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66-650: Agate is a semi-precious stone. Agate may also refer to: Agate The stone was given its name by Theophrastus , a Greek philosopher and naturalist , who discovered the stone along the shore line of the Dirillo River or Achates ( Ancient Greek : Ἀχάτης ) in Sicily , sometime between the 4th and 3rd centuries BCE. Agate minerals have the tendency to form on or within existing rocks, creating difficulties in accurately determining their time of formation. Their host rocks have been dated to have formed as early as

132-476: A polyhedron ( pl. : polyhedra or polyhedrons ; from Greek πολύ (poly-)  'many' and ἕδρον (-hedron)  'base, seat') is a three-dimensional figure with flat polygonal faces , straight edges and sharp corners or vertices . A convex polyhedron is a polyhedron that bounds a convex set . Every convex polyhedron can be constructed as the convex hull of its vertices, and for every finite set of points, not all on

198-518: A characteristic layering of concentric polygons . It has been suggested that growth is not crystallographically controlled but is due to the filling-in of spaces between pre-existing crystals which have since dissolved. Iris agate is a finely-banded and usually colorless agate, that when thinly sliced, exhibits spectral decomposition of white light into its constituent colors, requiring 400 to up to 30,000 bands per inch. Other forms of agate include Holley blue agate (also spelled "Holly blue agate"),

264-439: A convex polyhedron can be obtained by the process of polar reciprocation . Dual polyhedra exist in pairs, and the dual of a dual is just the original polyhedron again. Some polyhedra are self-dual, meaning that the dual of the polyhedron is congruent to the original polyhedron. Abstract polyhedra also have duals, obtained by reversing the partial order defining the polyhedron to obtain its dual or opposite order . These have

330-498: A convex polyhedron, or more generally any simply connected polyhedron with surface a topological sphere, it always equals 2. For more complicated shapes, the Euler characteristic relates to the number of toroidal holes, handles or cross-caps in the surface and will be less than 2. All polyhedra with odd-numbered Euler characteristic are non-orientable. A given figure with even Euler characteristic may or may not be orientable. For example,

396-509: A highly polished surface finish and resistance to chemical attack. It has traditionally been used to make knife-edge bearings for laboratory balances and precision pendulums, and sometimes to make mortars and pestles to crush and mix chemicals. Respiratory diseases such as silicosis , and a higher incidence of tuberculosis among workers involved in the agate industry, have been studied in India and China. Polyhedron In geometry ,

462-418: A list that includes many of these formulas.) Volumes of more complicated polyhedra may not have simple formulas. Volumes of such polyhedra may be computed by subdividing the polyhedron into smaller pieces (for example, by triangulation ). For example, the volume of a regular polyhedron can be computed by dividing it into congruent pyramids , with each pyramid having a face of the polyhedron as its base and

528-596: A moss-like pattern and is of a greenish colour. The coloration is not created by any vegetative growth, but rather through the mixture of chalcedony and oxidized iron hornblende. Dendritic agate also displays vegetative features, including fern-like patterns formed due to the presence of manganese and iron oxides. Turritella agate ( Elimia tenera ) is formed from the shells of fossilized freshwater Turritella gastropods with elongated spiral shells. Similarly, coral , petrified wood , porous rocks and other organic remains can also form agate. Coldwater agates , such as

594-423: A polyhedron into a single number χ {\displaystyle \chi } defined by the formula The same formula is also used for the Euler characteristic of other kinds of topological surfaces. It is an invariant of the surface, meaning that when a single surface is subdivided into vertices, edges, and faces in more than one way, the Euler characteristic will be the same for these subdivisions. For

660-399: A polyhedron is called its symmetry group . All the elements that can be superimposed on each other by symmetries are said to form a symmetry orbit . For example, all the faces of a cube lie in one orbit, while all the edges lie in another. If all the elements of a given dimension, say all the faces, lie in the same orbit, the figure is said to be transitive on that orbit. For example, a cube

726-411: A polyhedron that can be constructed by attaching more elementary polyhedrons. For example, triaugmented triangular prism is a composite polyhedron since it can be constructed by attaching three equilateral square pyramids onto the square faces of a triangular prism ; the square pyramids and the triangular prism are elementary. A midsphere of a convex polyhedron is a sphere tangent to every edge of

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792-411: A polyhedron to create new faces—or facets—without creating any new vertices). A facet of a polyhedron is any polygon whose corners are vertices of the polyhedron, and is not a face . The stellation and faceting are inverse or reciprocal processes: the dual of some stellation is a faceting of the dual to the original polyhedron. Polyhedra may be classified and are often named according to

858-422: A polyhedron, an intermediate sphere in radius between the insphere and circumsphere , for polyhedra for which all three of these spheres exist. Every convex polyhedron is combinatorially equivalent to a canonical polyhedron , a polyhedron that has a midsphere whose center coincides with the centroid of the polyhedron. The shape of the canonical polyhedron (but not its scale or position) is uniquely determined by

924-410: A rare dark blue ribbon agate found only near Holley, Oregon ; Lake Superior agate ; Carnelian agate (has reddish hues); Botswana agate ; plume agate ; condor agate ; tube agate containing visible flow channels or pinhole-sized "tubes"; fortification agate with contrasting concentric banding reminiscent of defensive ditches and walls around ancient forts; Binghamite , a variety found only on

990-431: A shared edge) and that every vertex is incident to a single alternating cycle of edges and faces (disallowing shapes like the union of two cubes sharing only a single vertex). For polyhedra defined in these ways, the classification of manifolds implies that the topological type of the surface is completely determined by the combination of its Euler characteristic and orientability. For example, every polyhedron whose surface

1056-424: A single symmetry orbit: Some classes of polyhedra have only a single main axis of symmetry. These include the pyramids , bipyramids , trapezohedra , cupolae , as well as the semiregular prisms and antiprisms. Regular polyhedra are the most highly symmetrical. Altogether there are nine regular polyhedra: five convex and four star polyhedra. The five convex examples have been known since antiquity and are called

1122-450: Is a polyhedron that forms a convex set as a solid. That being said, it is a three-dimensional solid whose every line segment connects two of its points lies its interior or on its boundary ; none of its faces are coplanar (they do not share the same plane) and none of its edges are collinear (they are not segments of the same line). A convex polyhedron can also be defined as a bounded intersection of finitely many half-spaces , or as

1188-456: Is a polyhedron with symmetries acting transitively on its faces. Their topology can be represented by a face configuration . All 5 Platonic solids and 13 Catalan solids are isohedra, as well as the infinite families of trapezohedra and bipyramids . Some definitions of isohedra allow geometric variations including concave and self-intersecting forms. Many of the symmetries or point groups in three dimensions are named after polyhedra having

1254-466: Is also still used today for decorative displays, cabochons, beads, carvings and Intarsia art as well as face-polished and tumble-polished specimens of varying size and origin. Idar-Oberstein was one of the centers which made use of agate on an industrial scale. Where in the beginning locally found agates were used to make all types of objects for the European market, this became a globalized business around

1320-435: Is an orientable manifold and whose Euler characteristic is 2 must be a topological sphere. A toroidal polyhedron is a polyhedron whose Euler characteristic is less than or equal to 0, or equivalently whose genus is 1 or greater. Topologically, the surfaces of such polyhedra are torus surfaces having one or more holes through the middle. For every convex polyhedron, there exists a dual polyhedron having The dual of

1386-466: Is called a lattice polyhedron or integral polyhedron . The Ehrhart polynomial of a lattice polyhedron counts how many points with integer coordinates lie within a scaled copy of the polyhedron, as a function of the scale factor. The study of these polynomials lies at the intersection of combinatorics and commutative algebra . There is a far-reaching equivalence between lattice polyhedra and certain algebraic varieties called toric varieties . This

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1452-460: Is common instead to slice the polyhedron by a small sphere centered at the vertex. Again, this produces a shape for the vertex figure that is invariant up to scaling. All of these choices lead to vertex figures with the same combinatorial structure, for the polyhedra to which they can be applied, but they may give them different geometric shapes. The surface area of a polyhedron is the sum of areas of its faces, for definitions of polyhedra for which

1518-414: Is face-transitive, while a truncated cube has two symmetry orbits of faces. The same abstract structure may support more or less symmetric geometric polyhedra. But where a polyhedral name is given, such as icosidodecahedron , the most symmetrical geometry is often implied. There are several types of highly symmetric polyhedron, classified by which kind of element – faces, edges, or vertices – belong to

1584-507: Is found in Africa and is especially hard. Crazy lace agate, typically found in Mexico, is often brightly colored with a complex pattern, demonstrating randomized distribution of contour lines and circular droplets, scattered throughout the rock. The stone is typically coloured red and white but is also seen to exhibit yellow and grey combinations as well. Moss agate , as the name suggests, exhibits

1650-483: Is known as the bellows theorem. A polyhedral compound is made of two or more polyhedra sharing a common centre. Symmetrical compounds often share the same vertices as other well-known polyhedra and may often also be formed by stellation. Some are listed in the list of Wenninger polyhedron models . An orthogonal polyhedron is one all of whose edges are parallel to axes of a Cartesian coordinate system. This implies that all faces meet at right angles , but this condition

1716-474: Is no universal agreement over which of these to choose. Some of these definitions exclude shapes that have often been counted as polyhedra (such as the self-crossing polyhedra ) or include shapes that are often not considered as valid polyhedra (such as solids whose boundaries are not manifolds ). As Branko Grünbaum observed, "The Original Sin in the theory of polyhedra goes back to Euclid, and through Kepler, Poinsot, Cauchy and many others ... at each stage ...

1782-644: Is one of the most common materials used in the art of hardstone carving , and has been recovered at a number of ancient sites, indicating its widespread use in the ancient world; for example, archaeological recovery at the Knossos site on Crete illustrates its role in Bronze Age Minoan culture. It has also been used for centuries for leather burnishing tools. The decorative arts use it to make ornaments such as pins , brooches or other types of jewellery , paper knives, inkstands , marbles and seals . Agate

1848-437: Is orientable or non-orientable by considering a topological cell complex with the same incidences between its vertices, edges, and faces. A more subtle distinction between polyhedron surfaces is given by their Euler characteristic , which combines the numbers of vertices V {\displaystyle V} , edges E {\displaystyle E} , and faces F {\displaystyle F} of

1914-456: Is weaker: Jessen's icosahedron has faces meeting at right angles, but does not have axis-parallel edges. Aside from the rectangular cuboids , orthogonal polyhedra are nonconvex. They are the 3D analogs of 2D orthogonal polygons, also known as rectilinear polygons . Orthogonal polyhedra are used in computational geometry , where their constrained structure has enabled advances on problems unsolved for arbitrary polyhedra, for example, unfolding

1980-455: The Archean Eon. Agates are most commonly found as nodules within the cavities of volcanic rocks . These cavities are formed from the gases trapped within the liquid volcanic material forming vesicles . Cavities are then filled in with silica-rich fluids from the volcanic material, layers are deposited on the walls of the cavity slowly working their way inwards. The first layer deposited on

2046-506: The Minkowski sums of line segments, and include several important space-filling polyhedra. A space-filling polyhedron packs with copies of itself to fill space. Such a close-packing or space-filling is often called a tessellation of space or a honeycomb. Space-filling polyhedra must have a Dehn invariant equal to zero. Some honeycombs involve more than one kind of polyhedron. A convex polyhedron in which all vertices have integer coordinates

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2112-509: The Platonic solids . These are the triangular pyramid or tetrahedron , cube , octahedron , dodecahedron and icosahedron : There are also four regular star polyhedra, known as the Kepler–Poinsot polyhedra after their discoverers. The dual of a regular polyhedron is also regular. Uniform polyhedra are vertex-transitive and every face is a regular polygon . They may be subdivided into

2178-712: The convex hull of finitely many points, in either case, restricted to intersections or hulls that have nonzero volume. Important classes of convex polyhedra include the family of prismatoid , the Platonic solids , the Archimedean solids and their duals the Catalan solids , and the regular polygonal faces polyhedron. The prismatoids are the polyhedron whose vertices lie on two parallel planes and their faces are likely to be trapezoids and triangles. Examples of prismatoids are pyramids , wedges , parallelipipeds , prisms , antiprisms , cupolas , and frustums . The Platonic solids are

2244-449: The deltahedron (whose faces are all equilateral triangles and Johnson solids (whose faces are arbitrary regular polygons). The convex polyhedron can be categorized into elementary polyhedron or composite polyhedron. An elementary polyhedron is a convex regular-faced polyhedron that cannot be produced into two or more polyhedrons by slicing it with a plane. Quite opposite to a composite polyhedron, it can be alternatively defined as

2310-575: The regular , quasi-regular , or semi-regular , and may be convex or starry. The duals of the uniform polyhedra have irregular faces but are face-transitive , and every vertex figure is a regular polygon. A uniform polyhedron has the same symmetry orbits as its dual, with the faces and vertices simply swapped over. The duals of the convex Archimedean polyhedra are sometimes called the Catalan solids . The uniform polyhedra and their duals are traditionally classified according to their degree of symmetry, and whether they are convex or not. An isohedron

2376-406: The tetrahemihexahedron , it is not possible to colour the two sides of each face with two different colours so that adjacent faces have consistent colours. In this case the polyhedron is said to be non-orientable. For polyhedra with self-crossing faces, it may not be clear what it means for adjacent faces to be consistently coloured, but for these polyhedra it is still possible to determine whether it

2442-560: The Cuyuna iron range (near Crosby) in Crow Wing County, Minnesota; fire agate showing an iridescent, internal flash or "fire", the result of a layer of clear agate over a layer of hydrothermally deposited hematite; Patuxent River stone , a red and yellow form of agate only found in Maryland ; and enhydro agate , which contains tiny inclusions of water, sometimes with air bubbles. Agate

2508-491: The Lake Michigan cloud agate, did not form under volcanic processes, but instead formed within the limestone and dolomite strata of marine origin. Like volcanic-origin agates, Coldwater agates formed from silica gels that lined pockets and seams within the bedrock. These agates are typically less colorful, with banded lines of grey and white chalcedony. Greek agate is a name given to pale white to tan colored agate found in

2574-399: The area of a face is well-defined. The geodesic distance between any two points on the surface of a polyhedron measures the length of the shortest curve that connects the two points, remaining within the surface. By Alexandrov's uniqueness theorem , every convex polyhedron is uniquely determined by the metric space of geodesic distances on its surface. However, non-convex polyhedra can have

2640-416: The associated symmetry. These include: Those with chiral symmetry do not have reflection symmetry and hence have two enantiomorphous forms which are reflections of each other. Examples include the snub cuboctahedron and snub icosidodecahedron . A zonohedron is a convex polyhedron in which every face is a polygon that is symmetric under rotations through 180°. Zonohedra can also be characterized as

2706-427: The cavity completely. Agate will form crystals within the reduced cavity, and the apex of each crystal may point towards the center of the cavity. The priming layer is often dark green, but can be modified by iron oxide resulting in a rust like appearance. Agate is very durable, and is often found detached from its host matrix, which has eroded. Once removed, the outer surface is usually pitted and rough from filling

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2772-448: The cavity of its former matrix. Agates have also been found in sedimentary rocks , normally in limestone or dolomite ; these sedimentary rocks acquire cavities often from decomposed branches or other buried organic material. If silica-rich fluids are able to penetrate into these cavities agates can be formed. Lace agate is a variety that exhibits a lace -like pattern with forms such as eyes, swirls, bands or zigzags. Blue lace agate

2838-443: The cavity walls is commonly known as the priming layer. Variations in the character of the solution or in the conditions of deposition may cause a corresponding variation in the successive layers. These variations in layers result in bands of chalcedony , often alternating with layers of crystalline quartz forming banded agate. Hollow agates can also form due to the deposition of liquid-rich silica not penetrating deep enough to fill

2904-439: The centre of the polyhedron as its apex. In general, it can be derived from the divergence theorem that the volume of a polyhedral solid is given by 1 3 | ∑ F ( Q F ⋅ N F ) area ⁡ ( F ) | , {\displaystyle {\frac {1}{3}}\left|\sum _{F}(Q_{F}\cdot N_{F})\operatorname {area} (F)\right|,} where

2970-458: The column for Greek cardinal numbers. The names of tetrahedra, hexahedra, octahedra (8-sided polyhedra), dodecahedra (12-sided polyhedra), and icosahedra (20-sided polyhedra) are sometimes used without additional qualification to refer to the Platonic solids , and sometimes used to refer more generally to polyhedra with the given number of sides without any assumption of symmetry. Some polyhedra have two distinct sides to their surface. For example,

3036-411: The combinatorial structure of the given polyhedron. Some polyhedrons do not have the property of convexity, and they are called non-convex polyhedrons . Such polyhedrons are star polyhedrons and Kepler–Poinsot polyhedrons , which constructed by either stellation (process of extending the faces—within their planes—so that they meet) or faceting (whose process of removing parts of

3102-456: The five ancientness polyhedrons— tetrahedron , octahedron , icosahedron , cube , and dodecahedron —classified by Plato in his Timaeus whose connecting four classical elements of nature. The Archimedean solids are the class of thirteen polyhedrons whose faces are all regular polygons and whose vertices are symmetric to each other; their dual polyhedrons are Catalan solids . The class of regular polygonal faces polyhedron are

3168-402: The former Greek colony of Sicily as early as 400 BCE. The Greeks used it for making jewelry and beads. Brazilian agate is found as sizable geodes of layered nodules. These occur in brownish tones inter-layered with white and gray. It is often dyed in various colors for ornamental purposes. Polyhedroid agate forms in a flat-sided shape similar to a polyhedron . When sliced, it often shows

3234-552: The inside and outside of a convex polyhedron paper model can each be given a different colour (although the inside colour will be hidden from view). These polyhedra are orientable . The same is true for non-convex polyhedra without self-crossings. Some non-convex self-crossing polyhedra can be coloured in the same way but have regions turned "inside out" so that both colours appear on the outside in different places; these are still considered to be orientable. However, for some other self-crossing polyhedra with simple-polygon faces, such as

3300-440: The literature on higher-dimensional geometry uses the term "polyhedron" to mean something else: not a three-dimensional polytope, but a shape that is different from a polytope in some way. For instance, some sources define a convex polyhedron to be the intersection of finitely many half-spaces , and a polytope to be a bounded polyhedron. The remainder of this article considers only three-dimensional polyhedra. A convex polyhedron

3366-565: The local structure of the polyhedron around the vertex. Precise definitions vary, but a vertex figure can be thought of as the polygon exposed where a slice through the polyhedron cuts off a vertex. For the Platonic solids and other highly-symmetric polyhedra, this slice may be chosen to pass through the midpoints of each edge incident to the vertex, but other polyhedra may not have a plane through these points. For convex polyhedra, and more generally for polyhedra whose vertices are in convex position , this slice can be chosen as any plane separating

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3432-566: The number of faces. The naming system is based on Classical Greek, and combines a prefix counting the faces with the suffix "hedron", meaning "base" or "seat" and referring to the faces. For example a tetrahedron is a polyhedron with four faces, a pentahedron is a polyhedron with five faces, a hexahedron is a polyhedron with six faces, etc. For a complete list of the Greek numeral prefixes see Numeral prefix § Table of number prefixes in English , in

3498-481: The one-holed toroid and the Klein bottle both have χ = 0 {\displaystyle \chi =0} , with the first being orientable and the other not. For many (but not all) ways of defining polyhedra, the surface of the polyhedron is required to be a manifold . This means that every edge is part of the boundary of exactly two faces (disallowing shapes like the union of two cubes that meet only along

3564-486: The polyhedron as a solid, whether they describe it as a surface, or whether they describe it more abstractly based on its incidence geometry . In all of these definitions, a polyhedron is typically understood as a three-dimensional example of the more general polytope in any number of dimensions. For example, a polygon has a two-dimensional body and no faces, while a 4-polytope has a four-dimensional body and an additional set of three-dimensional "cells". However, some of

3630-430: The same Euler characteristic and orientability as the initial polyhedron. However, this form of duality does not describe the shape of a dual polyhedron, but only its combinatorial structure. For some definitions of non-convex geometric polyhedra, there exist polyhedra whose abstract duals cannot be realized as geometric polyhedra under the same definition. For every vertex one can define a vertex figure , which describes

3696-601: The same plane, the convex hull is a convex polyhedron. Cubes and pyramids are examples of convex polyhedra. A polyhedron is a generalization of a 2-dimensional polygon and a 3-dimensional specialization of a polytope , a more general concept in any number of dimensions . Convex polyhedra are well-defined, with several equivalent standard definitions. However, the formal mathematical definition of polyhedra that are not required to be convex has been problematic. Many definitions of "polyhedron" have been given within particular contexts, some more rigorous than others, and there

3762-442: The same surface distances as each other, or the same as certain convex polyhedra. Polyhedral solids have an associated quantity called volume that measures how much space they occupy. Simple families of solids may have simple formulas for their volumes; for example, the volumes of pyramids, prisms, and parallelepipeds can easily be expressed in terms of their edge lengths or other coordinates. (See Volume § Volume formulas for

3828-465: The same volume that cannot be cut into smaller polyhedra and reassembled into each other. To prove this Dehn discovered another value associated with a polyhedron, the Dehn invariant , such that two polyhedra can only be dissected into each other when they have the same volume and the same Dehn invariant. It was later proven by Sydler that this is the only obstacle to dissection: every two Euclidean polyhedra with

3894-494: The same volumes and Dehn invariants can be cut up and reassembled into each other. The Dehn invariant is not a number, but a vector in an infinite-dimensional vector space, determined from the lengths and dihedral angles of a polyhedron's edges. Another of Hilbert's problems, Hilbert's 18th problem , concerns (among other things) polyhedra that tile space . Every such polyhedron must have Dehn invariant zero. The Dehn invariant has also been connected to flexible polyhedra by

3960-491: The strong bellows theorem, which states that the Dehn invariant of any flexible polyhedron remains invariant as it flexes. Many of the most studied polyhedra are highly symmetrical , that is, their appearance is unchanged by some reflection or rotation of space. Each such symmetry may change the location of a given vertex, face, or edge, but the set of all vertices (likewise faces, edges) is unchanged. The collection of symmetries of

4026-450: The sum is over faces F of the polyhedron, Q F is an arbitrary point on face F , N F is the unit vector perpendicular to F pointing outside the solid, and the multiplication dot is the dot product . In higher dimensions, volume computation may be challenging, in part because of the difficulty of listing the faces of a convex polyhedron specified only by its vertices, and there exist specialized algorithms to determine

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4092-533: The turn of the 20th century: Idar-Oberstein imported large quantities of agate from Brazil, as ship's ballast. Making use of a variety of proprietary chemical processes, they produced colored beads that were sold around the globe. Agates have long been used in arts and crafts. The sanctuary of a Presbyterian church in Yachats, Oregon , has six windows with panes made of agates collected from the local beaches. Industrial uses of agate exploit its hardness, ability to retain

4158-420: The vertex from the other vertices. When the polyhedron has a center of symmetry, it is standard to choose this plane to be perpendicular to the line through the given vertex and the center; with this choice, the shape of the vertex figure is determined up to scaling. When the vertices of a polyhedron are not in convex position, there will not always be a plane separating each vertex from the rest. In this case, it

4224-475: The volume in these cases. In two dimensions, the Bolyai–Gerwien theorem asserts that any polygon may be transformed into any other polygon of the same area by cutting it up into finitely many polygonal pieces and rearranging them . The analogous question for polyhedra was the subject of Hilbert's third problem . Max Dehn solved this problem by showing that, unlike in the 2-D case, there exist polyhedra of

4290-474: The writers failed to define what are the polyhedra". Nevertheless, there is general agreement that a polyhedron is a solid or surface that can be described by its vertices (corner points), edges (line segments connecting certain pairs of vertices), faces (two-dimensional polygons ), and that it sometimes can be said to have a particular three-dimensional interior volume . One can distinguish among these different definitions according to whether they describe

4356-520: Was used by Stanley to prove the Dehn–Sommerville equations for simplicial polytopes . It is possible for some polyhedra to change their overall shape, while keeping the shapes of their faces the same, by varying the angles of their edges. A polyhedron that can do this is called a flexible polyhedron. By Cauchy's rigidity theorem , flexible polyhedra must be non-convex. The volume of a flexible polyhedron must remain constant as it flexes; this result

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