Kissing number problem
In geometry, a kissing number is defined as the number of non-overlapping unit spheres that can be arranged such that they each touch a common unit sphere. For a lattice packing the kissing number is the same for every sphere, but for an arbitrary sphere packing the kissing number may vary from one sphere to another. Other names for kissing number that have been used are Newton number, contact number. In general, the kissing number problem seeks the maximum possible kissing number for n-dimensional spheres in -dimensional Euclidean space. Ordinary spheres correspond to two-dimensional closed surfaces in three-dimensional space. Finding the kissing number when centers of spheres are confined to a line or a plane is trivial. Proving a solution to the three-dimensional case, despite being easy to conceptualise and model in the physical world, eluded mathematicians until the mid-20th century. Solutions in higher dimensions are more challenging, only a handful of cases have been solved exactly. For others investigations have determined lower bounds, but not exact solutions.
In one dimension, the kissing number is 2: In two dimensions, the kissing number is 6: Proof: Consider a circle with center C, touched by circles with centers C1, C2.... Consider the rays C Ci; these rays all emanate from the same center C, so the sum of angles between adjacent rays is 360°. Assume by contradiction that there are more than six touching circles. At least two adjacent rays, say C C1 and C C2, are separated by an angle of less than 60°; the segments C Ci have the same length – 2r – for all i. Therefore, the triangle C C1 C2 is isosceles, its third side – C1 C2 – has a side length of less than 2r. Therefore, the circles 1 and 2 intersect – a contradiction. In three dimensions, the kissing number is 12, but the correct value was much more difficult to establish than in dimensions one and two, it is easy to arrange 12 spheres so that each touches a central sphere, but there is a lot of space left over, it is not obvious that there is no way to pack in a 13th sphere. This was the subject of a famous disagreement between mathematicians Isaac David Gregory.
Newton thought that the limit was 12. Some incomplete proofs that Newton was correct were offered in the nineteenth century, most notably one by Reinhold Hoppe, but the first correct proof did not appear until 1953; the twelve neighbors of the central sphere correspond to the maximum bulk coordination number of an atom in a crystal lattice in which all atoms have the same size. A coordination number of 12 is found in a hexagonal close-packed structure. In four dimensions, it was known for some time that the answer was either 24 or 25, it is easy to produce a packing of 24 spheres around a central sphere. As in the three-dimensional case, there is a lot of space left over—even more, in fact, than for n = 3—so the situation was less clear. In 2003, Oleg Musin proved the kissing number for n = 4 using a subtle trick; the kissing number in n dimensions is unknown for n > 4, except for n = 8, n = 24. The results in these dimensions stem from the existence of symmetrical lattices: the E8 lattice and the Leech lattice.
If arrangements are restricted to lattice arrangements, in which the centres of the spheres all lie on points in a lattice this restricted kissing number is known for n = 1 to 9 and n = 24 dimensions. For 5, 6, 7 dimensions the arrangement with the highest known kissing number found so far is the optimal lattice arrangement, but the existence of a non-lattice arrangement with a higher kissing number has not been excluded; the following table lists some known bounds on the kissing number in various dimensions. The dimensions in which the kissing number is known are listed in boldface; the kissing number problem can be generalized to the problem of finding the maximum number of non-overlapping congruent copies of any convex body that touch a given copy of the body. There are different versions of the problem depending on whether the copies are only required to be congruent to the original body, translates of the original body, or translated by a lattice. For the regular tetrahedron, for example, it is known that both the lattice kissing number and the translative kissing number are equal to 18, whereas the congruent kissing number is at least 56.
There are several approximation algorithms on intersection graphs where the approximation ratio depends on the kissing number. For example, there is a polynomial-time 10-approximation algorithm to find a maximum non-intersecting subset of a set of rotated unit squares; the kissing number problem can be stated as the existence of a solution to a set of inequalities. Let x n be a set of N D-dimensional position vectors of the centres of the spheres; the condition that this set of spheres can lie round the centre sphere without overlapping is: ∃ x { ∀ n ∧ ∀ m, n: m ≠ n { (
Triangle
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices A, B, C is denoted △ A B C. In Euclidean geometry any three points, when non-collinear, determine a unique triangle and a unique plane. In other words, there is only one plane that contains that triangle, every triangle is contained in some plane. If the entire geometry is only the Euclidean plane, there is only one plane and all triangles are contained in it; this article is about triangles in Euclidean geometry, in particular, the Euclidean plane, except where otherwise noted. Triangles can be classified according to the lengths of their sides: An equilateral triangle has all sides the same length. An equilateral triangle is a regular polygon with all angles measuring 60°. An isosceles triangle has two sides of equal length. An isosceles triangle has two angles of the same measure, namely the angles opposite to the two sides of the same length; some mathematicians define an isosceles triangle to have two equal sides, whereas others define an isosceles triangle as one with at least two equal sides.
The latter definition would make all equilateral triangles isosceles triangles. The 45–45–90 right triangle, which appears in the tetrakis square tiling, is isosceles. A scalene triangle has all its sides of different lengths. Equivalently, it has all angles of different measure. Hatch marks called tick marks, are used in diagrams of triangles and other geometric figures to identify sides of equal lengths. A side can be marked with a pattern of short line segments in the form of tally marks. In a triangle, the pattern is no more than 3 ticks. An equilateral triangle has the same pattern on all 3 sides, an isosceles triangle has the same pattern on just 2 sides, a scalene triangle has different patterns on all sides since no sides are equal. Patterns of 1, 2, or 3 concentric arcs inside the angles are used to indicate equal angles. An equilateral triangle has the same pattern on all 3 angles, an isosceles triangle has the same pattern on just 2 angles, a scalene triangle has different patterns on all angles since no angles are equal.
Triangles can be classified according to their internal angles, measured here in degrees. A right triangle has one of its interior angles measuring 90°; the side opposite to the right angle is the longest side of the triangle. The other two sides are called the catheti of the triangle. Right triangles obey the Pythagorean theorem: the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse: a2 + b2 = c2, where a and b are the lengths of the legs and c is the length of the hypotenuse. Special right triangles are right triangles with additional properties that make calculations involving them easier. One of the two most famous is the 3–4–5 right triangle, where 32 + 42 = 52. In this situation, 3, 4, 5 are a Pythagorean triple; the other one is an isosceles triangle. Triangles that do not have an angle measuring 90° are called oblique triangles. A triangle with all interior angles measuring less than 90° is an acute triangle or acute-angled triangle.
If c is the length of the longest side a2 + b2 > c2, where a and b are the lengths of the other sides. A triangle with one interior angle measuring more than 90° is an obtuse triangle or obtuse-angled triangle. If c is the length of the longest side a2 + b2 < c2, where a and b are the lengths of the other sides. A triangle with an interior angle of 180° is degenerate. A right degenerate triangle has collinear vertices. A triangle that has two angles with the same measure has two sides with the same length, therefore it is an isosceles triangle, it follows that in a triangle where all angles have the same measure, all three sides have the same length, such a triangle is therefore equilateral. Triangles are assumed to be two-dimensional plane figures. In rigorous treatments, a triangle is therefore called a 2-simplex. Elementary facts about triangles were presented by Euclid in books 1–4 of his Elements, around 300 BC; the sum of the measures of the interior angles of a triangle in Euclidean space is always 180 degrees.
This fact is equivalent to Euclid's parallel postulate. This allows determination of the measure of the third angle of any triangle given the measure of two angles. An exterior angle of a triangle is an angle, a linear pair to an interior angle; the measure of an exterior angle of a triangle is equal to the sum of the measures of the two interior angles that are not adjacent to it. The sum of the measures of the three exterior angles of any triangle is 360 degrees. Two triangles are said to be similar if every angle of one triangle has the same measure as the corresponding angle in the other triangle; the corresponding sides of similar triangles have lengths that are in the same proportion, this property is sufficient to establish similarity. Some basic theorems about similar triangles are: If and only if one pair of internal angles of two triangles have the sam
Dual polyhedron
In geometry, any polyhedron is associated with a second dual figure, where the vertices of one correspond to the faces of the other and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. Such dual figures remain combinatorial or abstract polyhedra, but not all are geometric polyhedra. Starting with any given polyhedron, the dual of its dual is the original polyhedron. Duality preserves the symmetries of a polyhedron. Therefore, for many classes of polyhedra defined by their symmetries, the duals belong to a symmetric class. Thus, the regular polyhedra – the Platonic solids and Kepler–Poinsot polyhedra – form dual pairs, where the regular tetrahedron is self-dual; the dual of an isogonal polyhedron, having equivalent vertices, is one, isohedral, having equivalent faces. The dual of an isotoxal polyhedron is isotoxal. Duality is related to reciprocity or polarity, a geometric transformation that, when applied to a convex polyhedron, realizes the dual polyhedron as another convex polyhedron.
There are many kinds of duality. The kinds most relevant to elementary polyhedra are polar reciprocity and topological or abstract duality; the duality of polyhedra is defined in terms of polar reciprocation about a concentric sphere. Here, each vertex is associated with a face plane so that the ray from the center to the vertex is perpendicular to the plane, the product of the distances from the center to each is equal to the square of the radius. In coordinates, for reciprocation about the sphere x 2 + y 2 + z 2 = r 2, the vertex is associated with the plane x 0 x + y 0 y + z 0 z = r 2; the vertices of the dual are the poles reciprocal to the face planes of the original, the faces of the dual lie in the polars reciprocal to the vertices of the original. Any two adjacent vertices define an edge, these will reciprocate to two adjacent faces which intersect to define an edge of the dual; this dual pair of edges are always orthogonal to each other. If r 0 is the radius of the sphere, r 1 and r 2 the distances from its centre to the pole and its polar, then: r 1.
R 2 = r 0 2 For the more symmetrical polyhedra having an obvious centroid, it is common to make the polyhedron and sphere concentric, as in the Dorman Luke construction described below. However, it is possible to reciprocate a polyhedron about any sphere, the resulting form of the dual will depend on the size and position of the sphere; the choice of center for the sphere is sufficient to define the dual up to similarity. If multiple symmetry axes are present, they will intersect at a single point, this is taken to be the centroid. Failing that, a circumscribed sphere, inscribed sphere, or midsphere is used. If a polyhedron in Euclidean space has an element passing through the center of the sphere, the corresponding element of its dual will go to infinity. Since Euclidean space never reaches infinity, the projective equivalent, called extended Euclidean space, may be formed by adding the required'plane at infinity'; some theorists prefer to say that there is no dual. Meanwhile, Wenninger found a way to represent these infinite duals, in a manner suitable for making models.
The concept of duality here is related to the duality in projective geometry, where lines and edges are interchanged. Projective polarity works well enough for convex polyhedra, but for non-convex figures such as star polyhedra, when we seek to rigorously define this form of polyhedral duality in terms of projective polarity, various problems appear. Because of the definitional issues for geometric duality of non-convex polyhedra, Grünbaum argues that any proper definition of a non-convex polyhedron should include a notion of a dual polyhedron. Any convex polyhedron can be distorted into a canonical form, in which a unit midsphere exists tangent to every edge, such that the average position of the points of tangency is the center of the sphere; this form is unique up to congruences. If we reciprocate such a canonical polyhedron about its midsphere, the dual polyhedron will share the same edge-tangency points and so must be canonical, it is the canonical dual, the two together form a canonical dual pair.
When a pair of polyhedra cannot be obtained by reciprocation from each other, they may be called duals of each other as long as the vertices of one correspond to the faces of the other, the edges of one correspond to the edges of the other, in an incidence-preserving way. Such pairs of polyhedra are abstractly dual; the vertices and edges of a convex polyhedron form a graph, embedded on a topological sphere, the surface of the polyhedron. The same graph can be projected to form
Seville
Seville is the capital and largest city of the autonomous community of Andalusia and the province of Seville, Spain. It is situated on the plain of the river Guadalquivir; the inhabitants of the city are known as sevillanos or hispalenses, after the Roman name of the city, Hispalis. Seville has a municipal population of about 690,000 as of 2016, a metropolitan population of about 1.5 million, making it the fourth-largest city in Spain and the 30th most populous municipality in the European Union. Its Old Town, with an area of 4 square kilometres, contains three UNESCO World Heritage Sites: the Alcázar palace complex, the Cathedral and the General Archive of the Indies; the Seville harbour, located about 80 kilometres from the Atlantic Ocean, is the only river port in Spain. Seville is the hottest major metropolitan area in the geographical Southwestern Europe, with summer average high temperatures of above 35 °C. Seville was founded as the Roman city of Hispalis, it became known as Ishbiliyya after the Muslim conquest in 712.
During the Muslim rule in Spain, Seville came under the jurisdiction of the Caliphate of Córdoba before becoming the independent Taifa of Seville. After the discovery of the Americas, Seville became one of the economic centres of the Spanish Empire as its port monopolised the trans-oceanic trade and the Casa de Contratación wielded its power, opening a Golden Age of arts and literature. In 1519, Ferdinand Magellan departed from Seville for the first circumnavigation of the Earth. Coinciding with the Baroque period of European history, the 17th century in Seville represented the most brilliant flowering of the city's culture; the 20th century in Seville saw the tribulations of the Spanish Civil War, decisive cultural milestones such as the Ibero-American Exposition of 1929 and Expo'92, the city's election as the capital of the Autonomous Community of Andalusia. Hisbaal is the oldest name for Seville, it appears to have originated during the Phoenician colonisation of the Tartessian culture in south-western Iberia and it refers to the God Baal.
According to Manuel Pellicer Catalán, the ancient name was Spal, it meant "lowland" in the Phoenician language. During Roman rule, the name was Latinised as Hispal and as Hispalis. After the Umayyad invasion, this name was adapted into Arabic as Ishbiliyya: since p does not exist in Arabic, it was replaced by b. NO8DO is the official motto of Seville, popularly believed to be a rebus signifying the Spanish No me ha dejado, meaning "She has not abandoned me"; the phrase, pronounced with synalepha as, is spelled with an eight in the middle representing the word madeja "skein ". Legend states that the title was given by King Alfonso X, resident in the city's Alcázar and supported by the citizens when his son Sancho IV of Castile, tried to usurp the throne from him; the emblem is present on Seville's municipal flag, features on city property such as manhole covers, Christopher Columbus's tomb in the Cathedral. Seville is 2,200 years old; the passage of the various civilizations instrumental in its growth has left the city with a distinct personality, a large and well-preserved historical centre.
The mythological founder of the city is Hercules identified with the Phoenician god Melqart, who the myth says sailed through the Strait of Gibraltar to the Atlantic, founded trading posts at the current sites of Cádiz and of Seville. The original core of the city, in the neighbourhood of the present-day street, Cuesta del Rosario, dates to the 8th century BC, when Seville was on an island in the Guadalquivir. Archaeological excavations in 1999 found anthropic remains under the north wall of the Real Alcázar dating to the 8th–7th century BC; the town was called Hisbaal by the Phoenicians and by the Tartessians, the indigenous pre-Roman Iberian people of Tartessos, who controlled the Guadalquivir Valley at the time. The city was known from Roman times as Hispal and as Hispalis. Hispalis developed into one of the great market and industrial centres of Hispania, while the nearby Roman city of Italica remained a Roman residential city. Large-scale Roman archaeological remains can be seen there and at the nearby town of Carmona as well.
Existing Roman features in Seville itself include the remains exposed in situ in the underground Antiquarium of the Metropol Parasol building, the remnants of an aqueduct, three pillars of a temple in Mármoles Street, the columns of La Alameda de Hércules and the remains in the Patio de Banderas square near the Seville Cathedral. The walls surrounding the city were built during the rule of Julius Caesar, but their current course and design were the result of Moorish reconstructions. Following Roman rule, there were successive conquests of the Roman province of Hispania Baetica by the Vandals, the Suebi and the Visigoths during the 5th and 6th centuries. Seville was taken by the Moors, during the conquest of Hispalis in 712, it was the capital for the kings of the Umayyad Caliphate, the Almoravid dynasty first and
Overlapping circles grid
An overlapping circles grid is a geometric pattern of repeating, overlapping circles of equal radii in two-dimensional space. Designs are based on circles centered on triangles or on the square lattice pattern of points. Patterns of seven overlapping circles appear in historical artefacts from the 7th century BC onwards; the name "Flower of Life" is given to the overlapping circles pattern in New Age publications. Of special interest is the six petal rosette derived from the "seven overlapping circles" pattern known as "Sun of the Alps" from its frequent use in alpine folk art in the 17th and 18th century; the triangular lattice form, with circle radii equal to their separation is called a seven overlapping circles grid. It contains 6 circles intersecting with a 7th circle centered on that intersection. Overlapping circles with similar geometrical constructions have been used infrequently in various of the decorative arts since ancient times; the pattern has found a wide range of usage in popular culture, in fashion, jewelry and decorative products.
The oldest known occurrence of the "overlapping circles" pattern is dated to the 7th or 6th century BCE, found on the threshold of the palace of Assyrian king Aššur-bāni-apli in Dur Šarrukin. The design becomes more widespread in the early centuries of the Common Era. One early example are five patterns of 19 overlapping circles drawn on the granite columns at the Temple of Osiris in Abydos, a further five on column opposite the building, they are drawn in red ochre and some are faint and difficult to distinguish. The patterns are graffiti, not found in natively Egyptian ornaments, they are dated to the early centuries of the Christian Era although medieval or modern origin cannot be ruled out with certainty, as the drawings are not mentioned in the extensive listings of graffiti at the temple compiled by Margaret Murray in 1904. Similar patterns were sometimes used in England as apotropaic marks to keep witches from entering buildings. Consecration crosses indicating points in churches anointed with holy water during a churches dedication take the form of overlapping circles.
In Islamic art, the pattern is one of several arrangements of circles used to construct grids for Islamic geometric patterns. It is used to design patterns with 6- and 12-pointed stars as well as hexagons in the style called girih; the resulting patterns however characteristically conceal the construction grid, presenting instead a design of interlaced strapwork. Patterns of seven overlapping circles are found on a Cypro-Archaic I cup of the 8th-7th century BC in Cyprus and Roman mosaics, for example at Herod's palace in the 1st century BC, they are found in the Hindu temple at Prambanan in Java. The design is found on one of the silver plaques of the Late Roman hoard of Kaiseraugst, it is found as an ornament in Gothic architecture, still in European folk art of the early modern period. High medieval examples include the Cosmati pavements in Westminster Abbey. Leonardo da Vinci explicitly discussed the mathematical proportions of the design; the name "Flower of Life" is modern, associated with the New Age movement, attributed to Drunvalo Melchizedek in his book The Ancient Secret of the Flower of Life.
The pattern and modern name have propagated into wide range of usage in popular culture, in fashion, jewelry and decorative products. The pattern in quilting has been called diamond wedding ring or triangle wedding ring to contrast it from the square pattern. Besides an occasional use in fashion, it is used in the decorative arts. For example, the album Sempiternal by Bring Me the Horizon uses the 61 overlapping circles grid as the main feature of its album cover, whereas the album A Head Full of Dreams by Coldplay features the 19 overlapping circles grid as the central part of its album cover. Teaser posters illustrating the cover art to A Head Full of Dreams were displayed on the London Underground in the last week of October 2015; the "Sun of the Alps" symbol has been used as the emblem of Padanian nationalism in northern Italy since the 1990s. It resembles a pattern found in that area on buildings. 1, 7, 19-circle hexagonal variantIn the examples below the pattern has a hexagonal outline, is further circumscribed.
Similar patternsIn the examples below the pattern does not have a hexagonal outline. Martha Bartfeld, author of geometric art tutorial books, described her independent discovery of the design in 1968, her original definition said, "This design consists of circles having a 1- radius, with each point of intersection serving as a new center. The design can be expanded ad infinitum depending upon the number of times the odd-numbered points are marked off." The pattern figure can be drawn by pen and compass, by creating multiple series of interlinking circles of the same diameter touching the previous circle's center. The second circle is centered at any point on the first circle. All following circles are centered on the intersection of two other circles; the pattern can be extended outwards in concentric hexagonal rings of circles. The first row shows rings of circles; the second row shows a three-dimensional interpretation of a set of n×n×n cube of spheres viewed from a diagonal axis. The third row shows the pattern completed with partial circle arcs within a set of completed circles.
Expan
Wallpaper group
A wallpaper group is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetries in the pattern. Such patterns occur in architecture and decorative art in textiles and tiles as well as wallpaper. A proof that there were only 17 distinct groups of possible patterns was first carried out by Evgraf Fedorov in 1891 and derived independently by George Pólya in 1924; the proof that the list of wallpaper groups was complete only came after the much harder case of space groups had been done. The seventeen possible wallpaper groups are listed below in § The seventeen groups. Wallpaper groups are two-dimensional symmetry groups, intermediate in complexity between the simpler frieze groups and the three-dimensional space groups. Wallpaper groups categorize patterns by their symmetries. Subtle differences may place similar patterns in different groups, while patterns that are different in style, scale or orientation may belong to the same group. Consider the following examples: Examples A and B have the same wallpaper group.
Example C has a different wallpaper group, called p4g or 4*2. The fact that A and B have the same wallpaper group means that they have the same symmetries, regardless of details of the designs, whereas C has a different set of symmetries despite any superficial similarities. A symmetry of a pattern is, loosely speaking, a way of transforming the pattern so that it looks the same after the transformation. For example, translational symmetry is present when the pattern can be translated some finite distance and appear unchanged. Think of shifting a set of vertical stripes horizontally by one stripe; the pattern is unchanged. Speaking, a true symmetry only exists in patterns that repeat and continue indefinitely. A set of only, 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, classification is applied to finite patterns, small imperfections may be ignored. Sometimes two categorizations are meaningful, one based on shapes alone and one including colors.
When colors are ignored there may be more symmetry. In black and white there are 17 wallpaper groups; the types of transformations that are relevant here are called Euclidean plane isometries. For example: If we shift example B one unit to the right, so that each square covers the square, adjacent to it the resulting pattern is the same as the pattern we started with; this type of symmetry is called a translation. Examples A and C are similar. If we turn example B clockwise by 90°, around the centre of one of the squares, again we obtain the same pattern; this is called a rotation. Examples A and C have 90° rotations, although it requires a little more ingenuity to find the correct centre of rotation for C. We can flip example B across a horizontal axis that runs across the middle of the image; this is called a reflection. Example B has reflections across a vertical axis, across two diagonal axes; the same can be said for A. However, example C is different, it only has reflections in vertical directions, not across diagonal axes.
If we flip across a diagonal line, we do not get the same pattern back. This is part of the reason that the wallpaper group of A and B is different from the wallpaper group of C. Another transformation is "Glide", a combination of reflection and translation parallel to the line of reflection. Mathematically, a wallpaper group or plane crystallographic group is a type of topologically discrete group of isometries of the Euclidean plane that contains two linearly independent translations. Two such isometry groups are of the same type if they are the same up to an affine transformation of the plane, thus e.g. a translation of the plane does not affect the wallpaper group. The same applies for a change of angle between translation vectors, provided that it does not add or remove any symmetry. Unlike in the three-dimensional case, we can equivalently restrict the affine transformations to those that preserve orientation, it follows from the Bieberbach theorem that all wallpaper groups are different as abstract groups.
2D patterns with double translational symmetry can be categorized according to their symmetry group type. Isometries of the Euclidean plane fall into four categories. Translations, denoted by Tv, where v is a vector in R2; this has the effect of shifting the plane applying displacement vector v. Rotations, denoted by Rc,θ, where c is a point in the plane, θ is the angle of rotation. Reflections, or mirror isometries, denoted by FL, where L is a line in R2.. This has the effect of reflecting the plane in the line L, called the reflection axis or the associated mirror. Glide reflections, denoted by GL,d, where L is a line in R2 and d is a distance; this is a combination of a reflection in the line L and a translation along L by a distance d. The condition