1.
Patterns in nature
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Patterns in nature are visible regularities of form found in the natural world. These patterns recur in different contexts and can sometimes be modelled mathematically, Natural patterns include symmetries, trees, spirals, meanders, waves, foams, tessellations, cracks and stripes. Early Greek philosophers studied pattern, with Plato, Pythagoras and Empedocles attempting to order in nature. The modern understanding of patterns developed gradually over time. In the 19th century, Belgian physicist Joseph Plateau examined soap films, german biologist and artist Ernst Haeckel painted hundreds of marine organisms to emphasise their symmetry. Scottish biologist DArcy Thompson pioneered the study of patterns in both plants and animals, showing that simple equations could explain spiral growth. In the 20th century, British mathematician Alan Turing predicted mechanisms of morphogenesis which give rise to patterns of spots, Hungarian biologist Aristid Lindenmayer and French American mathematician Benoît Mandelbrot showed how the mathematics of fractals could create plant growth patterns. Mathematics, physics and chemistry can explain patterns in nature at different levels, Patterns in living things are explained by the biological processes of natural selection and sexual selection. Studies of pattern formation make use of models to simulate a wide range of patterns. Early Greek philosophers attempted to order in nature, anticipating modern concepts. Plato — looking only at his work on natural patterns — argued for the existence of universals and he considered these to consist of ideal forms of which physical objects are never more than imperfect copies. Thus, a flower may be circular, but it is never a perfect mathematical circle. Pythagoras explained patterns in nature like the harmonies of music as arising from number, Empedocles to an extent anticipated Darwins evolutionary explanation for the structures of organisms. In 1202, Leonardo Fibonacci introduced the Fibonacci number sequence to the world with his book Liber Abaci. Fibonacci gave an example, on the growth in numbers of a theoretical rabbit population. The discourses central chapter features examples and observations of the quincunx in botany, in 1917, DArcy Wentworth Thompson published his book On Growth and Form. His description of phyllotaxis and the Fibonacci sequence, the relationships in the spiral growth patterns of plants, is classic. He showed that simple equations could describe all the apparently complex spiral growth patterns of animal horns, the Belgian physicist Joseph Plateau formulated the mathematical problem of the existence of a minimal surface with a given boundary, which is now named after him

2.
Fracture
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A fracture is the separation of an object or material into two or more pieces under the action of stress. The fracture of a solid usually occurs due to the development of certain displacement discontinuity surfaces within the solid, fracture strength or breaking strength is the stress when a specimen fails or fractures. The word fracture is often applied to bones of living creatures, or to crystalline materials, sometimes, individual crystals fracture without the structure actually separating into two or more pieces. Depending on the substance, a fracture reduces strength or inhibits transmission of waves, a detailed understanding of how fracture occurs in materials may be assisted by the study of fracture mechanics. Fracture strength, also known as breaking strength, is the stress at which a specimen fails via fracture and this is usually determined for a given specimen by a tensile test, which charts the stress-strain curve. The final recorded point is the fracture strength, ductile materials have a fracture strength lower than the ultimate tensile strength, whereas in brittle materials the fracture strength is equivalent to the UTS. If a ductile material reaches its ultimate strength in a load-controlled situation, it will continue to deform, with no additional load application. However, if the loading is displacement-controlled, the deformation of the material may relieve the load, in brittle fracture, no apparent plastic deformation takes place before fracture. In brittle crystalline materials, fracture can occur by cleavage as the result of tensile stress acting normal to crystallographic planes with low bonding. In amorphous solids, by contrast, the lack of a crystalline structure results in a conchoidal fracture, the sinking of RMS Titanic in 1912 from an iceberg collision is widely reported to have been due to brittle fracture of the hulls steel plates. Putting these two together, we get σ f r a c t u r e = E γ ρ4 a r o. Looking closely, we can see that sharp cracks and large defects both lower the strength of the material. Recently, scientists have discovered supersonic fracture, the phenomenon of crack propagation faster than the speed of sound in a material and this phenomenon was recently also verified by experiment of fracture in rubber-like materials. Brittle fracture may be avoided by limiting pressure and temperature within limits, each system has a brittle fracture prevention limit curve defined by the weakest components at given temperatures and pressures allowing for the largest undetected preexisting flaw in each component. In ductile fracture, extensive plastic deformation takes place before fracture, the terms rupture or ductile rupture describe the ultimate failure of ductile materials loaded in tension. Rather than cracking, the material pulls apart, generally leaving a rough surface, in this case there is slow propagation and an absorption of a large amount energy before fracture. The ductility of a material is referred to as toughness. Many ductile metals, especially materials with high purity, can sustain very large deformation of 50–100% or more strain before fracture under favorable loading condition, the strain at which the fracture happens is controlled by the purity of the materials

3.
Dune
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In physical geography, a dune is a hill of loose sand built by wind or the flow of water. Dunes occur in different shapes and sizes, formed by interaction with the flow of air or water, most kinds of dunes are longer on the windward side where the sand is pushed up the dune and have a shorter slip face in the lee of the wind. The valley or trough between dunes is called a slack, a dune field is an area covered by extensive sand dunes. Dunes occur, for example, in deserts and along some coasts. Some coastal areas have one or more sets of dunes running parallel to the shoreline directly inland from the beach, in most cases, the dunes are important in protecting the land against potential ravages by storm waves from the sea. Although the most widely distributed dunes are those associated with coastal regions, Dunes can form under the action of water flow, and on sand or gravel beds of rivers, estuaries and the sea-bed. The modern word dune came into English from French c,1790, which in turn came from Middle Dutch dūne. Crescent-shaped mounds are generally wider than they are long, the slipfaces are on the concave sides of the dunes. These dunes form under winds that blow consistently from one direction, some types of crescentic dunes move more quickly over desert surfaces than any other type of dune. A group of dunes moved more than 100 metres per year between 1954 and 1959 in Chinas Ningxia Province, and similar speeds have been recorded in the Western Desert of Egypt. The largest crescentic dunes on Earth, with mean crest-to-crest widths of more than three kilometres, are in Chinas Taklamakan Desert. They may be composed of clay, silt, sand, or gypsum, eroded from the floor or shore, transported up the concave side of the dune. Examples in Australia are up to 6.5 km long,1 km wide and they also occur in southern and West Africa, and in parts of the western United States, especially Texas. Straight or slightly sinuous sand ridges typically much longer than they are wide are known as linear dunes and they may be more than 160 kilometres long. Some linear dunes merge to form Y-shaped compound dunes, many form in bidirectional wind regimes. The long axes of these dunes extend in the resultant direction of sand movement, linear loess hills known as pahas are superficially similar. These hills appear to have formed during the last ice age under permafrost conditions dominated by sparse tundra vegetation. Radially symmetrical, star dunes are pyramidal sand mounds with slipfaces on three or more arms that radiate from the center of the mound

4.
Foam
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A foam is a substance formed by trapping pockets of gas in a liquid or solid. A bath sponge and the head on a glass of beer are examples of foams, in most foams, the volume of gas is large, with thin films of liquid or solid separating the regions of gas. Solid foams can be closed-cell or open-cell, in closed-cell foam, the gas forms discrete pockets, each completely surrounded by the solid material. In open-cell foam, gas pockets connect to each other, a bath sponge is an example of an open-cell foam, water easily flows through the entire structure, displacing the air. A camping mat is an example of a foam, gas pockets are sealed from each other so the mat cannot soak up water. Foams are examples of dispersed media, when the principal scale is small, i. e. for a very fine foam, this dispersed medium can be considered a type of colloid. Foam can also refer to something that is analogous to foam, such as foam, polyurethane foam, XPS foam, polystyrene, phenolic. A foam is, in cases, a multi-scale system. One scale is the bubble, material foams are typically disordered and have a variety of bubble sizes, at larger sizes, the study of idealized foams is closely linked to the mathematical problems of minimal surfaces and three-dimensional tessellations, also called honeycombs. The Weaire–Phelan structure is considered the best possible unit cell of a perfectly ordered foam, at lower scale than the bubble is the thickness of the film for metastable foams, which can be considered a network of interconnected films called lamellae. Ideally, the lamellae connect in triads and radiate 120° outward from the connection points, an even lower scale is the liquid–air interface at the surface of the film. Most of the time this interface is stabilized by a layer of amphiphilic structure, often made of surfactants, particles, or more complex associations. Several conditions are needed to produce foam, there must be mechanical work, surface active components that reduce the tension. To create foam, work is needed to increase the surface area, one of the ways foam is created is through dispersion, where a large amount of gas is mixed with a liquid. A more specific method of dispersion involves injecting a gas through a hole in a solid into a liquid, if this process is completed very slowly, then one bubble can be emitted from the orifice at a time as shown in the picture below. The force working against the force is the surface tension force, which is F s =2 r π γ, where γ is the surface tension. As more air is pushed into the bubble, the buoyancy force grows quicker than the tension force. Thus, detachment occurs when the force is large enough to overcome the surface tension force

5.
Meander
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A meander, in general, is a bend in a sinuous watercourse or river. A meander forms when moving water in a stream erodes the outer banks and widens its valley, a stream of any volume may assume a meandering course, alternately eroding sediments from the outside of a bend and depositing them on the inside. The result is a pattern as the stream meanders back. When a meander gets cut off from the stream, an oxbow lake forms. Over time meanders migrate downstream, sometimes in such a time as to create civil engineering problems for local municipalities attempting to maintain stable roads. There is not yet full consistency or standardization of scientific terminology used to describe watercourses, a variety of symbols and schemes exist. Parameters based on mathematical formulae or numerical data vary as well, unless otherwise defined in a specific scheme meandering and sinuosity here are synonymous and mean any repetitious pattern of bends, or waveforms. In some schemes, meandering applies only to rivers with exaggerated circular loops or secondary meanders, sinuosity is one of the channel types that a stream may assume over all or part of its course. All streams are sinuous at some time in their history over some part of their length. The term derives from the Meander River located in present-day Turkey and known to the Ancient Greeks as Μαίανδρος Maiandros and its course is so exceedingly winding that everything winding is called meandering. The Meander River is located south of Izmir, east of the ancient Greek town of Miletus, now, Milet and it flows through a graben in the Menderes Massif, but has a flood plain much wider than the meander zone in its lower reach. Its modern Turkish name is the Büyük Menderes River, when a fluid is introduced to an initially straight channel which then bends, the sidewalls induce a pressure gradient that causes the fluid to alter course and follow the bend. From here, two opposing processes occur, irrotational flow and secondary flow, for a river to meander, secondary flow must dominate. Irrotational flow, From Bernoullis equations, high pressure results in low velocity, therefore, in the absence of secondary flow we would expect low fluid velocity at the outside bend and high fluid velocity at the inside bend. This classic fluid mechanics result is irrotational vortex flow, in the context of meandering rivers, its effects are dominated by those of secondary flow. Secondary flow, A force balance exists between pressure forces pointing to the bend of the river and centrifugal forces pointing to the outside bend of the river. In the context of meandering rivers, a boundary layer exists within the layer of fluid that interacts with the river bed. Inside that layer and following standard boundary-layer theory, the velocity of the fluid is effectively zero, centrifugal force, which depends on velocity, is also therefore effectively zero

6.
Phyllotaxis
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In botany, phyllotaxis or phyllotaxy is the arrangement of leaves on a plant stem. Phyllotactic spirals form a class of patterns in nature. The basic arrangements of leaves on a stem are opposite, or alternate = spiral, leaves may also be whorled if several leaves arise, or appear to arise, from the same level on a stem. This arrangement is unusual on plants except for those with particularly short internodes. Examples of trees with whorled phyllotaxis are Brabejum stellatifolium and the related Macadamia genus, with an opposite leaf arrangement, two leaves arise from the stem at the same level, on opposite sides of the stem. An opposite leaf pair can be thought of as a whorl of two leaves, with an alternate pattern, each leaf arises at a different point on the stem. Examples include various bulbous plants such as Boophone and it also occurs in other plant habits such as those of Gasteria or Aloe seedlings, and also in some mature Aloe species such as Aloe plicatilis. In an opposite pattern, if successive leaf pairs are 90 degrees apart and it is common in members of the family Crassulaceae Decussate phyllotaxis also occurs in the Aizoaceae. A whorl can occur as a structure where all the leaves are attached at the base of the shoot. A basal whorl with a number of leaves spread out in a circle is called a rosette. A repeating spiral can be represented by a fraction describing the angle of windings leaf per leaf, alternate distichous leaves will have an angle of 1/2 of a full rotation. In beech and hazel the angle is 1/3, in oak and apricot it is 2/5, in sunflowers, poplar, and pear, it is 3/8, the numerator and denominator normally consist of a Fibonacci number and its second successor. The number of leaves is called rank, in the case of simple Fibonacci ratios. With larger Fibonacci pairs, the pattern becomes complex and non-repeating and this tends to occur with a basal configuration. Examples can be found in flowers and seed heads. The most famous example is the sunflower head and this phyllotactic pattern creates an optical effect of criss-crossing spirals. In the botanical literature, these designs are described by the number of counter-clockwise spirals and these also turn out to be Fibonacci numbers. In some cases, the appear to be multiples of Fibonacci numbers because the spirals consist of whorls

7.
Soap bubble
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A soap bubble is an extremely thin film of soapy water enclosing air that forms a hollow sphere with an iridescent surface. Soap bubbles usually last for only a few seconds before bursting and they are often used for childrens enjoyment, but they are also used in artistic performances. Assembling several bubbles results in a foam, when light shines onto a bubble it appears to change colour. Depending on the thickness of the film, different colours interfere constructively and destructively, Soap bubbles are physical examples of the complex mathematical problem of minimal surface. They will assume the shape of least surface area possible containing a given volume, a true minimal surface is more properly illustrated by a soap film, which has equal pressure on inside as outside, hence is a surface with zero mean curvature. A soap bubble is a soap film, due to the difference in outside and inside pressure. This has been dubbed the double bubble conjecture, due to these qualities soap bubbles films have been used with practical problem solving application. A famous example is his West German Pavilion at Expo 67 in Montreal, when two bubbles merge, they adopt a shape which makes the sum of their surface areas as small as possible, compatible with the volume of air each bubble encloses. If the bubbles are of size, their common wall is flat. If they arent the same size, their common wall bulges into the bubble, since the smaller one has a higher internal pressure than the larger one. At a point where three or more bubbles meet, they sort themselves out so only three bubble walls meet along a line. Since the surface tension is the same in each of the three surfaces, the three angles between them must be equal to 120°, only four bubble walls can meet at a point, with the lines where triplets of bubble walls meet separated by cos−1 ≈109. 47°. All these rules, known as Plateaus laws, determine how a foam is built from bubbles, the longevity of a soap bubble is limited by the ease of rupture of the very thin layer of water which constitutes its surface, namely a micrometer-thick soap film. It is thus sensitive to, Drainage within the soap film and this can be slowed down by increasing the water viscosity, for instance by adding glycerol. Still, there is a height limit, which is the capillary length, very high for soap bubbles. In principle, there is no limit in the length it can reach, evaporation, This can be slowed down by blowing bubbles in a wet atmosphere, or by adding some sugar to the water. Dirt and fat, When the bubble touches the ground, a wall, or our skin and this can be prevented by wetting these surfaces with water. When a soap bubble is in contact with a solid or a liquid surface wetting is observed, on a solid surface, the contact angle of the bubble depends on the surface energy of the solid

8.
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

9.
Crystal structure
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In crystallography, crystal structure is a description of the ordered arrangement of atoms, ions or molecules in a crystalline material. Ordered structures occur from the nature of the constituent particles to form symmetric patterns that repeat along the principal directions of three-dimensional space in matter. The smallest group of particles in the material that constitutes the pattern is the unit cell of the structure. The unit cell completely defines the symmetry and structure of the crystal lattice. The repeating patterns are said to be located at the points of the Bravais lattice, the lengths of the principal axes, or edges, of the unit cell and the angles between them are the lattice constants, also called lattice parameters. The symmetry properties of the crystal are described by the concept of space groups, all possible symmetric arrangements of particles in three-dimensional space may be described by the 230 space groups. The crystal structure and symmetry play a role in determining many physical properties, such as cleavage, electronic band structure. The crystal structure of a material can be described in terms of its unit cell, the unit cell is a box containing one or more atoms arranged in three dimensions. The unit cells stacked in three-dimensional space describe the arrangement of atoms of the crystal. Commonly, atomic positions are represented in terms of fractional coordinates, the atom positions within the unit cell can be calculated through application of symmetry operations to the asymmetric unit. The asymmetric unit refers to the smallest possible occupation of space within the unit cell and this does not, however imply that the entirety of the asymmetric unit must lie within the boundaries of the unit cell. Symmetric transformations of atom positions are calculated from the group of the crystal structure. Vectors and planes in a lattice are described by the three-value Miller index notation. It uses the indices ℓ, m, and n as directional parameters, which are separated by 90°, by definition, the syntax denotes a plane that intercepts the three points a1/ℓ, a2/m, and a3/n, or some multiple thereof. That is, the Miller indices are proportional to the inverses of the intercepts of the plane with the unit cell, if one or more of the indices is zero, it means that the planes do not intersect that axis. A plane containing a coordinate axis is translated so that it no longer contains that axis before its Miller indices are determined, the Miller indices for a plane are integers with no common factors. Negative indices are indicated with horizontal bars, as in, in an orthogonal coordinate system for a cubic cell, the Miller indices of a plane are the Cartesian components of a vector normal to the plane. Likewise, the planes are geometric planes linking nodes

10.
Quasicrystal
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A quasiperiodic crystal, or quasicrystal, is a structure that is ordered but not periodic. A quasicrystalline pattern can continuously fill all space, but it lacks translational symmetry. Aperiodic tilings were discovered by mathematicians in the early 1960s, and, some twenty years later, the discovery of these aperiodic forms in nature has produced a paradigm shift in the fields of crystallography. Quasicrystals had been investigated and observed earlier, but, until the 1980s, in 2009, after a dedicated search, a mineralogical finding, icosahedrite, offered evidence for the existence of natural quasicrystals. Roughly, an ordering is non-periodic if it lacks translational symmetry, symmetrical diffraction patterns result from the existence of an indefinitely large number of elements with a regular spacing, a property loosely described as long-range order. Experimentally, the aperiodicity is revealed in the symmetry of the diffraction pattern. In 1982 materials scientist Dan Shechtman observed that certain aluminium-manganese alloys produced the unusual diffractograms which today are seen as revelatory of quasicrystal structures. Due to fear of the communitys reaction, it took him two years to publish the results for which he was awarded the Nobel Prize in Chemistry in 2011. In 1961, Hao Wang asked whether determining if a set of tiles admits a tiling of the plane is an unsolvable problem or not. He conjectured that it is solvable, relying on the hypothesis that every set of tiles that can tile the plane can do it periodically. Nevertheless, two later, his student Robert Berger constructed a set of some 20,000 square tiles that can tile the plane. As further aperiodic sets of tiles were discovered, sets with fewer and fewer shapes were found, in 1976 Roger Penrose discovered a set of just two tiles, now referred to as Penrose tiles, that produced only non-periodic tilings of the plane. These tilings displayed instances of fivefold symmetry, around the same time Robert Ammann created a set of aperiodic tiles that produced eightfold symmetry. Mathematically, quasicrystals have been shown to be derivable from a method that treats them as projections of a higher-dimensional lattice. Icosahedral quasicrystals in three dimensions were projected from a six-dimensional hypercubic lattice by Peter Kramer and Roberto Neri in 1984, the tiling is formed by two tiles with rhombohedral shape. Shechtman first observed ten-fold electron diffraction patterns in 1982, as described in his notebook, the observation was made during a routine investigation, by electron microscopy, of a rapidly cooled alloy of aluminium and manganese prepared at the US National Bureau of Standards. In the summer of the same year Shechtman visited Ilan Blech, Blech responded that such diffractions had been seen before. Around that time, Shechtman also related his finding to John Cahn of NIST who did not offer any explanation, Shechtman quoted Cahn as saying, Danny, this material is telling us something and I challenge you to find out what it is