1.
Hypercubic honeycomb
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In geometry, a hypercubic honeycomb is a family of regular honeycombs in n-dimensions with the Schläfli symbols and containing the symmetry of Coxeter group Rn for n>=3. The tessellation is constructed from 4 n-hypercubes per ridge, the vertex figure is a cross-polytope. Coxeter named this family as δn+1 for an n-dimensional honeycomb, there are two general forms of the hypercube honeycombs, the regular form with identical hypercubic facets and one semiregular, with alternating hypercube facets, like a checkerboard. A third form is generated by an operation applied to the regular form. The orthotopic honeycombs are a family topologically equivalent to the cubic honeycombs but with lower symmetry, the facets are hyperrectangles, also called orthotopes, in 2 and 3 dimensions the orthotopes are rectangles and cuboids respectively. Alternated hypercubic honeycomb Quarter hypercubic honeycomb Simplectic honeycomb Truncated simplectic honeycomb Omnitruncated simplectic honeycomb Coxeter, Regular Polytopes, Dover edition, ISBN 0-486-61480-8 pp. 122–123,1973. Pp. 154–156, Partial truncation or alternation, represented by h prefix, h=, h=, h= p.296, Table II, Regular honeycombs, δn+1
2.
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
3.
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
4.
Octahedron
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In geometry, an octahedron is a polyhedron with eight faces, twelve edges, and six vertices. A regular octahedron is a Platonic solid composed of eight equilateral triangles, a regular octahedron is the dual polyhedron of a cube. It is a square bipyramid in any of three orthogonal orientations and it is also a triangular antiprism in any of four orientations. An octahedron is the case of the more general concept of a cross polytope. A regular octahedron is a 3-ball in the Manhattan metric, the second and third correspond to the B2 and A2 Coxeter planes. The octahedron 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. An octahedron with edge length √2 can be placed with its center at the origin and its vertices on the coordinate axes, the Cartesian coordinates of the vertices are then. In an x–y–z Cartesian coordinate system, the octahedron with center coordinates, additionally the inertia tensor of the stretched octahedron is I =. These reduce to the equations for the regular octahedron when x m = y m = z m = a 22, the interior of the compound of two dual tetrahedra is an octahedron, and this compound, called the stella octangula, is its first and only stellation. Correspondingly, an octahedron is the result of cutting off from a regular tetrahedron. One can also divide the edges of an octahedron in the ratio of the mean to define the vertices of an icosahedron. There are five octahedra that define any given icosahedron in this fashion, octahedra and tetrahedra can be alternated to form a vertex, edge, and face-uniform tessellation of space, called the octet truss by Buckminster Fuller. This is the only such tiling save the regular tessellation of cubes, another is a tessellation of octahedra and cuboctahedra. The octahedron is unique among the Platonic solids in having a number of faces meeting at each vertex. Consequently, it is the member of that group to possess mirror planes that do not pass through any of the faces. Using the standard nomenclature for Johnson solids, an octahedron would be called a square bipyramid, truncation of two opposite vertices results in a square bifrustum. The octahedron is 4-connected, meaning that it takes the removal of four vertices to disconnect the remaining vertices and it is one of only four 4-connected simplicial well-covered polyhedra, meaning that all of the maximal independent sets of its vertices have the same size
5.
Space group
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In mathematics, physics and chemistry, a space group is the symmetry group of a configuration in space, usually in three dimensions. In three dimensions, there are 219 distinct types, or 230 if chiral copies are considered distinct, Space groups are also studied in dimensions other than 3 where they are sometimes called Bieberbach groups, and are discrete cocompact groups of isometries of an oriented Euclidean space. In crystallography, space groups are called the crystallographic or Fedorov groups. A definitive source regarding 3-dimensional space groups is the International Tables for Crystallography, in 1879 Leonhard Sohncke listed the 65 space groups whose elements preserve the orientation. More accurately, he listed 66 groups, but Fedorov and Schönflies both noticed that two of them were really the same, the space groups in 3 dimensions were first enumerated by Fedorov, and shortly afterwards were independently enumerated by Schönflies. The correct list of 230 space groups was found by 1892 during correspondence between Fedorov and Schönflies, burckhardt describes the history of the discovery of the space groups in detail. The space groups in three dimensions are made from combinations of the 32 crystallographic point groups with the 14 Bravais lattices, the combination of all these symmetry operations results in a total of 230 different space groups describing all possible crystal symmetries. The elements of the space group fixing a point of space are rotations, reflections, the identity element, the translations form a normal abelian subgroup of rank 3, called the Bravais lattice. There are 14 possible types of Bravais lattice, the quotient of the space group by the Bravais lattice is a finite group which is one of the 32 possible point groups. Translation is defined as the moves from one point to another point. A glide plane is a reflection in a plane, followed by a parallel with that plane. This is noted by a, b or c, depending on which axis the glide is along. There is also the n glide, which is a glide along the half of a diagonal of a face, and the d glide, the latter is called the diamond glide plane as it features in the diamond structure. In 17 space groups, due to the centering of the cell, the glides occur in two directions simultaneously, i. e. the same glide plane can be called b or c, a or b. For example, group Abm2 could be also called Acm2, group Ccca could be called Cccb, in 1992, it was suggested to use symbol e for such planes. The symbols for five groups have been modified, A screw axis is a rotation about an axis. These are noted by a number, n, to describe the degree of rotation, the degree of translation is then added as a subscript showing how far along the axis the translation is, as a portion of the parallel lattice vector. So,21 is a rotation followed by a translation of 1/2 of the lattice vector
6.
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
7.
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
8.
Isogonal figure
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In geometry, a polytope is isogonal or vertex-transitive if, loosely speaking, all its vertices are equivalent. That implies that each vertex is surrounded by the kinds of face in the same or reverse order. Technically, we say that for any two vertices there exists a symmetry of the polytope mapping the first isometrically onto the second. Other ways of saying this are that the group of automorphisms of the polytope is transitive on its vertices, all vertices of a finite n-dimensional isogonal figure exist on an -sphere. The term isogonal has long used for polyhedra. Vertex-transitive is a synonym borrowed from modern ideas such as symmetry groups, all regular polygons, apeirogons and regular star polygons are isogonal. The dual of a polygon is an isotoxal polygon. Some even-sided polygons and apeirogons which alternate two edge lengths, for example a rectangle, are isogonal, all planar isogonal 2n-gons have dihedral symmetry with reflection lines across the mid-edge points. An isogonal polyhedron and 2D tiling has a kind of vertex. An isogonal polyhedron with all faces is also a uniform polyhedron. Geometrically distorted variations of uniform polyhedra and tilings can also be given the vertex configuration, isogonal polyhedra and 2D tilings may be further classified, Regular if it is also isohedral and isotoxal, this implies that every face is the same kind of regular polygon. Quasi-regular if it is also isotoxal but not isohedral, semi-regular if every face is a regular polygon but it is not isohedral or isotoxal. Uniform if every face is a polygon, i. e. it is regular, quasiregular or semi-regular. Noble if it is also isohedral and these definitions can be extended to higher-dimensional polytopes and tessellations. Most generally, all uniform polytopes are isogonal, for example, the dual of an isogonal polytope is called an isotope which is transitive on its facets. A polytope or tiling may be called if its vertices form k transitivity classes. A more restrictive term, k-uniform is defined as a figure constructed only from regular polygons. They can be represented visually with colors by different uniform colorings, edge-transitive Face-transitive Peter R. Cromwell, Polyhedra, Cambridge University Press 1997, ISBN 0-521-55432-2, p.369 Transitivity Grünbaum, Branko, Shephard, G. C
9.
Quasiregular polyhedron
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In geometry, a quasiregular polyhedron is a semiregular polyhedron that has exactly two kinds of regular faces, which alternate around each vertex. They are edge-transitive and hence a step closer to regular polyhedra than the semiregular which are merely vertex-transitive, there are only two convex quasiregular polyhedra, the cuboctahedron and the icosidodecahedron. These forms representing a pair of a figure and its dual can be given a vertical Schläfli symbol or r to represent their containing the faces of both the regular and dual regular. A quasiregular polyhedron with this symbol will have a vertex configuration p. q. p. q, more generally, a quasiregular figure can have a vertex configuration r, representing r instances of the faces around the vertex. Tilings of the plane can also be quasiregular, specifically the trihexagonal tiling, other quasiregular tilings exist on the hyperbolic plane, like the triheptagonal tiling,2. Or more generally,2, with 1/p+1/q<1/2, a regular figure with Schläfli symbol can be quasiregular, with vertex configuration q/2, if q is even. The octahedron can be considered quasiregular as a tetratetrahedron,2, similarly the square tiling 2 can be considered quasiregular, colored as a checkerboard. Also the triangular tiling can have alternately colored triangle faces,3, Coxeter defines a quasiregular polyhedron as one having a Wythoff symbol in the form p | q r, and it is regular if q=2 or q=r. In this form it is known as the tetratetrahedron. The remaining convex polyhedra have an odd number of faces at each vertex so cannot be colored in a way that preserves edge transitivity. It has Coxeter-Dynkin diagram Each of these forms the core of a dual pair of regular polyhedra. The names of two of these clues to the associated dual pair, respectively the cube + octahedron. The octahedron is the core of a pair of tetrahedra. Each of these quasiregular polyhedra can be constructed by an operation on either regular parent, truncating the edges fully. This sequence continues as the tiling, vertex figure 2 - a quasiregular tiling based on the triangular tiling. But not everybody uses this terminology and these duals are transitive on their edges and faces, they are the edge-transitive Catalan solids. The convex ones are, in corresponding order as above, The rhombic dodecahedron, the rhombic triacontahedron, with two types of alternating vertices,20 with three rhombic faces, and 12 with five rhombic faces. In addition, by duality with the octahedron, the cube and their face configuration are of the form V3. n.3. n, and Coxeter-Dynkin diagram These three quasiregular duals are also characterised by having rhombic faces
10.
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
11.
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
12.
Three-dimensional space
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Three-dimensional space is a geometric setting in which three values are required to determine the position of an element. This is the meaning of the term dimension. In physics and mathematics, a sequence of n numbers can be understood as a location in n-dimensional space, when n =3, the set of all such locations is called three-dimensional Euclidean space. It is commonly represented by the symbol ℝ3 and this serves as a three-parameter model of the physical universe in which all known matter exists. However, this space is one example of a large variety of spaces in three dimensions called 3-manifolds. Furthermore, in case, these three values can be labeled by any combination of three chosen from the terms width, height, depth, and breadth. In mathematics, analytic geometry describes every point in space by means of three coordinates. Three coordinate axes are given, each perpendicular to the two at the origin, the point at which they cross. They are usually labeled x, y, and z, below are images of the above-mentioned systems. Two distinct points determine a line. Three distinct points are either collinear or determine a unique plane, four distinct points can either be collinear, coplanar or determine the entire space. Two distinct lines can intersect, be parallel or be skew. Two parallel lines, or two intersecting lines, lie in a plane, so skew lines are lines that do not meet. Two distinct planes can either meet in a line or are parallel. Three distinct planes, no pair of which are parallel, can meet in a common line. In the last case, the three lines of intersection of each pair of planes are mutually parallel, a line can lie in a given plane, intersect that plane in a unique point or be parallel to the plane. In the last case, there will be lines in the plane that are parallel to the given line, a hyperplane is a subspace of one dimension less than the dimension of the full space. The hyperplanes of a space are the two-dimensional subspaces, that is
13.
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
14.
Self-dual tessellation
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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
15.
John Horton Conway
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John Horton Conway FRS is an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He has also contributed to many branches of mathematics, notably the invention of the cellular automaton called the Game of Life. Conway is currently Professor Emeritus of Mathematics at Princeton University in New Jersey, Conway was born in Liverpool, the son of Cyril Horton Conway and Agnes Boyce. He became interested in mathematics at an early age, his mother has recalled that he could recite the powers of two when he was four years old. By the age of eleven his ambition was to become a mathematician, after leaving secondary school, Conway entered Gonville and Caius College, Cambridge to study mathematics. Conway, who was a terribly introverted adolescent in school, interpreted his admission to Cambridge as an opportunity to transform himself into a new person and he was awarded his Bachelor of Arts degree in 1959 and began to undertake research in number theory supervised by Harold Davenport. Having solved the problem posed by Davenport on writing numbers as the sums of fifth powers. It appears that his interest in games began during his years studying the Cambridge Mathematical Tripos and he was awarded his doctorate in 1964 and was appointed as College Fellow and Lecturer in Mathematics at the University of Cambridge. After leaving Cambridge in 1986, he took up the appointment to the John von Neumann Chair of Mathematics at Princeton University, Conway is especially known for the invention of the Game of Life, one of the early examples of a cellular automaton. His initial experiments in that field were done with pen and paper, since the game was introduced by Martin Gardner in Scientific American in 1970, it has spawned hundreds of computer programs, web sites, and articles. It is a staple of recreational mathematics, there is an extensive wiki devoted to curating and cataloging the various aspects of the game. From the earliest days it has been a favorite in computer labs, at times Conway has said he hates the game of life–largely because it has come to overshadow some of the other deeper and more important things he has done. Nevertheless, the game did help launch a new branch of mathematics, the Game of Life is now known to be Turing complete. Conways career is intertwined with mathematics popularizer and Scientific American columnist Martin Gardner, when Gardner featured Conways Game of Life in his Mathematical Games column in October 1970, it became the most widely read of all his columns and made Conway an instant celebrity. Gardner and Conway had first corresponded in the late 1950s, for instance, he discussed Conways game of Sprouts, Hackenbush, and his angel and devil problem. In the September 1976 column he reviewed Conways book On Numbers and Games, Conway is widely known for his contributions to combinatorial game theory, a theory of partisan games. This he developed with Elwyn Berlekamp and Richard Guy, and with them also co-authored the book Winning Ways for your Mathematical Plays and he also wrote the book On Numbers and Games which lays out the mathematical foundations of CGT. He is also one of the inventors of sprouts, as well as philosophers football and he developed detailed analyses of many other games and puzzles, such as the Soma cube, peg solitaire, and Conways soldiers
16.
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
17.
Euclidean geometry
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Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry, the Elements. Euclids method consists in assuming a set of intuitively appealing axioms. Although many of Euclids results had been stated by earlier mathematicians, Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system. The Elements begins with plane geometry, still taught in school as the first axiomatic system. It goes on to the geometry of three dimensions. Much of the Elements states results of what are now called algebra and number theory, for more than two thousand years, the adjective Euclidean was unnecessary because no other sort of geometry had been conceived. Euclids axioms seemed so obvious that any theorem proved from them was deemed true in an absolute, often metaphysical. Today, however, many other self-consistent non-Euclidean geometries are known, Euclidean geometry is an example of synthetic geometry, in that it proceeds logically from axioms to propositions without the use of coordinates. This is in contrast to analytic geometry, which uses coordinates, the Elements is mainly a systematization of earlier knowledge of geometry. Its improvement over earlier treatments was recognized, with the result that there was little interest in preserving the earlier ones. There are 13 total books in the Elements, Books I–IV, Books V and VII–X deal with number theory, with numbers treated geometrically via their representation as line segments with various lengths. Notions such as numbers and rational and irrational numbers are introduced. The infinitude of prime numbers is proved, a typical result is the 1,3 ratio between the volume of a cone and a cylinder with the same height and base. Euclidean geometry is a system, in which all theorems are derived from a small number of axioms. To produce a straight line continuously in a straight line. To describe a circle with any centre and distance and that all right angles are equal to one another. Although Euclids statement of the only explicitly asserts the existence of the constructions. The Elements also include the five common notions, Things that are equal to the same thing are also equal to one another
18.
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 +
19.
Non-Euclidean geometry
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In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those specifying Euclidean geometry. In the latter case one obtains hyperbolic geometry and elliptic geometry, when the metric requirement is relaxed, then there are affine planes associated with the planar algebras which give rise to kinematic geometries that have also been called non-Euclidean geometry. The essential difference between the geometries is the nature of parallel lines. In hyperbolic geometry, by contrast, there are many lines through A not intersecting ℓ, while in elliptic geometry. In elliptic geometry the lines curve toward each other and intersect, the debate that eventually led to the discovery of the non-Euclidean geometries began almost as soon as Euclids work Elements was written. In the Elements, Euclid began with a number of assumptions. Other mathematicians have devised simpler forms of this property, regardless of the form of the postulate, however, it consistently appears to be more complicated than Euclids other postulates,1. To draw a line from any point to any point. To produce a straight line continuously in a straight line. To describe a circle with any centre and distance and that all right angles are equal to one another. For at least a thousand years, geometers were troubled by the complexity of the fifth postulate. Many attempted to find a proof by contradiction, including Ibn al-Haytham, Omar Khayyám, Nasīr al-Dīn al-Tūsī and these theorems along with their alternative postulates, such as Playfairs axiom, played an important role in the later development of non-Euclidean geometry. These early attempts did, however, provide some early properties of the hyperbolic and elliptic geometries. Another example is al-Tusis son, Sadr al-Din, who wrote a book on the subject in 1298, based on al-Tusis later thoughts and he essentially revised both the Euclidean system of axioms and postulates and the proofs of many propositions from the Elements. His work was published in Rome in 1594 and was studied by European geometers and he finally reached a point where he believed that his results demonstrated the impossibility of hyperbolic geometry. His claim seems to have based on Euclidean presuppositions, because no logical contradiction was present. In this attempt to prove Euclidean geometry he instead unintentionally discovered a new viable geometry, in 1766 Johann Lambert wrote, but did not publish, Theorie der Parallellinien in which he attempted, as Saccheri did, to prove the fifth postulate. He worked with a figure that today we call a Lambert quadrilateral and he quickly eliminated the possibility that the fourth angle is obtuse, as had Saccheri and Khayyam, and then proceeded to prove many theorems under the assumption of an acute angle
20.
Uniform honeycombs in hyperbolic space
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In hyperbolic geometry, a uniform honeycomb in hyperbolic space is a uniform tessellation of uniform polyhedral cells. Honeycombs are divided between compact and paracompact forms defined by Coxeter groups, the first category only including finite cells and vertex figures, the nine compact Coxeter groups are listed here with their Coxeter diagrams, in order of the relative volumes of their fundamental simplex domains. These 9 families generate a total of 76 unique uniform honeycombs, the full list of hyperbolic uniform honeycombs has not been proven and an unknown number of non-Wythoffian forms exist. One known example is cited with the family below, only two families are related as a mirror-removal halving, ↔. There are just two radical subgroups with nonsimplectic domains that can be generated by removing a set of two or more mirrors separated by all other mirrors by even-order branches. One is, represented by Coxeter diagrams an index 6 subgroup with a trigonal trapezohedron fundamental domain ↔, the other is, index 120 with a dodecahedral fundamental domain. There are also 23 paracompact Coxeter groups of rank 4 that produce paracompact uniform honeycombs with infinite or unbounded facets or vertex figure, other paracompact Coxeter groups exists as Vinberg polytope fundamental domains, including these triangular bipyramid fundamental domains as rank 5 graphs including parallel mirrors. Uniform honeycombs exist as all permutations of rings in these graphs, the bitruncated and runcinated forms contain the faces of two regular skew polyhedrons, and. There are 15 forms, generated by ring permutations of the Coxeter group and this family is related to the group by a half symmetry, or ↔, when the last mirror after the order-4 branch is inactive, or as an alternation if the third mirror is inactive ↔. There are 9 forms, generated by ring permutations of the Coxeter group, there are 11 forms, generated by ring permutations of the Coxeter group, or. If the branch ring states match, a symmetry can double into the family. There are 9 forms, generated by ring permutations of the Coxeter group, there are 9 forms, generated by ring permutations of the Coxeter group, The bitruncated and runcinated forms contain the faces of two regular skew polyhedrons, and. There are 6 forms, generated by ring permutations of the Coxeter group, there are 4 extended symmetries possible based on the symmetry of the rings, and. This symmetry family is related to a radical subgroup, index 6, ↔, constructed by. The truncated forms contain the faces of two regular skew polyhedrons, and, there are 9 forms, generated by ring permutations of the Coxeter group, The truncated forms contain the faces of two regular skew polyhedrons, and. There are 6 forms, generated by ring permutations of the Coxeter group, there are 4 extended symmetries possible based on the symmetry of the rings, and. The truncated forms contain the faces of two regular skew polyhedrons, and and this is the complete enumeration of the 76 Wythoffian uniform honeycombs. The alternations are listed for completeness, but most are non-uniform, uniform tilings in hyperbolic plane List of regular polytopes#Tessellations of hyperbolic 3-space James E. ed. Dover Publications,1973
21.
Uniform polytope
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A uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets. The uniform polytopes in two dimensions are the regular polygons and this is a generalization of the older category of semiregular polytopes, but also includes the regular polytopes. Further, star regular faces and vertex figures are allowed, which expand the possible solutions. A strict definition requires uniform polytopes to be finite, while a more expansive definition allows uniform honeycombs of Euclidean, nearly every uniform polytope can be generated by a Wythoff construction, and represented by a Coxeter diagram. Notable exceptions include the antiprism in four dimensions. Equivalently, the Wythoffian polytopes can be generated by applying basic operations to the regular polytopes in that dimension and this approach was first used by Johannes Kepler, and is the basis of the Conway polyhedron notation. Regular n-polytopes have n orders of rectification, the zeroth rectification is the original form. The th rectification is the dual, an extended Schläfli symbol can be used for representing rectified forms, with a single subscript, k-th rectification = tk = kr. Truncation operations that can be applied to regular n-polytopes in any combination, the resulting Coxeter diagram has two ringed nodes, and the operation is named for the distance between them. Truncation cuts vertices, cantellation cuts edges, runcination cuts faces, each higher operation also cuts lower ones too, so a cantellation also truncates vertices. T0,1 or t, Truncation - applied to polygons, a truncation removes vertices, and inserts a new facet in place of each former vertex. Faces are truncated, doubling their edges and it can be seen as rectifying its rectification. A cantellation truncates both vertices and edges and replaces them with new facets, cells are replaced by topologically expanded copies of themselves. There are higher cantellations also, bicantellation t1,3 or r2r, tricantellation t2,4 or r3r, quadricantellation t3,5 or r4r, etc. t0,1,2 or tr, Cantitruncation - applied to polyhedra and higher. It can be seen as a truncation of its rectification, a cantitruncation truncates both vertices and edges and replaces them with new facets. Cells are replaced by topologically expanded copies of themselves, runcination truncates vertices, edges, and faces, replacing them each with new facets. 4-faces are replaced by topologically expanded copies of themselves, There are higher runcinations also, biruncination t1,4, triruncination t2,5, etc. t0,4 or 2r2r, Sterication - applied to Uniform 5-polytopes and higher. It can be seen as birectifying its birectification, Sterication truncates vertices, edges, faces, and cells, replacing each with new facets
22.
Cubic crystal system
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In crystallography, the cubic crystal system is a crystal system where the unit cell is in the shape of a cube. This is one of the most common and simplest shapes found in crystals and minerals, there are three main varieties of these crystals, Primitive cubic Body-centered cubic, Face-centered cubic Each is subdivided into other variants listed below. Note that although the cell in these crystals is conventionally taken to be a cube. This is related to the fact that in most cubic crystal systems, a classic isometric crystal has square or pentagonal faces. The three Bravais lattices in the crystal system are, The primitive cubic system consists of one lattice point on each corner of the cube. Each atom at a point is then shared equally between eight adjacent cubes, and the unit cell therefore contains in total one atom. The body-centered cubic system has one point in the center of the unit cell in addition to the eight corner points. It has a net total of 2 lattice points per unit cell, Each sphere in a cF lattice has coordination number 12. The face-centered cubic system is related to the hexagonal close packed system. The plane of a cubic system is a hexagonal grid. Attempting to create a C-centered cubic crystal system would result in a simple tetragonal Bravais lattice, there are a total 36 cubic space groups. Other terms for hexoctahedral are, normal class, holohedral, ditesseral central class, a simple cubic unit cell has a single cubic void in the center. Additionally, there are 24 tetrahedral voids located in a square spacing around each octahedral void and these tetrahedral voids are not local maxima and are not technically voids, but they do occasionally appear in multi-atom unit cells. A face-centered cubic unit cell has eight tetrahedral voids located midway between each corner and the center of the cell, for a total of eight net tetrahedral voids. One important characteristic of a structure is its atomic packing factor. This is calculated by assuming all the atoms are identical spheres. The atomic packing factor is the proportion of space filled by these spheres, assuming one atom per lattice point, in a primitive cubic lattice with cube side length a, the sphere radius would be a⁄2 and the atomic packing factor turns out to be about 0.524. Similarly, in a bcc lattice, the atomic packing factor is 0.680, as a rule, since atoms in a solid attract each other, the more tightly packed arrangements of atoms tend to be more common
23.
Cartesian coordinates
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Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, usually at ordered pair. The coordinates can also be defined as the positions of the projections of the point onto the two axis, expressed as signed distances from the origin. One can use the principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes. In general, n Cartesian coordinates specify the point in an n-dimensional Euclidean space for any dimension n and these coordinates are equal, up to sign, to distances from the point to n mutually perpendicular hyperplanes. The invention of Cartesian coordinates in the 17th century by René Descartes revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra. Using the Cartesian coordinate system, geometric shapes can be described by Cartesian equations, algebraic equations involving the coordinates of the points lying on the shape. For example, a circle of radius 2, centered at the origin of the plane, a familiar example is the concept of the graph of a function. Cartesian coordinates are also tools for most applied disciplines that deal with geometry, including astronomy, physics, engineering. They are the most common system used in computer graphics, computer-aided geometric design. Nicole Oresme, a French cleric and friend of the Dauphin of the 14th Century, used similar to Cartesian coordinates well before the time of Descartes. The adjective Cartesian refers to the French mathematician and philosopher René Descartes who published this idea in 1637 and it was independently discovered by Pierre de Fermat, who also worked in three dimensions, although Fermat did not publish the discovery. Both authors used a single axis in their treatments and have a length measured in reference to this axis. The concept of using a pair of axes was introduced later, after Descartes La Géométrie was translated into Latin in 1649 by Frans van Schooten and these commentators introduced several concepts while trying to clarify the ideas contained in Descartes work. Many other coordinate systems have developed since Descartes, such as the polar coordinates for the plane. The development of the Cartesian coordinate system would play a role in the development of the Calculus by Isaac Newton. The two-coordinate description of the plane was later generalized into the concept of vector spaces. Choosing a Cartesian coordinate system for a one-dimensional space – that is, for a straight line—involves choosing a point O of the line, a unit of length, and an orientation for the line. An orientation chooses which of the two half-lines determined by O is the positive, and which is negative, we say that the line is oriented from the negative half towards the positive half
24.
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
25.
Uniform polyhedron
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A uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive. It follows that all vertices are congruent, Uniform polyhedra may be regular, quasi-regular or semi-regular. The faces and vertices need not be convex, so many of the uniform polyhedra are also star polyhedra, there are two infinite classes of uniform polyhedra together with 75 others. Dual polyhedra to uniform polyhedra are face-transitive and have regular vertex figures, the dual of a regular polyhedron is regular, while the dual of an Archimedean solid is a Catalan solid. The concept of uniform polyhedron is a case of the concept of uniform polytope. Coxeter, Longuet-Higgins & Miller define uniform polyhedra to be vertex-transitive polyhedra with regular faces, by a polygon they implicitly mean a polygon in 3-dimensional Euclidean space, these are allowed to be non-convex and to intersect each other. There are some generalizations of the concept of a uniform polyhedron, if the connectedness assumption is dropped, then we get uniform compounds, which can be split as a union of polyhedra, such as the compound of 5 cubes. If we drop the condition that the realization of the polyhedron is non-degenerate and these require a more general definition of polyhedra. Some of the ways they can be degenerate are as follows, some polyhedra have faces that are hidden, in the sense that no points of their interior can be seen from the outside. These are usually not counted as uniform polyhedra, some polyhedra have multiple edges and their faces are the faces of two or more polyhedra, though these are not compounds in the previous sense since the polyhedra share edges. There are some non-orientable polyhedra that have double covers satisfying the definition of a uniform polyhedron, there double covers have doubled faces, edges and vertices. They are usually not counted as uniform polyhedra, there are several polyhedra with doubled faces produced by Wythoffs construction. Most authors do not allow doubled faces and remove them as part of the construction, skillings figure has the property that it has double edges but its faces cannot be written as a union of two uniform polyhedra. Regular convex polyhedra, The Platonic solids date back to the classical Greeks and were studied by the Pythagoreans, Plato, Theaetetus, Timaeus of Locri, the Etruscans discovered the regular dodecahedron before 500 BC. Nonregular uniform convex polyhedra, The cuboctahedron was known by Plato, Archimedes discovered all of the 13 Archimedean solids. His original book on the subject was lost, but Pappus of Alexandria mentioned Archimedes listed 13 polyhedra, piero della Francesca rediscovered the five truncation of the Platonic solids, truncated tetrahedron, truncated octahedron, truncated cube, truncated dodecahedron, and truncated icosahedron. Luca Pacioli republished Francescas work in De divina proportione in 1509, adding the rhombicuboctahedron, calling it a icosihexahedron for its 26 faces, which was drawn by Leonardo da Vinci. Johannes Kepler was the first to publish the complete list of Archimedean solids, in 1619, regular star polyhedra, Kepler discovered two of the regular Kepler–Poinsot polyhedra and Louis Poinsot discovered the other two
26.
Crystal system
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In crystallography, the terms crystal system, crystal family and lattice system each refer to one of several classes of space groups, lattices, point groups or crystals. Informally, two crystals are in the crystal system if they have similar symmetries, though there are many exceptions to this. Space groups and crystals are divided into seven crystal systems according to their point groups, five of the crystal systems are essentially the same as five of the lattice systems, but the hexagonal and trigonal crystal systems differ from the hexagonal and rhombohedral lattice systems. The six crystal families are formed by combining the hexagonal and trigonal crystal systems into one hexagonal family, a lattice system is a class of lattices with the same set of lattice point groups, which are subgroups of the arithmetic crystal classes. The 14 Bravais lattices are grouped into seven lattice systems, triclinic, monoclinic, orthorhombic, tetragonal, rhombohedral, hexagonal, in a crystal system, a set of point groups and their corresponding space groups are assigned to a lattice system. Of the 32 point groups that exist in three dimensions, most are assigned to only one system, in which case both the crystal and lattice systems have the same name. However, five point groups are assigned to two systems, rhombohedral and hexagonal, because both exhibit threefold rotational symmetry. These point groups are assigned to the crystal system. In total there are seven crystal systems, triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, a crystal family is determined by lattices and point groups. It is formed by combining crystal systems which have space groups assigned to a lattice system. In three dimensions, the families and systems are identical, except the hexagonal and trigonal crystal systems. In total there are six families, triclinic, monoclinic, orthorhombic, tetragonal, hexagonal. Spaces with less than three dimensions have the number of crystal systems, crystal families and lattice systems. In one-dimensional space, there is one crystal system, in 2D space, there are four crystal systems, oblique, rectangular, square and hexagonal. The relation between three-dimensional crystal families, crystal systems and lattice systems is shown in the table, Note. To avoid confusion of terminology, the term trigonal lattice is not used, if the original structure and inverted structure are identical, then the structure is centrosymmetric. Still, even for non-centrosymmetric case, inverted structure in some cases can be rotated to align with the original structure and this is the case of non-centrosymmetric achiral structure. If the inverted structure cannot be rotated to align with the structure, then the structure is chiral
27.
Monoclinic crystal system
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In crystallography, the monoclinic crystal system is one of the 7 crystal systems. A crystal system is described by three vectors, in the monoclinic system, the crystal is described by vectors of unequal lengths, as in the orthorhombic system. They form a rectangular prism with a parallelogram as its base, hence two vectors are perpendicular, while the third vector meets the other two at an angle other than 90°. There is only one monoclinic Bravais lattice in two dimensions, the oblique lattice, two monoclinic Bravais lattices exist, the primitive monoclinic and the centered monoclinic lattices. In this axis setting, the primitive and base-centered lattices interchange in centering type, sphenoidal is also monoclinic hemimorphic, Domatic is also monoclinic hemihedral, Prismatic is also monoclinic normal. Crystal structure Hurlbut, Cornelius S. Klein, Cornelis, hahn, Theo, ed. International Tables for Crystallography, Volume A, Space Group Symmetry
28.
Triclinic crystal system
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In crystallography, the triclinic crystal system is one of the 7 crystal systems. A crystal system is described by three basis vectors, in the triclinic system, the crystal is described by vectors of unequal length, as in the orthorhombic system. In addition, no vector is at right angles orthogonal to another, the triclinic lattice is the least symmetric of the 14 three-dimensional Bravais lattices. It is the lattice type that itself has no mirror planes. There are a total 2 space groups, with each only one space group is associated. Pinacoidal is also known as triclinic normal, pedial is also triclinic hemihedral Mineral examples include plagioclase, microcline, rhodonite, turquoise, wollastonite and amblygonite, all in triclinic normal. Crystal structure Hurlbut, Cornelius S. Klein, Cornelis,1985, Manual of Mineralogy, 20th ed. pp.64 –65, ISBN 0-471-80580-7
29.
Orthorhombic crystal system
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In crystallography, the orthorhombic crystal system is one of the 7 crystal systems. All three bases intersect at 90° angles, so the three lattice vectors remain mutually orthogonal, there are two orthorhombic Bravais lattices in two dimensions, Primitive rectangular and centered rectangular. The primitive rectangular lattice can also be described by a centered rhombic unit cell, there are four orthorhombic Bravais lattices, primitive orthorhombic, base-centered orthorhombic, body-centered orthorhombic, and face-centered orthorhombic. In this axis setting, the primitive and base-centered lattices interchange in centering type, crystal structure Overview of all space groups Hurlbut, Cornelius S. Klein, Cornelis. Hahn, Theo, ed. International Tables for Crystallography, Volume A, Space Group Symmetry
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Tetragonal crystal system
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In crystallography, the tetragonal crystal system is one of the 7 crystal systems. Tetragonal crystal lattices result from stretching a cubic lattice along one of its vectors, so that the cube becomes a rectangular prism with a square base. There is only one tetragonal Bravais lattice in two dimensions, the square lattice, there are two tetragonal Bravais lattices, the simple tetragonal and the centered tetragonal. One might suppose stretching face-centered cubic would result in face-centered tetragonal, BCT is considered more fundamental, so that is the standard terminology. Crystal structure point groups Bravais lattices
31.
Hexagonal crystal family
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In crystallography, the hexagonal crystal family is one of the 6 crystal families. In the hexagonal family, the crystal is described by a right rhombic prism unit cell with two equal axes, an included angle of 120° and a height perpendicular to the two base axes. There are 52 space groups associated with it, which are exactly those whose Bravais lattice is either hexagonal or rhombohedral, the hexagonal crystal family consists of two lattice systems, hexagonal and rhombohedral. Each lattice system consists of one Bravais lattice, hence, there are 3 lattice points per unit cell in total and the lattice is non-primitive. The Bravais lattices in the hexagonal crystal family can also be described by rhombohedral axes, the unit cell is a rhombohedron. This is a cell with parameters a = b = c, α = β = γ ≠ 90°. In practice, the description is more commonly used because it is easier to deal with a coordinate system with two 90° angles. However, the axes are often shown in textbooks because this cell reveals 3m symmetry of crystal lattice. However, such a description is rarely used, the hexagonal crystal family consists of two crystal systems, trigonal and hexagonal. A crystal system is a set of point groups in which the point groups themselves, the trigonal crystal system consists of the 5 point groups that have a single three-fold rotation axis. The crystal structures of alpha-quartz in the example are described by two of those 18 space groups associated with the hexagonal lattice system. The hexagonal crystal system consists of the seven point groups such that all their groups have the hexagonal lattice as underlying lattice. Graphite is an example of a crystal that crystallizes in the crystal system. Note that the atom in the center of the HCP unit cell in the hexagonal lattice system does not appear in the unit cell of the hexagonal lattice. It is part of the two atom motif associated with each point in the underlying lattice. The trigonal crystal system is the crystal system whose point groups have more than one lattice system associated with their space groups. The 5 point groups in this system are listed below, with their international number and notation, their space groups in name. The point groups in this system are listed below, followed by their representations in Hermann–Mauguin or international notation and Schoenflies notation
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Crystal structure
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In crystallography, crystal structure is a description of the ordered arrangement of atoms, ions or molecules in a crystalline material. Ordered structures occur from the nature of the constituent particles to form symmetric patterns that repeat along the principal directions of three-dimensional space in matter. The smallest group of particles in the material that constitutes the pattern is the unit cell of the structure. The unit cell completely defines the symmetry and structure of the crystal lattice. The repeating patterns are said to be located at the points of the Bravais lattice, the lengths of the principal axes, or edges, of the unit cell and the angles between them are the lattice constants, also called lattice parameters. The symmetry properties of the crystal are described by the concept of space groups, all possible symmetric arrangements of particles in three-dimensional space may be described by the 230 space groups. The crystal structure and symmetry play a role in determining many physical properties, such as cleavage, electronic band structure. The crystal structure of a material can be described in terms of its unit cell, the unit cell is a box containing one or more atoms arranged in three dimensions. The unit cells stacked in three-dimensional space describe the arrangement of atoms of the crystal. Commonly, atomic positions are represented in terms of fractional coordinates, the atom positions within the unit cell can be calculated through application of symmetry operations to the asymmetric unit. The asymmetric unit refers to the smallest possible occupation of space within the unit cell and this does not, however imply that the entirety of the asymmetric unit must lie within the boundaries of the unit cell. Symmetric transformations of atom positions are calculated from the group of the crystal structure. Vectors and planes in a lattice are described by the three-value Miller index notation. It uses the indices ℓ, m, and n as directional parameters, which are separated by 90°, by definition, the syntax denotes a plane that intercepts the three points a1/ℓ, a2/m, and a3/n, or some multiple thereof. That is, the Miller indices are proportional to the inverses of the intercepts of the plane with the unit cell, if one or more of the indices is zero, it means that the planes do not intersect that axis. A plane containing a coordinate axis is translated so that it no longer contains that axis before its Miller indices are determined, the Miller indices for a plane are integers with no common factors. Negative indices are indicated with horizontal bars, as in, in an orthogonal coordinate system for a cubic cell, the Miller indices of a plane are the Cartesian components of a vector normal to the plane. Likewise, the planes are geometric planes linking nodes
33.
Parallelepiped
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In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms. By analogy, it relates to a parallelogram just as a cube relates to a square or as a cuboid to a rectangle, in Euclidean geometry, its definition encompasses all four concepts. In this context of geometry, in which angles are not differentiated, its definition admits only parallelograms. The rectangular cuboid, cube, and the rhombohedron are all specific cases of parallelepiped, parallelepipeds are a subclass of the prismatoids. Any of the three pairs of faces can be viewed as the base planes of the prism. A parallelepiped has three sets of four edges, the edges within each set are of equal length. Parallelepipeds result from linear transformations of a cube, since each face has point symmetry, a parallelepiped is a zonohedron. Also the whole parallelepiped has point symmetry Ci, each face is, seen from the outside, the mirror image of the opposite face. The faces are in general chiral, but the parallelepiped is not, a space-filling tessellation is possible with congruent copies of any parallelepiped. The volume of a parallelepiped is the product of the area of its base A, the base is any of the six faces of the parallelepiped. The height is the distance between the base and the opposite face. An alternative method defines the vectors a =, b = and c = to represent three edges that meet at one vertex, from the figure, we can deduce that the magnitude of α is limited to 0° ≤ α < 90°. On the contrary, the vector b × c may form with a an internal angle β larger than 90°, namely, since b × c is parallel to h, the value of β is either β = α or β = 180° − α. So cos α = ± cos β = | cos β |, and h = | a | | cos β |. We conclude that V = A h = | a | | b × c | | cos β |, which is, by definition of the scalar product, equivalent to the absolute value of a ·, Q. E. D. The latter expression is equivalent to the absolute value of the determinant of a three dimensional matrix built using a, b and c as rows, V = | det |. This is found using Cramers Rule on three reduced two dimensional matrices found from the original, the volume of any tetrahedron that shares three converging edges of a parallelepiped has a volume equal to one sixth of the volume of that parallelepiped. For parallelepipeds with a symmetry plane there are two cases, it has four rectangular faces it has two faces, while of the other faces, two adjacent ones are equal and the other two also
34.
Cuboid
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In geometry, a cuboid is a convex polyhedron bounded by six quadrilateral faces, whose polyhedral graph is the same as that of a cube. By Eulers formula the numbers of faces F, of vertices V, in the case of a cuboid this gives 6 +8 =12 +2, that is, like a cube, a cuboid has 6 faces,8 vertices, and 12 edges. Along with the rectangular cuboids, any parallelepiped is a cuboid of this type, in a rectangular cuboid, all angles are right angles, and opposite faces of a cuboid are equal. By definition this makes it a rectangular prism, and the terms rectangular parallelepiped or orthogonal parallelepiped are also used to designate this polyhedron. The terms rectangular prism and oblong prism, however, are ambiguous, the square cuboid, square box, or right square prism is a special case of the cuboid in which at least two faces are squares. It has Schläfli symbol ×, and its symmetry is doubled from to, the cube is a special case of the square cuboid in which all six faces are squares. It has Schläfli symbol, and its symmetry is raised from, to, if the dimensions of a rectangular cuboid are a, b and c, then its volume is abc and its surface area is 2. The length of the diagonal is d = a 2 + b 2 + c 2. Cuboid shapes are used for boxes, cupboards, rooms, buildings. Cuboids are among those solids that can tessellate 3-dimensional space, the shape is fairly versatile in being able to contain multiple smaller cuboids, e. g. sugar cubes in a box, boxes in a cupboard, cupboards in a room, and rooms in a building. A cuboid with integer edges as well as integer face diagonals is called an Euler brick, a perfect cuboid is an Euler brick whose space diagonal is also an integer. It is currently unknown whether a perfect cuboid actually exists, the number of different nets for a simple cube is 11, however this number increases significantly to 54 for a rectangular cuboid of 3 different lengths. Hyperrectangle Trapezohedron Weisstein, Eric W. Cuboid, rectangular prism and cuboid Paper models and pictures
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Trapezohedron
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The n-gonal trapezohedron, antidipyramid, antibipyramid or deltohedron is the dual polyhedron of an n-gonal antiprism. Its 2n faces are congruent kites, the n-gon part of the name does not reference the faces here but arrangement of vertices around an axis of symmetry. The dual n-gonal antiprism has two actual n-gon faces, an n-gonal trapezohedron can be decomposed into two equal n-gonal pyramids and an n-gonal antiprism. These figures, sometimes called deltohedra, must not be confused with deltahedra, in texts describing the crystal habits of minerals, the word trapezohedron is often used for the polyhedron properly known as a deltoidal icositetrahedron. In the case of the dual of a triangular antiprism the kites are rhombi and they are cubes scaled in the direction of a body diagonal. Also they are the parallelepipeds with congruent rhombic faces, a special case of a rhombohedron is one in the which the rhombi which form the faces have angles of 60° and 120°. It can be decomposed into two regular tetrahedra and a regular octahedron. Since parallelepipeds can fill space, so can a combination of regular tetrahedra, a degenerate form, n =2, form a geometric tetrahedron with 6 vertices,8 edges, and 4 degenerate kite faces that are degenerated into triangles. Its dual is a form of antiprism, also a tetrahedron. The symmetry group of an n-gonal trapezohedron is Dnd of order 4n, except in the case of a cube, which has the symmetry group Od of order 48. The rotation group is Dn of order 2n, except in the case of a cube, which has the larger rotation group O of order 24, if the kites surrounding the two peaks are of different shapes, it can only have Cnv symmetry, order 2n. Crystal arrangements of atoms can repeat in space with trapezohedral cells, the pentagonal trapezohedron is the only polyhedron other than the Platonic solids commonly used as a die in roleplaying games such as Dungeons & Dragons. Having 10 sides, it can be used in repetition to generate any decimal-based uniform probability desired, two dice of different colors are typically used for the two digits to represent numbers from 00 to 99. Self-intersecting trapezohedron exist with a star polygon central figure, defined by kite faces connecting each polygon edge to two points. Diminished trapezohedron Rhombic dodecahedron Rhombic triacontahedron Bipyramid Conway polyhedron notation Anthony Pugh, California, University of California Press Berkeley. Chapter 4, Duals of the Archimedean polyhedra, prisma and antiprisms Weisstein, virtual Reality Polyhedra The Encyclopedia of Polyhedra VRML models <3> <4> <5> <6> <7> <8> <9> <10> Conway Notation for Polyhedra Try, dAn, where n=3,4,5. Example dA5 is a pentagonal trapezohedron
36.
Point group
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In geometry, a point group is a group of geometric symmetries that keep at least one point fixed. Point groups can exist in a Euclidean space with any dimension, point groups can be realized as sets of orthogonal matrices M that transform point x into point y, y = Mx where the origin is the fixed point. Point-group elements can either be rotations or else reflections, or improper rotations and these are the crystallographic point groups. Point groups can be classified into groups and achiral groups. The chiral groups are subgroups of the orthogonal group SO, they contain only orientation-preserving orthogonal transformations. The achiral groups contain also transformations of determinant −1, in an achiral group, the orientation-preserving transformations form a subgroup of index 2. Finite Coxeter groups or reflection groups are those point groups that are generated purely by a set of reflectional mirrors passing through the same point, a rank n Coxeter group has n mirrors and is represented by a Coxeter-Dynkin diagram. Coxeter notation offers a bracketed notation equivalent to the Coxeter diagram, with symbols for rotational. There are only two one-dimensional point groups, the identity group and the reflection group, point groups in two dimensions, sometimes called rosette groups. The subset of pure reflectional point groups, defined by 1 or 2 mirrors, can also be given by their Coxeter group and related polygons. The symmetry of the groups can be doubled by an isomorphism. Point groups in three dimensions, sometimes called point groups after their wide use in studying the symmetries of small molecules. They come in 7 infinite families of axial or prismatic groups, the reflection point groups, defined by 1 to 3 mirror planes, can also be given by their Coxeter group and related polyhedra. The group can be doubled, written as, mapping the first and last mirrors onto each other, doubling the symmetry to 48, the four-dimensional point groups are listed in Conway and Smith, Section 4, Tables 4. 1-4.3. The following list gives the four-dimensional reflection groups, each group is specified as a Coxeter group, and like the polyhedral groups of 3D, it can be named by its related convex regular 4-polytope. Front-back symmetric groups like and can be doubled, shown as double brackets in Coxeters notation, the following table gives the five-dimensional reflection groups, by listing them as Coxeter groups. The following table gives the six-dimensional reflection groups, by listing them as Coxeter groups, the following table gives the seven-dimensional reflection groups, by listing them as Coxeter groups. The following table gives the eight-dimensional reflection groups, by listing them as Coxeter groups, S. M. Coxeter, Kaleidoscopes, Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C
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Coxeter notation
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The notation is named after H. S. M. Coxeter, and has been more comprehensively defined by Norman Johnson. For Coxeter groups defined by pure reflections, there is a correspondence between the bracket notation and Coxeter-Dynkin diagram. The numbers in the notation represent the mirror reflection orders in the branches of the Coxeter diagram. It uses the same simplification, suppressing 2s between orthogonal mirrors, the Coxeter notation is simplified with exponents to represent the number of branches in a row for linear diagram. So the An group is represented by, to imply n nodes connected by n-1 order-3 branches, example A2 = = or represents diagrams or. Coxeter initially represented bifurcating diagrams with vertical positioning of numbers, but later abbreviated with an exponent notation, like, Coxeter allowed for zeros as special cases to fit the An family, like A3 = = = =, like = =. Coxeter groups formed by cyclic diagrams are represented by parenthesese inside of brackets, if the branch orders are equal, they can be grouped as an exponent as the length the cycle in brackets, like =, representing Coxeter diagram or. More complicated looping diagrams can also be expressed with care, the paracompact complete graph diagram or, is represented as with the superscript as the symmetry of its regular tetrahedron coxeter diagram. The Coxeter diagram usually leaves order-2 branches undrawn, but the bracket notation includes an explicit 2 to connect the subgraphs, so the Coxeter diagram = A2×A2 = 2A2 can be represented by × =2 =. For the affine and hyperbolic groups, the subscript is one less than the number of nodes in each case, Coxeters notation represents rotational/translational symmetry by adding a + superscript operator outside the brackets which cuts the order of the group in half. This is called a direct subgroup because what remains are only direct isometries without reflective symmetry, + operators can also be applied inside of the brackets, and creates semidirect subgroups that include both reflective and nonreflective generators. Semidirect subgroups can only apply to Coxeter group subgroups that have even order branches next to it, the subgroup index is 2n for n + operators. So the snub cube, has symmetry +, and the tetrahedron, has symmetry. Johnson extends the + operator to work with a placeholder 1 nodes, in general this operation only applies to mirrors bounded by all even-order branches. The 1 represents a mirror so can be seen as, or, like diagram or, the effect of a mirror removal is to duplicate connecting nodes, which can be seen in the Coxeter diagrams, =, or in bracket notation, = =. Each of these mirrors can be removed so h = = = and this can be shown in a Coxeter diagram by adding a + symbol above the node, = =. If both mirrors are removed, a subgroup is generated, with the branch order becoming a gyration point of half the order, q = = +. For example, = = = ×, order 4. = +, the opposite to halving is doubling which adds a mirror, bisecting a fundamental domain, and doubling the group order
38.
Uniform coloring
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In geometry, a uniform coloring is a property of a uniform figure that is colored to be vertex-transitive. Different symmetries can be expressed on the geometric figure with the faces following different uniform color patterns. A uniform coloring can be specified by listing the different colors with indices around a vertex figure, in addition, an n-uniform coloring is a property of a uniform figure which has n types vertex figure, that are collectively vertex transitive. A related term is Archimedean color requires one vertex figure coloring repeated in a periodic arrangement, a more general term are k-Archimedean colorings which count k distinctly colored vertex figures. For example this Archimedean coloring of a tiling has two colors, but requires 4 unique colors by symmetry positions and become a 2-uniform coloring, Grünbaum, Branko, Shephard. CS1 maint, Multiple names, authors list Uniform and Archimedean colorings, pp. 102–107 Weisstein, Uniform Tessellations on the Euclid plane Tessellations of the Plane David Baileys World of Tessellations k-uniform tilings n-uniform tilings
39.
Triangular tiling
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In geometry, the triangular tiling or triangular tessellation is one of the three regular tilings of the Euclidean plane. Because the internal angle of the triangle is 60 degrees. The triangular tiling has Schläfli symbol of, Conway calls it a deltille, named from the triangular shape of the Greek letter delta. The triangular tiling can also be called a kishextille by a kis operation that adds a center point and it is one of three regular tilings of the plane. The other two are the square tiling and the hexagonal tiling, there are 9 distinct uniform colorings of a triangular tiling. Three of them can be derived from others by repeating colors,111212 and 111112 from 121213 by combining 1 and 3, there is one class of Archimedean colorings,111112, which is not 1-uniform, containing alternate rows of triangles where every third is colored. The example shown is 2-uniform, but there are many such Archimedean colorings that can be created by arbitrary horizontal shifts of the rows. The vertex arrangement of the tiling is called an A2 lattice. It is the 2-dimensional case of a simplectic honeycomb, the A*2 lattice can be constructed by the union of all three A2 lattices, and equivalent to the A2 lattice. + + = dual of = The vertices of the tiling are the centers of the densest possible circle packing. Every circle is in contact with 6 other circles in the packing, the packing density is π⁄√12 or 90. 69%. Since the union of 3 A2 lattices is also an A2 lattice, the voronoi cell of a triangular tiling is a hexagon, and so the voronoi tessellation, the hexagonal tiling has a direct correspondence to the circle packings. Triangular tilings can be made with the equivalent topology as the regular tiling, with identical faces and vertex-transitivity, there are 5 variations. Symmetry given assumes all faces are the same color, the planar tilings are related to polyhedra. Putting fewer triangles on a vertex leaves a gap and allows it to be folded into a pyramid and these can be expanded to Platonic solids, five, four and three triangles on a vertex define an icosahedron, octahedron, and tetrahedron respectively. This tiling is related as a part of sequence of regular polyhedra with Schläfli symbols. It is also related as a part of sequence of Catalan solids with face configuration Vn.6.6. Like the uniform there are eight uniform tilings that can be based from the regular hexagonal tiling
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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
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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
42.
Order-5 cubic honeycomb
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The order-5 cubic honeycomb is one of four compact regular space-filling tessellations in hyperbolic 3-space. With Schläfli symbol, it has five cubes around each edge and it is dual with the order-4 dodecahedral 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. It a radial subgroup symmetry construction with dodecahedral fundamental domains, Coxeter notation and it has a related alternation honeycomb, represented by ↔, having icosahedron and tetrahedron cells. The first polytope in the sequence is the tesseract, and the second is the Euclidean cubic honeycomb, the rectified order-5 cubic honeycomb, has alternating icosahedron and cuboctahedron cells, with a pentagonal prism vertex figure. There are four rectified compact regular honeycombs, The truncated order-5 cubic honeycomb, has truncated cube and icosahedron cells, same as Bitruncated order-4 dodecahedral honeycomb The cantellated order-5 cubic honeycomb, has rhombicuboctahedron and icosidodecahedron cells, with a wedge vertex figure. It is similar to the Euclidean runcitruncated cubic honeycomb, t0,1,3 and it is similar to the Euclidean omnitruncated cubic honeycomb, t0,1,2,3, In 3-dimensional hyperbolic geometry, the alternated order-5 cubic honeycomb is a uniform compact space-filling tessellation. With Schläfli symbol h, it can be considered a quasiregular honeycomb and it has 3 related forms, the cantic order-5 cubic honeycomb, the runcic order-5 cubic honeycomb, and the runcicantic order-5 cubic honeycomb. The cantic order-5 cubic honeycomb is a uniform compact space-filling tessellation and it has Schläfli symbol h2 and a rectangular pyramid vertex figure. The runcic order-5 cubic honeycomb is a uniform compact space-filling tessellation and it has Schläfli symbol h3 and a triangular prism vertex figure. The runcicantic order-5 cubic honeycomb is a uniform compact space-filling tessellation and it has Schläfli symbol h2,3 and a mirrored sphenoid vertex figure. Convex uniform honeycombs in hyperbolic space Coxeter, Regular Polytopes, 3rd, Coxeter, The Beauty of Geometry, Twelve Essays, Dover Publications,1999 ISBN 0-486-40919-8 Norman Johnson Uniform Polytopes, Manuscript N. W. 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
43.
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
44.
3-sphere
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In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It consists of the set of points equidistant from a central point in 4-dimensional Euclidean space. A 3-sphere is an example of a 3-manifold, in coordinates, a 3-sphere with center and radius r is the set of all points in real, 4-dimensional space such that ∑ i =032 =2 +2 +2 +2 = r 2. The 3-sphere centered at the origin with radius 1 is called the unit 3-sphere and is usually denoted S3 and it is often convenient to regard R4 as the space with 2 complex dimensions or the quaternions. The unit 3-sphere is then given by S3 = or S3 = and this description as the quaternions of norm one, identifies the 3-sphere with the versors in the quaternion division ring. Just as the circle is important for planar polar coordinates. See polar decomposition of a quaternion for details of development of the three-sphere. This view of the 3-sphere is the basis for the study of space as developed by Georges Lemaître. The 3-dimensional cubic hyperarea of a 3-sphere of radius r is 2 π2 r 3 while the 4-dimensional quartic hypervolume is 12 π2 r 4, every non-empty intersection of a 3-sphere with a three-dimensional hyperplane is a 2-sphere. Then the 2-sphere shrinks again down to a point as the 3-sphere leaves the hyperplane. A 3-sphere is a compact, connected, 3-dimensional manifold without boundary, what this means, in the broad sense, is that any loop, or circular path, on the 3-sphere can be continuously shrunk to a point without leaving the 3-sphere. The Poincaré conjecture, proved in 2003 by Grigori Perelman, provides that the 3-sphere is the only three-dimensional manifold with these properties, the 3-sphere is homeomorphic to the one-point compactification of R3. In general, any space that is homeomorphic to the 3-sphere is called a topological 3-sphere. The homology groups of the 3-sphere are as follows, H0, any topological space with these homology groups is known as a homology 3-sphere. Initially Poincaré conjectured that all homology 3-spheres are homeomorphic to S3, infinitely many homology spheres are now known to exist. For example, a Dehn filling with slope 1/n on any knot in the 3-sphere gives a homology sphere, as to the homotopy groups, we have π1 = π2 = and π3 is infinite cyclic. The higher-homotopy groups are all finite abelian but otherwise follow no discernible pattern, for more discussion see homotopy groups of spheres. The 3-sphere is naturally a smooth manifold, in fact, an embedded submanifold of R4
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Euclidean space
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In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces. It is named after the Ancient Greek mathematician Euclid of Alexandria, the term Euclidean distinguishes these spaces from other types of spaces considered in modern geometry. Euclidean spaces also generalize to higher dimensions, classical Greek geometry defined the Euclidean plane and Euclidean three-dimensional space using certain postulates, while the other properties of these spaces were deduced as theorems. Geometric constructions are used to define rational numbers. It means that points of the space are specified with collections of real numbers and this approach brings the tools of algebra and calculus to bear on questions of geometry and has the advantage that it generalizes easily to Euclidean spaces of more than three dimensions. From the modern viewpoint, there is only one Euclidean space of each dimension. With Cartesian coordinates it is modelled by the coordinate space of the same dimension. In one dimension, this is the line, in two dimensions, it is the Cartesian plane, and in higher dimensions it is a coordinate space with three or more real number coordinates. One way to think of the Euclidean plane is as a set of points satisfying certain relationships, expressible in terms of distance, for example, there are two fundamental operations on the plane. One is translation, which means a shifting of the plane so that point is shifted in the same direction. The other is rotation about a point in the plane. In order to all of this mathematically precise, the theory must clearly define the notions of distance, angle, translation. Even when used in theories, Euclidean space is an abstraction detached from actual physical locations, specific reference frames, measurement instruments. The standard way to such space, as carried out in the remainder of this article, is to define the Euclidean plane as a two-dimensional real vector space equipped with an inner product. The reason for working with vector spaces instead of Rn is that it is often preferable to work in a coordinate-free manner. Once the Euclidean plane has been described in language, it is actually a simple matter to extend its concept to arbitrary dimensions. For the most part, the vocabulary, formulae, and calculations are not made any more difficult by the presence of more dimensions. Intuitively, the distinction says merely that there is no choice of where the origin should go in the space
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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