1.
Geometry
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Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer, Geometry arose independently in a number of early cultures as a practical way for dealing with lengths, areas, and volumes. Geometry began to see elements of mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into a form by Euclid, whose treatment, Euclids Elements. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC, islamic scientists preserved Greek ideas and expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid footing by mathematicians such as René Descartes. Since then, and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, while geometry has evolved significantly throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, lines, planes, surfaces, angles, contemporary geometry has many subfields, Euclidean geometry is geometry in its classical sense. The mandatory educational curriculum of the majority of nations includes the study of points, lines, planes, angles, triangles, congruence, similarity, solid figures, circles, Euclidean geometry also has applications in computer science, crystallography, and various branches of modern mathematics. Differential geometry uses techniques of calculus and linear algebra to problems in geometry. It has applications in physics, including in general relativity, topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this often means dealing with large-scale properties of spaces, convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues, often using techniques of real analysis. It has close connections to convex analysis, optimization and functional analysis, algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques. It has applications in areas, including cryptography and string theory. Discrete geometry is concerned mainly with questions of relative position of simple objects, such as points. It shares many methods and principles with combinatorics, Geometry has applications to many fields, including art, architecture, physics, as well as to other branches of mathematics. The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia, the earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets such as Plimpton 322. For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, later clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiters position and motion within time-velocity space
2.
Sphere
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A sphere is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball. This distance r is the radius of the ball, and the point is the center of the mathematical ball. The longest straight line through the ball, connecting two points of the sphere, passes through the center and its length is twice the radius. While outside mathematics the terms sphere and ball are used interchangeably. The ball and the share the same radius, diameter. The surface area of a sphere is, A =4 π r 2, at any given radius r, the incremental volume equals the product of the surface area at radius r and the thickness of a shell, δ V ≈ A ⋅ δ r. The total volume is the summation of all volumes, V ≈ ∑ A ⋅ δ r. In the limit as δr approaches zero this equation becomes, V = ∫0 r A d r ′, substitute V,43 π r 3 = ∫0 r A d r ′. Differentiating both sides of equation with respect to r yields A as a function of r,4 π r 2 = A. Which is generally abbreviated as, A =4 π r 2, alternatively, the area element on the sphere is given in spherical coordinates by dA = r2 sin θ dθ dφ. In Cartesian coordinates, the element is d S = r r 2 − ∑ i ≠ k x i 2 ∏ i ≠ k d x i, ∀ k. For more generality, see area element, the total area can thus be obtained by integration, A = ∫02 π ∫0 π r 2 sin θ d θ d φ =4 π r 2. In three dimensions, the volume inside a sphere is derived to be V =43 π r 3 where r is the radius of the sphere, archimedes first derived this formula, which shows that the volume inside a sphere is 2/3 that of a circumscribed cylinder. In modern mathematics, this formula can be derived using integral calculus, at any given x, the incremental volume equals the product of the cross-sectional area of the disk at x and its thickness, δ V ≈ π y 2 ⋅ δ x. The total volume is the summation of all volumes, V ≈ ∑ π y 2 ⋅ δ x. In the limit as δx approaches zero this equation becomes, V = ∫ − r r π y 2 d x. At any given x, a right-angled triangle connects x, y and r to the origin, hence, applying the Pythagorean theorem yields, thus, substituting y with a function of x gives, V = ∫ − r r π d x. Which can now be evaluated as follows, V = π − r r = π − π =43 π r 3
3.
Lattice (group)
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In geometry and group theory, a lattice in R n is a subgroup of R n which is isomorphic to Z n, and which spans the real vector space R n. In other words, for any basis of R n, the subgroup of all linear combinations with integer coefficients of the basis vectors forms a lattice, a lattice may be viewed as a regular tiling of a space by a primitive cell. Lattices have many significant applications in mathematics, particularly in connection to Lie algebras, number theory. More generally, lattice models are studied in physics, often by the techniques of computational physics, a lattice is the symmetry group of discrete translational symmetry in n directions. A pattern with this lattice of translational symmetry cannot have more, as a group a lattice is a finitely-generated free abelian group, and thus isomorphic to Z n. A lattice in the sense of a 3-dimensional array of regularly spaced points coinciding with e. g, a simple example of a lattice in R n is the subgroup Z n. More complicated examples include the E8 lattice, which is a lattice in R8, the period lattice in R2 is central to the study of elliptic functions, developed in nineteenth century mathematics, it generalises to higher dimensions in the theory of abelian functions. Lattices called root lattices are important in the theory of simple Lie algebras, for example, a typical lattice Λ in R n thus has the form Λ = where is a basis for R n. Different bases can generate the lattice, but the absolute value of the determinant of the vectors vi is uniquely determined by Λ. If one thinks of a lattice as dividing the whole of R n into equal polyhedra and this is why d is sometimes called the covolume of the lattice. If this equals 1, the lattice is called unimodular, minkowskis theorem relates the number d and the volume of a symmetric convex set S to the number of lattice points contained in S. The number of lattice points contained in an all of whose vertices are elements of the lattice is described by the polytopes Ehrhart polynomial. Formulas for some of the coefficients of this polynomial involve d as well, Lattice basis reduction is the problem of finding a short and nearly orthogonal lattice basis. The Lenstra-Lenstra-Lovász lattice basis reduction algorithm approximates such a basis in polynomial time, it has found numerous applications. There are five 2D lattice types as given by the crystallographic restriction theorem, below, the wallpaper group of the lattice is given in IUC notation, Orbifold notation, and Coxeter notation, along with a wallpaper diagram showing the symmetry domains. Note that a pattern with this lattice of translational symmetry cannot have more, a full list of subgroups is available. For example below the hexagonal/triangular lattice is given twice, with full 6-fold, if the symmetry group of a pattern contains an n-fold rotation then the lattice has n-fold symmetry for even n and 2n-fold for odd n. For the classification of a lattice, start with one point
4.
Carl Friedrich Gauss
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Johann Carl Friedrich Gauss was born on 30 April 1777 in Brunswick, in the Duchy of Brunswick-Wolfenbüttel, as the son of poor working-class parents. Gauss later solved this puzzle about his birthdate in the context of finding the date of Easter and he was christened and confirmed in a church near the school he attended as a child. A contested story relates that, when he was eight, he figured out how to add up all the numbers from 1 to 100, there are many other anecdotes about his precocity while a toddler, and he made his first ground-breaking mathematical discoveries while still a teenager. He completed Disquisitiones Arithmeticae, his opus, in 1798 at the age of 21. This work was fundamental in consolidating number theory as a discipline and has shaped the field to the present day, while at university, Gauss independently rediscovered several important theorems. Gauss was so pleased by this result that he requested that a regular heptadecagon be inscribed on his tombstone, the stonemason declined, stating that the difficult construction would essentially look like a circle. The year 1796 was most productive for both Gauss and number theory and he discovered a construction of the heptadecagon on 30 March. He further advanced modular arithmetic, greatly simplifying manipulations in number theory, on 8 April he became the first to prove the quadratic reciprocity law. This remarkably general law allows mathematicians to determine the solvability of any quadratic equation in modular arithmetic, the prime number theorem, conjectured on 31 May, gives a good understanding of how the prime numbers are distributed among the integers. Gauss also discovered that every integer is representable as a sum of at most three triangular numbers on 10 July and then jotted down in his diary the note, ΕΥΡΗΚΑ. On October 1 he published a result on the number of solutions of polynomials with coefficients in finite fields, in 1831 Gauss developed a fruitful collaboration with the physics professor Wilhelm Weber, leading to new knowledge in magnetism and the discovery of Kirchhoffs circuit laws in electricity. It was during this time that he formulated his namesake law and they constructed the first electromechanical telegraph in 1833, which connected the observatory with the institute for physics in Göttingen. In 1840, Gauss published his influential Dioptrische Untersuchungen, in which he gave the first systematic analysis on the formation of images under a paraxial approximation. Among his results, Gauss showed that under a paraxial approximation an optical system can be characterized by its cardinal points and he derived the Gaussian lens formula. In 1845, he became associated member of the Royal Institute of the Netherlands, in 1854, Gauss selected the topic for Bernhard Riemanns Habilitationvortrag, Über die Hypothesen, welche der Geometrie zu Grunde liegen. On the way home from Riemanns lecture, Weber reported that Gauss was full of praise, Gauss died in Göttingen, on 23 February 1855 and is interred in the Albani Cemetery there. Two individuals gave eulogies at his funeral, Gausss son-in-law Heinrich Ewald and Wolfgang Sartorius von Waltershausen and his brain was preserved and was studied by Rudolf Wagner who found its mass to be 1,492 grams and the cerebral area equal to 219,588 square millimeters. Highly developed convolutions were also found, which in the early 20th century were suggested as the explanation of his genius, Gauss was a Lutheran Protestant, a member of the St. Albans Evangelical Lutheran church in Göttingen
5.
Kepler conjecture
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The Kepler conjecture, named after the 17th-century mathematician and astronomer Johannes Kepler, is a mathematical conjecture about sphere packing in three-dimensional Euclidean space. It says that no arrangement of equally sized spheres filling space has an average density than that of the cubic close packing. The density of these arrangements is around 74. 05%, in 1998 Thomas Hales, following an approach suggested by Fejes Tóth, announced that he had a proof of the Kepler conjecture. Hales proof is a proof by exhaustion involving the checking of many cases using complex computer calculations. Referees have said that they are 99% certain of the correctness of Hales proof, in 2014, the Flyspeck project team, headed by Hales, announced the completion of a formal proof of the Kepler conjecture using a combination of the Isabelle and HOL Light proof assistants. Imagine filling a container with small equal-sized spheres. The density of the arrangement is equal to the volume of the spheres divided by the volume of the container. To maximize the number of spheres in the means to create an arrangement with the highest possible density. Experiment shows that dropping the spheres in randomly will achieve a density of around 65%, however, a higher density can be achieved by carefully arranging the spheres as follows. Start with a layer of spheres in a lattice, then put the next layer of spheres in the lowest points you can find above the first layer. The conjecture was first stated by Johannes Kepler in his paper On the six-cornered snowflake and he had started to study arrangements of spheres as a result of his correspondence with the English mathematician and astronomer Thomas Harriot in 1606. Harriot was a friend and assistant of Sir Walter Raleigh, who had set Harriot the problem of determining how best to stack cannonballs on the decks of his ships. Harriot published a study of various stacking patterns in 1591, and this meant that any packing arrangement that disproved the Kepler conjecture would have to be an irregular one. But eliminating all possible arrangements is very difficult, and this is what made the Kepler conjecture so hard to prove. After Gauss, no progress was made towards proving the Kepler conjecture in the nineteenth century. In 1900 David Hilbert included it in his list of twenty three unsolved problems of mathematics—it forms part of Hilberts eighteenth problem, the next step toward a solution was taken by László Fejes Tóth. Fejes Tóth showed that the problem of determining the density of all arrangements could be reduced to a finite number of calculations. This meant that a proof by exhaustion was, in principle, possible, as Fejes Tóth realised, a fast enough computer could turn this theoretical result into a practical approach to the problem
6.
Thomas Callister Hales
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Thomas Callister Hales is an American mathematician working on the Langlands program. He is known in the area for having worked on the fundamental lemma, many of his ideas were incorporated into the final proof, due to Ngô Bảo Châu. He is also known for his proof of the Kepler conjecture on sphere packing and he received his Ph. D. from Princeton University in 1986, his dissertation was entitled The Subregular Germ of Orbital Integrals. Between 1993 and 2002 he worked at the University of Michigan and he was aided by graduate student Samuel Ferguson. In 1999, Hales proved the conjecture, he also stated that the conjecture may have been present in the minds of mathematicians before Marcus Terentius Varro. After 2002, Hales became the University of Pittsburghs Mellon Professor of mathematics, in 2003, Hales started work on Flyspeck to vindicate his proof of the Kepler conjecture. His proof relied on computer calculation to verify conjectures, the project used two proof assistants, HOL Light and Isabelle. Annals of Mathematics accepted the proof in 2005, but was only 99% sure of the proof, in August 2014, the Flyspeck teams software finally verified the proof to be correct. Hales won the Chauvenet Prize in 2003 and a Lester R. Ford Award in 2008, in 2012 he became a fellow of the American Mathematical Society. The status of the Kepler conjecture, The Mathematical Intelligencer,16, 47–58, historical overview of the Kepler conjecture, Discrete & Computational Geometry,36, 5–20, doi,10. 1007/s00454-005-1210-2, ISSN 0179-5376, MR2229657 Hales, Thomas C. A formulation of the Kepler conjecture, Discrete & Computational Geometry,36, 21–69, doi,10. 1007/s00454-005-1211-1, ISSN 0179-5376, MR2229658 Hales, Thomas C
7.
Crystal
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A crystal or crystalline solid is a solid material whose constituents are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macroscopic single crystals are usually identifiable by their geometrical shape, the scientific study of crystals and crystal formation is known as crystallography. The process of crystal formation via mechanisms of crystal growth is called crystallization or solidification, the word crystal derives from the Ancient Greek word κρύσταλλος, meaning both ice and rock crystal, from κρύος, icy cold, frost. Examples of large crystals include snowflakes, diamonds, and table salt, most inorganic solids are not crystals but polycrystals, i. e. many microscopic crystals fused together into a single solid. Examples of polycrystals include most metals, rocks, ceramics, a third category of solids is amorphous solids, where the atoms have no periodic structure whatsoever. Examples of amorphous solids include glass, wax, and many plastics, Crystals are often used in pseudoscientific practices such as crystal therapy, and, along with gemstones, are sometimes associated with spellwork in Wiccan beliefs and related religious movements. The scientific definition of a crystal is based on the arrangement of atoms inside it. A crystal is a solid where the form a periodic arrangement. For example, when liquid water starts freezing, the change begins with small ice crystals that grow until they fuse. Most macroscopic inorganic solids are polycrystalline, including almost all metals, ceramics, ice, rocks, solids that are neither crystalline nor polycrystalline, such as glass, are called amorphous solids, also called glassy, vitreous, or noncrystalline. These have no periodic order, even microscopically, there are distinct differences between crystalline solids and amorphous solids, most notably, the process of forming a glass does not release the latent heat of fusion, but forming a crystal does. A crystal structure is characterized by its cell, a small imaginary box containing one or more atoms in a specific spatial arrangement. The unit cells are stacked in three-dimensional space to form the crystal, the symmetry of a crystal is constrained by the requirement that the unit cells stack perfectly with no gaps. There are 219 possible crystal symmetries, called space groups. These are grouped into 7 crystal systems, such as cubic crystal system or hexagonal crystal system, Crystals are commonly recognized by their shape, consisting of flat faces with sharp angles. Euhedral crystals are those with obvious, well-formed flat faces, anhedral crystals do not, usually because the crystal is one grain in a polycrystalline solid. The flat faces of a crystal are oriented in a specific way relative to the underlying atomic arrangement of the crystal. This occurs because some surface orientations are more stable than others, as a crystal grows, new atoms attach easily to the rougher and less stable parts of the surface, but less easily to the flat, stable surfaces
8.
Cuboctahedron
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In geometry, a cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, as such, it is a quasiregular polyhedron, i. e. an Archimedean solid that is not only vertex-transitive but also edge-transitive. Its dual polyhedron is the rhombic dodecahedron, the cuboctahedron was probably known to Plato, Herons Definitiones quotes Archimedes as saying that Plato knew of a solid made of 8 triangles and 6 squares. Heptaparallelohedron Fuller applied the name Dymaxion to this shape, used in a version of the Dymaxion map. He also called it the Vector Equilibrium and he called a cuboctahedron consisting of rigid struts connected by flexible vertices a jitterbug. With Oh symmetry, order 48, it is a cube or rectified octahedron With Td symmetry, order 24. With D3d symmetry, order 12, it is a triangular gyrobicupola. The area A and the volume V of the cuboctahedron of edge length a are, the cuboctahedron has four special orthogonal projections, centered on a vertex, an edge, and the two types of faces, triangular and square. The last two correspond to the B2 and A2 Coxeter planes, the skew projections show a square and hexagon passing through the center of the cuboctahedron. The cuboctahedron can also be represented as a tiling. This projection is conformal, preserving angles but not areas or lengths, straight lines on the sphere are projected as circular arcs on the plane. The cuboctahedrons 12 vertices can represent the vectors of the simple Lie group A3. With the addition of 6 vertices of the octahedron, these represent the 18 root vectors of the simple Lie group B3. The cuboctahedron can be dissected into two triangular cupolas by a common hexagon passing through the center of the cuboctahedron, if these two triangular cupolas are twisted so triangles and squares line up, Johnson solid J27, the triangular orthobicupola, is created. The cuboctahedron can also be dissected into 6 square pyramids and 8 tetrahedra meeting at a central point and this dissection is expressed in the alternated cubic honeycomb where pairs of square pyramids are combined into octahedra. A cuboctahedron can be obtained by taking a cross section of a four-dimensional 16-cell. Its first stellation is the compound of a cube and its dual octahedron, the cuboctahedron is a rectified cube and also a rectified octahedron. It is also a cantellated tetrahedron, with this construction it is given the Wythoff symbol,33 |2
9.
Triangular orthobicupola
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In geometry, the triangular orthobicupola is one of the Johnson solids. As the name suggests, it can be constructed by attaching two triangular cupolas along their bases and it has an equal number of squares and triangles at each vertex, however, it is not vertex-transitive. It is also called an anticuboctahedron, twisted cuboctahedron or disheptahedron, a Johnson solid is one of 92 strictly convex polyhedra that have regular faces but are not uniform. They were named by Norman Johnson, who first listed these polyhedra in 1966, the triangular orthobicupola is the first in an infinite set of orthobicupolae. Given a triangular orthobicupola, a 60-degree rotation of one cupola before the joining yields a cuboctahedron, hence, another name for the triangular orthobicupola is the anticuboctahedron. The elongated triangular orthobicupola, which is constructed by elongating this solid, has a relationship with the rhombicuboctahedron. The dual of the triangular orthobicupola is the trapezo-rhombic dodecahedron and it has 6 rhombic and 6 trapezoidal faces, and is similar to the rhombic dodecahedron. The following formulae for volume, surface area, and circumradius can be used if all faces are regular, a 2 The circumradius of a triangular orthobicupola is the same as the edge length. The rectified cubic honeycomb can be dissected and rebuilt as a lattice of triangular orthobicupolae. Eric W. Weisstein, Johnson solid at MathWorld
10.
Cubic crystal system
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In crystallography, the cubic crystal system is a crystal system where the unit cell is in the shape of a cube. This is one of the most common and simplest shapes found in crystals and minerals, there are three main varieties of these crystals, Primitive cubic Body-centered cubic, Face-centered cubic Each is subdivided into other variants listed below. Note that although the cell in these crystals is conventionally taken to be a cube. This is related to the fact that in most cubic crystal systems, a classic isometric crystal has square or pentagonal faces. The three Bravais lattices in the crystal system are, The primitive cubic system consists of one lattice point on each corner of the cube. Each atom at a point is then shared equally between eight adjacent cubes, and the unit cell therefore contains in total one atom. The body-centered cubic system has one point in the center of the unit cell in addition to the eight corner points. It has a net total of 2 lattice points per unit cell, Each sphere in a cF lattice has coordination number 12. The face-centered cubic system is related to the hexagonal close packed system. The plane of a cubic system is a hexagonal grid. Attempting to create a C-centered cubic crystal system would result in a simple tetragonal Bravais lattice, there are a total 36 cubic space groups. Other terms for hexoctahedral are, normal class, holohedral, ditesseral central class, a simple cubic unit cell has a single cubic void in the center. Additionally, there are 24 tetrahedral voids located in a square spacing around each octahedral void and these tetrahedral voids are not local maxima and are not technically voids, but they do occasionally appear in multi-atom unit cells. A face-centered cubic unit cell has eight tetrahedral voids located midway between each corner and the center of the cell, for a total of eight net tetrahedral voids. One important characteristic of a structure is its atomic packing factor. This is calculated by assuming all the atoms are identical spheres. The atomic packing factor is the proportion of space filled by these spheres, assuming one atom per lattice point, in a primitive cubic lattice with cube side length a, the sphere radius would be a⁄2 and the atomic packing factor turns out to be about 0.524. Similarly, in a bcc lattice, the atomic packing factor is 0.680, as a rule, since atoms in a solid attract each other, the more tightly packed arrangements of atoms tend to be more common
11.
Hexagonal crystal family
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In crystallography, the hexagonal crystal family is one of the 6 crystal families. In the hexagonal family, the crystal is described by a right rhombic prism unit cell with two equal axes, an included angle of 120° and a height perpendicular to the two base axes. There are 52 space groups associated with it, which are exactly those whose Bravais lattice is either hexagonal or rhombohedral, the hexagonal crystal family consists of two lattice systems, hexagonal and rhombohedral. Each lattice system consists of one Bravais lattice, hence, there are 3 lattice points per unit cell in total and the lattice is non-primitive. The Bravais lattices in the hexagonal crystal family can also be described by rhombohedral axes, the unit cell is a rhombohedron. This is a cell with parameters a = b = c, α = β = γ ≠ 90°. In practice, the description is more commonly used because it is easier to deal with a coordinate system with two 90° angles. However, the axes are often shown in textbooks because this cell reveals 3m symmetry of crystal lattice. However, such a description is rarely used, the hexagonal crystal family consists of two crystal systems, trigonal and hexagonal. A crystal system is a set of point groups in which the point groups themselves, the trigonal crystal system consists of the 5 point groups that have a single three-fold rotation axis. The crystal structures of alpha-quartz in the example are described by two of those 18 space groups associated with the hexagonal lattice system. The hexagonal crystal system consists of the seven point groups such that all their groups have the hexagonal lattice as underlying lattice. Graphite is an example of a crystal that crystallizes in the crystal system. Note that the atom in the center of the HCP unit cell in the hexagonal lattice system does not appear in the unit cell of the hexagonal lattice. It is part of the two atom motif associated with each point in the underlying lattice. The trigonal crystal system is the crystal system whose point groups have more than one lattice system associated with their space groups. The 5 point groups in this system are listed below, with their international number and notation, their space groups in name. The point groups in this system are listed below, followed by their representations in Hermann–Mauguin or international notation and Schoenflies notation
12.
Symmetry
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Symmetry in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, symmetry has a precise definition, that an object is invariant to any of various transformations. Although these two meanings of symmetry can sometimes be told apart, they are related, so they are discussed together. The opposite of symmetry is asymmetry, a geometric shape or object is symmetric if it can be divided into two or more identical pieces that are arranged in an organized fashion. This means that an object is symmetric if there is a transformation that moves individual pieces of the object, an object has rotational symmetry if the object can be rotated about a fixed point without changing the overall shape. An object has symmetry if it can be translated without changing its overall shape. An object has symmetry if it can be simultaneously translated and rotated in three-dimensional space along a line known as a screw axis. An object has symmetry if it does not change shape when it is expanded or contracted. Fractals also exhibit a form of symmetry, where small portions of the fractal are similar in shape to large portions. Other symmetries include glide reflection symmetry and rotoreflection symmetry, a dyadic relation R is symmetric if and only if, whenever its true that Rab, its true that Rba. Thus, is the age as is symmetrical, for if Paul is the same age as Mary. Symmetric binary logical connectives are and, or, biconditional, nand, xor, the set of operations that preserve a given property of the object form a group. In general, every kind of structure in mathematics will have its own kind of symmetry, examples include even and odd functions in calculus, the symmetric group in abstract algebra, symmetric matrices in linear algebra, and the Galois group in Galois theory. In statistics, it appears as symmetric probability distributions, and as skewness, symmetry in physics has been generalized to mean invariance—that is, lack of change—under any kind of transformation, for example arbitrary coordinate transformations. This concept has one of the most powerful tools of theoretical physics. See Noethers theorem, and also, Wigners classification, which says that the symmetries of the laws of physics determine the properties of the found in nature. Important symmetries in physics include continuous symmetries and discrete symmetries of spacetime, internal symmetries of particles, in biology, the notion of symmetry is mostly used explicitly to describe body shapes. Bilateral animals, including humans, are more or less symmetric with respect to the plane which divides the body into left
13.
Root system
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In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, further, the classification scheme for root systems, by Dynkin diagrams, occurs in parts of mathematics with no overt connection to Lie theory. Finally, root systems are important for their own sake, as in graph theory. As a first example, consider the six vectors in 2-dimensional Euclidean space, R2, as shown in the image at the right and these vectors span the whole space. If you consider the line perpendicular to any root, say β, then the reflection of R2 in that line sends any other root, say α, moreover, the root to which it is sent equals α + nβ, where n is an integer. These six vectors satisfy the definition, and therefore they form a root system. Let V be a finite-dimensional Euclidean vector space, with the standard Euclidean inner product denoted by, in this context, a root system that also satisfies the integrality condition is known as a crystallographic root system. Other authors omit condition 2, then they call root systems satisfying condition 2 reduced, in this article, all root systems are assumed to be reduced and crystallographic. In view of property 3, the integrality condition is equivalent to stating that β, Note that the operator ⟨ ⋅, ⋅ ⟩, Φ × Φ → Z defined by property 4 is not an inner product. It is not necessarily symmetric and is only in the first argument. The rank of a root system Φ is the dimension of V, two root systems may be combined by regarding the Euclidean spaces they span as mutually orthogonal subspaces of a common Euclidean space. A root system which does not arise from such a combination, such as the systems A2, B2, and G2 pictured to the right, is said to be irreducible. Two root systems and are called if there is an invertible linear transformation E1 → E2 which sends Φ1 to Φ2 such that for each pair of roots. The group of isometries of V generated by reflections through hyperplanes associated to the roots of Φ is called the Weyl group of Φ, as it acts faithfully on the finite set Φ, the Weyl group is always finite. The root lattice of a root system Φ is the Z-submodule of V generated by Φ, there is only one root system of rank 1, consisting of two nonzero vectors. This root system is called A1, in rank 2 there are four possibilities, corresponding to σ α = β + n α, where n =0,1,2,3. Whenever Φ is a system in V, and U is a subspace of V spanned by Ψ = Φ ∩ U. Thus, the exhaustive list of four systems of rank 2 shows the geometric possibilities for any two roots chosen from a root system of arbitrary rank
14.
Cannonball problem
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François Édouard Anatole Lucas was a French mathematician. Lucas is known for his study of the Fibonacci sequence, the related Lucas sequences and Lucas numbers are named after him. Lucas was educated at the École Normale Supérieure and he worked in the Paris observatory and later became a professor of mathematics in Paris. In the meantime he served in the army, in 1875, Lucas posed a challenge to prove that the only solution of the Diophantine equation, ∑ n =1 N n 2 = M2 with N >1 is when N =24 and M =70. This is known as the problem, since it can be visualized as the problem of taking a square arrangement of cannonballs on the ground. It was not until 1918 that a proof was found for this remarkable fact, more recently, elementary proofs have been published. He devised methods for testing the primality of numbers, in 1857, at age 15, Lucas began testing the primality of 2127 −1 by hand, using Lucas Sequences. In 1876, after 19 years of testing, he proved that 2127 −1 was prime. This may stand forever as the largest prime number proven by hand, later Derrick Henry Lehmer refined Lucas primality tests and obtained the Lucas–Lehmer primality test. He worked on the development of the umbral calculus, Lucas was also interested in recreational mathematics. He found an elegant binary solution to the Baguenaudier puzzle, at the banquet of the annual congress of the Association française pour lavancement des sciences, a waiter dropped some crockery and a piece of broken plate cut Lucas on the cheek. He died a few days later of a skin inflammation probably caused by septicemia. He was only 49 years old, canadian Mathematical Society series of monographs and advanced texts. OConnor, John J. Robertson, Edmund F. Édouard Lucas, MacTutor History of Mathematics archive, scans of Lucass original Tower of Hanoi puzzle in French, with translations Édouard Lucas, by Clark Kimberling Édouard Lucas
15.
Thomas Harriot
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Thomas Harriot — or spelled Harriott, Hariot, or Heriot — was an English astronomer, mathematician, ethnographer, and translator. He is sometimes credited with the introduction of the potato to the British Isles, Harriot was the first person to make a drawing of the Moon through a telescope, on 26 July 1609, over four months before Galileo. After graduating from St Mary Hall, Oxford, Harriot travelled to the Americas, accompanying the 1585 expedition to Roanoke island funded by Sir Walter Raleigh and led by Sir Ralph Lane. Harriot was a member of the venture, having translated and learned the Carolina Algonquian language from two Native Americans, Wanchese and Manteo. On his return to England he worked for the 9th Earl of Northumberland, at the Earls house, he became a prolific mathematician and astronomer to whom the theory of refraction is attributed. Born in 1560 in Oxford, England, Thomas Harriot attended St Mary Hall and his name appears in the halls registry dating from 1577. Prior to his expedition with Raleigh, Harriot wrote a treatise on navigation, in addition, he made efforts to communicate with Manteo and Wanchese, two Native Americans who had been brought to England. Harriot devised an alphabet to transcribe their Carolina Algonquian language. Harriot and Manteo spent many days in one company, Harriot interrogated Manteo closely about life in the New World. In addition, he recorded the sense of awe with which the Native Americans viewed European technology, Many things they sawe with us. as mathematical instruments, as the only Englishman who had learned Algonkin prior to the voyage, Harriot was vital to the success of the expedition. His account of the voyage, named A Briefe and True Report of the New Found Land of Virginia, was published in 1588. The True Report contains an account of the Native American population encountered by the expedition, it proved very influential upon later English explorers. He wrote, Whereby it may be hoped, if means of government be used, that they may in short time be brought to civility. At the same time, his views of Native Americans industry and capacity to learn were later largely ignored in favour of the parts of the True Report about extractable minerals and resources. As a scientific adviser during the voyage, Harriot was asked by Raleigh to find the most efficient way to stack cannonballs on the deck of the ship. His ensuing theory about the close-packing of spheres shows a resemblance to atomism and modern atomic theory. His correspondence about optics with Johannes Kepler, in which he described some of his ideas, Harriott was employed for many years by Henry Percy, 9th Earl of Northumberland, with whom he resided at Syon House, which was run by Henry Percys cousin Thomas Percy. Harriot himself was interrogated and briefly imprisoned but was soon released, Walter Warner, Robert Hues, William Lower, and other scientists were present around the Earl of Northumberlands mansion as they worked for him and assisted in the teaching of the familys children
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Walter Raleigh
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Sir Walter Raleigh was an English landed gentleman, writer, poet, soldier, politician, courtier, spy and explorer. He was cousin to Sir Richard Grenville and younger half-brother of Sir Humphrey Gilbert and he is also well known for popularising tobacco in England. Raleigh was born to a Protestant family in Devon, the son of Walter Raleigh, later, he became a landlord of property confiscated from the native Irish. He rose rapidly in the favour of Queen Elizabeth I and was knighted in 1585, Raleigh was instrumental in the English colonisation of North America and was granted a royal patent to explore Virginia, which paved the way for future English settlements. In 1591, he secretly married Elizabeth Throckmorton, one of the Queens ladies-in-waiting, without the Queens permission, for which he, after his release, they retired to his estate at Sherborne, Dorset. In 1594, Raleigh heard of a City of Gold in South America and sailed to find it, publishing an exaggerated account of his experiences in a book that contributed to the legend of El Dorado. After Queen Elizabeth died in 1603, Raleigh was again imprisoned in the Tower, this time for being involved in the Main Plot against King James I, in 1616, he was released to lead a second expedition in search of El Dorado. During the expedition, men led by his top commander ransacked a Spanish outpost and he returned to England and, to appease the Spanish, was arrested and executed in 1618. Raleigh was one of the most notable figures of the Elizabethan era, in 2002, he was featured in the BBC poll of the 100 Greatest Britons. Little is known about Raleighs birth, some historians believe that he was born on 22 January 1552, although the Oxford Dictionary of National Biography currently favours a date of 1554. He grew up in the house of Hayes Barton, a farmhouse near the village of East Budleigh and he was the youngest of five sons born to Catherine Champernowne in two successive marriages. His half-brothers John Gilbert, Humphrey Gilbert, and Adrian Gilbert, Catherine Champernowne was a niece of Kat Ashley, Elizabeths governess, who introduced the young men at court. Raleighs family was highly Protestant in religious orientation and had a number of near escapes during the reign of Roman Catholic Queen Mary I of England, in the most notable of these, his father had to hide in a tower to avoid execution. As a result, Raleigh developed a hatred of Roman Catholicism during his childhood, in matters of religion, Elizabeth was more moderate than her half sister Mary. In 1569, Raleigh left for France to serve with the Huguenots in the French religious civil wars, in 1572, Raleigh was registered as an undergraduate at Oriel College, Oxford, but he left a year later without a degree. Raleigh proceeded to finish his education in the Inns of Court, in 1575, he was registered at the Middle Temple. At his trial in 1603, he stated that he had never studied law and his life is uncertain between 1569 and 1575, but in his History of the World he claimed to have been an eyewitness at the Battle of Moncontour in France. In 1575 or 1576, Raleigh returned to England, between 1579 and 1583, Raleigh took part in the suppression of the Desmond Rebellions
17.
Snowball fight
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A snowball fight is a physical game in which balls of snow are thrown with the intention of hitting somebody else. The game is similar to dodgeball in its major factors, though typically less organized and this activity is primarily played during months when there is sufficient snowfall. Today, the activity is notable for its prominence in the western world, modern snowball fights tend to have very loose official regulation or constant properties, and so can only loosely be referred to as games. However, a snowball fight played for fun will often have these characteristics, There is crude formation of teams. Those in a fight often do not behave malevolently, a target is not viciously assaulted by snowballs. There is minimal physical contact, aside from perhaps wrestling, in contrast to other forms of fighting, there is usually no intention of bodily harm. Construction and use of snow forts is usually permitted, on February 8,2013, nearly 2,500 students of the Boston University participated in a snowball fight on Bostons Esplanade facilitated by historic winter storm Nemo. On January 12,2013,5,834 people officially took part in Seattle, Washington set the Guinness World Records record for the worlds largest snowball fight, during Seattles Snow Day. Seattles world record was broken on January 31,2016 in Saskatoon, Saskatchewan Canada, on January 22,2010,5,387 people in Taebaek, Republic of Korea, set the world record for most people engaged in a snowball fight. However, historical studies of snowball fights point to Leuven, Belgium as the actual capital of the world. A recent snowball fight there broke the record for the largest snowball fight ever recorded in history. Students from the University of Pennsylvania helped create and fund this fight which reached 5,768 participants, the event was promoted via Facebook and Twitter. At least a half-dozen D. C. and U. S. Park police cars were positioned around Dupont Circle throughout the snowball fight. On December 9,2009, a crowd of over 4,000 students at the University of Wisconsin-Madison participated in a snowball fight on Bascom Hill. There were reports of injuries, mainly broken noses. The snowball fight was scheduled weeks in advance, and was helped by the fact that the University canceled all classes due to 12-16 inches of snow that fell the night before. However, this snowball fight failed to break the set in October of the same year in Leuven. During the American Civil War, on January 29,1863, what began as a few hundred men from Texas plotting a friendly fight against their Arkansas camp mates soon escalated into a brawl that involved 9,000 soldiers of the Army of Northern Virginia
18.
Diophantine equation
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In mathematics, a Diophantine equation is a polynomial equation, usually in two or more unknowns, such that only the integer solutions are sought or studied. A linear Diophantine equation is an equation between two sums of monomials of degree zero or one, an exponential Diophantine equation is one in which exponents on terms can be unknowns. Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations, in more technical language, they define an algebraic curve, algebraic surface, or more general object, and ask about the lattice points on it. The mathematical study of Diophantine problems that Diophantus initiated is now called Diophantine analysis, the solutions are described by the following theorem, This Diophantine equation has a solution if and only if c is a multiple of the greatest common divisor of a and b. Moreover, if is a solution, then the solutions have the form, where k is an arbitrary integer. Proof, If d is this greatest common divisor, Bézouts identity asserts the existence of integers e and f such that ae + bf = d, If c is a multiple of d, then c = dh for some integer h, and is a solution. On the other hand, for pair of integers x and y. Thus, if the equation has a solution, then c must be a multiple of d. If a = ud and b = vd, then for every solution, we have a + b = ax + by + k = ax + by + k = ax + by, showing that is another solution. Finally, given two solutions such that ax1 + by1 = ax2 + by2 = c, one deduces that u + v =0. As u and v are coprime, Euclids lemma shows that exists a integer k such that x2 − x1 = kv. Therefore, x2 = x1 + kv and y2 = y1 − ku, the system to be solved may thus be rewritten as B = UC. Calling yi the entries of V−1X and di those of D = UC and it follows that the system has a solution if and only if bi, i divides di for i ≤ k and di =0 for i > k. If this condition is fulfilled, the solutions of the system are V. Hermite normal form may also be used for solving systems of linear Diophantine equations, however, Hermite normal form does not directly provide the solutions, to get the solutions from the Hermite normal form, one has to successively solve several linear equations. Nevertheless, Richard Zippel wrote that the Smith normal form is more than is actually needed to solve linear diophantine equations. Instead of reducing the equation to diagonal form, we only need to make it triangular, the Hermite normal form is substantially easier to compute than the Smith normal form. Integer linear programming amounts to finding some integer solutions of systems that include also inequations
19.
Coordination number
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In chemistry, crystallography, and materials science the coordination number of a central atom in a molecule or crystal is the number of its near neighbors. This number is determined differently for molecules than for crystals. For molecules and polyatomic ions the coordination number of an atom is determined by counting the other atoms to which it is bonded. For example, − has Cr3+ as its central cation, and has a number of 6. However the solid-state structures of crystals often have less clearly defined bonds, the simplest method is one used in materials science. The usual value of the number for a given structure refers to an atom in the interior of a crystal lattice with neighbors in all directions. In chemistry, coordination number, defined originally in 1893 by Alfred Werner, is the number of neighbors of a central atom in a molecule or ion. Although a carbon atom has four chemical bonds in most stable molecules, in effect we count the first bond to each neighboring atom, but not the other bonds. In coordination complexes, only the first or sigma bond between each ligand and the central atom counts, but not any pi bonds to the same ligands. In tungsten hexacarbonyl, W6, the number of tungsten is counted as six although pi as well as sigma bonding is important in such metal carbonyls. The most common number for d-block transition metal complexes is 6. The observed range is 2 to 9, metals in the f-block can accommodate higher coordination number due to their greater ionic radii and availability of more orbitals for bonding. Coordination numbers of 8 to 12 are commonly observed for f-block elements, for example, with bidentate nitrate ions as ligands, CeIV and ThIV form the 12-coordinate ions 2− and 2−. When the surrounding ligands are much smaller than the central atom, one computational chemistry study predicted a particularly stable PbHe2+15 ion composed of a central lead ion coordinated with no fewer than 15 helium atoms. At the opposite extreme, steric shielding can give rise to low coordination numbers. An extremely rare instance of a metal adopting a number of 1 occurs in the terphenyl-based arylthallium complex 2, 6-Tipp2C6H3Tl. In ferrocene the hapticity, η, of each cyclopentadienide anion is five, there are various ways of assigning the contribution made to the coordination number of the central iron atom by each cyclopentadienide ligand. The contribution could be assigned as one since there is one ligand, or as five since there are five neighbouring atoms, normally the count of electron pairs is taken
20.
Atomic packing factor
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In crystallography, atomic packing factor, packing efficiency or packing fraction is the fraction of volume in a crystal structure that is occupied by constituent particles. It is a quantity and always less than unity. In atomic systems, by convention, the APF is determined by assuming that atoms are rigid spheres, the radius of the spheres is taken to be the maximal value such that the atoms do not overlap. It can be proven mathematically that for one-component structures, the most dense arrangement of atoms has an APF of about 0.74, for multiple-component structures, the APF can exceed 0.74. Common sphere packings taken on by atomic systems are listed below with their corresponding packing fraction. Hexagonal close-packed,0.74 Face-centered cubic,0.74 Body-centered cubic,0.68 Simple cubic,0.52 Diamond cubic,0.34 The majority of metals take on either the hcp, fcc, or bcc structure. The primitive unit cell for the cubic crystal structure contains several fractions taken from nine atoms, one on each corner of the cube. Because the volume of each of the eight corner atoms is shared between eight adjacent cells, each BCC cell contains the equivalent volume of two atoms, each corner atom touches the center atom. A line that is drawn from one corner of the cube through the center and to the other passes through 4r. By geometry, the length of the diagonal is a√3, therefore, the length of each side of the BCC structure can be related to the radius of the atom by a =4 r 3. For the hexagonal close-packed structure the derivation is similar, here the unit cell is a hexagonal prism containing six atoms. Let a be the length of its base and c be its height. The volume of the cell of hcp can be taken as 24√2r3. Then, a =2 r, c =423 r.74048048, crystal Packing density Cubic crystal system Diamond cubic Percolation threshold Schaffer, Saxena, Antolovich, Sanders, Warner. The Science and Design of Engineering Materials, callister, W. Materials Science and Engineering. San Francisco, CA, John Wiley and Sons
21.
Bravais lattice
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This discrete set of vectors must be closed under vector addition and subtraction. For any choice of position vector R, the lattice looks exactly the same, when the discrete points are atoms, ions, or polymer strings of solid matter, the Bravais lattice concept is used to formally define a crystalline arrangement and its frontiers. A crystal is made up of an arrangement of one or more atoms repeated at each lattice point. Consequently, the crystal looks the same when viewed from any equivalent lattice point, two Bravais lattices are often considered equivalent if they have isomorphic symmetry groups. In this sense, there are 14 possible Bravais lattices in three-dimensional space, the 14 possible symmetry groups of Bravais lattices are 14 of the 230 space groups. In two-dimensional space, there are 5 Bravais lattices, grouped into four crystal families, the unit cells are specified according to the relative lengths of the cell edges and the angle between them. The area of the cell can be calculated by evaluating the norm || a × b ||. The properties of the families are given below, In three-dimensional space. These are obtained by combining one of the six families with one of the centering types. For example, the monoclinic I lattice can be described by a monoclinic C lattice by different choice of crystal axes, similarly, all A- or B-centred lattices can be described either by a C- or P-centering. This reduces the number of combinations to 14 conventional Bravais lattices, the unit cells are specified according to the relative lengths of the cell edges and the angles between them. The volume of the cell can be calculated by evaluating the triple product a ·, where a, b. The properties of the families are given below, In four dimensions. Of these,23 are primitive and 41 are centered, ten Bravais lattices split into enantiomorphic pairs. Bravais, A. Mémoire sur les systèmes formés par les points distribués régulièrement sur un plan ou dans lespace, hahn, Theo, ed. International Tables for Crystallography, Volume A, Space Group Symmetry. Catalogue of Lattices Smith, Walter Fox
22.
Tetrahedron
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In geometry, a tetrahedron, also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra, the tetrahedron is the three-dimensional case of the more general concept of a Euclidean simplex. The tetrahedron is one kind of pyramid, which is a polyhedron with a polygon base. In the case of a tetrahedron the base is a triangle, like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper. For any tetrahedron there exists a sphere on which all four vertices lie, a regular tetrahedron is one in which all four faces are equilateral triangles. It is one of the five regular Platonic solids, which have known since antiquity. In a regular tetrahedron, not only are all its faces the same size and shape, regular tetrahedra alone do not tessellate, but if alternated with regular octahedra they form the alternated cubic honeycomb, which is a tessellation. The regular tetrahedron is self-dual, which means that its dual is another regular tetrahedron, the compound figure comprising two such dual tetrahedra form a stellated octahedron or stella octangula. This form has Coxeter diagram and Schläfli symbol h, the tetrahedron in this case has edge length 2√2. Inverting these coordinates generates the dual tetrahedron, and the together form the stellated octahedron. In other words, if C is the centroid of the base and this follows from the fact that the medians of a triangle intersect at its centroid, and this point divides each of them in two segments, one of which is twice as long as the other. The vertices of a cube can be grouped into two groups of four, each forming a regular tetrahedron, the symmetries of a regular tetrahedron correspond to half of those of a cube, those that map the tetrahedra to themselves, and not to each other. The tetrahedron is the only Platonic solid that is not mapped to itself by point inversion, the regular tetrahedron has 24 isometries, forming the symmetry group Td, isomorphic to the symmetric group, S4. The first corresponds to the A2 Coxeter plane, the two skew perpendicular opposite edges of a regular tetrahedron define a set of parallel planes. When one of these intersects the tetrahedron the resulting cross section is a rectangle. When the intersecting plane is one of the edges the rectangle is long. When halfway between the two edges the intersection is a square, the aspect ratio of the rectangle reverses as you pass this halfway point. For the midpoint square intersection the resulting boundary line traverses every face of the tetrahedron similarly, if the tetrahedron is bisected on this plane, both halves become wedges
23.
Cartesian coordinate system
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Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, usually at ordered pair. The coordinates can also be defined as the positions of the projections of the point onto the two axis, expressed as signed distances from the origin. One can use the principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes. In general, n Cartesian coordinates specify the point in an n-dimensional Euclidean space for any dimension n and these coordinates are equal, up to sign, to distances from the point to n mutually perpendicular hyperplanes. The invention of Cartesian coordinates in the 17th century by René Descartes revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra. Using the Cartesian coordinate system, geometric shapes can be described by Cartesian equations, algebraic equations involving the coordinates of the points lying on the shape. For example, a circle of radius 2, centered at the origin of the plane, a familiar example is the concept of the graph of a function. Cartesian coordinates are also tools for most applied disciplines that deal with geometry, including astronomy, physics, engineering. They are the most common system used in computer graphics, computer-aided geometric design. Nicole Oresme, a French cleric and friend of the Dauphin of the 14th Century, used similar to Cartesian coordinates well before the time of Descartes. The adjective Cartesian refers to the French mathematician and philosopher René Descartes who published this idea in 1637 and it was independently discovered by Pierre de Fermat, who also worked in three dimensions, although Fermat did not publish the discovery. Both authors used a single axis in their treatments and have a length measured in reference to this axis. The concept of using a pair of axes was introduced later, after Descartes La Géométrie was translated into Latin in 1649 by Frans van Schooten and these commentators introduced several concepts while trying to clarify the ideas contained in Descartes work. Many other coordinate systems have developed since Descartes, such as the polar coordinates for the plane. The development of the Cartesian coordinate system would play a role in the development of the Calculus by Isaac Newton. The two-coordinate description of the plane was later generalized into the concept of vector spaces. Choosing a Cartesian coordinate system for a one-dimensional space – that is, for a straight line—involves choosing a point O of the line, a unit of length, and an orientation for the line. An orientation chooses which of the two half-lines determined by O is the positive, and which is negative, we say that the line is oriented from the negative half towards the positive half
24.
Sphere packing
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In geometry, a sphere packing is an arrangement of non-overlapping spheres within a containing space. The spheres considered are usually all of identical size, and the space is usually three-dimensional Euclidean space, however, sphere packing problems can be generalised to consider unequal spheres, n-dimensional Euclidean space or to non-Euclidean spaces such as hyperbolic space. A typical sphere packing problem is to find an arrangement in which the fill as large a proportion of the space as possible. The proportion of space filled by the spheres is called the density of the arrangement, for equal spheres in three dimensions the densest packing uses approximately 74% of the volume. A random packing of equal spheres generally has a density around 64%, a lattice arrangement is one in which the centers of the spheres form a very symmetric pattern which only needs n vectors to be uniquely defined. Arrangements in which the spheres do not form a lattice can still be periodic, lattice arrangements are easier to handle than irregular ones—their high degree of symmetry makes it easier to classify them and to measure their densities. In three-dimensional Euclidean space, the densest packing of spheres is achieved by a family of structures called close-packed structures. One method for generating such a structure is as follows, consider a plane with a compact arrangement of spheres on it. For any three neighbouring spheres, a sphere can be placed on top in the hollow between the three bottom spheres. If we do this everywhere in a plane above the first. A third layer can be placed directly above the first one, or the spheres can be offset, there are thus three types of planes, called A, B and C. Two simple arrangements within the close-packed family correspond to regular lattices, one is called cubic close packing — where the layers are alternated in the ABCABC… sequence. The other is called hexagonal close packing — where the layers are alternated in the ABAB… sequence, but many layer stacking sequences are possible, and still generate a close-packed structure. In all of these arrangements each sphere is surrounded by 12 other spheres, carl Friedrich Gauss proved in 1831 that these packings have the highest density amongst all possible lattice packings. In 1611 Johannes Kepler had conjectured that this is the maximum possible density amongst both regular and irregular arrangements — this became known as the Kepler conjecture. In 1998, Thomas Callister Hales, following the approach suggested by László Fejes Tóth in 1953, Hales proof is a proof by exhaustion involving checking of many individual cases using complex computer calculations. Referees said that they were 99% certain of the correctness of Hales proof, on 10 August 2014 Hales announced the completion of a formal proof using automated proof checking, removing any doubt. Some other lattice packings are often found in physical systems, Packings where all spheres are constrained by their neighbours to stay in one location are called rigid or jammed
25.
Honeycomb (geometry)
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In geometry, a honeycomb is a space filling or close packing of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the general mathematical tiling or tessellation in any number of dimensions. Its dimension can be clarified as n-honeycomb for a honeycomb of n-dimensional space, honeycombs are usually constructed in ordinary Euclidean space. They may also be constructed in non-Euclidean spaces, such as hyperbolic honeycombs, any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space. There are infinitely many honeycombs, which have only been partially classified, the more regular ones have attracted the most interest, while a rich and varied assortment of others continue to be discovered. The simplest honeycombs to build are formed from stacked layers or slabs of prisms based on some tessellations of the plane, in particular, for every parallelepiped, copies can fill space, with the cubic honeycomb being special because it is the only regular honeycomb in ordinary space. Another interesting family is the Hill tetrahedra and their generalizations, which can tile the space. A 3-dimensional uniform honeycomb is a honeycomb in 3-space composed of polyhedral cells. There are 28 convex examples in Euclidean 3-space, also called the Archimedean honeycombs, a honeycomb is called regular if the group of isometries preserving the tiling acts transitively on flags, where a flag is a vertex lying on an edge lying on a face lying on a cell. Every regular honeycomb is automatically uniform, however, there is just one regular honeycomb in Euclidean 3-space, the cubic honeycomb. An infinite number of unique honeycombs can be created by order of patterns of repeating these slab layers. A honeycomb having all cells identical within its symmetries is said to be cell-transitive or isochoric, in the 3-dimensional euclidean space, a cell of such a honeycomb is said to be a space-filling polyhedron. A necessary condition for a polyhedron to be a space-filling polyhedron is that its Dehn invariant must be zero, five space-filling polyhedra can tessellate 3-dimensional euclidean space using translations only. They are called parallelohedra, Cubic honeycomb Hexagonal prismatic honeycomb Rhombic dodecahedral honeycomb Elongated dodecahedral honeycomb, bitruncated cubic honeycomb Other known examples of space-filling polyhedra include, The Triangular prismatic honeycomb. The gyrated triangular prismatic honeycomb The triakis truncated tetrahedral honeycomb, the Voronoi cells of the carbon atoms in diamond are this shape. The trapezo-rhombic dodecahedral honeycomb Isohedral tilings, sometimes, two or more different polyhedra may be combined to fill space. Two classes can be distinguished, Non-convex cells which pack without overlapping and these include a packing of the small stellated rhombic dodecahedron, as in the Yoshimoto Cube. Overlapping of cells whose positive and negative densities cancel out to form a uniformly dense continuum, in 3-dimensional hyperbolic space, the dihedral angle of a polyhedron depends on its size
26.
Tetrahedral-octahedral honeycomb
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The tetrahedral-octahedral honeycomb, alternated cubic honeycomb is a quasiregular space-filling tessellation in Euclidean 3-space. It is composed of alternating octahedra and tetrahedra in a ratio of 1,2, other names include half cubic honeycomb, half cubic cellulation, or tetragonal disphenoidal cellulation. John Horton Conway calls this honeycomb a tetroctahedrille, and its dual dodecahedrille and it is vertex-transitive with 8 tetrahedra and 6 octahedra around each vertex. It is edge-transitive with 2 tetrahedra and 2 octahedra alternating on each edge, a geometric honeycomb is a space-filling of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the general mathematical tiling or tessellation in any number of dimensions. Honeycombs are usually constructed in ordinary Euclidean space, like the uniform honeycombs. They may also be constructed in non-Euclidean spaces, such as hyperbolic uniform honeycombs, any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space. It is also part of another family of uniform honeycombs called simplectic honeycombs. In this case of 3-space, the honeycomb is alternated, reducing the cubic cells to tetrahedra. As such it can be represented by an extended Schläfli symbol h as containing half the vertices of the cubic honeycomb, theres a similar honeycomb called gyrated tetrahedral-octahedral honeycomb which has layers rotated 60 degrees so half the edges have neighboring rather than alternating tetrahedra and octahedra. Each slice will contain up and downward facing square pyramids and tetrahedra sitting on their edges, a second slice direction needs no new faces and includes alternating tetrahedral and octahedral. This slab honeycomb is a scaliform honeycomb rather than uniform because it has nonuniform cells, the alternated cubic honeycomb can be orthogonally projected into the planar square tiling by a geometric folding operation that maps one pairs of mirrors into each other. The projection of the cubic honeycomb creates two offset copies of the square tiling vertex arrangement of the plane, Its vertex arrangement represents an A3 lattice or D3 lattice. It is the 3-dimensional case of a simplectic honeycomb and its Voronoi cell is a rhombic dodecahedron, the dual of the cuboctahedron vertex figure for the tet-oct honeycomb. The D+3 packing can be constructed by the union of two D3 lattices, the D+ n packing is only a lattice for even dimensions. The kissing number of the D*3 lattice is 8 and its Voronoi tessellation is a cubic honeycomb. The, Coxeter group generates 15 permutations of uniform honeycombs,9 with distinct geometry including the cubic honeycomb. The expanded cubic honeycomb is geometrically identical to the cubic honeycomb, the, Coxeter group generates 9 permutations of uniform honeycombs,4 with distinct geometry including the alternated cubic honeycomb
27.
Gyrated tetrahedral-octahedral honeycomb
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The tetrahedral-octahedral honeycomb, alternated cubic honeycomb is a quasiregular space-filling tessellation in Euclidean 3-space. It is composed of alternating octahedra and tetrahedra in a ratio of 1,2, other names include half cubic honeycomb, half cubic cellulation, or tetragonal disphenoidal cellulation. John Horton Conway calls this honeycomb a tetroctahedrille, and its dual dodecahedrille and it is vertex-transitive with 8 tetrahedra and 6 octahedra around each vertex. It is edge-transitive with 2 tetrahedra and 2 octahedra alternating on each edge, a geometric honeycomb is a space-filling of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the general mathematical tiling or tessellation in any number of dimensions. Honeycombs are usually constructed in ordinary Euclidean space, like the uniform honeycombs. They may also be constructed in non-Euclidean spaces, such as hyperbolic uniform honeycombs, any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space. It is also part of another family of uniform honeycombs called simplectic honeycombs. In this case of 3-space, the honeycomb is alternated, reducing the cubic cells to tetrahedra. As such it can be represented by an extended Schläfli symbol h as containing half the vertices of the cubic honeycomb, theres a similar honeycomb called gyrated tetrahedral-octahedral honeycomb which has layers rotated 60 degrees so half the edges have neighboring rather than alternating tetrahedra and octahedra. Each slice will contain up and downward facing square pyramids and tetrahedra sitting on their edges, a second slice direction needs no new faces and includes alternating tetrahedral and octahedral. This slab honeycomb is a scaliform honeycomb rather than uniform because it has nonuniform cells, the alternated cubic honeycomb can be orthogonally projected into the planar square tiling by a geometric folding operation that maps one pairs of mirrors into each other. The projection of the cubic honeycomb creates two offset copies of the square tiling vertex arrangement of the plane, Its vertex arrangement represents an A3 lattice or D3 lattice. It is the 3-dimensional case of a simplectic honeycomb and its Voronoi cell is a rhombic dodecahedron, the dual of the cuboctahedron vertex figure for the tet-oct honeycomb. The D+3 packing can be constructed by the union of two D3 lattices, the D+ n packing is only a lattice for even dimensions. The kissing number of the D*3 lattice is 8 and its Voronoi tessellation is a cubic honeycomb. The, Coxeter group generates 15 permutations of uniform honeycombs,9 with distinct geometry including the cubic honeycomb. The expanded cubic honeycomb is geometrically identical to the cubic honeycomb, the, Coxeter group generates 9 permutations of uniform honeycombs,4 with distinct geometry including the alternated cubic honeycomb
28.
Rhombic dodecahedral honeycomb
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The rhombic dodecahedral honeycomb is a space-filling tessellation in Euclidean 3-space. It is the Voronoi diagram of the face-centered cubic sphere-packing, which has the densest possible packing of spheres in ordinary space. It consists of copies of a cell, the rhombic dodecahedron. All faces are rhombi, with diagonals in the ratio 1, three cells meet at each edge. The honeycomb is thus cell-transitive, face-transitive and edge-transitive, but it is not vertex-transitive, the vertices with the obtuse rhombic face angles have 4 cells. The vertices with the acute rhombic face angles have 6 cells, the trapezo-rhombic dodecahedral honeycomb is a space-filling tessellation in Euclidean 3-space. It consists of copies of a cell, the trapezo-rhombic dodecahedron. It is similar to the higher symmetric rhombic dodecahedral honeycomb which has all 12 faces as rhombi and it is a dual to the vertex-transitive gyrated tetrahedral-octahedral honeycomb. The rhombic pyramidal honeycomb is a uniform space-filling tessellation in Euclidean 3-space, john Horton Conway calls it a truncated tetraoctahedrille. It is dual to the cubic honeycomb, Williams, Robert. The Geometrical Foundation of Natural Structure, A Source Book of Design, examples of Housing Construction using this geometry
29.
Trapezo-rhombic dodecahedral honeycomb
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The rhombic dodecahedral honeycomb is a space-filling tessellation in Euclidean 3-space. It is the Voronoi diagram of the face-centered cubic sphere-packing, which has the densest possible packing of spheres in ordinary space. It consists of copies of a cell, the rhombic dodecahedron. All faces are rhombi, with diagonals in the ratio 1, three cells meet at each edge. The honeycomb is thus cell-transitive, face-transitive and edge-transitive, but it is not vertex-transitive, the vertices with the obtuse rhombic face angles have 4 cells. The vertices with the acute rhombic face angles have 6 cells, the trapezo-rhombic dodecahedral honeycomb is a space-filling tessellation in Euclidean 3-space. It consists of copies of a cell, the trapezo-rhombic dodecahedron. It is similar to the higher symmetric rhombic dodecahedral honeycomb which has all 12 faces as rhombi and it is a dual to the vertex-transitive gyrated tetrahedral-octahedral honeycomb. The rhombic pyramidal honeycomb is a uniform space-filling tessellation in Euclidean 3-space, john Horton Conway calls it a truncated tetraoctahedrille. It is dual to the cubic honeycomb, Williams, Robert. The Geometrical Foundation of Natural Structure, A Source Book of Design, examples of Housing Construction using this geometry
30.
Plateau's laws
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Plateaus laws describe the structure of soap films. These laws were formulated in the 19th century by the Belgian physicist Joseph Plateau from his experimental observations, many patterns in nature are based on foams obeying these laws. Plateaus laws describe the shape and configuration of soap films as follows, the mean curvature of a portion of a soap film is everywhere constant on any point on the same piece of soap film. Soap films always meet in threes along an edge called a Plateau border and these Plateau borders meet in fours at a vertex, and they do so at an angle of arccos ≈109. 47°. Configurations other than those of Plateaus laws are unstable, and the film will tend to rearrange itself to conform to these laws. That these laws hold for minimal surfaces was proved mathematically by Jean Taylor using geometric measure theory, young–Laplace equation, governing the curvature of surfaces in a soap film Ball, Philip. Natures Patterns, a tapestry in three parts
31.
ArXiv
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In many fields of mathematics and physics, almost all scientific papers are self-archived on the arXiv repository. Begun on August 14,1991, arXiv. org passed the half-million article milestone on October 3,2008, by 2014 the submission rate had grown to more than 8,000 per month. The arXiv was made possible by the low-bandwidth TeX file format, around 1990, Joanne Cohn began emailing physics preprints to colleagues as TeX files, but the number of papers being sent soon filled mailboxes to capacity. Additional modes of access were added, FTP in 1991, Gopher in 1992. The term e-print was quickly adopted to describe the articles and its original domain name was xxx. lanl. gov. Due to LANLs lack of interest in the rapidly expanding technology, in 1999 Ginsparg changed institutions to Cornell University and it is now hosted principally by Cornell, with 8 mirrors around the world. Its existence was one of the factors that led to the current movement in scientific publishing known as open access. Mathematicians and scientists regularly upload their papers to arXiv. org for worldwide access, Ginsparg was awarded a MacArthur Fellowship in 2002 for his establishment of arXiv. The annual budget for arXiv is approximately $826,000 for 2013 to 2017, funded jointly by Cornell University Library, annual donations were envisaged to vary in size between $2,300 to $4,000, based on each institution’s usage. As of 14 January 2014,174 institutions have pledged support for the period 2013–2017 on this basis, in September 2011, Cornell University Library took overall administrative and financial responsibility for arXivs operation and development. Ginsparg was quoted in the Chronicle of Higher Education as saying it was supposed to be a three-hour tour, however, Ginsparg remains on the arXiv Scientific Advisory Board and on the arXiv Physics Advisory Committee. The lists of moderators for many sections of the arXiv are publicly available, additionally, an endorsement system was introduced in 2004 as part of an effort to ensure content that is relevant and of interest to current research in the specified disciplines. Under the system, for categories that use it, an author must be endorsed by an established arXiv author before being allowed to submit papers to those categories. Endorsers are not asked to review the paper for errors, new authors from recognized academic institutions generally receive automatic endorsement, which in practice means that they do not need to deal with the endorsement system at all. However, the endorsement system has attracted criticism for allegedly restricting scientific inquiry, perelman appears content to forgo the traditional peer-reviewed journal process, stating, If anybody is interested in my way of solving the problem, its all there – let them go and read about it. The arXiv generally re-classifies these works, e. g. in General mathematics, papers can be submitted in any of several formats, including LaTeX, and PDF printed from a word processor other than TeX or LaTeX. The submission is rejected by the software if generating the final PDF file fails, if any image file is too large. ArXiv now allows one to store and modify an incomplete submission, the time stamp on the article is set when the submission is finalized
32.
John Horton Conway
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John Horton Conway FRS is an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He has also contributed to many branches of mathematics, notably the invention of the cellular automaton called the Game of Life. Conway is currently Professor Emeritus of Mathematics at Princeton University in New Jersey, Conway was born in Liverpool, the son of Cyril Horton Conway and Agnes Boyce. He became interested in mathematics at an early age, his mother has recalled that he could recite the powers of two when he was four years old. By the age of eleven his ambition was to become a mathematician, after leaving secondary school, Conway entered Gonville and Caius College, Cambridge to study mathematics. Conway, who was a terribly introverted adolescent in school, interpreted his admission to Cambridge as an opportunity to transform himself into a new person and he was awarded his Bachelor of Arts degree in 1959 and began to undertake research in number theory supervised by Harold Davenport. Having solved the problem posed by Davenport on writing numbers as the sums of fifth powers. It appears that his interest in games began during his years studying the Cambridge Mathematical Tripos and he was awarded his doctorate in 1964 and was appointed as College Fellow and Lecturer in Mathematics at the University of Cambridge. After leaving Cambridge in 1986, he took up the appointment to the John von Neumann Chair of Mathematics at Princeton University, Conway is especially known for the invention of the Game of Life, one of the early examples of a cellular automaton. His initial experiments in that field were done with pen and paper, since the game was introduced by Martin Gardner in Scientific American in 1970, it has spawned hundreds of computer programs, web sites, and articles. It is a staple of recreational mathematics, there is an extensive wiki devoted to curating and cataloging the various aspects of the game. From the earliest days it has been a favorite in computer labs, at times Conway has said he hates the game of life–largely because it has come to overshadow some of the other deeper and more important things he has done. Nevertheless, the game did help launch a new branch of mathematics, the Game of Life is now known to be Turing complete. Conways career is intertwined with mathematics popularizer and Scientific American columnist Martin Gardner, when Gardner featured Conways Game of Life in his Mathematical Games column in October 1970, it became the most widely read of all his columns and made Conway an instant celebrity. Gardner and Conway had first corresponded in the late 1950s, for instance, he discussed Conways game of Sprouts, Hackenbush, and his angel and devil problem. In the September 1976 column he reviewed Conways book On Numbers and Games, Conway is widely known for his contributions to combinatorial game theory, a theory of partisan games. This he developed with Elwyn Berlekamp and Richard Guy, and with them also co-authored the book Winning Ways for your Mathematical Plays and he also wrote the book On Numbers and Games which lays out the mathematical foundations of CGT. He is also one of the inventors of sprouts, as well as philosophers football and he developed detailed analyses of many other games and puzzles, such as the Soma cube, peg solitaire, and Conways soldiers
33.
Neil Sloane
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Neil James Alexander Sloane is a British-American mathematician. His major contributions are in the fields of combinatorics, error-correcting codes, Sloane is best known for being the creator and maintainer of the On-Line Encyclopedia of Integer Sequences. Sloane was born in Wales and brought up in Australia and he studied at Cornell University, New York state, under Nick DeClaris, Frank Rosenblatt, Frederick Jelinek and Wolfgang Heinrich Johannes Fuchs, receiving his Ph. D. in 1967. His doctoral dissertation was titled Lengths of Cycle Times in Random Neural Networks, Sloane joined AT&T Bell Labs in 1968 and retired from AT&T Labs in 2012. He became an AT&T Fellow in 1998 and he is also a Fellow of the Learned Society of Wales, an IEEE Fellow, a Fellow of the American Mathematical Society, and a member of the National Academy of Engineering. He is a winner of a Lester R. Ford Award in 1978, in 2005 Sloane received the IEEE Richard W. Hamming Medal. In 2008 he received the Mathematical Association of America David P. Robbins award, in 2014, to celebrate his 75th birthday, Neil Sloane shared some of his favorite integer sequences. Besides mathematics, he loves rock climbing and has authored two rock-climbing guides to New Jersey, N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, NY,1973. F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier/North-Holland, M. Harwit and N. J. A. Sloane, Hadamard Transform Optics, Academic Press, San Diego CA,1979. N. J. A. Sloane and A. D. Wyner, editors, Claude Elwood Shannon, Collected Papers, IEEE Press, N. J. A. Sloane and S. Plouffe, The Encyclopedia of Integer Sequences, Academic Press, San Diego,1995. J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, NY, 1st edn. A. S. Hedayat, N. J. A. Sloane and J. Stufken, Orthogonal Arrays, Theory and Applications, Springer-Verlag, NY,1999. G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer-Verlag,2006
34.
International Standard Book Number
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The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each edition and variation of a book, for example, an e-book, a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, the method of assigning an ISBN is nation-based and varies from country to country, often depending on how large the publishing industry is within a country. The initial ISBN configuration of recognition was generated in 1967 based upon the 9-digit Standard Book Numbering created in 1966, the 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108. Occasionally, a book may appear without a printed ISBN if it is printed privately or the author does not follow the usual ISBN procedure, however, this can be rectified later. Another identifier, the International Standard Serial Number, identifies periodical publications such as magazines, the ISBN configuration of recognition was generated in 1967 in the United Kingdom by David Whitaker and in 1968 in the US by Emery Koltay. The 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108, the United Kingdom continued to use the 9-digit SBN code until 1974. The ISO on-line facility only refers back to 1978, an SBN may be converted to an ISBN by prefixing the digit 0. For example, the edition of Mr. J. G. Reeder Returns, published by Hodder in 1965, has SBN340013818 -340 indicating the publisher,01381 their serial number. This can be converted to ISBN 0-340-01381-8, the check digit does not need to be re-calculated, since 1 January 2007, ISBNs have contained 13 digits, a format that is compatible with Bookland European Article Number EAN-13s. An ISBN is assigned to each edition and variation of a book, for example, an ebook, a paperback, and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, a 13-digit ISBN can be separated into its parts, and when this is done it is customary to separate the parts with hyphens or spaces. Separating the parts of a 10-digit ISBN is also done with either hyphens or spaces, figuring out how to correctly separate a given ISBN number is complicated, because most of the parts do not use a fixed number of digits. ISBN issuance is country-specific, in that ISBNs are issued by the ISBN registration agency that is responsible for country or territory regardless of the publication language. Some ISBN registration agencies are based in national libraries or within ministries of culture, in other cases, the ISBN registration service is provided by organisations such as bibliographic data providers that are not government funded. In Canada, ISBNs are issued at no cost with the purpose of encouraging Canadian culture. In the United Kingdom, United States, and some countries, where the service is provided by non-government-funded organisations. Australia, ISBNs are issued by the library services agency Thorpe-Bowker
35.
Packing problems
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Packing problems are a class of optimization problems in mathematics that involve attempting to pack objects together into containers. The goal is to pack a single container as densely as possible or pack all objects using as few containers as possible. Many of these problems can be related to real life packaging, storage, each packing problem has a dual covering problem, which asks how many of the same objects are required to completely cover every region of the container, where objects are allowed to overlap. In a Bin packing problem, you are given, containers A set of some or all of which must be packed into one or more containers. The set may contain different objects with their sizes specified, or an object of a fixed dimension that can be used repeatedly. Usually the packing must be without overlaps between goods and other goods or the container walls, in some variants, the aim is to find the configuration that packs a single container with the maximal density. More commonly, the aim is to all the objects into as few containers as possible. In some variants the overlapping is allowed but should be minimized, many of these problems, when the container size is increased in all directions, become equivalent to the problem of packing objects as densely as possible in infinite Euclidean space. This problem is relevant to a number of disciplines, and has received significant attention. The Kepler conjecture postulated an optimal solution for packing spheres hundreds of years before it was correct by Thomas Callister Hales. Many other shapes have received attention, including ellipsoids, Platonic and Archimedean solids including tetrahedra and these problems are mathematically distinct from the ideas in the circle packing theorem. The related circle packing problem deals with packing circles, possibly of different sizes, on a surface, the counterparts of a circle in other dimensions can never be packed with complete efficiency in dimensions larger than one. That is, there always be unused space if you are only packing circles. The most efficient way of packing circles, hexagonal packing, produces approximately 91% efficiency, in three dimensions, the face-centered cubic lattice offers the best lattice packing of spheres, and is believed to be the optimal of all packings. With simple sphere packings in three dimensions there are nine possible defineable packings, the 8-dimensional E8 lattice and 24-dimensional Leech lattice have also been proven to be optimal in their respective real dimensional space. Cubes can easily be arranged to fill space completely, the most natural packing being the cubic honeycomb. No other Platonic solid can tile space on its own, tetrahedra can achieve a packing of at least 85%. One of the best packings of regular dodecahedra is based on the aforementioned face-centered cubic lattice, tetrahedra and octahedra together can fill all of space in an arrangement known as the tetrahedral-octahedral honeycomb
36.
Circle packing
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This article describes the packing of circles on surfaces. For the related article on circle packing with an intersection graph. In geometry, circle packing is the study of the arrangement of circles on a surface such that no overlapping occurs. The associated packing density, η, of an arrangement is the proportion of the covered by the circles. Generalisations can be made to higher dimensions – this is called sphere packing, while the circle has a relatively low maximum packing density of 0.9069 on the Euclidean plane, it does not have the lowest possible. Packing densities of concave shapes such as star polygons can be arbitrarily small, the density of this arrangement is η h = π23 ≈0.9069. Axel Thue provided the first proof that this was optimal in 1890, showing that the lattice is the densest of all possible circle packings. However, his proof was considered by some to be incomplete, the first rigorous proof is attributed to László Fejes Tóth in 1940. At the other extreme, very low density arrangements of rigidly packed circles have been identified, there are 11 circle packings based on the 11 uniform tilings of the plane. In these packings, every circle can be mapped to every other circle by reflections and rotations, the hexagonal gaps can be filled by one circle and the dodecagonal gaps can be filled with 7 circles, creating 3-uniform packings. The truncated trihexagonal tiling with both types of gaps can be filled as a 4-uniform packing, the snub hexagonal tiling has two mirror-image forms. A related problem is to determine the arrangement of identically interacting points that are constrained to lie within a given surface. The Thomson problem deals with the lowest energy distribution of electric charges on the surface of a sphere. The Tammes problem is a generalisation of this, dealing with maximising the minimum distance between circles on sphere and this is analogous to distributing non-point charges on a sphere. Packing circles in simple bounded shapes is a type of problem in recreational mathematics. The influence of the walls is important, and hexagonal packing is generally not optimal for small numbers of circles. There are also a range of problems which permit the sizes of the circles to be non-uniform, one such extension is to find the maximum possible density of a system with two specific sizes of circle. Only nine particular radius ratios permit compact packing, which is when every pair of circles in contact is in contact with two other circles
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