Portal:Patterns in nature
Portal maintenance status: (October 2018)
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Introduction
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 explain order in nature. The modern understanding of visible patterns developed gradually over time.
In the 19th century, Belgian physicist Joseph Plateau examined soap films, leading him to formulate the concept of a minimal surface. German biologist and artist Ernst Haeckel painted hundreds of marine organisms to emphasise their symmetry. Scottish biologist D'Arcy Thompson pioneered the study of growth 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 and stripes. Hungarian biologist Aristid Lindenmayer and French American mathematician Benoît Mandelbrot showed how the mathematics of fractals could create plant growth patterns.
Selected general articles
A page of the Liber Abaci from the Biblioteca Nazionale di Firenze showing (on right) the numbers of the Fibonacci sequence. Note in particular the 2,8, and 9 which resemble Arabic numerals more than Eastern Arabic numerals or Indian numerals
Liber Abaci (1202, also spelled as Liber Abbaci) is a historic book on arithmetic by Leonardo of Pisa, known later by his nickname Fibonacci.
Liber Abaci was among the first Western books to describe the Hindu–Arabic numeral system and to use symbols traditionally described as "Arabic numerals". By addressing the applications of both commercial tradesmen and mathematicians, it contributed to convincing the public of the superiority of the system, and the use of these glyphs. Read more...- Visualisation of the vortex street behind a circular cylinder in air; the flow is made visible through release of oil vapour in the air near the cylinder
In fluid dynamics, a Kármán vortex street (or a von Kármán vortex street) is a repeating pattern of swirling vortices, caused by a process known as vortex shedding, which is responsible for the unsteady separation of flow of a fluid around blunt bodies. It is named after the engineer and fluid dynamicist Theodore von Kármán, and is responsible for such phenomena as the "singing" of suspended telephone or power lines and the vibration of a car antenna at certain speeds. Read more...
A meander is one of a series of regular sinuous curves, bends, loops, turns, or windings in the channel of a river, stream, or other watercourse. It is produced by a stream or river swinging from side to side as it flows across its floodplain or shifts its channel within a valley. A meander is produced by a stream or river as it erodes the sediments comprising an outer, concave bank (cut bank) and deposits this and other sediment downstream on an inner, convex bank which is typically a point bar. The result of sediments being eroded from the outside concave bank and their deposition on an inside convex bank is the formation of a sinuous course as a channel migrates back and forth across the down-valley axis of a floodplain. The zone within which a meandering stream shifts its channel across either its floodplain or valley floor from time to time is known as a meander belt. It typically ranges from 15 to 18 times the width of the channel. Over time, meanders migrate downstream, sometimes in such a short time as to create civil engineering problems for local municipalities attempting to maintain stable roads and bridges.
The degree of meandering of the channel of a river, stream, or other watercourse is measured by its sinuosity. The sinuosity of a watercourse is the ratio of the length of the channel to the straight line down-valley distance. Streams or rivers with a single channel and sinuosities of 1.5 or more are defined as meandering streams or rivers. Read more...
Fibonacci (c. 1175 – c. 1250) was an Italian mathematician from the Republic of Pisa, considered to be "the most talented Western mathematician of the Middle Ages".
The name he is commonly called, "Fibonacci" (Italian: [fiboˈnattʃi]), was made up in 1838 by the Franco-Italian historian Guillaume Libri and is short for filius Bonacci ("son of (the) Bonacci") and he is also known as Leonardo Bonacci, Leonardo of Pisa, Leonardo Pisano Bigollo, or Leonardo Fibonacci.
Fibonacci popularized the Hindu–Arabic numeral system in the Western World primarily through his composition in 1202 of Liber Abaci (Book of Calculation). He also introduced Europe to the sequence of Fibonacci numbers, which he used as an example in Liber Abaci. Read more...
Natural selection is the differential survival and reproduction of individuals due to differences in phenotype. It is a key mechanism of evolution, the change in the heritable traits characteristic of a population over generations. Charles Darwin popularised the term "natural selection", contrasting it with artificial selection, which is intentional, whereas natural selection is not.
Variation exists within all populations of organisms. This occurs partly because random mutations arise in the genome of an individual organism, and offspring can inherit such mutations. Throughout the lives of the individuals, their genomes interact with their environments to cause variations in traits. The environment of a genome includes the molecular biology in the cell, other cells, other individuals, populations, species, as well as the abiotic environment. Because individuals with certain variants of the trait tend to survive and reproduce more than individuals with other, less successful variants, the population evolves. Other factors affecting reproductive success include sexual selection (now often included in natural selection) and fecundity selection. Read more...
Self-organization in micron-sized Nb3O7(OH) cubes during a hydrothermal treatment at 200 °C. Initially amorphous cubes gradually transform into ordered 3D meshes of crystalline nanowires as summarized in the model below.
Self-organization, also called (in the social sciences) spontaneous order, is a process where some form of overall order arises from local interactions between parts of an initially disordered system. The process is spontaneous, not needing control by any external agent. It is often triggered by random fluctuations, amplified by positive feedback. The resulting organization is wholly decentralized, distributed over all the components of the system. As such, the organization is typically robust and able to survive or self-repair substantial perturbation. Chaos theory discusses self-organization in terms of islands of predictability in a sea of chaotic unpredictability.
Self-organization occurs in many physical, chemical, biological, robotic, and cognitive systems. Examples of self-organization include crystallization, thermal convection of fluids, chemical oscillation, animal swarming, neural circuits, and artificial neural networks. Read more...
The formation of complex symmetrical and fractal patterns in snowflakes exemplifies emergence in a physical system
In philosophy, systems theory, science, and art, emergence is the condition of an entity having properties its parts do not have, due to interactions among the parts.
Emergence plays a central role in theories of integrative levels and of complex systems. For instance, the phenomenon of life as studied in biology is an emergent property of chemistry, and psychological phenomena emerge from the neurobiological phenomena of living things. Read more...
Turing's paper explained how natural patterns such as stripes, spots and spirals, like those of the giant pufferfish, may arise naturally.
"The Chemical Basis of Morphogenesis" is an article written by the English mathematician Alan Turing in 1952 describing the way in which natural patterns such as stripes, spots and spirals may arise naturally out of a homogeneous, uniform state. The theory, which can be called a reaction–diffusion theory of morphogenesis, has served as a basic model in theoretical biology. Read more...
The (3-D) crystal structure of H2O ice Ih (c) consists of bases of H2O ice molecules (b) located on lattice points within the (2-D) hexagonal space lattice (a). The values for the H–O–H angle and O–H distance have come from Physics of Ice with uncertainties of ±1.5° and ±0.005 Å, respectively. The white box in (c) is the unit cell defined by Bernal and Fowler
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 intrinsic 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 repeating pattern is the unit cell of the structure. The unit cell completely defines the symmetry and structure of the entire crystal lattice, which is built up by repetitive translation of the unit cell along its principal axes. The repeating patterns are said to be located at the points of the Bravais lattice. Read more...- The science of pattern formation deals with the visible, (statistically) orderly outcomes of self-organization and the common principles behind similar patterns in nature.
In developmental biology, pattern formation refers to the generation of complex organizations of cell fates in space and time. Pattern formation is controlled by genes. The role of genes in pattern formation is an aspect of morphogenesis, the creation of diverse anatomies from similar genes, now being explored in the science of evolutionary developmental biology or evo-devo. The mechanisms involved are well seen in the anterior-posterior patterning of embryos from the model organism Drosophila melanogaster (a fruit fly), one of the first organisms to have its morphogenesis studied, and in the eyespots of butterflies, whose development is a variant of the standard (fruit fly) mechanism. Read more...
Empedocles (/ɛmˈpɛdəkliːz/; Greek: Ἐμπεδοκλῆς [empedoklɛ̂ːs], Empedoklēs; c. 490 – c. 430 BC) was a Greek pre-Socratic philosopher and a citizen of Akragas, a Greek city in Sicily. Empedocles' philosophy is best known for originating the cosmogonic theory of the four classical elements. He also proposed forces he called Love and Strife which would mix and separate the elements, respectively. These physical speculations were part of a history of the universe which also dealt with the origin and development of life.
Influenced by the Pythagoreans, Empedocles was a vegetarian who supported the doctrine of reincarnation. He is generally considered the last Greek philosopher to have recorded his ideas in verse. Some of his work survives, more than is the case for any other pre-Socratic philosopher. Empedocles' death was mythologized by ancient writers, and has been the subject of a number of literary treatments. Read more...
Sexual selection creates colourful differences between sexes (sexual dimorphism) in Goldie's bird-of-paradise. Male above; female below. Painting by John Gerrard Keulemans (d.1912)
Sexual selection is a mode of natural selection where members of one biological sex choose mates of the other sex to mate with (intersexual selection), and compete with members of the same sex for access to members of the opposite sex (intrasexual selection). These two forms of selection mean that some individuals have better reproductive success than others within a population, either from being more attractive or preferring more attractive partners to produce offspring. For instance, in the breeding season, sexual selection in frogs occurs with the males first gathering at the water's edge and making their mating calls: croaking. The females then arrive and choose the males with the deepest croaks and best territories. Generalizing, males benefit from frequent mating and monopolizing access to a group of fertile females. Females have a limited number of offspring they can have and they maximize the return on the energy they invest in reproduction.
The concept was first articulated by Charles Darwin and Alfred Russel Wallace who described it as driving species adaptations and that many organisms had evolved features whose function was deleterious to their individual survival, and then developed by Ronald Fisher in the early 20th century. Sexual selection can, typically, lead males to extreme efforts to demonstrate their fitness to be chosen by females, producing sexual dimorphism in secondary sexual characteristics, such as the ornate plumage of birds such as birds of paradise and peafowl, or the antlers of deer, or the manes of lions, caused by a positive feedback mechanism known as a Fisherian runaway, where the passing-on of the desire for a trait in one sex is as important as having the trait in the other sex in producing the runaway effect. Although the sexy son hypothesis indicates that females would prefer male offspring, Fisher's principle explains why the sex ratio is 1:1 almost without exception. Sexual selection is also found in plants and fungi. Read more...
A tessellation of a flat surface is the tiling of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps. In mathematics, tessellations can be generalized to higher dimensions and a variety of geometries.
A periodic tiling has a repeating pattern. Some special kinds include regular tilings with regular polygonal tiles all of the same shape, and semiregular tilings with regular tiles of more than one shape and with every corner identically arranged. The patterns formed by periodic tilings can be categorized into 17 wallpaper groups. A tiling that lacks a repeating pattern is called "non-periodic". An aperiodic tiling uses a small set of tile shapes that cannot form a repeating pattern. In the geometry of higher dimensions, a space-filling or honeycomb is also called a tessellation of space. Read more...
Ernst Heinrich Philipp August Haeckel (German: [ˈʔɛɐ̯nst ˈhɛkl̩]; 16 February 1834 – 9 August 1919) was a German biologist, naturalist, philosopher, physician, professor, marine biologist, and artist who discovered, described and named thousands of new species, mapped a genealogical tree relating all life forms, and coined many terms in biology, including anthropogeny, ecology, phylum, phylogeny, and Protista. Haeckel promoted and popularised Charles Darwin's work in Germany and developed the influential but no longer widely held recapitulation theory ("ontogeny recapitulates phylogeny") claiming that an individual organism's biological development, or ontogeny, parallels and summarises its species' evolutionary development, or phylogeny.
The published artwork of Haeckel includes over 100 detailed, multi-colour illustrations of animals and sea creatures, collected in his Kunstformen der Natur ("Art Forms of Nature"). As a philosopher, Ernst Haeckel wrote Die Welträthsel (1895–1899; in English: The Riddle of the Universe, 1901), the genesis for the term "world riddle" (Welträtsel); and Freedom in Science and Teaching to support teaching evolution. Read more...- In psychology and cognitive neuroscience, pattern recognition describes a cognitive process that matches information from a stimulus with information retrieved from memory.
Pattern recognition occurs when information from the environment is received and entered into short-term memory, causing automatic activation of a specific content of long-term memory. An early example of this is learning the alphabet in order. When a carer repeats ‘A, B, C’ multiple times to a child, utilizing the pattern recognition, the child says ‘C’ after he/she hears ‘A, B’ in order. Recognizing patterns allow us to predict and expect what is coming. The process of pattern recognition involves matching the information received with the information already stored in the brain. Making the connection between memories and information perceived is a step of pattern recognition called identification. Pattern recognition requires repetition of experience. Semantic memory, which is used implicitly and subconsciously is the main type of memory involved with recognition. Read more...
Biology is the natural science that studies life and living organisms, including their physical structure, chemical processes, molecular interactions, physiological mechanisms, development and evolution. Despite the complexity of the science, there are certain unifying concepts that consolidate it into a single, coherent field. Biology recognizes the cell as the basic unit of life, genes as the basic unit of heredity, and evolution as the engine that propels the creation and extinction of species. Living organisms are open systems that survive by transforming energy and decreasing their local entropy to maintain a stable and vital condition defined as homeostasis.
Sub-disciplines of biology are defined by the research methods employed and the kind of system studied: theoretical biology uses mathematical methods to formulate quantitative models while experimental biology performs empirical experiments to test the validity of proposed theories and understand the mechanisms underlying life and how it appeared and evolved from non-living matter about 4 billion years ago through a gradual increase in the complexity of the system. See branches of biology. Read more...
Soap foam bubbles
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. Soap foams are also known as suds.
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 closed-cell foam: gas pockets are sealed from each other so the mat cannot soak up water. Read more...
Typical aerodynamic teardrop shape, assuming a viscous medium passing from left to right, the diagram shows the pressure distribution as the thickness of the black line and shows the velocity in the boundary layer as the violet triangles. The green vortex generators prompt the transition to turbulent flow and prevent back-flow also called flow separation from the high-pressure region in the back. The surface in front is as smooth as possible or even employs shark-like skin, as any turbulence here increases the energy of the airflow. The truncation on the right, known as a Kammback, also prevents backflow from the high-pressure region in the back across the spoilers to the convergent part.
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including aerodynamics (the study of air and other gases in motion) and hydrodynamics (the study of liquids in motion). Fluid dynamics has a wide range of applications, including calculating forces and moments on aircraft, determining the mass flow rate of petroleum through pipelines, predicting weather patterns, understanding nebulae in interstellar space and modelling fission weapon detonation,
Fluid dynamics offers a systematic structure—which underlies these practical disciplines—that embraces empirical and semi-empirical laws derived from flow measurement and used to solve practical problems. The solution to a fluid dynamics problem typically involves the calculation of various properties of the fluid, such as flow velocity, pressure, density, and temperature, as functions of space and time. Read more...
Daguerrotype portrait of Belgian physicist Joseph Plateau dated 1843
Joseph Antoine Ferdinand Plateau (14 October 1801 – 15 September 1883) was a Belgian physicist. He was one of the first people to demonstrate the illusion of a moving image. To do this he used counter rotating disks with repeating drawn images in small increments of motion on one and regularly spaced slits in the other. He called this device of 1832 the phenakistiscope. Read more...- Sir D'Arcy Wentworth Thompson CB FRS FRSE (2 May 1860 – 21 June 1948) was a Scottish biologist, mathematician and classics scholar. He was a pioneer of mathematical biology, travelled on expeditions to the Bering Strait and held the position of Professor of Natural History at University College, Dundee for 32 years, then at St Andrews for 31 years. He was elected a Fellow of the Royal Society, was knighted, and received the Darwin Medal and the Daniel Giraud Elliot Medal.
Thompson is remembered as the author of the 1917 book On Growth and Form, which led the way for the scientific explanation of morphogenesis, the process by which patterns and body structures are formed in plants and animals. Read more... - 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, either on their own or on contact with another object. They are often used for children's enjoyment, but they are also used in artistic performances. Assembling several bubbles results in foam.
When light shines onto a bubble it appears to change colour. Unlike those seen in a rainbow, which arise from differential refraction, the colours seen in a soap bubble arise from interference of light reflecting off the front and back surfaces of the thin soap film. Depending on the thickness of the film, different colours interfere constructively and destructively. Read more... - A logarithmic spiral, equiangular spiral or growth spiral is a self-similar spiral curve which often appears in nature. The logarithmic spiral was first described by Descartes and later extensively investigated by Jacob Bernoulli, who called it Spira mirabilis, "the marvelous spiral". Read more...
On Growth and Form is a book by the Scottish mathematical biologist D'Arcy Wentworth Thompson (1860–1948). The book is long – 793 pages in the first edition of 1917, 1116 pages in the second edition of 1942.
The book covers many topics including the effects of scale on the shape of animals and plants, large ones necessarily being relatively thick in shape; the effects of surface tension in shaping soap films and similar structures such as cells; the logarithmic spiral as seen in mollusc shells and ruminant horns; the arrangement of leaves and other plant parts (phyllotaxis); and Thompson's own method of transformations, showing the changes in shape of animal skulls and other structures on a Cartesian grid. Read more...
"How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension" is a paper by mathematician Benoît Mandelbrot, first published in Science in 5 May 1967. In this paper, Mandelbrot discusses self-similar curves that have Hausdorff dimension between 1 and 2. These curves are examples of fractals, although Mandelbrot does not use this term in the paper, as he did not coin it until 1975. The paper is one of Mandelbrot's first publications on the topic of fractals. Read more...
The peacock flounder can change its pattern and colours to match its environment.
Camouflage is the use of any combination of materials, coloration, or illumination for concealment, either by making animals or objects hard to see (crypsis), or by disguising them as something else (mimesis). Examples include the leopard's spotted coat, the battledress of a modern soldier, and the leaf-mimic katydid's wings. A third approach, motion dazzle, confuses the observer with a conspicuous pattern, making the object visible but momentarily harder to locate. The majority of camouflage methods aim for crypsis, often through a general resemblance to the background, high contrast disruptive coloration, eliminating shadow, and countershading. In the open ocean, where there is no background, the principal methods of camouflage are transparency, silvering, and countershading, while the ability to produce light is among other things used for counter-illumination on the undersides of cephalopods such as squid. Some animals, such as chameleons and octopuses, are capable of actively changing their skin pattern and colours, whether for camouflage or for signalling. It is possible that some plants use camouflage to evade being eaten by herbivores.
Military camouflage was spurred by the increasing range and accuracy of firearms in the 19th century. In particular the replacement of the inaccurate musket with the rifle made personal concealment in battle a survival skill. In the 20th century, military camouflage developed rapidly, especially during the First World War. On land, artists such as André Mare designed camouflage schemes and observation posts disguised as trees. At sea, merchant ships and troop carriers were painted in dazzle patterns that were highly visible, but designed to confuse enemy submarines as to the target's speed, range, and heading. During and after the Second World War, a variety of camouflage schemes were used for aircraft and for ground vehicles in different theatres of war. The use of radar since the mid-20th century has largely made camouflage for fixed-wing military aircraft obsolete. Read more...
In mathematics, a fractal is a detailed, recursive, and infinitely self-similar mathematical set whose Hausdorff dimension strictly exceeds its topological dimension. Fractals are encountered ubiquitously in nature due to their tendency to appear nearly the same at different levels, as is illustrated here in the successively small magnifications of the Mandelbrot set. Fractals exhibit similar patterns at increasingly small scales, also known as expanding symmetry or unfolding symmetry; If this replication is exactly the same at every scale, as in the Menger sponge, it is called a self-similar pattern.
One way that fractals are different from finite geometric figures is the way in which they scale. Doubling the edge lengths of a polygon multiplies its area by four, which is two (the ratio of the new to the old side length) raised to the power of two (the dimension of the space the polygon resides in). Likewise, if the radius of a sphere is doubled, its volume scales by eight, which is two (the ratio of the new to the old radius) to the power of three (the dimension that the sphere resides in). However, if a fractal's one-dimensional lengths are all doubled, the spatial content of the fractal scales by a power that is not necessarily an integer. This power is called the fractal dimension of the fractal, and it usually exceeds the fractal's topological dimension. Read more...
[Left] Normal Streptocarpus flower (zygomorphic or mirror-symmetric), and [right] peloric (radially symmetric) flower on the same plant
Floral symmetry describes whether, and how, a flower, in particular its perianth, can be divided into two or more identical or mirror-image parts.
Uncommonly, flowers may have no axis of symmetry at all, typically because their parts are spirally arranged. Read more...
Symmetry (from Greek συμμετρία symmetria "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definition, that an object is invariant to any of various transformations; including reflection, rotation or scaling. Although these two meanings of "symmetry" can sometimes be told apart, they are related, so in this article they are discussed together.
Mathematical symmetry may be observed with respect to the passage of time; as a spatial relationship; through geometric transformations; through other kinds of functional transformations; and as an aspect of abstract objects, theoretic models, language, music and even knowledge itself. Read more...
Pythagoras of Samos (c. 570 – c. 495 BC) was an Ionian Greek philosopher and the eponymous founder of the Pythagoreanism movement. His political and religious teachings were well known in Magna Graecia and influenced the philosophies of Plato, Aristotle, and, through them, Western philosophy. Knowledge of his life is clouded by legend, but he appears to have been the son of Mnesarchus, a seal engraver on the island of Samos. Modern scholars disagree regarding Pythagoras's education and influences, but they do agree that, around 530 BC, he travelled to Croton, where he founded a school in which initiates were sworn to secrecy and lived a communal, ascetic lifestyle. This lifestyle entailed a number of dietary prohibitions, traditionally said to have included vegetarianism, although modern scholars doubt that he ever advocated for complete vegetarianism.
The teaching most securely identified with Pythagoras is metempsychosis, or the "transmigration of souls", which holds that every soul is immortal and, upon death, enters into a new body. He may have also devised the doctrine of musica universalis, which holds that the planets move according to mathematical equations and thus resonate to produce an inaudible symphony of music. Scholars debate whether Pythagoras developed the numerological and musical teachings attributed to him, or if those teachings were developed by his later followers, particularly Philolaus of Croton. Following Croton's decisive victory over Sybaris in around 510 BC, Pythagoras's followers came into conflict with supporters of democracy and Pythagorean meeting houses were burned. Pythagoras may have been killed during this persecution, or escaped to Metapontum, where he eventually died. Read more...
Bubbles in a foam of soapy water obey Plateau's laws. At every vertex the angle is close to 109.47 degrees, the tetrahedral angle
Plateau's 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. Read more...
Widmanstätten patterns, also called Thomson structures, are figures of long nickel-iron crystals, found in the octahedrite iron meteorites and some pallasites. They consist of a fine interleaving of kamacite and taenite bands or ribbons called lamellae. Commonly, in gaps between the lamellae, a fine-grained mixture of kamacite and taenite called plessite can be found. Widmanstätten patterns describe features in modern steels, titanium and zirconium alloys. Read more...
At a TED conference in 2010.
Benoit B. Mandelbrot (20 November 1924 – 14 October 2010) was a Polish-born, French and American mathematician and polymath with broad interests in the practical sciences, especially regarding what he labeled as "the art of roughness" of physical phenomena and "the uncontrolled element in life". He referred to himself as a "fractalist" and is recognized for his contribution to the field of fractal geometry, which included coining the word "fractal", as well as developing a theory of "roughness and self-similarity" in nature.
In 1936, while he was a child, Mandelbrot's family emigrated to France from Warsaw, Poland. After World War II ended, Mandelbrot studied mathematics, graduating from universities in Paris and the United States and receiving a master's degree in aeronautics from the California Institute of Technology. He spent most of his career in both the United States and France, having dual French and American citizenship. In 1958, he began a 35-year career at IBM, where he became an IBM Fellow, and periodically took leaves of absence to teach at Harvard University. At Harvard, following the publication of his study of U.S. commodity markets in relation to cotton futures, he taught economics and applied sciences. Read more...
A selection of animals showing a range of possible body symmetries, including both asymmetry, radial and bilateral body plans
Symmetry in biology is the balanced distribution of duplicate body parts or shapes within the body of an organism. In nature and biology, symmetry is always approximate. For example, plant leaves – while considered symmetrical – rarely match up exactly when folded in half. Symmetry creates a class of patterns in nature, where the near-repetition of the pattern element is by reflection or rotation.
The body plans of most multicellular organisms exhibit some form of symmetry, whether radial, bilateral, or spherical. A small minority, notably among the sponges, exhibit no symmetry (i.e., are asymmetric). Symmetry was once important in animal taxonomy; the Radiata, animals with radial symmetry, formed one of the four branches of Georges Cuvier's classification of the animal kingdom. Read more...
Erg Chebbi, Morocco
In physical geography, a dune is a hill of loose sand built by aeolian processes (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 stoss (upflow) side, where the sand is pushed up the dune, and have a shorter "slip face" in the lee side. The valley or trough between dunes is called a slack. A "dune field" or erg is an area covered by extensive dunes.
Dunes occur in some 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, the largest complexes of dunes are found inland in dry regions and associated with ancient lake or sea beds. Dunes can form under the action of water flow (fluvial processes), and on sand or gravel beds of rivers, estuaries and the sea-bed. Read more...
Euclid (holding calipers), Greek mathematician, 3rd century BC, as imagined by Raphael in this detail from The School of Athens.
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") includes the study of such topics as quantity, structure, space, and change.
Mathematicians seek and use patterns to formulate new conjectures; they resolve the truth or falsity of conjectures by mathematical proof. When mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry. Read more...
Physics (from Ancient Greek: φυσική (ἐπιστήμη), translit. physikḗ (epistḗmē), lit. 'knowledge of nature', from φύσις phýsis "nature") is the natural science that studies matter and its motion and behavior through space and time and that studies the related entities of energy and force. Physics is one of the most fundamental scientific disciplines, and its main goal is to understand how the universe behaves.
Physics is one of the oldest academic disciplines and, through its inclusion of astronomy, perhaps the oldest. Over the last two millennia, physics, chemistry, biology, and certain branches of mathematics were a part of natural philosophy, but during the scientific revolution in the 17th century, these natural sciences emerged as unique research endeavors in their own right. Physics intersects with many interdisciplinary areas of research, such as biophysics and quantum chemistry, and the boundaries of physics are not rigidly defined. New ideas in physics often explain the fundamental mechanisms studied by other sciences and suggest new avenues of research in academic disciplines such as mathematics and philosophy. Read more...
In physics, a wave is a disturbance that transfers energy through matter or space, with little or no associated mass transport. Waves consist of oscillations or vibrations of a physical medium or a field, around relatively fixed locations. From the perspective of mathematics, waves, as functions of time and space, are a class of signals.
There are two main types of waves: mechanical and electromagnetic. Mechanical waves propagate through a physical matter, whose substance is being deformed. Restoring forces then reverse the deformation. For example, sound waves propagate via air molecules colliding with their neighbours. When the molecules collide, they also bounce away from each other (a restoring force). This keeps the molecules from continuing to travel in the direction of the wave. Electromagnetic waves do not require a medium. Instead, they consist of periodic oscillations of electrical and magnetic fields originally generated by charged particles, and can therefore travel through a vacuum. These types vary in wavelength, and include radio waves, microwaves, infrared radiation, visible light, ultraviolet radiation, X-rays and gamma rays. Read more...
Wilson Alwyn "Snowflake" Bentley (February 9, 1865 – December 23, 1931) is one of the first known photographers of snowflakes. He perfected a process of catching flakes on black velvet in such a way that their images could be captured before they either melted or sublimated.
Kenneth G. Libbrecht notes that the techniques used by Bentley to photograph snowflakes are essentially the same as used today, and that while the quality of his photographs reflects the technical limitations of the equipment of the era, "he did it so well that hardly anybody bothered to photograph snowflakes for almost 100 years". The broadest collection of Bentley's photographs is held by the Jericho Historical Society in his home town, Jericho, Vermont. Read more...
Plate from Henry Walter Bates (1862) illustrating Batesian mimicry between Dismorphia species (top row, third row) and various Ithomiini (Nymphalidae, second row, bottom row)
In evolutionary biology, mimicry is an evolved resemblance between an organism and another object, often an organism of another species. Mimicry may evolve between different species, or between individuals of the same species. Often, mimicry functions to protect a species from predators, making it an antipredator adaptation. Mimicry evolves if a receiver (such as a predator) perceives the similarity between a mimic (the organism that has a resemblance) and a model (the organism it resembles) and as a result changes its behaviour in a way that provides a selective advantage to the mimic. The resemblances that evolve in mimicry can be visual, acoustic, chemical, tactile, or electric, or combinations of these sensory modalities. Mimicry may be to the advantage of both organisms that share a resemblance, in which case it is a form of mutualism; or mimicry can be to the detriment of one, making it parasitic or competitive. The evolutionary convergence between groups is driven by the selective action of a signal-receiver or dupe. Birds, for example, use sight to identify palatable insects, whilst avoiding the noxious ones. Over time, palatable insects may evolve to resemble noxious ones, making them mimics and the noxious ones models. In the case of mutualism, sometimes both groups are referred to as "co-mimics". It is often thought that models must be more abundant than mimics, but this is not so. Mimicry may involve numerous species; many harmless species such as hoverflies are Batesian mimics of strongly defended species such as wasps, while many such well-defended species form Mullerian mimicry rings, all resembling each other. Mimicry between prey species and their predators often involves three or more species.
In its broadest definition, mimicry can include non-living models. The specific terms masquerade and mimesis are sometimes used when the models are inanimate. For example, animals such as flower mantises, planthoppers, comma and geometer moth caterpillars resemble twigs, bark, leaves, bird droppings or flowers. Many animals bear eyespots, which are hypothesized to resemble the eyes of larger animals. They may not resemble any specific organism's eyes, and whether or not animals respond to them as eyes is also unclear. Nonetheless, eyespots are the subject of a rich contemporary literature. The model is usually another species, except in automimicry, where members of the species mimic other members, or other parts of their own bodies, and in inter-sexual mimicry, where members of one sex mimic members of the other. Read more...
Crisscrossing spirals of Aloe polyphylla
In botany, phyllotaxis or phyllotaxy is the arrangement of leaves on a plant stem (from Ancient Greek phýllon "leaf" and táxis "arrangement"). Phyllotactic spirals form a distinctive class of patterns in nature. Read more...
Mathematics in art: Albrecht Dürer's copper plate engraving Melencolia I, 1514. Mathematical references include a compass for geometry, a magic square and a truncated rhombohedron, while measurement is indicated by the scales and hourglass.
Mathematics and art are related in a variety of ways. Mathematics has itself been described as an art motivated by beauty. Mathematics can be discerned in arts such as music, dance, painting, architecture, sculpture, and textiles. This article focuses, however, on mathematics in the visual arts.
Mathematics and art have a long historical relationship. Artists have used mathematics since the 4th century BC when the Greek sculptor Polykleitos wrote his Canon, prescribing proportions based on the ratio 1:√2 for the ideal male nude. Persistent popular claims have been made for the use of the golden ratio in ancient art and architecture, without reliable evidence. In the Italian Renaissance, Luca Pacioli wrote the influential treatise De Divina Proportione (1509), illustrated with woodcuts by Leonardo da Vinci, on the use of the golden ratio in art. Another Italian painter, Piero della Francesca, developed Euclid's ideas on perspective in treatises such as De Prospectiva Pingendi, and in his paintings. The engraver Albrecht Dürer made many references to mathematics in his work Melencolia I. In modern times, the graphic artist M. C. Escher made intensive use of tessellation and hyperbolic geometry, with the help of the mathematician H. S. M. Coxeter, while the De Stijl movement led by Theo van Doesburg and Piet Mondrian explicitly embraced geometrical forms. Mathematics has inspired textile arts such as quilting, knitting, cross-stitch, crochet, embroidery, weaving, Turkish and other carpet-making, as well as kilim. In Islamic art, symmetries are evident in forms as varied as Persian girih and Moroccan zellige tilework, Mughal jali pierced stone screens, and widespread muqarnas vaulting. Read more...
Alan Mathison Turing OBE FRS (/ˈtjʊərɪŋ/; 23 June 1912 – 7 June 1954) was an English mathematician, computer scientist, logician, cryptanalyst, philosopher, and theoretical biologist. Turing was highly influential in the development of theoretical computer science, providing a formalisation of the concepts of algorithm and computation with the Turing machine, which can be considered a model of a general-purpose computer. Turing is widely considered to be the father of theoretical computer science and artificial intelligence. Despite these accomplishments, he was never fully recognized in his home country during his lifetime due to his homosexuality, which was then a crime in the UK.
During the Second World War, Turing worked for the Government Code and Cypher School (GC&CS) at Bletchley Park, Britain's codebreaking centre that produced Ultra intelligence. For a time he led Hut 8, the section that was responsible for German naval cryptanalysis. Here he devised a number of techniques for speeding the breaking of German ciphers, including improvements to the pre-war Polish bombe method, an electromechanical machine that could find settings for the Enigma machine. Turing played a pivotal role in cracking intercepted coded messages that enabled the Allies to defeat the Nazis in many crucial engagements, including the Battle of the Atlantic, and in so doing helped win the war. Counterfactual history is difficult with respect to the effect Ultra intelligence had on the length of the war, but at the upper end it has been estimated that this work shortened the war in Europe by more than two years and saved over 14 million lives. Read more...
Plato (/ˈpleɪtoʊ/;[a] Greek: Πλάτων[a] Plátōn, pronounced [plá.tɔːn] in Classical Attic; 428/427 or 424/423[b] – 348/347 BC) was a philosopher in Classical Greece and the founder of the Academy in Athens, the first institution of higher learning in the Western world. He is widely considered the pivotal figure in the development of Western philosophy. Unlike nearly all of his philosophical contemporaries, Plato's entire work is believed to have survived intact for over 2,400 years.
Along with his teacher, Socrates, and his most famous student, Aristotle, Plato laid the foundations of Western philosophy and science. Alfred North Whitehead once noted: "the safest general characterization of the European philosophical tradition is that it consists of a series of footnotes to Plato." In addition to being a foundational figure for Western science, philosophy, and mathematics, Plato has also often been cited as one of the founders of Western religion and spirituality. Read more...
A quasiperiodic crystal, or quasicrystal, is a structure that is ordered but not periodic. A quasicrystalline pattern can continuously fill all available space, but it lacks translational symmetry. While crystals, according to the classical crystallographic restriction theorem, can possess only two, three, four, and six-fold rotational symmetries, the Bragg diffraction pattern of quasicrystals shows sharp peaks with other symmetry orders, for instance five-fold.
Aperiodic tilings were discovered by mathematicians in the early 1960s, and, some twenty years later, they were found to apply to the study of quasicrystals. 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, they were disregarded in favor of the prevailing views about the atomic structure of matter. In 2009, after a dedicated search, a mineralogical finding, icosahedrite, offered evidence for the existence of natural quasicrystals. Read more...
A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) 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, consisting of flat faces with specific, characteristic orientations. 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 (krustallos), meaning both "ice" and "rock crystal", from κρύος (kruos), "icy cold, frost". Read more...- 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. If a displacement develops perpendicular to the surface of displacement, it is called a normal tensile crack or simply a crack; if a displacement develops tangentially to the surface of displacement, it is called a shear crack, slip band, or dislocation.
Brittle fractures occur with no apparent deformation before fracture; ductile fractures occur when visible deformation does occur before separation. Fracture strength or breaking strength is the stress when a specimen fails or fractures. A detailed understanding of how fracture occurs in materials may be assisted by the study of fracture mechanics. Read more...
Chaos theory is a branch of mathematics focusing on the behavior of dynamical systems that are highly sensitive to initial conditions. "Chaos" is an interdisciplinary theory stating that within the apparent randomness of chaotic complex systems, there are underlying patterns, constant feedback loops, repetition, self-similarity, fractals, self-organization, and reliance on programming at the initial point known as sensitive dependence on initial conditions. The butterfly effect describes how a small change in one state of a deterministic nonlinear system can result in large differences in a later state, e.g. a butterfly flapping its wings in Brazil can cause a hurricane in Texas.
Small differences in initial conditions, such as those due to rounding errors in numerical computation, yield widely diverging outcomes for such dynamical systems, rendering long-term prediction of their behavior impossible in general. This happens even though these systems are deterministic, meaning that their future behavior is fully determined by their initial conditions, with no random elements involved. In other words, the deterministic nature of these systems does not make them predictable. This behavior is known as deterministic chaos, or simply chaos. The theory was summarized by Edward Lorenz as: Read more...
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Selected images
Fibonacci patterns occur widely in plant structures, including this cone of queen sago, Cycas circinalis
Natural patterns form as wind blows sand in the dunes of the Namib Desert. The crescent shaped dunes and the ripples on their surfaces repeat wherever there are suitable conditions.
Echinoderms like this starfish have fivefold symmetry.
Snowflakes have sixfold symmetry.
Patterns of the veiled chameleon, Chamaeleo calyptratus, provide camouflage and signal mood as well as breeding condition.
Volvox has spherical symmetry.
Animals often show mirror or bilateral symmetry, like this tiger.
D'Arcy Thompson pioneered the study of growth and form in his 1917 book
Garnet showing rhombic dodecahedral crystal habit.
Water splash approximates radial symmetry.
The growth patterns of certain trees resemble these Lindenmayer system fractals.
Fluorite showing cubic crystal habit.
Fivefold symmetry can be seen in many flowers and some fruits like this medlar.
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