1.
16-cell honeycomb
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In four-dimensional Euclidean geometry, the 16-cell honeycomb is one of the three regular space-filling tessellations in Euclidean 4-space. The other two are its dual the 24-cell honeycomb, and the tesseractic honeycomb and this honeycomb is constructed from 16-cell facets, three around every face. It has a 24-cell vertex figure and this vertex arrangement or lattice is called the B4, D4, or F4 lattice. Vertices can be placed at all integer coordinates, such that the sum of the coordinates is even, the vertex arrangement of the 16-cell honeycomb is called the D4 lattice or F4 lattice. The D+4 lattice can be constructed by the union of two D4 lattices, and is identical to the honeycomb, ∪ = = This packing is only a lattice for even dimensions. The kissing number is 23 =8, the kissing number of the D*4 lattice is 24 and its Voronoi tessellation is a 24-cell honeycomb, containing all rectified 16-cells Voronoi cells, or. There are three different symmetry constructions of this tessellation, each symmetry can be represented by different arrangements of colored 16-cell facets. It is related to the regular hyperbolic 5-space 5-orthoplex honeycomb, with 5-orthoplex facets, the regular 4-polytope 24-cell, with octahedral cell, and cube, with square faces. S. M. Regular Polytopes, Dover edition, ISBN 0-486-61480-8 pp. 154–156, Partial truncation or alternation, represented by h prefix, coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 H. S. M, coxeter, Regular and Semi-Regular Polytopes III, George Olshevsky, Uniform Panoploid Tetracombs, Manuscript Klitzing, Richard. X3o3o4o3o - hext - O104 Conway JH, Sloane NJH

2.
16-cell
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In four-dimensional geometry, a 16-cell is a regular convex 4-polytope. It is one of the six regular convex 4-polytopes first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century and it is also called C16, hexadecachoron, or hexdecahedroid. It is a part of an family of polytopes, called cross-polytopes or orthoplexes. The dual polytope is the tesseract, conways name for a cross-polytope is orthoplex, for orthant complex. The 16-cell has 16 cells as the tesseract has 16 vertices and it is bounded by 16 cells, all of which are regular tetrahedra. It has 32 triangular faces,24 edges, and 8 vertices, the 24 edges bound 6 squares lying in the 6 coordinate planes. The eight vertices of the 16-cell are, all vertices are connected by edges except opposite pairs. The Schläfli symbol of the 16-cell is and its vertex figure is a regular octahedron. There are 8 tetrahedra,12 triangles, and 6 edges meeting at every vertex and its edge figure is a square. There are 4 tetrahedra and 4 triangles meeting at every edge, the 16-cell can be decomposed into two similar disjoint circular chains of eight tetrahedrons each, four edges long. Each chain, when stretched out straight, forms a Boerdijk–Coxeter helix and this decomposition can be seen in a 4-4 duoantiprism construction of the 16-cell, or, Schläfli symbol ⨂ or ss, symmetry, order 64. The 16-cell can be dissected into two octahedral pyramids, which share a new octahedron base through the 16-cell center, one can tessellate 4-dimensional Euclidean space by regular 16-cells. This is called the 16-cell honeycomb and has Schläfli symbol, hence, the 16-cell has a dihedral angle of 120°. The dual tessellation, 24-cell honeycomb, is made of by regular 24-cells, together with the tesseractic honeycomb, these are the only three regular tessellations of R4. Each 16-cell has 16 neighbors with which it shares a tetrahedron,24 neighbors with which it only an edge. Twenty-four 16-cells meet at any vertex in this tessellation. A 16-cell can constructed from two Boerdijk–Coxeter helixes of eight chained tetrahedra, each folded into a 4-dimensional ring, the 16 triangle faces can be seen in a 2D net within a triangular tiling, with 6 triangles around every vertex. The purple edges represent the Petrie polygon of the 16-cell, the cell-first parallel projection of the 16-cell into 3-space has a cubical envelope

3.
Tetrahedron
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In geometry, a tetrahedron, also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra, the tetrahedron is the three-dimensional case of the more general concept of a Euclidean simplex. The tetrahedron is one kind of pyramid, which is a polyhedron with a polygon base. In the case of a tetrahedron the base is a triangle, like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper. For any tetrahedron there exists a sphere on which all four vertices lie, a regular tetrahedron is one in which all four faces are equilateral triangles. It is one of the five regular Platonic solids, which have known since antiquity. In a regular tetrahedron, not only are all its faces the same size and shape, regular tetrahedra alone do not tessellate, but if alternated with regular octahedra they form the alternated cubic honeycomb, which is a tessellation. The regular tetrahedron is self-dual, which means that its dual is another regular tetrahedron, the compound figure comprising two such dual tetrahedra form a stellated octahedron or stella octangula. This form has Coxeter diagram and Schläfli symbol h, the tetrahedron in this case has edge length 2√2. Inverting these coordinates generates the dual tetrahedron, and the together form the stellated octahedron. In other words, if C is the centroid of the base and this follows from the fact that the medians of a triangle intersect at its centroid, and this point divides each of them in two segments, one of which is twice as long as the other. The vertices of a cube can be grouped into two groups of four, each forming a regular tetrahedron, the symmetries of a regular tetrahedron correspond to half of those of a cube, those that map the tetrahedra to themselves, and not to each other. The tetrahedron is the only Platonic solid that is not mapped to itself by point inversion, the regular tetrahedron has 24 isometries, forming the symmetry group Td, isomorphic to the symmetric group, S4. The first corresponds to the A2 Coxeter plane, the two skew perpendicular opposite edges of a regular tetrahedron define a set of parallel planes. When one of these intersects the tetrahedron the resulting cross section is a rectangle. When the intersecting plane is one of the edges the rectangle is long. When halfway between the two edges the intersection is a square, the aspect ratio of the rectangle reverses as you pass this halfway point. For the midpoint square intersection the resulting boundary line traverses every face of the tetrahedron similarly, if the tetrahedron is bisected on this plane, both halves become wedges

4.
Triangle
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A triangle is a polygon with three edges and three vertices. It is one of the shapes in geometry. A triangle with vertices A, B, and C is denoted △ A B C, in Euclidean geometry any three points, when non-collinear, determine a unique triangle and a unique plane. This article is about triangles in Euclidean geometry except where otherwise noted, triangles can be classified according to the lengths of their sides, An equilateral triangle has all sides the same length. An equilateral triangle is also a polygon with all angles measuring 60°. An isosceles triangle has two sides of equal length, some mathematicians define an isosceles triangle to have exactly two equal sides, whereas others define an isosceles triangle as one with at least two equal sides. The latter definition would make all equilateral triangles isosceles triangles, the 45–45–90 right triangle, which appears in the tetrakis square tiling, is isosceles. A scalene triangle has all its sides of different lengths, equivalently, it has all angles of different measure. Hatch marks, also called tick marks, are used in diagrams of triangles, a side can be marked with a pattern of ticks, short line segments in the form of tally marks, two sides have equal lengths if they are both marked with the same pattern. In a triangle, the pattern is no more than 3 ticks. Similarly, patterns of 1,2, or 3 concentric arcs inside the angles are used to indicate equal angles, triangles can also be classified according to their internal angles, measured here in degrees. A right triangle has one of its interior angles measuring 90°, the side opposite to the right angle is the hypotenuse, the longest side of the triangle. The other two sides are called the legs or catheti of the triangle, special right triangles are right triangles with additional properties that make calculations involving them easier. One of the two most famous is the 3–4–5 right triangle, where 32 +42 =52, in this situation,3,4, and 5 are a Pythagorean triple. The other one is a triangle that has 2 angles that each measure 45 degrees. Triangles that do not have an angle measuring 90° are called oblique triangles, a triangle with all interior angles measuring less than 90° is an acute triangle or acute-angled triangle. If c is the length of the longest side, then a2 + b2 > c2, a triangle with one interior angle measuring more than 90° is an obtuse triangle or obtuse-angled triangle. If c is the length of the longest side, then a2 + b2 < c2, a triangle with an interior angle of 180° is degenerate

5.
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

6.
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

7.
24-cell honeycomb
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In four-dimensional Euclidean geometry, the 24-cell honeycomb, or icositetrachoric honeycomb is a regular space-filling tessellation of 4-dimensional Euclidean space by regular 24-cells. It can be represented by Schläfli symbol, the dual tessellation by regular 16-cell honeycomb has Schläfli symbol. Together with the tesseractic honeycomb these are the regular tessellations of Euclidean 4-space. If a 3-sphere is inscribed in each hypercell of this tessellation, the packing density of this arrangement is π216 ≅0.61685. The 24-cell honeycomb can be constructed as the Voronoi tessellation of the D4 or F4 root lattice, each 24-cell is then centered at a D4 lattice point, i. e. one of. These points can also be described as Hurwitz quaternions with even square norm, the vertices of the honeycomb lie at the deep holes of the D4 lattice. These are the Hurwitz quaternions with odd square norm and it can be constructed as a birectified tesseractic honeycomb, by taking a tesseractic honeycomb and placing vertices at the centers of all the square faces. The 24-cell facets exist between these vertices as rectified 16-cells, if the coordinates of the tesseractic honeycomb are integers, the birectified tesseractic honeycomb vertices can be placed at all permutations of half-unit shifts in two of the four dimensions, thus. Each 24-cell in the 24-cell honeycomb has 24 neighboring 24-cells, with each neighbor it shares exactly one octahedral cell. It has 24 more neighbors such that each of these it shares a single vertex. It has no neighbors with which it only an edge or only a face. The vertex figure of the 24-cell honeycomb is a tesseract, so there are 16 edges,32 triangles,24 octahedra, and 8 24-cells meeting at every vertex. The edge figure is a tetrahedron, so there are 4 triangles,6 octahedra, finally, the face figure is a triangle, so there are 3 octahedra and 3 24-cells meeting at every face. One way to visualize a 4-dimensional figure is to consider various 3-dimensional cross-sections and that is, the intersection of various hyperplanes with the figure in question. Applying this technique to the 24-cell honeycomb gives rise to various 3-dimensional honeycombs with varying degrees of regularity, a vertex-first cross-section uses some hyperplane orthogonal to a line joining opposite vertices of one of the 24-cells. For instance, one could take any of the hyperplanes in the coordinate system given above. The cross-section of by one of these gives a rhombic dodecahedral honeycomb. Each of the rhombic dodecahedra corresponds to a maximal cross-section of one of the 24-cells intersecting the hyperplane, accordingly, the rhombic dodecahedral honeycomb is the Voronoi tessellation of the D3 root lattice

8.
Coxeter group
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In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups, however, not all Coxeter groups are finite, and not all can be described in terms of symmetries and Euclidean reflections. Coxeter groups were introduced as abstractions of reflection groups, and finite Coxeter groups were classified in 1935, Coxeter groups find applications in many areas of mathematics. Examples of finite Coxeter groups include the groups of regular polytopes. The condition m i j = ∞ means no relation of the form m should be imposed, the pair where W is a Coxeter group with generators S = is called a Coxeter system. Note that in general S is not uniquely determined by W, for example, the Coxeter groups of type B3 and A1 × A3 are isomorphic but the Coxeter systems are not equivalent. A number of conclusions can be drawn immediately from the above definition, the relation m i i =1 means that 1 =2 =1 for all i, as such the generators are involutions. If m i j =2, then the r i and r j commute. This follows by observing that x x = y y =1, in order to avoid redundancy among the relations, it is necessary to assume that m i j = m j i. This follows by observing that y y =1, together with m =1 implies that m = m y y = y m y = y y =1. Alternatively, k and k are elements, as y k y −1 = k y y −1 = k. The Coxeter matrix is the n × n, symmetric matrix with entries m i j, indeed, every symmetric matrix with positive integer and ∞ entries and with 1s on the diagonal such that all nondiagonal entries are greater than 1 serves to define a Coxeter group. The Coxeter matrix can be encoded by a Coxeter diagram. The vertices of the graph are labelled by generator subscripts, vertices i and j are adjacent if and only if m i j ≥3. An edge is labelled with the value of m i j whenever the value is 4 or greater, in particular, two generators commute if and only if they are not connected by an edge. Furthermore, if a Coxeter graph has two or more connected components, the group is the direct product of the groups associated to the individual components. Thus the disjoint union of Coxeter graphs yields a product of Coxeter groups. The Coxeter matrix, M i j, is related to the n × n Schläfli matrix C with entries C i j = −2 cos , but the elements are modified, being proportional to the dot product of the pairwise generators

9.
Geometry
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Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer, Geometry arose independently in a number of early cultures as a practical way for dealing with lengths, areas, and volumes. Geometry began to see elements of mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into a form by Euclid, whose treatment, Euclids Elements. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC, islamic scientists preserved Greek ideas and expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid footing by mathematicians such as René Descartes. Since then, and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, while geometry has evolved significantly throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, lines, planes, surfaces, angles, contemporary geometry has many subfields, Euclidean geometry is geometry in its classical sense. The mandatory educational curriculum of the majority of nations includes the study of points, lines, planes, angles, triangles, congruence, similarity, solid figures, circles, Euclidean geometry also has applications in computer science, crystallography, and various branches of modern mathematics. Differential geometry uses techniques of calculus and linear algebra to problems in geometry. It has applications in physics, including in general relativity, topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this often means dealing with large-scale properties of spaces, convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues, often using techniques of real analysis. It has close connections to convex analysis, optimization and functional analysis, algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques. It has applications in areas, including cryptography and string theory. Discrete geometry is concerned mainly with questions of relative position of simple objects, such as points. It shares many methods and principles with combinatorics, Geometry has applications to many fields, including art, architecture, physics, as well as to other branches of mathematics. The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia, the earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets such as Plimpton 322. For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, later clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiters position and motion within time-velocity space

10.
Hyperbolic space
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In mathematics, hyperbolic space is a homogeneous space that has a constant negative curvature, where in this case the curvature is the sectional curvature. When embedded to a Euclidean space, every point of a space is a saddle point. Hyperbolic n-space, denoted Hn, is the symmetric, simply connected. Hyperbolic space is a space exhibiting hyperbolic geometry and it is the negative-curvature analogue of the n-sphere. Although hyperbolic space Hn is diffeomorphic to Rn, its negative-curvature metric gives it very different geometric properties, hyperbolic 2-space, H2, is also called the hyperbolic plane. Instead, the postulate is replaced by the following alternative, Given any line L and point P not on L. It is then a theorem that there are many such lines through P. This axiom still does not uniquely characterize the hyperbolic plane up to isometry, there is a constant, the curvature K <0. However, it does uniquely characterize it up to homothety, meaning up to bijections which only change the notion of distance by an overall constant, by choosing an appropriate length scale, one can thus assume, without loss of generality, that K = −1. Models of hyperbolic spaces that can be embedded in a flat spaces may be constructed, in particular, the existence of model spaces implies that the parallel postulate is logically independent of the other axioms of Euclidean geometry. There are several important models of space, the Klein model, the hyperboloid model, the Poincaré ball model. These all model the same geometry in the sense that any two of them can be related by a transformation that preserves all the properties of the space. The hyperboloid model realizes hyperbolic space as a hyperboloid in Rn+1 =, the hyperboloid is the locus Hn of points whose coordinates satisfy x 02 − x 12 − ⋯ − x n 2 =1, x 0 >0. In this model a line is the curve formed by the intersection of Hn with a plane through the origin in Rn+1, the hyperboloid model is closely related to the geometry of Minkowski space. The space Rn+1, equipped with the bilinear form B, is an -dimensional Minkowski space Rn,1, one can associate a distance on the hyperboloid model by defining the distance between two points x and y on H to be d = arcosh B. This function satisfies the axioms of a metric space and it is preserved by the action of the Lorentz group on Rn,1. Hence the Lorentz group acts as a group preserving isometry on Hn. An alternative model of geometry is on a certain domain in projective space

11.
Regular polytope
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In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or j-faces — cells, faces and so on — are also transitive on the symmetries of the polytope, Regular polytopes are the generalized analog in any number of dimensions of regular polygons and regular polyhedra. The strong symmetry of the regular polytopes gives them an aesthetic quality that interests both non-mathematicians and mathematicians, classically, a regular polytope in n dimensions may be defined as having regular facets and regular vertex figures. These two conditions are sufficient to ensure that all faces are alike and all vertices are alike, note, however, that this definition does not work for abstract polytopes. A regular polytope can be represented by a Schläfli symbol of the form, with regular facets as, Regular polytopes are classified primarily according to their dimensionality. They can be classified according to symmetry. For example, the cube and the regular octahedron share the same symmetry, indeed, symmetry groups are sometimes named after regular polytopes, for example the tetrahedral and icosahedral symmetries. Three special classes of regular polytope exist in every dimensionality, Regular simplex Measure polytope Cross polytope In two dimensions there are many regular polygons. In three and four dimensions there are more regular polyhedra and 4-polytopes besides these three. In five dimensions and above, these are the only ones, see also the list of regular polytopes. The idea of a polytope is sometimes generalised to include related kinds of geometrical object, some of these have regular examples, as discussed in the section on historical discovery below. A concise symbolic representation for regular polytopes was developed by Ludwig Schläfli in the 19th Century, the notation is best explained by adding one dimension at a time. A convex regular polygon having n sides is denoted by, so an equilateral triangle is, a square, and so on indefinitely. A regular star polygon which winds m times around its centre is denoted by the fractional value, a regular polyhedron having faces with p faces joining around a vertex is denoted by. The nine regular polyhedra are and. is the figure of the polyhedron. A regular 4-polytope having cells with q cells joining around an edge is denoted by, the vertex figure of the 4-polytope is a. A five-dimensional regular polytope is an, the dual of a regular polytope is also a regular polytope. The Schläfli symbol for the dual polytope is just the original written backwards, is self-dual, is dual to, to

12.
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

13.
Honeycomb (geometry)
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In geometry, a honeycomb is a space filling or close packing 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. Its dimension can be clarified as n-honeycomb for a honeycomb of n-dimensional space, honeycombs are usually constructed in ordinary Euclidean space. They may also be constructed in non-Euclidean spaces, such as hyperbolic honeycombs, any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space. There are infinitely many honeycombs, which have only been partially classified, the more regular ones have attracted the most interest, while a rich and varied assortment of others continue to be discovered. The simplest honeycombs to build are formed from stacked layers or slabs of prisms based on some tessellations of the plane, in particular, for every parallelepiped, copies can fill space, with the cubic honeycomb being special because it is the only regular honeycomb in ordinary space. Another interesting family is the Hill tetrahedra and their generalizations, which can tile the space. A 3-dimensional uniform honeycomb is a honeycomb in 3-space composed of polyhedral cells. There are 28 convex examples in Euclidean 3-space, also called the Archimedean honeycombs, a honeycomb is called regular if the group of isometries preserving the tiling acts transitively on flags, where a flag is a vertex lying on an edge lying on a face lying on a cell. Every regular honeycomb is automatically uniform, however, there is just one regular honeycomb in Euclidean 3-space, the cubic honeycomb. An infinite number of unique honeycombs can be created by order of patterns of repeating these slab layers. A honeycomb having all cells identical within its symmetries is said to be cell-transitive or isochoric, in the 3-dimensional euclidean space, a cell of such a honeycomb is said to be a space-filling polyhedron. A necessary condition for a polyhedron to be a space-filling polyhedron is that its Dehn invariant must be zero, five space-filling polyhedra can tessellate 3-dimensional euclidean space using translations only. They are called parallelohedra, Cubic honeycomb Hexagonal prismatic honeycomb Rhombic dodecahedral honeycomb Elongated dodecahedral honeycomb, bitruncated cubic honeycomb Other known examples of space-filling polyhedra include, The Triangular prismatic honeycomb. The gyrated triangular prismatic honeycomb The triakis truncated tetrahedral honeycomb, the Voronoi cells of the carbon atoms in diamond are this shape. The trapezo-rhombic dodecahedral honeycomb Isohedral tilings, sometimes, two or more different polyhedra may be combined to fill space. Two classes can be distinguished, Non-convex cells which pack without overlapping and these include a packing of the small stellated rhombic dodecahedron, as in the Yoshimoto Cube. Overlapping of cells whose positive and negative densities cancel out to form a uniformly dense continuum, in 3-dimensional hyperbolic space, the dihedral angle of a polyhedron depends on its size

14.
Dual polyhedron
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Such dual figures remain combinatorial or abstract polyhedra, but not all are also geometric polyhedra. Starting with any given polyhedron, the dual of its dual is the original polyhedron, duality preserves the symmetries of a polyhedron. Therefore, for classes of polyhedra defined by their symmetries. Thus, the regular polyhedra – the Platonic solids and Kepler-Poinsot polyhedra – form dual pairs, the dual of an isogonal polyhedron, having equivalent vertices, is one which is isohedral, having equivalent faces. The dual of a polyhedron is also isotoxal. Duality is closely related to reciprocity or polarity, a transformation that. There are many kinds of duality, the kinds most relevant to elementary polyhedra are polar reciprocity and topological or abstract duality. The duality of polyhedra is often defined in terms of polar reciprocation about a concentric sphere. In coordinates, for reciprocation about the sphere x 2 + y 2 + z 2 = r 2, the vertex is associated with the plane x 0 x + y 0 y + z 0 z = r 2. The vertices of the dual are the reciprocal to the face planes of the original. Also, any two adjacent vertices define an edge, and these will reciprocate to two adjacent faces which intersect to define an edge of the dual and this dual pair of edges are always orthogonal to each other. If r 0 is the radius of the sphere, and r 1 and r 2 respectively the distances from its centre to the pole and its polar, then, r 1. R2 = r 02 For the more symmetrical polyhedra having an obvious centroid, it is common to make the polyhedron and sphere concentric, the choice of center for the sphere is sufficient to define the dual up to similarity. If multiple symmetry axes are present, they will intersect at a single point. Failing that, a sphere, inscribed sphere, or midsphere is commonly used. If a polyhedron in Euclidean space has an element passing through the center of the sphere, since Euclidean space never reaches infinity, the projective equivalent, called extended Euclidean space, may be formed by adding the required plane at infinity. Some theorists prefer to stick to Euclidean space and say there is no dual. Meanwhile, Wenninger found a way to represent these infinite duals, the concept of duality here is closely related to the duality in projective geometry, where lines and edges are interchanged

15.
Harold Scott MacDonald Coxeter
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Harold Scott MacDonald Donald Coxeter, FRS, FRSC, CC was a British-born Canadian geometer. Coxeter is regarded as one of the greatest geometers of the 20th century and he was born in London but spent most of his adult life in Canada. He was always called Donald, from his third name MacDonald, in his youth, Coxeter composed music and was an accomplished pianist at the age of 10. He felt that mathematics and music were intimately related, outlining his ideas in a 1962 article on Mathematics and he worked for 60 years at the University of Toronto and published twelve books. He was most noted for his work on regular polytopes and higher-dimensional geometries and he was a champion of the classical approach to geometry, in a period when the tendency was to approach geometry more and more via algebra. Coxeter went up to Trinity College, Cambridge in 1926 to read mathematics, there he earned his BA in 1928, and his doctorate in 1931. In 1932 he went to Princeton University for a year as a Rockefeller Fellow, where he worked with Hermann Weyl, Oswald Veblen, returning to Trinity for a year, he attended Ludwig Wittgensteins seminars on the philosophy of mathematics. In 1934 he spent a year at Princeton as a Procter Fellow. In 1936 Coxeter moved to the University of Toronto, flather, and John Flinders Petrie published The Fifty-Nine Icosahedra with University of Toronto Press. In 1940 Coxeter edited the eleventh edition of Mathematical Recreations and Essays and he was elevated to professor in 1948. Coxeter was elected a Fellow of the Royal Society of Canada in 1948 and he also inspired some of the innovations of Buckminster Fuller. Coxeter, M. S. Longuet-Higgins and J. C. P. Miller were the first to publish the full list of uniform polyhedra, since 1978, the Canadian Mathematical Society have awarded the Coxeter–James Prize in his honor. He was made a Fellow of the Royal Society in 1950, in 1990, he became a Foreign Member of the American Academy of Arts and Sciences and in 1997 was made a Companion of the Order of Canada. In 1973 he got the Jeffery–Williams Prize,1940, Regular and Semi-Regular Polytopes I, Mathematische Zeitschrift 46, 380-407, MR2,10 doi,10. 1007/BF011814491942, Non-Euclidean Geometry, University of Toronto Press, MAA. 1954, Uniform Polyhedra, Philosophical Transactions of the Royal Society A246, arthur Sherk, Peter McMullen, Anthony C. Thompson and Asia Ivić Weiss, editors, Kaleidoscopes — Selected Writings of H. S. M. John Wiley and Sons ISBN 0-471-01003-01999, The Beauty of Geometry, Twelve Essays, Dover Publications, LCCN 99-35678, ISBN 0-486-40919-8 Davis, Chandler, Ellers, Erich W, the Coxeter Legacy, Reflections and Projections. King of Infinite Space, Donald Coxeter, the Man Who Saved Geometry, www. donaldcoxeter. com www. math. yorku. ca/dcoxeter webpages dedicated to him Jarons World, Shapes in Other Dimensions, Discover mag. Apr 2007 The Mathematics in the Art of M. C, escher video of a lecture by H. S. M

16.
Regular Polytopes (book)
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Regular Polytopes is a mathematical geometry book written by Canadian mathematician Harold Scott MacDonald Coxeter. Originally published in 1947, the book was updated and republished in 1963 and 1973, the book is a comprehensive survey of the geometry of regular polytopes, the generalisation of regular polygons and regular polyhedra to higher dimensions. Originating with an essay entitled Dimensional Analogy written in 1923, the first edition of the book took Coxeter twenty-four years to complete, regular Polytopes is a standard reference work on regular polygons, polyhedra and their higher dimensional analogues. It is unusual in the breadth of its coverage, its combination of mathematical rigour with geometric insight, Coxeter starts by introducing two-dimensional polygons and three-dimensional polyhedra. He then gives a rigorous definition of regularity and uses it to show that there are no other convex regular polyhedra apart from the five Platonic solids. The concept of regularity is extended to non-convex shapes such as star polygons and star polyhedra, to tessellations and honeycombs, Coxeter introduces and uses the groups generated by reflections that became known as Coxeter groups. The book combines algebraic rigour with clear explanations, many of which are illustrated with diagrams, the black and white plates in the book show solid models of three-dimensional polyhedra, and wire-frame models of projections of some higher-dimensional polytopes. At the end of each chapter Coxeter includes an Historical remarks section which provides a perspective of the development of the subject. The first two have been expounded by Sommerville and Neville, and we shall presuppose some familiarity with such treatises. Concerning the third, Poincaré wrote, A man who pursues it, will end up holding on to the fourth dimension. ”The original 1948 edition received a more complete review by M. Goldberg in MR0027148

17.
International Standard Book Number
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The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each edition and variation of a book, for example, an e-book, a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, the method of assigning an ISBN is nation-based and varies from country to country, often depending on how large the publishing industry is within a country. The initial ISBN configuration of recognition was generated in 1967 based upon the 9-digit Standard Book Numbering created in 1966, the 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108. Occasionally, a book may appear without a printed ISBN if it is printed privately or the author does not follow the usual ISBN procedure, however, this can be rectified later. Another identifier, the International Standard Serial Number, identifies periodical publications such as magazines, the ISBN configuration of recognition was generated in 1967 in the United Kingdom by David Whitaker and in 1968 in the US by Emery Koltay. The 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108, the United Kingdom continued to use the 9-digit SBN code until 1974. The ISO on-line facility only refers back to 1978, an SBN may be converted to an ISBN by prefixing the digit 0. For example, the edition of Mr. J. G. Reeder Returns, published by Hodder in 1965, has SBN340013818 -340 indicating the publisher,01381 their serial number. This can be converted to ISBN 0-340-01381-8, the check digit does not need to be re-calculated, since 1 January 2007, ISBNs have contained 13 digits, a format that is compatible with Bookland European Article Number EAN-13s. An ISBN is assigned to each edition and variation of a book, for example, an ebook, a paperback, and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, a 13-digit ISBN can be separated into its parts, and when this is done it is customary to separate the parts with hyphens or spaces. Separating the parts of a 10-digit ISBN is also done with either hyphens or spaces, figuring out how to correctly separate a given ISBN number is complicated, because most of the parts do not use a fixed number of digits. ISBN issuance is country-specific, in that ISBNs are issued by the ISBN registration agency that is responsible for country or territory regardless of the publication language. Some ISBN registration agencies are based in national libraries or within ministries of culture, in other cases, the ISBN registration service is provided by organisations such as bibliographic data providers that are not government funded. In Canada, ISBNs are issued at no cost with the purpose of encouraging Canadian culture. In the United Kingdom, United States, and some countries, where the service is provided by non-government-funded organisations. Australia, ISBNs are issued by the library services agency Thorpe-Bowker