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92-632: A tessellation or tiling is the covering of a surface , often a plane , using one or more geometric shapes , called tiles , with no overlaps and no gaps. In mathematics , tessellation can be generalized to higher dimensions and a variety of geometries. A periodic tiling has a repeating pattern. Some special kinds include regular tilings with regular polygonal tiles all of the same shape, and semiregular tilings with regular tiles of more than one shape and with every corner identically arranged. The patterns formed by periodic tilings can be categorized into 17 wallpaper groups . A tiling that lacks

184-466: A lap —at the beginning of a course . In some cases these special shapes or sizes are manufactured. In the diagrams below, some of the cuts most commonly used for generating a lap are coloured as follows: Less frequently used cuts are all coloured as follows: A nearly universal rule in brickwork is that perpends should not be contiguous across courses . Walls, running linearly and extending upwards, can be of varying depth or thickness. Typically,

276-635: A parallelogram subtended by a minimal set of translation vectors, starting from a rotational centre. We can divide this by one diagonal, and take one half (a triangle) as fundamental domain. Such a triangle has the same area as the quadrilateral and can be constructed from it by cutting and pasting. If only one shape of tile is allowed, tilings exist with convex N -gons for N equal to 3, 4, 5, and 6. For N = 5 , see Pentagonal tiling , for N = 6 , see Hexagonal tiling , for N = 7 , see Heptagonal tiling and for N = 8 , see octagonal tiling . With non-convex polygons, there are far fewer limitations in

368-527: A brick wall . Bricks may be differentiated from blocks by size. For example, in the UK a brick is defined as a unit having dimensions less than 337.5 mm × 225 mm × 112.5 mm (13.3 in × 8.9 in × 4.4 in) and a block is defined as a unit having one or more dimensions greater than the largest possible brick. Brick is a popular medium for constructing buildings, and examples of brickwork are found through history as far back as

460-458: A cavity, load-bearing requirements, expense, and the era during which the architect was or is working. Wall thickness specification has proven considerably various, and while some non-load-bearing brick walls may be as little as half a brick thick, or even less when shiners are laid stretcher bond in partition walls, others brick walls are much thicker. The Monadnock Building in Chicago, for example,

552-534: A challenging task for the bricklayer to correctly maintain while constructing a wall whose courses are partially obscured by scaffold, and interrupted by door or window openings, or other bond-disrupting obstacles. If the bricklayer frequently stops to check that bricks are correctly arranged, then masonry in a raking monk bond can be expensive to build. Occasionally, brickwork in such a raking monk bond may contain minor errors of header and stretcher alignment some of which may have been silently corrected by incorporating

644-408: A compensating irregularity into the brickwork in a course further up the wall. In spite of these complexities and their associated costs, the bond has proven a common choice for constructing brickwork in the north of Europe. Raking courses in monk bond may—for instance—be staggered in such a way as to generate the appearance of diagonal lines of stretchers. One method of achieving this effect relies on

736-434: A course begins with a quoin stretcher, the course will ordinarily terminate with a quoin stretcher at the other end. The next course up will begin with a quoin header. For the course's second brick, a queen closer is laid, generating the lap of the bond. The third brick along is a stretcher, and is—on account of the lap—centred above the header below. This second course then resumes its paired run of stretcher and header, until

828-498: A curve of positive length. The colouring guaranteed by the four colour theorem does not generally respect the symmetries of the tessellation. To produce a colouring that does, it is necessary to treat the colours as part of the tessellation. Here, as many as seven colours may be needed, as demonstrated in the image at left. Next to the various tilings by regular polygons , tilings by other polygons have also been studied. Any triangle or quadrilateral (even non-convex ) can be used as

920-595: A degree of self-organisation being observed using micro and nanotechnologies . The honeycomb is a well-known example of tessellation in nature with its hexagonal cells. In botany, the term "tessellate" describes a checkered pattern, for example on a flower petal, tree bark, or fruit. Flowers including the fritillary , and some species of Colchicum , are characteristically tessellate. Many patterns in nature are formed by cracks in sheets of materials. These patterns can be described by Gilbert tessellations , also known as random crack networks. The Gilbert tessellation

1012-411: A fitting aesthetic finish. Despite there being no masonry connection between the leaves, their transverse rigidity still needs to be guaranteed. The device used to satisfy this need is the insertion at regular intervals of wall ties into the cavity wall's mortar beds. Flemish bond has one stretcher between headers, with the headers centred over the stretchers in the courses below. Where

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1104-418: A full three bricks thick: Overhead sections of alternate (odd and even) courses of double Flemish bond of three bricks' thickness The colour-coded plans highlight facing bricks in the east–west wall. An elevation for this east–west wall is shown to the right. This bond has two stretchers between every header with the headers centred over the perpend between the two stretchers in the course below in

1196-431: A half bricks' thickness The colour-coded plans highlight facing bricks in the east–west wall. An elevation for this east–west wall is shown to the right. For a still more substantial wall, two headers may be laid directly behind the face header, a further two pairs of headers laid at 90° behind the face stretcher, and then finally a stretcher laid to the rear of these four headers. This pattern generates brickwork

1288-431: A header following a quoin stretcher at the corner of the wall. This fact has no bearing on the appearance of the wall; the choice of brick appears to the spectator like any ordinary header: Overhead plans of alternate (odd and even) courses of double Flemish bond of one and a half bricks' thickness For a more substantial wall, a header may be laid directly behind the face header, a further two headers laid at 90° behind

1380-412: A hobbyist mathematician. The discovery is under professional review and, upon confirmation, will be credited as solving a longstanding mathematical problem . Sometimes the colour of a tile is understood as part of the tiling; at other times arbitrary colours may be applied later. When discussing a tiling that is displayed in colours, to avoid ambiguity, one needs to specify whether the colours are part of

1472-419: A main genre in origami (paper folding), where pleats are used to connect molecules, such as twist folds, together in a repeating fashion. Tessellation is used in manufacturing industry to reduce the wastage of material (yield losses) such as sheet metal when cutting out shapes for objects such as car doors or drink cans . Tessellation is apparent in the mudcrack -like cracking of thin films – with

1564-467: A pencil and ink study showing the required geometry. Escher explained that "No single component of all the series, which from infinitely far away rise like rockets perpendicularly from the limit and are at last lost in it, ever reaches the boundary line." Tessellated designs often appear on textiles, whether woven, stitched in, or printed. Tessellation patterns have been used to design interlocking motifs of patch shapes in quilts . Tessellations are also

1656-413: A plane without any gaps, according to a given set of rules. These rules can be varied. Common ones are that there must be no gaps between tiles, and that no corner of one tile can lie along the edge of another. The tessellations created by bonded brickwork do not obey this rule. Among those that do, a regular tessellation has both identical regular tiles and identical regular corners or vertices, having

1748-545: A prototile to create a tessellation, the shape is said to tessellate or to tile the plane . The Conway criterion is a sufficient, but not necessary, set of rules for deciding whether a given shape tiles the plane periodically without reflections: some tiles fail the criterion, but still tile the plane. No general rule has been found for determining whether a given shape can tile the plane or not, which means there are many unsolved problems concerning tessellations. Mathematically, tessellations can be extended to spaces other than

1840-399: A prototile to form a monohedral tessellation, often in more than one way. Copies of an arbitrary quadrilateral can form a tessellation with translational symmetry and 2-fold rotational symmetry with centres at the midpoints of all sides. For an asymmetric quadrilateral this tiling belongs to wallpaper group p2 . As fundamental domain we have the quadrilateral. Equivalently, we can construct

1932-751: A repeating pattern is called "non-periodic". An aperiodic tiling uses a small set of tile shapes that cannot form a repeating pattern (an aperiodic set of prototiles ). A tessellation of space , also known as a space filling or honeycomb, can be defined in the geometry of higher dimensions. A real physical tessellation is a tiling made of materials such as cemented ceramic squares or hexagons. Such tilings may be decorative patterns , or may have functions such as providing durable and water-resistant pavement , floor, or wall coverings. Historically, tessellations were used in Ancient Rome and in Islamic art such as in

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2024-571: A result of contraction forces causing cracks as the lava cools. The extensive crack networks that develop often produce hexagonal columns of lava. One example of such an array of columns is the Giant's Causeway in Northern Ireland. Tessellated pavement , a characteristic example of which is found at Eaglehawk Neck on the Tasman Peninsula of Tasmania , is a rare sedimentary rock formation where

2116-401: A square is {4}. The Schläfli notation makes it possible to describe tilings compactly. For example, a tiling of regular hexagons has three six-sided polygons at each vertex, so its Schläfli symbol is {6,3}. Other methods also exist for describing polygonal tilings. When the tessellation is made of regular polygons, the most common notation is the vertex configuration , which is simply a list of

2208-400: A surface at all if, at the subatomic level, they never actually come in contact with other objects. The surface of an object is the part of the object that is primarily perceived. Humans equate seeing the surface of an object with seeing an object. For example, in looking at an automobile, it is normally not possible to see the engine, electronics, and other internal structures, but the object

2300-411: A surface may be the idealized limit between two fluids , liquid and gas (the surface of the sea in air) or the idealized boundary of a solid (the surface of a ball). In fluid dynamics , the shape of a free surface may be defined by surface tension . However, they are surfaces only at macroscopic scale . At microscopic scale , they may have some thickness. At atomic scale , they do not look at all as

2392-736: A surface, because of holes formed by spaces between atoms or molecules . Other surfaces considered in physics are wavefronts . One of these, discovered by Fresnel , is called wave surface by mathematicians. The surface of the reflector of a telescope is a paraboloid of revolution . Other occurrences: One of the main challenges in computer graphics is creating realistic simulations of surfaces. In technical applications of 3D computer graphics ( CAx ) such as computer-aided design and computer-aided manufacturing , surfaces are one way of representing objects. The other ways are wireframe (lines and curves) and solids. Point clouds are also sometimes used as temporary ways to represent an object, with

2484-458: A tessellation or tiling is a cover of the Euclidean plane by a countable number of closed sets, called tiles , such that the tiles intersect only on their boundaries . These tiles may be polygons or any other shapes. Many tessellations are formed from a finite number of prototiles in which all tiles in the tessellation are congruent to the given prototiles. If a geometric shape can be used as

2576-416: Is a pentagon tiling using irregular pentagons: regular pentagons cannot tile the Euclidean plane as the internal angle of a regular pentagon, ⁠ 3 π / 5 ⁠ , is not a divisor of 2 π . An isohedral tiling is a special variation of a monohedral tiling in which all tiles belong to the same transitivity class, that is, all tiles are transforms of the same prototile under the symmetry group of

2668-521: Is a convex polyhedron with the property of tiling space only aperiodically. A Schwarz triangle is a spherical triangle that can be used to tile a sphere . It is possible to tessellate in non-Euclidean geometries such as hyperbolic geometry . A uniform tiling in the hyperbolic plane (that may be regular, quasiregular, or semiregular) is an edge-to-edge filling of the hyperbolic plane, with regular polygons as faces ; these are vertex-transitive ( transitive on its vertices ), and isogonal (there

2760-457: Is a mathematical model for the formation of mudcracks , needle-like crystals , and similar structures. The model, named after Edgar Gilbert , allows cracks to form starting from being randomly scattered over the plane; each crack propagates in two opposite directions along a line through the initiation point, its slope chosen at random, creating a tessellation of irregular convex polygons. Basaltic lava flows often display columnar jointing as

2852-522: Is a small cubical piece of clay , stone , or glass used to make mosaics. The word "tessella" means "small square" (from tessera , square, which in turn is from the Greek word τέσσερα for four ). It corresponds to the everyday term tiling , which refers to applications of tessellations, often made of glazed clay. Tessellation in two dimensions, also called planar tiling, is a topic in geometry that studies how shapes, known as tiles , can be arranged to fill

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2944-417: Is a tessellation for which every tile is topologically equivalent to a disk , the intersection of any two tiles is a connected set or the empty set , and all tiles are uniformly bounded . This means that a single circumscribing radius and a single inscribing radius can be used for all the tiles in the whole tiling; the condition disallows tiles that are pathologically long or thin. A monohedral tiling

3036-456: Is a tessellation in which all tiles are congruent ; it has only one prototile. A particularly interesting type of monohedral tessellation is the spiral monohedral tiling. The first spiral monohedral tiling was discovered by Heinz Voderberg in 1936; the Voderberg tiling has a unit tile that is a nonconvex enneagon . The Hirschhorn tiling , published by Michael D. Hirschhorn and D. C. Hunt in 1985,

3128-404: Is a tiling where every vertex point is identical; that is, the arrangement of polygons about each vertex is the same. The fundamental region is a shape such as a rectangle that is repeated to form the tessellation. For example, a regular tessellation of the plane with squares has a meeting of four squares at every vertex . The sides of the polygons are not necessarily identical to the edges of

3220-414: Is a very tall masonry building, and has load-bearing brick walls nearly two metres thick at the base. The majority of brick walls are however usually between one and three bricks thick. At these more modest wall thicknesses, distinct patterns have emerged allowing for a structurally sound layout of bricks internal to each particular specified thickness of wall. The advent during the mid twentieth century of

3312-479: Is also given separate names with respect to their position. Mortar placed horizontally below or top of a brick is called a bed , and mortar placed vertically between bricks is called a perpend . A brick made with just rectilinear dimensions is called a solid brick . Bricks might have a depression on both beds or on a single bed. The depression is called a frog , and the bricks are known as frogged bricks . Frogs can be deep or shallow but should never exceed 20% of

3404-461: Is also of fundamental interest. Synchrotron x-ray and neutron scattering measurements are used to provide experimental data on the structure and motion of molecular adsorbates adsorbed on surfaces. The aim of such methods is to provide the data needed to benchmark the latest developments in the modelling of surface systems, their electronic and physical structures and the energetics and friction associated with surface motion. Current projects focus on

3496-690: Is an isometry mapping any vertex onto any other). A uniform honeycomb in hyperbolic space is a uniform tessellation of uniform polyhedral cells . In three-dimensional (3-D) hyperbolic space there are nine Coxeter group families of compact convex uniform honeycombs , generated as Wythoff constructions , and represented by permutations of rings of the Coxeter diagrams for each family. In architecture, tessellations have been used to create decorative motifs since ancient times. Mosaic tilings often had geometric patterns. Later civilisations also used larger tiles, either plain or individually decorated. Some of

3588-408: Is as thick as the width of one brick, but a wall is said to be one brick thick if it as wide as the length of a brick. Accordingly, a single-leaf wall is a half brick thickness; a wall with the simplest possible masonry transverse bond is said to be one brick thick, and so on. The thickness specified for a wall is determined by such factors as damp proofing considerations, whether or not the wall has

3680-399: Is generated by a queen closer on every alternate course: Double Flemish bond of one brick's thickness: overhead sections of alternate (odd and even) courses, and side elevation The colour-coded plans highlight facing bricks in the east–west wall. An elevation for this east–west wall is shown to the right. A simple way to add some width to the wall would be to add stretching bricks at

3772-485: Is given a classification based on how it is laid, and how the exposed face is oriented relative to the face of the finished wall. The practice of laying uncut full-sized bricks wherever possible gives brickwork its maximum possible strength. In the diagrams below, such uncut full-sized bricks are coloured as follows: Occasionally though a brick must be cut to fit a given space, or to be the right shape for fulfilling some particular purpose such as generating an offset—called

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3864-458: Is just one regular honeycomb, which has eight cubes at each polyhedron vertex. Similarly, in three dimensions there is just one quasiregular honeycomb, which has eight tetrahedra and six octahedra at each polyhedron vertex. However, there are many possible semiregular honeycombs in three dimensions. Uniform honeycombs can be constructed using the Wythoff construction . The Schmitt-Conway biprism

3956-708: Is squaring the plane, tiling it by squares whose sizes are all natural numbers without repetitions; James and Frederick Henle proved that this was possible. Surface The concept of surface has been abstracted and formalized in mathematics , specifically in geometry . Depending on the properties on which the emphasis is given, there are several non equivalent such formalizations, that are all called surface , sometimes with some qualifier, such as algebraic surface , smooth surface or fractal surface . The concept of surface and its mathematical abstraction are both widely used in physics , engineering , computer graphics , and many other disciplines, primarily in representing

4048-431: Is still recognized as an automobile because the surface identifies it as one. Conceptually, the "surface" of an object can be defined as the topmost layer of atoms. Many objects and organisms have a surface that is in some way distinct from their interior. For example, the peel of an apple has very different qualities from the interior of the apple, and the exterior surface of a radio may have very different components from

4140-446: Is the rep-tiles ; these tilings have unexpected self-replicating properties. Pinwheel tilings are non-periodic, using a rep-tile construction; the tiles appear in infinitely many orientations. It might be thought that a non-periodic pattern would be entirely without symmetry, but this is not so. Aperiodic tilings, while lacking in translational symmetry , do have symmetries of other types, by infinite repetition of any bounded patch of

4232-504: Is the mortar upon which a brick is laid. A perpend is a vertical joint between any two bricks and is usually—but not always—filled with mortar. A "face brick" is a higher-quality brick, designed for use in visible external surfaces in face-work , as opposed to a "filler brick" for internal parts of the wall, or where the surface is to be covered with stucco or a similar coating, or where the filler bricks will be concealed by other bricks (in structures more than two bricks thick). A brick

4324-617: The Alhambra palace in Granada , Spain . Although this is disputed, the variety and sophistication of the Alhambra tilings have interested modern researchers. Of the three regular tilings two are in the p6m wallpaper group and one is in p4m . Tilings in 2-D with translational symmetry in just one direction may be categorized by the seven frieze groups describing the possible frieze patterns . Orbifold notation can be used to describe wallpaper groups of

4416-700: The Bronze Age . The fired-brick faces of the ziggurat of ancient Dur-Kurigalzu in Iraq date from around 1400 BC, and the brick buildings of ancient Mohenjo-daro in modern day Pakistan were built around 2600 BC. Much older examples of brickwork made with dried (but not fired) bricks may be found in such ancient locations as Jericho in Palestine, Çatal Höyük in Anatolia, and Mehrgarh in Pakistan. These structures have survived from

4508-453: The Moroccan architecture and 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 hyperbolic geometry , for artistic effect. Tessellations are sometimes employed for decorative effect in quilting . Tessellations form a class of patterns in nature , for example in

4600-457: The Stone Age to the modern day. Brick dimensions are expressed in construction or technical documents in two ways as co-ordinating dimensions and working dimensions. Brick size may be slightly different due to shrinkage or distortion due to firing, etc. An example of a co-ordinating metric commonly used for bricks in the UK is as follows: In this case the co-ordinating metric works because

4692-714: The Weaire–Phelan structure , which uses less surface area to separate cells of equal volume than Kelvin's foam. Tessellations have given rise to many types of tiling puzzle , from traditional jigsaw puzzles (with irregular pieces of wood or cardboard) and the tangram , to more modern puzzles that often have a mathematical basis. For example, polyiamonds and polyominoes are figures of regular triangles and squares, often used in tiling puzzles. Authors such as Henry Dudeney and Martin Gardner have made many uses of tessellation in recreational mathematics . For example, Dudeney invented

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4784-431: The cavity wall saw the popularisation and development of another method of strengthening brickwork—the wall tie. A cavity wall comprises two totally discrete walls, separated by an air gap, which serves both as barrier to moisture and heat. Typically the main loads taken by the foundations are carried there by the inner leaf, and the major functions of the external leaf are to protect the whole from weather, and to provide

4876-496: The hinged dissection , while Gardner wrote about the " rep-tile ", a shape that can be dissected into smaller copies of the same shape. Inspired by Gardner's articles in Scientific American , the amateur mathematician Marjorie Rice found four new tessellations with pentagons. Squaring the square is the problem of tiling an integral square (one whose sides have integer length) using only other integral squares. An extension

4968-445: The rhombic dodecahedron , the truncated octahedron , and triangular, quadrilateral, and hexagonal prisms , among others. Any polyhedron that fits this criterion is known as a plesiohedron , and may possess between 4 and 38 faces. Naturally occurring rhombic dodecahedra are found as crystals of andradite (a kind of garnet ) and fluorite . Tessellations in three or more dimensions are called honeycombs . In three dimensions there

5060-417: The Euclidean plane. Penrose tilings , which use two different quadrilateral prototiles, are the best known example of tiles that forcibly create non-periodic patterns. They belong to a general class of aperiodic tilings , which use tiles that cannot tessellate periodically. The recursive process of substitution tiling is a method of generating aperiodic tilings. One class that can be generated in this way

5152-509: The Euclidean plane. The Swiss geometer Ludwig Schläfli pioneered this by defining polyschemes , which mathematicians nowadays call polytopes . These are the analogues to polygons and polyhedra in spaces with more dimensions. He further defined the Schläfli symbol notation to make it easy to describe polytopes. For example, the Schläfli symbol for an equilateral triangle is {3}, while that for

5244-560: The arrays of hexagonal cells found in honeycombs . Tessellations were used by the Sumerians (about 4000 BC) 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 an early documented study of tessellations. He wrote about regular and semiregular tessellations in his Harmonices Mundi ; he

5336-411: The behavior of real-world materials. PBR has found practical applications beyond entertainment, extending its impact to architectural design , product prototyping , and scientific simulations. Bond (brick) Brickwork is masonry produced by a bricklayer , using bricks and mortar . Typically, rows of bricks called courses are laid on top of one another to build up a structure such as

5428-485: The bond's most symmetric form. The great variety of monk bond patterns allow for many possible layouts at the quoins, and many possible arrangements for generating a lap. A quoin brick may be a stretcher, a three-quarter bat, or a header. Queen closers may be used next to the quoins, but the practice is not mandatory. Monk bond may however take any of a number of arrangements for course staggering. The disposal of bricks in these often highly irregular raking patterns can be

5520-425: The bricks are laid also running linearly and extending upwards, forming wythes or leafs . It is as important as with the perpends to bond these leaves together. Historically, the dominant method for consolidating the leaves together was to lay bricks across them, rather than running linearly. Brickwork observing either or both of these two conventions is described as being laid in one or another bond . A leaf

5612-402: The bricks behind the facing bricks may be laid in groups of four bricks and a half-bat. The half-bat sits at the centre of the group and the four bricks are placed about the half-bat, in a square formation. These groups are laid next to each other for the length of a course, making brickwork one and a half bricks thick. To preserve the bond, it is necessary to lay a three-quarter bat instead of

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5704-423: The defining points, Delaunay triangulations maximize the minimum of the angles formed by the edges. Voronoi tilings with randomly placed points can be used to construct random tilings of the plane. Tessellation can be extended to three dimensions. Certain polyhedra can be stacked in a regular crystal pattern to fill (or tile) three-dimensional space, including the cube (the only Platonic polyhedron to do so),

5796-478: The face stretcher, and then finally a stretcher laid to the rear of these two headers. This pattern generates brickwork a full two bricks thick: Overhead sections of alternate (odd and even) courses of double Flemish bond of two bricks' thickness The colour-coded plans highlight facing bricks in the east–west wall. An elevation for this east–west wall is shown to the right. Overhead sections of alternate (odd and even) courses of double Flemish bond of two and

5888-412: The final pair is reached, whereupon a second and final queen closer is inserted as the penultimate brick, mirroring the arrangement at the beginning of the course, and duly closing the bond. Some examples of Flemish bond incorporate stretchers of one colour and headers of another. This effect is commonly a product of treating the header face of the heading bricks while the bricks are being baked as part of

5980-399: The front and rear duplication of the pattern. If the wall is arranged such that the bricks at the rear do not have this pattern, then the brickwork is said to be single Flemish bond . Flemish bond brickwork with a thickness of one brick is the repeating pattern of a stretcher laid immediately to the rear of the face stretcher, and then next along the course, a header. A lap (correct overlap)

6072-480: The goal of using the points to create one or more of the three permanent representations. One technique used for enhancing surface realism in computer graphics is the use of physically-based rendering (PBR) algorithms which simulate the interaction of light with surfaces based on their physical properties, such as reflectance , roughness, and transparency . By incorporating mathematical models and algorithms, PBR can generate highly realistic renderings that resemble

6164-417: The interior. Peeling the apple constitutes removal of the surface, ultimately leaving a different surface with a different texture and appearance, identifiable as a peeled apple. Removing the exterior surface of an electronic device may render its purpose unrecognizable. By contrast, removing the outermost layer of a rock or the topmost layer of liquid contained in a glass would leave a substance or material with

6256-402: The length of a single brick (215 mm) is equal to the total of the width of a brick (102.5 mm) plus a perpend (10 mm) plus the width of a second brick (102.5 mm). There are many other brick sizes worldwide, and many of them use this same co-ordinating principle. As the most common bricks are rectangular prisms, six surfaces are named as follows: Mortar placed between bricks

6348-443: The manufacturing process. Some of the header faces are exposed to wood smoke, generating a grey-blue colour, while other simply vitrified until they reach a deeper blue colour. Some headers have a glazed face, caused by using salt in the firing. Sometimes Staffordshire Blue bricks are used for the heading bricks. Brickwork that appears as Flemish bond from both the front and the rear is double Flemish bond , so called on account of

6440-465: The mirror-image pair of tilings counts as two). These can be described by their vertex configuration ; for example, a semi-regular tiling using squares and regular octagons has the vertex configuration 4.8 (each vertex has one square and two octagons). Many non-edge-to-edge tilings of the Euclidean plane are possible, including the family of Pythagorean tilings , tessellations that use two (parameterised) sizes of square, each square touching four squares of

6532-595: The most decorative were the Moorish wall tilings of Islamic architecture , using Girih and Zellige tiles in buildings such as the Alhambra and La Mezquita . Tessellations frequently appeared in the graphic art of M. C. Escher ; he was inspired by the Moorish use of symmetry in places such as the Alhambra when he visited Spain in 1936. Escher made four " Circle Limit " drawings of tilings that use hyperbolic geometry. For his woodcut "Circle Limit IV" (1960), Escher prepared

6624-464: The number of sides of the polygons around a vertex. The square tiling has a vertex configuration of 4.4.4.4, or 4. The tiling of regular hexagons is noted 6.6.6, or 6. Mathematicians use some technical terms when discussing tilings. An edge is the intersection between two bordering tiles; it is often a straight line. A vertex is the point of intersection of three or more bordering tiles. Using these terms, an isogonal or vertex-transitive tiling

6716-427: The number of sides, even if only one shape is allowed. Polyominoes are examples of tiles that are either convex of non-convex, for which various combinations, rotations, and reflections can be used to tile a plane. For results on tiling the plane with polyominoes , see Polyomino § Uses of polyominoes . Voronoi or Dirichlet tilings are tessellations where each tile is defined as the set of points closest to one of

6808-406: The other size. An edge tessellation is one in which each tile can be reflected over an edge to take up the position of a neighbouring tile, such as in an array of equilateral or isosceles triangles. Tilings with translational symmetry in two independent directions can be categorized by wallpaper groups , of which 17 exist. It has been claimed that all seventeen of these groups are represented in

6900-434: The plane is also undecidable. Truchet tiles are square tiles decorated with patterns so they do not have rotational symmetry ; in 1704, Sébastien Truchet used a square tile split into two triangles of contrasting colours. These can tile the plane either periodically or randomly. An einstein tile is a single shape that forces aperiodic tiling. The first such tile, dubbed a "hat", was discovered in 2023 by David Smith,

6992-443: The points in a discrete set of defining points. (Think of geographical regions where each region is defined as all the points closest to a given city or post office.) The Voronoi cell for each defining point is a convex polygon. The Delaunay triangulation is a tessellation that is the dual graph of a Voronoi tessellation. Delaunay triangulations are useful in numerical simulation, in part because among all possible triangulations of

7084-403: The rear, making a Single Flemish bond one and a half bricks thick: Overhead sections of alternate (odd and even) courses of single Flemish bond of one and a half bricks' thickness The colour-coded plans highlight facing bricks in the east–west wall. An elevation for this east–west wall is shown to the right. For a double Flemish bond of one and a half bricks' thickness, facing bricks and

7176-426: The rock has fractured into rectangular blocks. Other natural patterns occur in foams ; these are packed according to Plateau's laws , which require minimal surfaces . Such foams present a problem in how to pack cells as tightly as possible: in 1887, Lord Kelvin proposed a packing using only one solid, the bitruncated cubic honeycomb with very slightly curved faces. In 1993, Denis Weaire and Robert Phelan proposed

7268-485: The same angle between adjacent edges for every tile. There are only three shapes that can form such regular tessellations: the equilateral triangle , square and the regular hexagon . Any one of these three shapes can be duplicated infinitely to fill a plane with no gaps. Many other types of tessellation are possible under different constraints. For example, there are eight types of semi-regular tessellation, made with more than one kind of regular polygon but still having

7360-533: The same arrangement of polygons at every corner. Irregular tessellations can also be made from other shapes such as pentagons , polyominoes and in fact almost any kind of geometric shape. The artist M. C. Escher is famous for making tessellations with irregular interlocking tiles, shaped like animals and other natural objects. If suitable contrasting colours are chosen for the tiles of differing shape, striking patterns are formed, and these can be used to decorate physical surfaces such as church floors. More formally,

7452-452: The same colour; hence they are sometimes called Wang dominoes . A suitable set of Wang dominoes can tile the plane, but only aperiodically. This is known because any Turing machine can be represented as a set of Wang dominoes that tile the plane if, and only if, the Turing machine does not halt. Since the halting problem is undecidable, the problem of deciding whether a Wang domino set can tile

7544-407: The same composition, only slightly reduced in volume. In mathematics , a surface is a mathematical model of the common concept of a surface. It is a generalization of a plane , but, unlike a plane, it may be curved ; this is analogous to a curve generalizing a straight line . There are several more precise definitions, depending on the context and the mathematical tools that are used for

7636-603: The study. The simplest mathematical surfaces are planes and spheres in the Euclidean 3-space . The exact definition of a surface may depend on the context. Typically, in algebraic geometry , a surface may cross itself (and may have other singularities ), while, in topology and differential geometry , it may not. The concept of a surface in the physical sciences encompasses the structures and dynamics of and occurring at surfaces. The field underlies many practical disciplines such as semiconductor physics and applied nanotechnology but

7728-466: The surface adsorption of polyaromatic hydrocarbons (PAHs), a class of molecules key to the refinement of the modelling of dispersive forces through approaches such as density functional theory, and build on our complementary work applying helium atom scattering and scanning tunnelling microscopy to small molecules with aromatic functionality. Many surfaces considered in physics and chemistry ( physical sciences in general) are interfaces . For example,

7820-433: The surfaces of physical objects. For example, in analyzing the aerodynamic properties of an airplane , the central consideration is the flow of air along its surface. The concept also raises certain philosophical questions—for example, how thick is the layer of atoms or molecules that can be considered part of the surface of an object (i.e., where does the "surface" end and the "interior" begin), and do objects really have

7912-442: The tiles. An edge-to-edge tiling is any polygonal tessellation where adjacent tiles only share one full side, i.e., no tile shares a partial side or more than one side with any other tile. In an edge-to-edge tiling, the sides of the polygons and the edges of the tiles are the same. The familiar "brick wall" tiling is not edge-to-edge because the long side of each rectangular brick is shared with two bordering bricks. A normal tiling

8004-467: The tiling and in certain finite groups of rotations or reflections of those patches. A substitution rule, such as can be used to generate Penrose patterns using assemblies of tiles called rhombs, illustrates scaling symmetry. A Fibonacci word can be used to build an aperiodic tiling, and to study quasicrystals , which are structures with aperiodic order. Wang tiles are squares coloured on each edge, and placed so that abutting edges of adjacent tiles have

8096-399: The tiling or just part of its illustration. This affects whether tiles with the same shape, but different colours, are considered identical, which in turn affects questions of symmetry. The four colour theorem states that for every tessellation of a normal Euclidean plane , with a set of four available colours, each tile can be coloured in one colour such that no tiles of equal colour meet at

8188-621: The tiling. If a prototile admits a tiling, but no such tiling is isohedral, then the prototile is called anisohedral and forms anisohedral tilings . A regular tessellation is a highly symmetric , edge-to-edge tiling made up of regular polygons , all of the same shape. There are only three regular tessellations: those made up of equilateral triangles , squares , or regular hexagons . All three of these tilings are isogonal and monohedral. A semi-regular (or Archimedean) tessellation uses more than one type of regular polygon in an isogonal arrangement. There are eight semi-regular tilings (or nine if

8280-413: The total volume of the brick. Cellular bricks have depressions exceeding 20% of the volume of the brick. Perforated bricks have holes through the brick from bed to bed, cutting it all the way. Most of the building standards and good construction practices recommend the volume of holes should not exceed 20% of the total volume of the brick. Parts of brickwork include bricks , beds and perpends . The bed

8372-443: The use of a repeating sequence of courses with back-and-forth header staggering. In this grouping, a header appears at a given point in the group's first course. In the next course up, a header is offset one and a half stretcher lengths to the left of the header in the course below, and then in the third course, a header is offset one stretcher length to the right of the header in the middle course. This accented swing of headers, one and

8464-639: Was possibly the first to explore and to explain the hexagonal structures of honeycomb and snowflakes . Some two hundred years later in 1891, the Russian crystallographer Yevgraf Fyodorov proved that every periodic tiling of the plane features one of seventeen different groups of isometries. Fyodorov's work marked the unofficial beginning of the mathematical study of tessellations. Other prominent contributors include Alexei Vasilievich Shubnikov and Nikolai Belov in their book Colored Symmetry (1964), and Heinrich Heesch and Otto Kienzle (1963). In Latin, tessella

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