1.
Tessellation
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A tessellation of a flat surface is the tiling of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps. In mathematics, tessellations can be generalized to higher dimensions and a variety of geometries, a periodic tiling has a repeating pattern. The patterns formed by periodic tilings can be categorized into 17 wallpaper groups, a tiling that lacks a repeating pattern is called non-periodic. An aperiodic tiling uses a set of tile shapes that cannot form a repeating pattern. In the geometry of higher dimensions, a space-filling or honeycomb is called a tessellation of space. A real physical tessellation is a made of materials such as cemented ceramic squares or hexagons. Such tilings may be decorative patterns, or may have such as providing durable and water-resistant pavement. Historically, tessellations were used in Ancient Rome and in Islamic art such as in the decorative geometric tiling of the Alhambra palace, in the twentieth century, the work of M. C. Escher often made use of tessellations, both in ordinary Euclidean geometry and in geometry, for artistic effect. Tessellations are sometimes employed for decorative effect in quilting, Tessellations form a class of patterns in nature, for example in the arrays of hexagonal cells found in honeycombs. Tessellations were used by the Sumerians in building wall decorations formed by patterns of clay tiles, decorative mosaic tilings made of small squared blocks called tesserae were widely employed in classical antiquity, sometimes displaying geometric patterns. In 1619 Johannes Kepler made a documented study of tessellations. He wrote about regular and semiregular tessellations in his Harmonices Mundi, he was possibly the first to explore and to explain the structures of honeycomb. Some two hundred years later in 1891, the Russian crystallographer Yevgraf Fyodorov proved that every periodic tiling of the features one of seventeen different groups of isometries. Fyodorovs work marked the beginning of the mathematical study of tessellations. Other prominent contributors include Shubnikov and Belov, and Heinrich Heesch, in Latin, tessella is a small cubical piece of clay, stone or glass used to make mosaics. The word tessella means small square and it corresponds to the everyday term tiling, which refers to applications of tessellations, often made of glazed clay. Tessellation or tiling in two dimensions is a topic in geometry that studies how shapes, known as tiles, can be arranged to fill a plane without any gaps, according to a given set of rules
2.
Johannes Kepler
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Johannes Kepler was a German mathematician, astronomer, and astrologer. A key figure in the 17th-century scientific revolution, he is best known for his laws of motion, based on his works Astronomia nova, Harmonices Mundi. These works also provided one of the foundations for Isaac Newtons theory of universal gravitation, Kepler was a mathematics teacher at a seminary school in Graz, where he became an associate of Prince Hans Ulrich von Eggenberg. Later he became an assistant to the astronomer Tycho Brahe in Prague and he was also a mathematics teacher in Linz, and an adviser to General Wallenstein. Kepler lived in an era when there was no distinction between astronomy and astrology, but there was a strong division between astronomy and physics. Kepler was born on December 27, the feast day of St John the Evangelist,1571 and his grandfather, Sebald Kepler, had been Lord Mayor of the city. By the time Johannes was born, he had two brothers and one sister and the Kepler family fortune was in decline and his father, Heinrich Kepler, earned a precarious living as a mercenary, and he left the family when Johannes was five years old. He was believed to have died in the Eighty Years War in the Netherlands and his mother Katharina Guldenmann, an innkeepers daughter, was a healer and herbalist. Born prematurely, Johannes claimed to have weak and sickly as a child. Nevertheless, he often impressed travelers at his grandfathers inn with his phenomenal mathematical faculty and he was introduced to astronomy at an early age, and developed a love for it that would span his entire life. At age six, he observed the Great Comet of 1577, in 1580, at age nine, he observed another astronomical event, a lunar eclipse, recording that he remembered being called outdoors to see it and that the moon appeared quite red. However, childhood smallpox left him with vision and crippled hands. In 1589, after moving through grammar school, Latin school, there, he studied philosophy under Vitus Müller and theology under Jacob Heerbrand, who also taught Michael Maestlin while he was a student, until he became Chancellor at Tübingen in 1590. He proved himself to be a mathematician and earned a reputation as a skilful astrologer. Under the instruction of Michael Maestlin, Tübingens professor of mathematics from 1583 to 1631 and he became a Copernican at that time. In a student disputation, he defended heliocentrism from both a theoretical and theological perspective, maintaining that the Sun was the source of motive power in the universe. Despite his desire to become a minister, near the end of his studies, Kepler was recommended for a position as teacher of mathematics and he accepted the position in April 1594, at the age of 23. Keplers first major work, Mysterium Cosmographicum, was the first published defense of the Copernican system
3.
Hexagon
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In geometry, a hexagon is a six sided polygon or 6-gon. The total of the angles of any hexagon is 720°. A regular hexagon has Schläfli symbol and can also be constructed as an equilateral triangle, t. A regular hexagon is defined as a hexagon that is both equilateral and equiangular and it is bicentric, meaning that it is both cyclic and tangential. The common length of the sides equals the radius of the circumscribed circle, all internal angles are 120 degrees. A regular hexagon has 6 rotational symmetries and 6 reflection symmetries, the longest diagonals of a regular hexagon, connecting diametrically opposite vertices, are twice the length of one side. Like squares and equilateral triangles, regular hexagons fit together without any gaps to tile the plane, the cells of a beehive honeycomb are hexagonal for this reason and because the shape makes efficient use of space and building materials. The Voronoi diagram of a triangular lattice is the honeycomb tessellation of hexagons. It is not usually considered a triambus, although it is equilateral, the maximal diameter, D is twice the maximal radius or circumradius, R, which equals the side length, t. The minimal diameter or the diameter of the circle, d, is twice the minimal radius or inradius. If a regular hexagon has successive vertices A, B, C, D, E, F, the regular hexagon has Dih6 symmetry, order 12. There are 3 dihedral subgroups, Dih3, Dih2, and Dih1, and 4 cyclic subgroups, Z6, Z3, Z2 and these symmetries express 9 distinct symmetries of a regular hexagon. John Conway labels these by a letter and group order, r12 is full symmetry, and a1 is no symmetry. These two forms are duals of each other and have half the order of the regular hexagon. The i4 forms are regular hexagons flattened or stretched along one symmetry direction and it can be seen as an elongated rhombus, while d2 and p2 can be seen as horizontally and vertically elongated kites. G2 hexagons, with sides parallel are also called hexagonal parallelogons. Each subgroup symmetry allows one or more degrees of freedom for irregular forms, only the g6 subgroup has no degrees of freedom but can seen as directed edges. Hexagons of symmetry g2, i4, and r12, as parallelogons can tessellate the Euclidean plane by translation, other hexagon shapes can tile the plane with different orientations
4.
Wallpaper group
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A wallpaper group is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art, There are 17 possible distinct groups. Wallpaper groups are two-dimensional symmetry groups, intermediate in complexity between the simpler frieze groups and the space groups. Wallpaper groups categorize patterns by their symmetries, subtle differences may place similar patterns in different groups, while patterns that are very different in style, color, scale or orientation may belong to the same group. Consider the following examples, Examples A and B have the same group, it is called p4m in the IUC notation. Example C has a different wallpaper group, called p4g or 4*2, a complete list of all seventeen possible wallpaper groups can be found below. A symmetry of a pattern is, loosely speaking, a way of transforming the pattern so that it exactly the same after the transformation. For example, translational symmetry is present when the pattern can be translated some finite distance, think of shifting a set of vertical stripes horizontally by one stripe. Strictly speaking, a true symmetry only exists in patterns that repeat exactly, a set of only, say, five stripes does not have translational symmetry—when shifted, the stripe on one end disappears and a new stripe is added at the other end. In practice, however, classification is applied to finite patterns, sometimes two categorizations are meaningful, one based on shapes alone and one also including colors. When colors are ignored there may be more symmetry, the types of transformations that are relevant here are called Euclidean plane isometries. This type of symmetry is called a translation, Examples A and C are similar, except that the smallest possible shifts are in diagonal directions. If we turn example B clockwise by 90°, around the centre of one of the squares, Examples A and C also have 90° rotations, although it requires a little more ingenuity to find the correct centre of rotation for C. We can also flip example B across a horizontal axis that runs across the middle of the image, example B also has reflections across a vertical axis, and across two diagonal axes. The same can be said for A. However, example C is different and it only has reflections in horizontal and vertical directions, not across diagonal axes. If we flip across a line, we do not get the same pattern back. This is part of the reason that the group of A and B is different from the wallpaper group of C. A proof that there were only 17 possible patterns was first carried out by Evgraf Fedorov in 1891, the proof that the list of wallpaper groups was complete only came after the much harder case of space groups had been done
5.
Triangular tiling
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In geometry, the triangular tiling or triangular tessellation is one of the three regular tilings of the Euclidean plane. Because the internal angle of the triangle is 60 degrees. The triangular tiling has Schläfli symbol of, Conway calls it a deltille, named from the triangular shape of the Greek letter delta. The triangular tiling can also be called a kishextille by a kis operation that adds a center point and it is one of three regular tilings of the plane. The other two are the square tiling and the hexagonal tiling, there are 9 distinct uniform colorings of a triangular tiling. Three of them can be derived from others by repeating colors,111212 and 111112 from 121213 by combining 1 and 3, there is one class of Archimedean colorings,111112, which is not 1-uniform, containing alternate rows of triangles where every third is colored. The example shown is 2-uniform, but there are many such Archimedean colorings that can be created by arbitrary horizontal shifts of the rows. The vertex arrangement of the tiling is called an A2 lattice. It is the 2-dimensional case of a simplectic honeycomb, the A*2 lattice can be constructed by the union of all three A2 lattices, and equivalent to the A2 lattice. + + = dual of = The vertices of the tiling are the centers of the densest possible circle packing. Every circle is in contact with 6 other circles in the packing, the packing density is π⁄√12 or 90. 69%. Since the union of 3 A2 lattices is also an A2 lattice, the voronoi cell of a triangular tiling is a hexagon, and so the voronoi tessellation, the hexagonal tiling has a direct correspondence to the circle packings. Triangular tilings can be made with the equivalent topology as the regular tiling, with identical faces and vertex-transitivity, there are 5 variations. Symmetry given assumes all faces are the same color, the planar tilings are related to polyhedra. Putting fewer triangles on a vertex leaves a gap and allows it to be folded into a pyramid and these can be expanded to Platonic solids, five, four and three triangles on a vertex define an icosahedron, octahedron, and tetrahedron respectively. This tiling is related as a part of sequence of regular polyhedra with Schläfli symbols. It is also related as a part of sequence of Catalan solids with face configuration Vn.6.6. Like the uniform there are eight uniform tilings that can be based from the regular hexagonal tiling
6.
Mirror image
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A mirror image is a reflected duplication of an object that appears almost identical, but is reversed in the direction perpendicular to the mirror surface. As an optical effect it results from reflection off of such as a mirror or water. It is also a concept in geometry and can be used as a process for 3-D structures. Two-dimensional mirror images can be seen in the reflections of mirrors or other reflecting surfaces, or on a printed surface seen inside out. If we look at an object that is effectively two-dimensional and then turn it towards a mirror, in this example, it is the change in orientation rather than the mirror itself that causes the observed reversal. Another example is when we stand with our backs to the mirror, then we compare the object with its reflection by turning ourselves 180 degrees, towards the mirror. Again we perceive a left-right reversal due to a change in orientation, so, in these examples the mirror does not actually cause the observed reversals. The concept of reflection can be extended to three-dimensional objects, including the inside parts, the term then relates to structural as well as visual aspects. A three-dimensional object is reversed in the perpendicular to the mirror surface. In physics, mirror images are investigated in the subject called geometrical optics, in chemistry, two versions of a molecule, one a mirror image of the other, are called enantiomers if they are not superposable on each other. That is an example of chirality, in general, an object and its mirror image are called enantiomorphs. If a point of an object has coordinates then the image of this point has coordinates, thus reflection is a reversal of the coordinate axis perpendicular to the mirrors surface. In everyday use, a mirror does not reverse right and left, however, there is often a perception of left-right reversal, probably because the left and right of an object are defined by its top and front. Reflection in a mirror does result in a change in chirality, as a consequence, if one looks in a mirror and lets two axes coincide with those in the mirror, then this gives a reversal of the third axis. Similarly, if you stand side-on to a mirror your left and its important to realise there are only two enantiomorphs, the object and its image. So, no matter how the object is oriented towards the mirror, all the images are fundamentally identical. In the photograph of the urn and mirror, the urn is fairly symmetrical front-back, so, its not surprising that no obvious reversal of the urn can be seen in the mirror image. A mirror image appears more obviously three-dimensional if the observer moves and this is because the relative position of objects changes as the observers perspective changes, or is different viewed with each eye
7.
Hexagonal tiling
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In geometry, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which three hexagons meet at each vertex. It has Schläfli symbol of or t, english mathematician Conway calls it a hextille. The internal angle of the hexagon is 120 degrees so three hexagons at a point make a full 360 degrees and it is one of three regular tilings of the plane. The other two are the triangular tiling and the square tiling, the hexagonal tiling is the densest way to arrange circles in two dimensions. The Honeycomb conjecture states that the tiling is the best way to divide a surface into regions of equal area with the least total perimeter. The optimal three-dimensional structure for making honeycomb was investigated by Lord Kelvin, however, the less regular Weaire–Phelan structure is slightly better. This structure exists naturally in the form of graphite, where each sheet of graphene resembles chicken wire, tubular graphene sheets have been synthesised, these are known as carbon nanotubes. They have many applications, due to their high tensile strength. Chicken wire consists of a lattice of wires. The hexagonal tiling appears in many crystals, in three dimensions, the face-centered cubic and hexagonal close packing are common crystal structures. They are the densest known sphere packings in three dimensions, and are believed to be optimal, structurally, they comprise parallel layers of hexagonal tilings, similar to the structure of graphite. They differ in the way that the layers are staggered from each other, pure copper, amongst other materials, forms a face-centered cubic lattice. There are three distinct uniform colorings of a tiling, all generated from reflective symmetry of Wythoff constructions. The represent the periodic repeat of one colored tile, counting hexagonal distances as h first, the 3-color tiling is a tessellation generated by the order-3 permutohedrons. A chamferred hexagonal tiling replacing edges with new hexagons and transforms into another hexagonal tiling, in the limit, the original faces disappear, and the new hexagons degenerate into rhombi, and it becomes a rhombic tiling. The hexagons can be dissected into sets of 6 triangles and this is similar to the relation of the rhombic dodecahedron and the rhombo-hexagonal dodecahedron tessellations in 3 dimensions. This tiling is related to regular polyhedra with vertex figure n3. It is similarly related to the uniform truncated polyhedra with vertex figure n.6.6 and this tiling is also a part of a sequence of truncated rhombic polyhedra and tilings with Coxeter group symmetry
8.
Heptagon
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In geometry, a heptagon is a seven-sided polygon or 7-gon. The heptagon is also referred to as the septagon, using sept- together with the Greek suffix -agon meaning angle. A regular heptagon, in all sides and all angles are equal, has internal angles of 5π/7 radians. The area of a regular heptagon of side length a is given by, the apothem is half the cotangent of π /7, and the area of each of the 14 small triangles is one-fourth of the apothem. This expression cannot be rewritten without complex components, since the indicated cubic function is casus irreducibilis. As 7 is a Pierpont prime but not a Fermat prime and this type of construction is called a neusis construction. It is also constructible with compass, straightedge and angle trisector, the impossibility of straightedge and compass construction follows from the observation that 2 cos 2 π7 ≈1.247 is a zero of the irreducible cubic x3 + x2 − 2x −1. Consequently, this polynomial is the polynomial of 2cos, whereas the degree of the minimal polynomial for a constructible number must be a power of 2. An approximation for practical use with an error of about 0. 2% is shown in the drawing and it is attributed to Albrecht Dürer. Let A lie on the circumference of the circumcircle, then B D =12 B C gives an approximation for the edge of the heptagon. Example to illustrate the error, At a circumscribed circle radius r =1 m, since 7 is a prime number there is one subgroup with dihedral symmetry, Dih1, and 2 cyclic group symmetries, Z7, and Z1. These 4 symmetries can be seen in 4 distinct symmetries on the heptagon, john Conway labels these by a letter and group order. Full symmetry of the form is r14 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices or edges, cyclic symmetries in the middle column are labeled as g for their central gyration orders. Each subgroup symmetry allows one or more degrees of freedom for irregular forms, only the g7 subgroup has no degrees of freedom but can seen as directed edges. However, no algebraic expressions with purely real terms exist for the solutions of this equation, because it is an example of casus irreducibilis. A heptagonal triangle has vertices coinciding with the first, second, and fourth vertices of a regular heptagon and angles π /7,2 π /7, thus its sides coincide with one side and two particular diagonals of the regular heptagon. Two kinds of star heptagons can be constructed from regular heptagons, labeled by Schläfli symbols, blue, and green star heptagons inside a red heptagon
9.
Rhombitrihexagonal tiling
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In geometry, the rhombitrihexagonal tiling is a semiregular tiling of the Euclidean plane. There are one triangle, two squares, and one hexagon on each vertex and it has Schläfli symbol of rr. John Conway calls it a rhombihexadeltille and it can be considered a cantellated by Norman Johnsons terminology or an expanded hexagonal tiling by Alicia Boole Stotts operational language. There are 3 regular and 8 semiregular tilings in the plane, there is only one uniform coloring in a rhombitrihexagonal tiling. With edge-colorings there is a half symmetry form orbifold notation, the hexagons can be considered as truncated triangles, t with two types of edges. It has Coxeter diagram, Schläfli symbol s2, the bicolored square can be distorted into isosceles trapezoids. In the limit, where the rectangles degenerate into edges, a triangular tiling results, there is one related 2-uniform tilings, having hexagons dissected into 6 triangles. Every circle is in contact with 4 other circles in the packing, the translational lattice domain contains 6 distinct circles. The gap inside each hexagon allows for one circle, related to a 2-uniform tiling with the hexagons divided into 6 triangles, there are eight uniform tilings that can be based from the regular hexagonal tiling. Drawing the tiles colored as red on the faces, yellow at the original vertices. This tiling is related as a part of sequence of cantellated polyhedra with vertex figure. These vertex-transitive figures have reflectional symmetry, the deltoidal trihexagonal tiling is a dual of the semiregular tiling known as the rhombitrihexagonal tiling. The edges of this tiling can be formed by the overlay of the regular triangular tiling. Each kite face of this tiling has angles 120°, 90°, 60° and 90° and it is one of only eight tilings of the plane in which every edge lies on a line of symmetry of the tiling. The deltoidal trihexagonal tiling is a dual of the semiregular tiling rhombitrihexagonal tiling and its faces are deltoids or kites. It is one of 7 dual uniform tilings in hexagonal symmetry and this tiling has face transitive variations, that can distort the kites into bilateral trapezoids or more general quadrillaterals. Ignoring the face colors below, the symmetry is p6m, and the lower symmetry is p31m with 3 mirrors meeting at a point. This tiling is related to the tiling by dividing the triangles and hexagons into central triangles
10.
Vertex configuration
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In geometry, a vertex configuration is a shorthand notation for representing the vertex figure of a polyhedron or tiling as the sequence of faces around a vertex. For uniform polyhedra there is one vertex type and therefore the vertex configuration fully defines the polyhedron. A vertex configuration is given as a sequence of numbers representing the number of sides of the faces going around the vertex, the notation a. b. c describes a vertex that has 3 faces around it, faces with a, b, and c sides. For example,3.5.3.5 indicates a vertex belonging to 4 faces, alternating triangles and this vertex configuration defines the vertex-transitive icosidodecahedron. The notation is cyclic and therefore is equivalent with different starting points, the order is important, so 3.3.5.5 is different from 3.5.3.5. Repeated elements can be collected as exponents so this example is represented as 2. It has variously called a vertex description, vertex type, vertex symbol, vertex arrangement, vertex pattern. It is also called a Cundy and Rollett symbol for its usage for the Archimedean solids in their 1952 book Mathematical Models, a vertex configuration can also be represented as a polygonal vertex figure showing the faces around the vertex. Different notations are used, sometimes with a comma and sometimes a period separator, the period operator is useful because it looks like a product and an exponent notation can be used. For example,3.5.3.5 is sometimes written as 2, the notation can also be considered an expansive form of the simple Schläfli symbol for regular polyhedra. The Schläfli notation means q p-gons around each vertex, so can be written as p. p. p. or pq. For example, an icosahedron is =3.3.3.3.3 or 35 and this notation applies to polygonal tilings as well as polyhedra. A planar vertex configuration denotes a uniform tiling just like a nonplanar vertex configuration denotes a uniform polyhedron, the notation is ambiguous for chiral forms. For example, the cube has clockwise and counterclockwise forms which are identical across mirror images. Both have a 3.3.3.3.4 vertex configuration, the notation also applies for nonconvex regular faces, the star polygons. For example, a pentagram has the symbol, meaning it has 5 sides going around the centre twice, for example, there are 4 regular star polyhedra with regular polygon or star polygon vertex figures. The small stellated dodecahedron has the Schläfli symbol of which expands to a vertex configuration 5/2. 5/2. 5/2. 5/2. 5/2 or combined as 5. The great stellated dodecahedron, has a vertex figure and configuration or 3