1.
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
2.
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
3.
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
4.
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
5.
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
6.
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
7.
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
8.
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
9.
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
10.
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