1.
Deltahedron
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In geometry, a deltahedron is a polyhedron whose faces are all equilateral triangles. The name is taken from the Greek majuscule delta, which has the shape of an equilateral triangle, There are infinitely many deltahedra, but of these only eight are convex, having 4,6,8,10,12,14,16 and 20 faces. The number of faces, edges, and vertices is listed below for each of the eight convex deltahedra, There are only eight strictly-convex deltahedra, three are regular polyhedra, and five are Johnson solids. In the 6-faced deltahedron, some vertices have degree 3 and some degree 4, in the 10-, 12-, 14-, and 16-faced deltahedra, some vertices have degree 4 and some degree 5. These five irregular deltahedra belong to the class of Johnson solids, Deltahedra retain their shape, even if the edges are free to rotate around their vertices so that the angles between edges are fluid. Not all polyhedra have this property, for example, if you relax some of the angles of a cube, There is no 18-faced convex deltahedron. There are infinitely many cases with coplanar triangles, allowing for sections of the infinite triangular tilings, if the sets of coplanar triangles are considered a single face, a smaller set of faces, edges, and vertices can be counted. The coplanar triangular faces can be merged into rhombic, trapezoidal, hexagonal, each face must be a convex polyiamond such as, and. Some smaller examples include, There are a number of nonconvex forms. Konvexe pseudoreguläre Polyeder, Zeitschrift für mathematischen und naturwissenschaftlichen Unterricht,46, cundy, H. Martyn, Deltahedra, Mathematical Gazette,36, 263–266. Cundy, H. Martyn, Rollett, A.3.11, Deltahedra, Mathematical Models, Stradbroke, England, Tarquin Pub. pp. 142–144. Gardner, Martin, Fractal Music, Hypercards, and More, Mathematical Recreations from Scientific American, New York, W. H. Freeman, pp.40,53, and 58–60. Pugh, Anthony, Polyhedra, A visual approach, California, University of California Press Berkeley, ISBN 0-520-03056-7 pp. 35–36 Weisstein, the eight convex deltahedra Deltahedron Deltahedron
2.
Conway polyhedron notation
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In geometry, Conway polyhedron notation, invented by John Horton Conway and promoted by George W. Hart, is used to describe polyhedra based on a seed polyhedron modified by various prefix operations. Conway and Hart extended the idea of using operators, like truncation defined by Kepler, the basic descriptive operators can generate all the Archimedean solids and Catalan solids from regular seeds. For example tC represents a cube, and taC, parsed as t, is a truncated cuboctahedron. The simplest operator dual swaps vertex and face elements, like a cube is an octahedron. Applied in a series, these allow many higher order polyhedra to be generated. A resulting polyhedron will have a fixed topology, while exact geometry is not constrained, the seed polyhedra are the Platonic solids, represented by the first letter of their name, the prisms for n-gonal forms, antiprisms, cupolae and pyramids. Any polyhedron can serve as a seed, as long as the operations can be executed on it, for example regular-faced Johnson solids can be referenced as Jn, for n=1.92. In general, it is difficult to predict the appearance of the composite of two or more operations from a given seed polyhedron. For instance ambo applied twice becomes the same as the operation, aa=e, while a truncation after ambo produces bevel. There has been no general theory describing what polyhedra can be generated in by any set of operators, instead all results have been discovered empirically. Elements are given from the seed to the new forms, assuming seed is a polyhedron, An example image is given for each operation. The basic operations are sufficient to generate the reflective uniform polyhedra, some basic operations can be made as composites of others. Special forms The kis operator has a variation, kn, which only adds pyramids to n-sided faces, the truncate operator has a variation, tn, which only truncates order-n vertices. The operators are applied like functions from right to left, for example, a cuboctahedron is an ambo cube, i. e. t = aC, and a truncated cuboctahedron is t = t = taC. Chirality operator r – reflect – makes the image of the seed. Alternately an overline can be used for picking the other chiral form, the operations are visualized here on cube seed examples, drawn on the surface of the cube, with blue faces that cross original edges, and pink faces that center at original vertices. The first row generates the Archimedean solids and the row the Catalan solids. Comparing each new polyhedron with the cube, each operation can be visually understood, the truncated icosahedron, tI or zD, which is Goldberg polyhedron G, creates more polyhedra which are neither vertex nor face-transitive
3.
Kite (geometry)
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In Euclidean geometry, a kite is a quadrilateral whose four sides can be grouped into two pairs of equal-length sides that are adjacent to each other. In contrast, a parallelogram also has two pairs of sides, but they are opposite to each other rather than adjacent. Kite quadrilaterals are named for the wind-blown, flying kites, which often have this shape, kites are also known as deltoids, but the word deltoid may also refer to a deltoid curve, an unrelated geometric object. A kite, as defined above, may be convex or concave. A concave kite is called a dart or arrowhead, and is a type of pseudotriangle. If all four sides of a kite have the same length, if a kite is equiangular, meaning that all four of its angles are equal, then it must also be equilateral and thus a square. A kite with three equal 108° angles and one 36° angle forms the hull of the lute of Pythagoras. The kites that are cyclic quadrilaterals are exactly the ones formed from two congruent right triangles. That is, for these kites the two angles on opposite sides of the symmetry axis are each 90 degrees. These shapes are called right kites and they are in fact bicentric quadrilaterals, among all the bicentric quadrilaterals with a given two circle radii, the one with maximum area is a right kite. The tiling that it produces by its reflections is the deltoidal trihexagonal tiling, among all quadrilaterals, the shape that has the greatest ratio of its perimeter to its diameter is an equidiagonal kite with angles π/3, 5π/12, 5π/6, 5π/12. Its four vertices lie at the three corners and one of the midpoints of the Reuleaux triangle. In non-Euclidean geometry, a Lambert quadrilateral is a kite with three right angles. A quadrilateral is a if and only if any one of the following conditions is true. One diagonal is the bisector of the other diagonal. One diagonal is a line of symmetry, one diagonal bisects a pair of opposite angles. The kites are the quadrilaterals that have an axis of symmetry along one of their diagonals, if crossings are allowed, the list of quadrilaterals with axes of symmetry must be expanded to also include the antiparallelograms. Every kite is orthodiagonal, meaning that its two diagonals are at angles to each other
4.
Face configuration
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In geometry, a vertex configuration is a shorthand notation for representing the vertex figure of a polyhedron or tiling as the sequence of faces around a vertex. For uniform polyhedra there is one vertex type and therefore the vertex configuration fully defines the polyhedron. A vertex configuration is given as a sequence of numbers representing the number of sides of the faces going around the vertex, the notation a. b. c describes a vertex that has 3 faces around it, faces with a, b, and c sides. For example,3.5.3.5 indicates a vertex belonging to 4 faces, alternating triangles and this vertex configuration defines the vertex-transitive icosidodecahedron. The notation is cyclic and therefore is equivalent with different starting points, the order is important, so 3.3.5.5 is different from 3.5.3.5. Repeated elements can be collected as exponents so this example is represented as 2. It has variously called a vertex description, vertex type, vertex symbol, vertex arrangement, vertex pattern. It is also called a Cundy and Rollett symbol for its usage for the Archimedean solids in their 1952 book Mathematical Models, a vertex configuration can also be represented as a polygonal vertex figure showing the faces around the vertex. Different notations are used, sometimes with a comma and sometimes a period separator, the period operator is useful because it looks like a product and an exponent notation can be used. For example,3.5.3.5 is sometimes written as 2, the notation can also be considered an expansive form of the simple Schläfli symbol for regular polyhedra. The Schläfli notation means q p-gons around each vertex, so can be written as p. p. p. or pq. For example, an icosahedron is =3.3.3.3.3 or 35 and this notation applies to polygonal tilings as well as polyhedra. A planar vertex configuration denotes a uniform tiling just like a nonplanar vertex configuration denotes a uniform polyhedron, the notation is ambiguous for chiral forms. For example, the cube has clockwise and counterclockwise forms which are identical across mirror images. Both have a 3.3.3.3.4 vertex configuration, the notation also applies for nonconvex regular faces, the star polygons. For example, a pentagram has the symbol, meaning it has 5 sides going around the centre twice, for example, there are 4 regular star polyhedra with regular polygon or star polygon vertex figures. The small stellated dodecahedron has the Schläfli symbol of which expands to a vertex configuration 5/2. 5/2. 5/2. 5/2. 5/2 or combined as 5. The great stellated dodecahedron, has a vertex figure and configuration or 3
5.
Point groups in three dimensions
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In geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere. It is a subgroup of the orthogonal group O, the group of all isometries that leave the origin fixed, or correspondingly, O itself is a subgroup of the Euclidean group E of all isometries. Symmetry groups of objects are isometry groups, accordingly, analysis of isometry groups is analysis of possible symmetries. All isometries of a bounded 3D object have one or more fixed points. We choose the origin as one of them, the rotation group of an object is equal to its full symmetry group if and only if the object is chiral. Finite Coxeter groups are a set of point groups 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 markup symbols for rotational and other subsymmetry point groups. SO is a subgroup of E+, which consists of direct isometries, i. e. isometries preserving orientation, it contains those that leave the origin fixed. O is the product of SO and the group generated by inversion. An example would be C4 for H and S4 for M, Thus M is obtained from H by inverting the isometries in H ∖ L. This is clarifying when categorizing isometry groups, see below, in 2D the cyclic group of k-fold rotations Ck is for every positive integer k a normal subgroup of O and SO. Accordingly, in 3D, for every axis the cyclic group of rotations about that axis is a normal subgroup of the group of all rotations about that axis. e. See also the similar overview including translations, when comparing the symmetry type of two objects, the origin is chosen for each separately, i. e. they need not have the same center. Moreover, two objects are considered to be of the symmetry type if their symmetry groups are conjugate subgroups of O. The conjugacy definition would allow a mirror image of the structure, but this is not needed. For example, if a symmetry group contains a 3-fold axis of rotation, there are many infinite isometry groups, for example, the cyclic group generated by a rotation by an irrational number of turns about an axis. We may create non-cyclical abelian groups by adding more rotations around the same axis, there are also non-abelian groups generated by rotations around different axes. They will be infinite unless the rotations are specially chosen, all the infinite groups mentioned so far are not closed as topological subgroups of O
6.
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
7.
Antiprism
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In geometry, an n-sided antiprism is a polyhedron composed of two parallel copies of some particular n-sided polygon, connected by an alternating band of triangles. Antiprisms are a subclass of the prismatoids and are a type of snub polyhedra, Antiprisms are similar to prisms except the bases are twisted relative to each other, and that the side faces are triangles, rather than quadrilaterals. In the case of a regular n-sided base, one considers the case where its copy is twisted by an angle 180°/n. Extra regularity is obtained when the line connecting the centers is perpendicular to the base planes. As faces, it has the two bases and, connecting those bases, 2n isosceles triangles. A uniform antiprism has, apart from the faces, 2n equilateral triangles as faces. As a class, the uniform antiprisms form a series of vertex-uniform polyhedra. For n =2 we have as degenerate case the regular tetrahedron as a digonal antiprism, the dual polyhedra of the antiprisms are the trapezohedra. Let a be the edge-length of a uniform antiprism, then the volume is V = n 4 cos 2 π2 n −1 sin 3 π2 n 12 sin 2 π n a 3 and the surface area is A = n 2 a 2. There are a set of truncated antiprisms, including a lower-symmetry form of the truncated octahedron. These can be alternated to create snub antiprisms, two of which are Johnson solids, and the snub triangular antiprism is a lower form of the icosahedron. The symmetry group contains inversion if and only if n is odd, uniform star antiprisms are named by their star polygon bases, and exist in prograde and retrograde solutions. Crossed forms have intersecting vertex figures, and are denoted by inverted fractions, p/ instead of p/q, in the retrograde forms but not in the prograde forms, the triangles joining the star bases intersect the axis of rotational symmetry. Some retrograde star antiprisms with regular star polygon bases cannot be constructed with equal edge lengths, star antiprism compounds also can be constructed where p and q have common factors, thus a 10/4 antiprism is the compound of two 5/2 star antiprisms. Prism Apeirogonal antiprism Grand antiprism – a four-dimensional polytope One World Trade Center, California, University of California Press Berkeley. Chapter 2, Archimedean polyhedra, prisma and antiprisms Weisstein, Eric W. Antiprism, archived from the original on 4 February 2007. Archived from the original on 4 February 2007, nonconvex Prisms and Antiprisms Paper models of prisms and antiprisms
8.
Isohedral figure
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In geometry, a polytope of dimension 3 or higher is isohedral or face-transitive when all its faces are the same. More specifically, all faces must be not merely congruent but must be transitive, in other words, for any faces A and B, there must be a symmetry of the entire solid by rotations and reflections that maps A onto B. For this reason, convex polyhedra are the shapes that will make fair dice. They can be described by their face configuration, a polyhedron which is isohedral has a dual polyhedron that is vertex-transitive. The Catalan solids, the bipyramids and the trapezohedra are all isohedral and they are the duals of the isogonal Archimedean solids, prisms and antiprisms, respectively. The Platonic solids, which are either self-dual or dual with another Platonic solid, are vertex, edge, a polyhedron which is isohedral and isogonal is said to be noble. A polyhedron is if it contains k faces within its symmetry fundamental domain. Similarly a k-isohedral tiling has k separate symmetry orbits, a monohedral polyhedron or monohedral tiling has congruent faces, as either direct or reflectively, which occur in one or more symmetry positions. An r-hedral polyhedra or tiling has r types of faces, a facet-transitive or isotopic figure is a n-dimensional polytopes or honeycomb, with its facets congruent and transitive. The dual of an isotope is an isogonal polytope, by definition, this isotopic property is common to the duals of the uniform polytopes. An isotopic 2-dimensional figure is isotoxal, an isotopic 3-dimensional figure is isohedral. An isotopic 4-dimensional figure is isochoric, edge-transitive Anisohedral tiling Peter R. Cromwell, Polyhedra, Cambridge University Press 1997, ISBN 0-521-55432-2, p.367 Transitivity Olshevsky, George. Archived from the original on 4 February 2007
9.
Congruence (geometry)
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In geometry, two figures or objects are congruent if they have the same shape and size, or if one has the same shape and size as the mirror image of the other. This means that either object can be repositioned and reflected so as to coincide precisely with the other object, so two distinct plane figures on a piece of paper are congruent if we can cut them out and then match them up completely. Turning the paper over is permitted, in elementary geometry the word congruent is often used as follows. The word equal is often used in place of congruent for these objects, two line segments are congruent if they have the same length. Two angles are congruent if they have the same measure, two circles are congruent if they have the same diameter. The related concept of similarity applies if the objects differ in size, for two polygons to be congruent, they must have an equal number of sides. Two polygons with n sides are congruent if and only if they each have identical sequences side-angle-side-angle-. for n sides. Congruence of polygons can be established graphically as follows, First, match, second, draw a vector from one of the vertices of the one of the figures to the corresponding vertex of the other figure. Translate the first figure by this vector so that two vertices match. Third, rotate the translated figure about the matched vertex until one pair of corresponding sides matches, fourth, reflect the rotated figure about this matched side until the figures match. If at any time the step cannot be completed, the polygons are not congruent, two triangles are congruent if their corresponding sides are equal in length, in which case their corresponding angles are equal in measure. SSS, If three pairs of sides of two triangles are equal in length, then the triangles are congruent, ASA, If two pairs of angles of two triangles are equal in measurement, and the included sides are equal in length, then the triangles are congruent. The ASA Postulate was contributed by Thales of Miletus, in most systems of axioms, the three criteria—SAS, SSS and ASA—are established as theorems. In the School Mathematics Study Group system SAS is taken as one of 22 postulates, AAS, If two pairs of angles of two triangles are equal in measurement, and a pair of corresponding non-included sides are equal in length, then the triangles are congruent. For American usage, AAS is equivalent to an ASA condition, RHS, also known as HL, If two right-angled triangles have their hypotenuses equal in length, and a pair of shorter sides are equal in length, then the triangles are congruent. The SSA condition which specifies two sides and a non-included angle does not by itself prove congruence, in order to show congruence, additional information is required such as the measure of the corresponding angles and in some cases the lengths of the two pairs of corresponding sides. The opposite side is longer when the corresponding angles are acute. This is the case and two different triangles can be formed from the given information, but further information distinguishing them can lead to a proof of congruence
10.
Crystal habit
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In mineralogy, crystal habit is the characteristic external shape of an individual crystal or crystal group. A single crystals habit is a description of its shape and its crystallographic forms. Recognizing the habit may help in identifying a mineral, when the faces are well-developed due to uncrowded growth a crystal is called euhedral, one with partially developed faces is subhedral, and one with undeveloped crystal faces is called anhedral. The long axis of a quartz crystal typically has a six-sided prismatic habit with parallel opposite faces. Aggregates can be formed of individual crystals with euhedral to anhedral grains, the arrangement of crystals within the aggregate can be characteristic of certain minerals. For example, minerals used for asbestos insulation often grow in a fibrous habit, the terms used by mineralogists to report crystal habits describe the typical appearance of an ideal mineral. Recognizing the habit can aid in identification as some habits are characteristic, most minerals, however, do not display ideal habits due to conditions during crystallization. Minerals belonging to the crystal system do not necessarily exhibit the same habit. Some habits of a mineral are unique to its variety and locality, For example, while most sapphires form elongate barrel-shaped crystals, ordinarily, the latter habit is seen only in ruby. Sapphire and ruby are both varieties of the mineral, corundum. Some minerals may replace other existing minerals while preserving the originals habit, a classic example is tigers eye quartz, crocidolite asbestos replaced by silica. While quartz typically forms prismatic crystals, in tigers eye the original fibrous habit of crocidolite is preserved, the names of crystal habits are derived from, Predominant crystal faces. Abnormal grain growth Grain growth Crystallization
11.
Mineral
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A mineral is a naturally occurring chemical compound, usually of crystalline form and abiogenic in origin. A mineral has one specific chemical composition, whereas a rock can be an aggregate of different minerals or mineraloids, the study of minerals is called mineralogy. There are over 5,300 known mineral species, over 5,070 of these have been approved by the International Mineralogical Association, the silicate minerals compose over 90% of the Earths crust. The diversity and abundance of species is controlled by the Earths chemistry. Silicon and oxygen constitute approximately 75% of the Earths crust, which translates directly into the predominance of silicate minerals, minerals are distinguished by various chemical and physical properties. Differences in chemical composition and crystal structure distinguish the various species, changes in the temperature, pressure, or bulk composition of a rock mass cause changes in its minerals. Minerals can be described by their various properties, which are related to their chemical structure. Common distinguishing characteristics include crystal structure and habit, hardness, lustre, diaphaneity, colour, streak, tenacity, cleavage, fracture, parting, more specific tests for describing minerals include magnetism, taste or smell, radioactivity and reaction to acid. Minerals are classified by key chemical constituents, the two dominant systems are the Dana classification and the Strunz classification, the silicate class of minerals is subdivided into six subclasses by the degree of polymerization in the chemical structure. All silicate minerals have a unit of a 4− silica tetrahedron—that is, a silicon cation coordinated by four oxygen anions. These tetrahedra can be polymerized to give the subclasses, orthosilicates, disilicates, cyclosilicates, inosilicates, phyllosilicates, other important mineral groups include the native elements, sulfides, oxides, halides, carbonates, sulfates, and phosphates. The first criterion means that a mineral has to form by a natural process, stability at room temperature, in the simplest sense, is synonymous to the mineral being solid. More specifically, a compound has to be stable or metastable at 25 °C, modern advances have included extensive study of liquid crystals, which also extensively involve mineralogy. Minerals are chemical compounds, and as such they can be described by fixed or a variable formula, many mineral groups and species are composed of a solid solution, pure substances are not usually found because of contamination or chemical substitution. Finally, the requirement of an ordered atomic arrangement is usually synonymous with crystallinity, however, crystals are also periodic, an ordered atomic arrangement gives rise to a variety of macroscopic physical properties, such as crystal form, hardness, and cleavage. There have been recent proposals to amend the definition to consider biogenic or amorphous substances as minerals. The formal definition of an approved by the IMA in 1995, A mineral is an element or chemical compound that is normally crystalline. However, if geological processes were involved in the genesis of the compound, Mineral classification schemes and their definitions are evolving to match recent advances in mineral science
12.
Deltoidal icositetrahedron
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In geometry, a deltoidal icositetrahedron is a Catalan solid which looks a bit like an overinflated cube. Its dual polyhedron is the rhombicuboctahedron, the short and long edges of each kite are in the ratio 1, ≈1,1.292893. The shape is called a trapezohedron in mineral contexts, although in solid geometry that name has another meaning. The deltoidal icositetrahedron has three positions, all centered on vertices, The great triakis octahedron is a stellation of the deltoidal icositetrahedron. The deltoidal icositetrahedron is topologically equivalent to a cube whose faces are divided in quadrants and it can also be projected onto a regular octahedron, with kite faces, or more general quadrilaterals with pyritohedral symmetry. In Conway polyhedron notation, they represent an ortho operation to a cube or octahedron, in crystallography a rotational variation is called a dyakis dodecahedron or diploid. The deltoidal icositetrahedron is one of a family of duals to the uniform polyhedra related to the cube and this polyhedron is topologically related as a part of sequence of deltoidal polyhedra with face figure, and continues as tilings of the hyperbolic plane. These face-transitive figures have reflectional symmetry, deltoidal hexecontahedron Tetrakis hexahedron, another 24-face Catalan solid which looks a bit like an overinflated cube. The Haunter of the Dark, a story by H. P, lovecraft, whose plot involves this figure Williams, Robert. The Geometrical Foundation of Natural Structure, A Source Book of Design, deltoidal Icositetrahedron – Interactive Polyhedron model
13.
Symmetry group
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In abstract algebra, the symmetry group of an object is the group of all transformations under which the object is invariant with composition as the group operation. For a space with a metric, it is a subgroup of the group of the space concerned. If not stated otherwise, this article considers symmetry groups in Euclidean geometry, the objects may be geometric figures, images, and patterns, such as a wallpaper pattern. The definition can be more precise by specifying what is meant by image or pattern. For symmetry of objects, one may also want to take their physical composition into account. The group of isometries of space induces an action on objects in it. The symmetry group is also called full symmetry group in order to emphasize that it includes the orientation-reversing isometries under which the figure is invariant. The subgroup of orientation-preserving isometries that leave the figure invariant is called its symmetry group. The proper symmetry group of an object is equal to its symmetry group if. The proper symmetry group is then a subgroup of the orthogonal group SO. A discrete symmetry group is a group such that for every point of the space the set of images of the point under the isometries in the symmetry group is a discrete set. There are also continuous symmetry groups, which contain rotations of arbitrarily small angles or translations of arbitrarily small distances, the group of all symmetries of a sphere O is an example of this, and in general such continuous symmetry groups are studied as Lie groups. With a categorization of subgroups of the Euclidean group corresponds a categorization of symmetry groups, for example, two 3D figures have mirror symmetry, but with respect to different mirror planes. Two 3D figures have 3-fold rotational symmetry, but with respect to different axes, two 2D patterns have translational symmetry, each in one direction, the two translation vectors have the same length but a different direction. When considering isometry groups, one may restrict oneself to those where for all points the set of images under the isometries is topologically closed. This includes all discrete isometry groups and also involved in continuous symmetries. A figure with this group is non-drawable and up to arbitrarily fine detail homogeneous. The group generated by all translations, this group cannot be the group of a pattern, it would be homogeneous
14.
Bipyramid
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An n-gonal bipyramid or dipyramid is a polyhedron formed by joining an n-gonal pyramid and its mirror image base-to-base. An n-gonal bipyramid has 2n triangle faces, 3n edges, and 2 + n vertices, the referenced n-gon in the name of the bipyramids is not an external face but an internal one, existing on the primary symmetry plane which connects the two pyramid halves. A right bipyramid has two points above and below the centroid of its base, nonright bipyramids are called oblique bipyramids. A regular bipyramid has a regular polygon internal face and is implied to be a right bipyramid. A right bipyramid can be represented as + P for internal polygon P, a concave bipyramid has a concave interior polygon. The face-transitive regular bipyramids are the dual polyhedra of the uniform prisms, a bipyramid can be projected on a sphere or globe as n equally spaced lines of longitude going from pole to pole, and bisected by a line around the equator. Bipyramid faces, projected as spherical triangles, represent the fundamental domains in the dihedral symmetry Dnh, the volume of a bipyramid is V =2/3Bh where B is the area of the base and h the height from the base to the apex. This works for any location of the apex, provided that h is measured as the distance from the plane which contains the base. The volume of a bipyramid whose base is a regular n-sided polygon with side length s and whose height is h is therefore, only three kinds of bipyramids can have all edges of the same length, the triangular, tetragonal, and pentagonal bipyramids. The rotation group is Dn of order 2n, except in the case of an octahedron, which has the larger symmetry group O of order 24. The digonal faces of a spherical 2n-bipyramid represents the fundamental domains of symmetry in three dimensions, Dnh, order 4n. The reflection domains can be shown as alternately colored triangles as mirror images, a scalenohedron is topologically identical to a 2n-bipyramid, but contains congruent scalene triangles. In one type the 2n vertices around the center alternate in rings above, in the other type, the 2n vertices are on the same plane, but alternate in two radii. The first has 2-fold rotation axes mid-edge around the sides, reflection planes through the vertices, in crystallography, 8-sided and 12-sided scalenohedra exist. All of these forms are isohedra, the second has symmetry Dn, order 2n. The smallest scalenohedron has 8 faces and is identical to the regular octahedron. The second type is a rhombic bipyramid, the first type has 6 vertices can be represented as, where z is a parameter between 0 and 1, creating a regular octahedron at z =0, and becoming a disphenoid with merged coplanar faces at z =1. For z >1, it becomes concave, self-intersecting bipyramids exist with a star polygon central figure, defined by triangular faces connecting each polygon edge to these two points
15.
Tetrahedron
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In geometry, a tetrahedron, also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra, the tetrahedron is the three-dimensional case of the more general concept of a Euclidean simplex. The tetrahedron is one kind of pyramid, which is a polyhedron with a polygon base. In the case of a tetrahedron the base is a triangle, like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper. For any tetrahedron there exists a sphere on which all four vertices lie, a regular tetrahedron is one in which all four faces are equilateral triangles. It is one of the five regular Platonic solids, which have known since antiquity. In a regular tetrahedron, not only are all its faces the same size and shape, regular tetrahedra alone do not tessellate, but if alternated with regular octahedra they form the alternated cubic honeycomb, which is a tessellation. The regular tetrahedron is self-dual, which means that its dual is another regular tetrahedron, the compound figure comprising two such dual tetrahedra form a stellated octahedron or stella octangula. This form has Coxeter diagram and Schläfli symbol h, the tetrahedron in this case has edge length 2√2. Inverting these coordinates generates the dual tetrahedron, and the together form the stellated octahedron. In other words, if C is the centroid of the base and this follows from the fact that the medians of a triangle intersect at its centroid, and this point divides each of them in two segments, one of which is twice as long as the other. The vertices of a cube can be grouped into two groups of four, each forming a regular tetrahedron, the symmetries of a regular tetrahedron correspond to half of those of a cube, those that map the tetrahedra to themselves, and not to each other. The tetrahedron is the only Platonic solid that is not mapped to itself by point inversion, the regular tetrahedron has 24 isometries, forming the symmetry group Td, isomorphic to the symmetric group, S4. The first corresponds to the A2 Coxeter plane, the two skew perpendicular opposite edges of a regular tetrahedron define a set of parallel planes. When one of these intersects the tetrahedron the resulting cross section is a rectangle. When the intersecting plane is one of the edges the rectangle is long. When halfway between the two edges the intersection is a square, the aspect ratio of the rectangle reverses as you pass this halfway point. For the midpoint square intersection the resulting boundary line traverses every face of the tetrahedron similarly, if the tetrahedron is bisected on this plane, both halves become wedges
16.
Trigonal trapezohedron
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In geometry, a trigonal trapezohedron or trigonal deltohedron is a three-dimensional figure formed by six congruent rhombi. Six identical rhombic faces can construct two configurations of trigonal trapezohedra, the acute or prolate form has three acute angles corners of the rhombic faces meeting at two polar axis vertices. The obtuse or oblate or flat form has three obtuse angle corners of the rhombic faces meeting at the two polar axis vertices, the trigonal trapezohedra is a special case of a rhombohedron. A general rhombohedron allows up to three types of rhombic faces, a trigonal trapezohedron is a special kind of parallelepiped, and are the only parallelepipeds with six congruent faces. Since all of the edges must have the length, every trigonal trapezohedron is also a rhombohedron. It is the simplest of the trapezohedra, a sequence of polyhedra which are dual to the antiprisms. The dual of a trigonal trapezohedron is a triangular antiprism, a trigonal trapezohedron with square faces is a cube. A lower symmetry variation of the trigonal trapezohedron has only rotational symmetry, D3, a golden rhombohedron is one of two special case of the trigonal trapezohedron with golden rhombus faces. The acute or prolate form has three acute angles corners of the rhombic faces meeting at two polar axis vertices, the obtuse or oblate or flat form has three obtuse angle corners of the rhombic faces meeting at the two polar axis vertices. Cartesian coordinates for a golden rhombohedron with one pole at the origin are, and vector additions thereof, the rhombic hexecontahedron can be constructed by 20 acute golden rhombohedra meeting at a point. A regular octahedron augumented by 2 regular tetrahedra creates a trigonal trapezohedron, truncated triangular trapezohedron Weisstein, Eric W. Trapezohedron
17.
Tetragonal trapezohedron
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The tetragonal trapezohedron, or deltohedron, is the second in an infinite series of face-uniform polyhedra, which are dual to the antiprisms. It has eight faces, which are congruent kites, and is dual to the square antiprism, adding four cuboids to a mesh for the cubical octahedron would also give a mesh for Schneiders pyramid. However, it is whether a decomposition of this type can be obtained in which all the cuboids are convex polyhedra with flat faces. The tetragonal trapezohedron is first in a series of snub polyhedra. Paper model tetragonal trapezohedron Weisstein, Eric W. Trapezohedron
18.
Pentagonal trapezohedron
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The pentagonal trapezohedron or deltohedron is the third in an infinite series of face-transitive polyhedra which are dual polyhedra to the antiprisms. It has ten faces which are congruent kites and it can be decomposed into two pentagonal pyramids and a pentagonal antiprism in the middle. It can also be decomposed into two pentagonal pyramids and a dodecahedron in the middle, the pentagonal trapezohedron was patented for use as a gaming die in 1906. Subsequent patents on ten-sided dice have made minor refinements to the design by rounding or truncating the edges. This enables the die to tumble so that the outcome is less predictable, one such refinement became notorious at the 1980 Gen Con when the patent was incorrectly thought to cover ten-sided dice in general. Ten-sided dice are commonly numbered from 0 to 9, as this allows two to be rolled in order to obtain a percentile result. Where one die represents the tens, the other represents units therefore a result of 7 on the former and 0 on the latter would be combined to produce 70, a result of double-zero is commonly interpreted as 100. Ten-sided dice may also be numbered 1 to 10 for use in games where a number in this range is desirable. A fairly consistent arrangement of the faces on ten-digit dice has been observed, the even and odd digits are divided among the two opposing caps of the die, and each pair of opposite faces adds to nine. When casting a 10-sided die, if numbered from 0-9, two are used to obtain a percentage roll. Rolling 2 of these are attributed in the results 00-99, where 00 can be viewed as a 100 as the result in some games. Alone casting a 0-9 ten sided dice, the 0 face is valued at 10, cundy H. M and Rollett, A. P. Mathematical models, 2nd Edn. Oxford University Press, p.117 Generalized formula of uniform polyhedron having 2n congruent right kite faces from Academia. edu Weisstein, virtual Reality Polyhedra www. georgehart. com, The Encyclopedia of Polyhedra VRML model Conway Notation for Polyhedra Try, dA5
19.
Hexagonal trapezohedron
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The hexagonal trapezohedron or deltohedron is the fourth in an infinite series of face-uniform polyhedra which are dual polyhedron to the antiprisms. It has twelve faces which are congruent kites, virtual Reality Polyhedra The Encyclopedia of Polyhedra VRML model <6>
20.
Octagonal trapezohedron
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The octagonal trapezohedron or deltohedron is the sixth in an infinite series of face-uniform polyhedra which are dual polyhedron to the antiprisms. It has sixteen faces which are congruent kites, virtual Reality Polyhedra The Encyclopedia of Polyhedra VRML model <8>
21.
Decagonal trapezohedron
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The decagonal trapezohedron is the eighth in an infinite series of face-uniform polyhedra which are dual polyhedron to the antiprisms. It has twenty faces which are congruent kites, virtual Reality Polyhedra The Encyclopedia of Polyhedra VRML model <10>
22.
Apeirogonal antiprism
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In geometry, an apeirogonal antiprism or infinite antiprism is the arithmetic limit of the family of antiprisms, it can be considered an infinite polyhedron or a tiling of the plane. If the sides are equilateral triangles, it is a uniform tiling, in general, it can have two sets of alternating congruent isosceles triangles, surrounded by two half-planes. The apeirogonal antiprism is the limit of the family of antiprisms sr or p.3.3.3, as p tends to infinity. Similarly to the uniform polyhedra and the uniform tilings, eight uniform tilings may be based from the regular apeirogonal tiling, the Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ISBN 978-1-56881-220-5 Grünbaum, Branko, Shephard, G. C. T. Gosset, On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan,1900
23.
Zonohedron
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A zonohedron is a convex polyhedron where every face is a polygon with point symmetry or, equivalently, symmetry under rotations through 180°. Any zonohedron may equivalently be described as the Minkowski sum of a set of segments in three-dimensional space. Zonohedra were originally defined and studied by E. S. Fedorov, more generally, in any dimension, the Minkowski sum of line segments forms a polytope known as a zonotope. The original motivation for studying zonohedra is that the Voronoi diagram of any lattice forms a uniform honeycomb in which the cells are zonohedra. Any zonohedron formed in this way can tessellate 3-dimensional space and is called a primary parallelohedron, each primary parallelohedron is combinatorially equivalent to one of five types, the rhombohedron, hexagonal prism, truncated octahedron, rhombic dodecahedron, and the rhombo-hexagonal dodecahedron. Let be a collection of three-dimensional vectors, with each vector vi we may associate a line segment. The Minkowski sum forms a zonohedron, and all zonohedra that contain the origin have this form, the vectors from which the zonohedron is formed are called its generators. This characterization allows the definition of zonohedra to be generalized to higher dimensions, each edge in a zonohedron is parallel to at least one of the generators, and has length equal to the sum of the lengths of the generators to which it is parallel. Therefore, by choosing a set of generators with no parallel pairs of vectors, by choosing sets of vectors with high degrees of symmetry, we can form in this way, zonohedra with at least as much symmetry. Generators parallel to the edges of an octahedron form a truncated octahedron, the Minkowski sum of any two zonohedra is another zonohedron, generated by the union of the generators of the two given zonohedra. Both of these zonohedra are simple, as is the truncated small rhombicuboctahedron formed from the Minkowski sum of the cube, truncated octahedron, conversely any arrangement of great circles may be formed from the Gauss map of a zonohedron generated by vectors perpendicular to the planes through the circles. Any simple zonohedron corresponds in this way to a simplicial arrangement, simplicial arrangements of great circles correspond via central projection to simplicial arrangements of lines in the projective plane, which were studied by Grünbaum. There are also many examples that do not fit into these three families. Any prism over a polygon with an even number of sides forms a zonohedron. These prisms can be formed so that all faces are regular, two faces are equal to the regular polygon from which the prism was formed. Zonohedra of this type are the cube, hexagonal prism, octagonal prism, decagonal prism, dodecagonal prism, the truncated cuboctahedron, with 12 squares,8 hexagons, and 6 octagons. The truncated icosidodecahedron, with 30 squares,20 hexagons and 12 decagons, in addition, certain Catalan solids are again zonohedra, The rhombic dodecahedron is the dual of the cuboctahedron. The rhombic triacontahedron is the dual of the icosidodecahedron, zonohedrification is a process defined by George W. Hart for creating a zonohedron from another polyhedron
24.
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
25.
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
26.
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
27.
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
28.
Tetrahedral-octahedral honeycomb
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The tetrahedral-octahedral honeycomb, alternated cubic honeycomb is a quasiregular space-filling tessellation in Euclidean 3-space. It is composed of alternating octahedra and tetrahedra in a ratio of 1,2, other names include half cubic honeycomb, half cubic cellulation, or tetragonal disphenoidal cellulation. John Horton Conway calls this honeycomb a tetroctahedrille, and its dual dodecahedrille and it is vertex-transitive with 8 tetrahedra and 6 octahedra around each vertex. It is edge-transitive with 2 tetrahedra and 2 octahedra alternating on each edge, a geometric honeycomb is a space-filling of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the general mathematical tiling or tessellation in any number of dimensions. Honeycombs are usually constructed in ordinary Euclidean space, like the uniform honeycombs. They may also be constructed in non-Euclidean spaces, such as hyperbolic uniform honeycombs, any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space. It is also part of another family of uniform honeycombs called simplectic honeycombs. In this case of 3-space, the honeycomb is alternated, reducing the cubic cells to tetrahedra. As such it can be represented by an extended Schläfli symbol h as containing half the vertices of the cubic honeycomb, theres a similar honeycomb called gyrated tetrahedral-octahedral honeycomb which has layers rotated 60 degrees so half the edges have neighboring rather than alternating tetrahedra and octahedra. Each slice will contain up and downward facing square pyramids and tetrahedra sitting on their edges, a second slice direction needs no new faces and includes alternating tetrahedral and octahedral. This slab honeycomb is a scaliform honeycomb rather than uniform because it has nonuniform cells, the alternated cubic honeycomb can be orthogonally projected into the planar square tiling by a geometric folding operation that maps one pairs of mirrors into each other. The projection of the cubic honeycomb creates two offset copies of the square tiling vertex arrangement of the plane, Its vertex arrangement represents an A3 lattice or D3 lattice. It is the 3-dimensional case of a simplectic honeycomb and its Voronoi cell is a rhombic dodecahedron, the dual of the cuboctahedron vertex figure for the tet-oct honeycomb. The D+3 packing can be constructed by the union of two D3 lattices, the D+ n packing is only a lattice for even dimensions. The kissing number of the D*3 lattice is 8 and its Voronoi tessellation is a cubic honeycomb. The, Coxeter group generates 15 permutations of uniform honeycombs,9 with distinct geometry including the cubic honeycomb. The expanded cubic honeycomb is geometrically identical to the cubic honeycomb, the, Coxeter group generates 9 permutations of uniform honeycombs,4 with distinct geometry including the alternated cubic honeycomb
29.
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
30.
Platonic solid
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In three-dimensional space, a Platonic solid is a regular, convex polyhedron. It is constructed by congruent regular polygonal faces with the number of faces meeting at each vertex. Five solids meet those criteria, Geometers have studied the mathematical beauty and they are named for the ancient Greek philosopher Plato who theorized in his dialogue, the Timaeus, that the classical elements were made of these regular solids. The Platonic solids have been known since antiquity, dice go back to the dawn of civilization with shapes that predated formal charting of Platonic solids. The ancient Greeks studied the Platonic solids extensively, some sources credit Pythagoras with their discovery. In any case, Theaetetus gave a description of all five. The Platonic solids are prominent in the philosophy of Plato, their namesake, Plato wrote about them in the dialogue Timaeus c.360 B. C. in which he associated each of the four classical elements with a regular solid. Earth was associated with the cube, air with the octahedron, water with the icosahedron, there was intuitive justification for these associations, the heat of fire feels sharp and stabbing. Air is made of the octahedron, its components are so smooth that one can barely feel it. Water, the icosahedron, flows out of hand when picked up. By contrast, a highly nonspherical solid, the hexahedron represents earth and these clumsy little solids cause dirt to crumble and break when picked up in stark difference to the smooth flow of water. Moreover, the cubes being the regular solid that tessellates Euclidean space was believed to cause the solidity of the Earth. Of the fifth Platonic solid, the dodecahedron, Plato obscurely remarks. the god used for arranging the constellations on the whole heaven. Aristotle added an element, aithēr and postulated that the heavens were made of this element. Euclid completely mathematically described the Platonic solids in the Elements, the last book of which is devoted to their properties, propositions 13–17 in Book XIII describe the construction of the tetrahedron, octahedron, cube, icosahedron, and dodecahedron in that order. For each solid Euclid finds the ratio of the diameter of the sphere to the edge length. In Proposition 18 he argues there are no further convex regular polyhedra. Andreas Speiser has advocated the view that the construction of the 5 regular solids is the goal of the deductive system canonized in the Elements
31.
Dice
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Dice are small throwable objects with multiple resting positions, used for generating random numbers. Dice are suitable as gambling devices for games like craps and are used in non-gambling tabletop games. A traditional die is a cube, with each of its six faces showing a different number of dots from 1 to 6. When thrown or rolled, the die comes to rest showing on its surface a random integer from one to six. A variety of devices are also described as dice, such specialized dice may have polyhedral or irregular shapes. They may be used to produce other than one through six. Loaded and crooked dice are designed to favor some results over others for purposes of cheating or amusement. A dice tray, a used to contain thrown dice, is sometimes used for gambling or board games. Dice have been used since before recorded history, and it is uncertain where they originated, the oldest known dice were excavated as part of a backgammon-like game set at the Burnt City, an archeological site in south-eastern Iran, estimated to be from between 2800–2500 BCE. Other excavations from ancient tombs in the Indus Valley civilization indicate a South Asian origin, the Egyptian game of Senet was played with dice. Senet was played before 3000 BC and up to the 2nd century AD and it was likely a racing game, but there is no scholarly consensus on the rules of Senet. Dicing is mentioned as an Indian game in the Rigveda, Atharvaveda, there are several biblical references to casting lots, as in Psalm 22, indicating that dicing was commonplace when the psalm was composed. Knucklebones was a game of skill played by women and children, although gambling was illegal, many Romans were passionate gamblers who enjoyed dicing, which was known as aleam ludere. Dicing was even a popular pastime of emperors, letters by Augustus to Tacitus and his daughter recount his hobby of dicing. There were two sizes of Roman dice, tali were large dice inscribed with one, three, four, and six on four sides. Tesserae were smaller dice with sides numbered one to six. Twenty-sided dice date back to the 2nd century AD and from Ptolemaic Egypt as early as the 2nd century BC, dominoes and playing cards originated in China as developments from dice. The transition from dice to playing cards occurred in China around the Tang dynasty, in Japan, dice were used to play a popular game called sugoroku
32.
Role-playing game
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A role-playing game is a game in which players assume the roles of characters in a fictional setting. Players take responsibility for acting out these roles within a narrative, actions taken within many games succeed or fail according to a formal system of rules and guidelines. There are several forms of RPG, the original form, sometimes called the tabletop RPG, is conducted through discussion, whereas in live action role-playing games players physically perform their characters actions. In both of these forms, an arranger called a game master usually decides on the rules and setting to be used, acting as referee, while each of the other players plays the role of a single character. Several varieties of RPG also exist in media, such as multi-player text-based MUDs and their graphics-based successors. These games often share settings and rules with tabletop RPGs, despite this variety of forms, some game forms such as trading card games and wargames that are related to role-playing games may not be included. Role-playing activity may sometimes be present in games, but it is not the primary focus. The term is sometimes used to describe roleplay simulation games and exercises used in teaching, training. Both authors and major publishers of tabletop role-playing games consider them to be a form of interactive and collaborative storytelling, events, characters, and narrative structure give a sense of a narrative experience, and the game need not have a strongly-defined storyline. Interactivity is the difference between role-playing games and traditional fiction. Whereas a viewer of a show is a passive observer. Such role-playing games extend an older tradition of storytelling games where a party of friends collaborate to create a story. Participants in a game will generate specific characters and an ongoing plot. A consistent system of rules and a more or less realistic campaign setting in games aids suspension of disbelief, the level of realism in games ranges from just enough internal consistency to set up a believable story or credible challenge up to full-blown simulations of real-world processes. There is also a variety of systems of rules and game settings. Games that emphasize plot and character interaction over game mechanics and combat sometimes prefer the name storytelling game and these types of games tend to minimize or altogether eliminate the use of dice or other randomizing elements. Some games are played with characters created before the game by the GM and this type of game is typically played at gaming conventions, or in standalone games that do not form part of a campaign. Tabletop and pen-and-paper RPGs are conducted through discussion in a social gathering
33.
Dungeons & Dragons
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Dungeons & Dragons is a fantasy tabletop role-playing game originally designed by Gary Gygax and Dave Arneson, and first published in 1974 by Tactical Studies Rules, Inc. The game has been published by Wizards of the Coast since 1997 and it was derived from miniature wargames with a variation of the Chainmail game serving as the initial rule system. D&Ds publication is commonly recognized as the beginning of modern role-playing games, D&D departs from traditional wargaming and assigns each player a specific character to play instead of a military formation. These characters embark upon imaginary adventures within a fantasy setting, a Dungeon Master serves as the games referee and storyteller, while maintaining the setting in which the adventures occur and playing the role of the inhabitants. The characters form a party that interacts with the settings inhabitants, together they solve dilemmas, engage in battles and gather treasure and knowledge. In the process the characters experience points to become increasingly powerful over a series of sessions. The early success of Dungeons & Dragons led to a proliferation of game systems. Despite this competition, D&D remains the leader in the role-playing game industry. In 1977, the game was split into two branches, the relatively rules-light game system of Dungeons & Dragons and the more structured, AD&D 2nd Edition was published in 1989. In 2000, the line of the game was discontinued. These rules formed the basis of the d20 System which is available under the Open Game License for use by other publishers, Dungeons & Dragons version 3.5 was released in June 2003, with a 4th edition in June 2008. A 5th edition was released during the half of 2014. The game has been supplemented by many adventures as well as commercial campaign settings suitable for use by regular gaming groups. The game has won awards and has been translated into many languages beyond the original English. Dungeons & Dragons is a structured yet open-ended role-playing game and it is normally played indoors with the participants seated around a tabletop. Typically, each player only a single character, which represents an individual in a fictional setting. During the course of play, each player directs the actions of their character, a game often continues over a series of meetings to complete a single adventure, and longer into a series of related gaming adventures, called a campaign. The results of the choices and the overall storyline for the game are determined by the Dungeon Master according to the rules of the game
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Discrete uniform distribution
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Another way of saying discrete uniform distribution would be a known, finite number of outcomes equally likely to happen. A simple example of the uniform distribution is throwing a fair die. The possible values are 1,2,3,4,5,6, if two dice are thrown and their values added, the resulting distribution is no longer uniform since not all sums have equal probability. The discrete uniform distribution itself is inherently non-parametric and it is convenient, however, to represent its values generally by an integer interval, so that a, b become the main parameters of the distribution. This problem is known as the German tank problem, following the application of maximum estimation to estimates of German tank production during World War II. The UMVU estimator for the maximum is given by N ^ = k +1 k m −1 = m + m k −1 where m is the maximum and k is the sample size. This can be seen as a simple case of maximum spacing estimation. This has a variance of 1 k ≈ N2 k 2 for small samples k ≪ N so a standard deviation of approximately N k, the sample maximum is the maximum likelihood estimator for the population maximum, but, as discussed above, it is biased. If samples are not numbered but are recognizable or markable, one can instead estimate population size via the capture-recapture method, see rencontres numbers for an account of the probability distribution of the number of fixed points of a uniformly distributed random permutation
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Numerical digit
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A digit is a numeric symbol used in combinations to represent numbers in positional numeral systems. The name digit comes from the fact that the 10 digits of the hands correspond to the 10 symbols of the common base 10 numeral system, i. e. the decimal digits. In a given system, if the base is an integer. For example, the system has ten digits, whereas binary has two digits. In a basic system, a numeral is a sequence of digits. Each position in the sequence has a value, and each digit has a value. The value of the numeral is computed by multiplying each digit in the sequence by its place value, each digit in a number system represents an integer. For example, in decimal the digit 1 represents the one, and in the hexadecimal system. A positional number system must have a digit representing the integers from zero up to, but not including, thus in the positional decimal system, the numbers 0 to 9 can be expressed using their respective numerals 0 to 9 in the rightmost units position. The Hindu–Arabic numeral system uses a decimal separator, commonly a period in English, or a comma in other European languages, to denote the place or units place. Each successive place to the left of this has a value equal to the place value of the previous digit times the base. Similarly, each place to the right of the separator has a place value equal to the place value of the previous digit divided by the base. For example, in the numeral 10, the total value of the number is 1 ten,0 ones,3 tenths, and 4 hundredths. Note that the zero, which contributes no value to the number, the place value of any given digit in a numeral can be given by a simple calculation, which in itself is a compliment to the logic behind numeral systems. And to the right, the digit is multiplied by the base raised by a negative n, for example, in the number 10. This system was established by the 7th century in India, but was not yet in its modern form because the use of the digit zero had not yet widely accepted. Instead of a zero, a dot was left in the numeral as a placeholder, the first widely acknowledged use of zero was in 876. The original numerals were very similar to the ones, even down to the glyphs used to represent digits
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Star polygon
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In geometry, a star polygon is a type of non-convex polygon. Only the regular polygons have been studied in any depth. The first usage is included in polygrams which includes polygons like the pentagram, star polygon names combine a numeral prefix, such as penta-, with the Greek suffix -gram. The prefix is normally a Greek cardinal, but synonyms using other prefixes exist, for example, a nine-pointed polygon or enneagram is also known as a nonagram, using the ordinal nona from Latin. The -gram suffix derives from γραμμή meaning a line, alternatively for integers p and q, it can be considered as being constructed by connecting every qth point out of p points regularly spaced in a circular placement. A regular star polygon is denoted by its Schläfli symbol, where p and q are relatively prime, the symmetry group of is dihedral group Dn of order 2n, independent of k. A regular star polygon can also be obtained as a sequence of stellations of a regular core polygon. Regular star polygons were first studied systematically by Thomas Bradwardine, if p and q are not coprime, a degenerate polygon will result with coinciding vertices and edges. For example will appear as a triangle, but can be labeled with two sets of vertices 1-6 and this should be seen not as two overlapping triangles, but a double-winding of a single unicursal hexagon. For |n/d|, the vertices have an exterior angle, β. These polygons are often seen in tiling patterns, the parametric angle α can be chosen to match internal angles of neighboring polygons in a tessellation pattern. The interior of a polygon may be treated in different ways. Three such treatments are illustrated for a pentagram, branko Grunbaum and Geoffrey Shephard consider two of them, as regular star polygons and concave isogonal 2n-gons. These include, Where a side occurs, one side is treated as outside and this is shown in the left hand illustration and commonly occurs in computer vector graphics rendering. The number of times that the polygonal curve winds around a given region determines its density, the exterior is given a density of 0, and any region of density >0 is treated as internal. This is shown in the illustration and commonly occurs in the mathematical treatment of polyhedra. Where a line may be drawn between two sides, the region in which the line lies is treated as inside the figure and this is shown in the right hand illustration and commonly occurs when making a physical model. When the area of the polygon is calculated, each of these approaches yields a different answer, star polygons feature prominently in art and culture