types of regular 4-polytope facets. There is only one figure, called a demipenteract. A regular 5-polytope has all identical regular 4-polytope facets, a prismatic 5-polytope is constructed by a Cartesian product of two lower-dimensional polytopes. A prismatic 5-polytope is uniform if its factors are uniform, the hypercube is prismatic, but is considered separately because it has symmetries other than those inherited from its factors. A 4-space tessellation is the division of four-dimensional Euclidean space into a grid of polychoral facets. Strictly speaking, tessellations are not polytopes as they do not bound a 5D volume, a uniform 4-space tessellation is one whose vertices are related by a space group and whose facets are uniform 4-polytopes. Regular 5-polytopes can be represented by the Schläfli symbol, with s polychoral facets around each face, the 5-demicube honeycomb, vertex figure is a rectified 5-orthoplex and facets are the 5-orthoplex and 5-demicube. Pyramidal 5-polytopes, or 5-pyramids, can be generated by a 4-polytope base in a 4-space hyperplane connected to a point off the hyperplane, the 5-simplex is the simplest example with a 4-simplex base