1.
Caprona ransonnettii
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Caprona ransonnettii, commonly known as the golden angle, is a butterfly belonging to the family Hesperiidae. It occurs in Sri Lanka, Odisha and in the Nilgiri mountains, in 1891, Edward Yerbury Watson gave this detailed description, Upperside fuliginous ochreous-brown. Female, forewing with the spots and marginal lunules, and the band on hindwing more prominent. A similar variation has been noted by Mr. de Niceville in C. tissa, a not very distantly allied species, and in both cases it is the dry-season form which is the paler

2.
Golden ratio
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In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. The figure on the right illustrates the geometric relationship, expressed algebraically, for quantities a and b with a > b >0, a + b a = a b = def φ, where the Greek letter phi represents the golden ratio. Its value is, φ =1 +52 =1.6180339887 …, A001622 The golden ratio is also called the golden mean or golden section. Other names include extreme and mean ratio, medial section, divine proportion, divine section, golden proportion, golden cut, the golden ratio appears in some patterns in nature, including the spiral arrangement of leaves and other plant parts. The golden ratio has also used to analyze the proportions of natural objects as well as man-made systems such as financial markets. Two quantities a and b are said to be in the golden ratio φ if a + b a = a b = φ, one method for finding the value of φ is to start with the left fraction. Through simplifying the fraction and substituting in b/a = 1/φ, a + b a =1 + b a =1 +1 φ, multiplying by φ gives φ +1 = φ2 which can be rearranged to φ2 − φ −1 =0. First, the line segment A B ¯ is about doubled and then the semicircle with the radius A S ¯ around the point S is drawn, now the semicircle is drawn with the radius A B ¯ around the point B. The arising intersection point E corresponds 2 φ, next up, the perpendicular on the line segment A E ¯ from the point D will be establish. The subsequent parallel F S ¯ to the line segment C M ¯, produces, as it were and it is well recognizable, this triangle and the triangle M S C are similar to each other. The hypotenuse F S ¯ has due to the cathetuses S D ¯ =1 and D F ¯ =2 according the Pythagorean theorem, finally, the circle arc is drawn with the radius 5 around the point F. The golden ratio has been claimed to have held a fascination for at least 2,400 years. But the fascination with the Golden Ratio is not confined just to mathematicians, biologists, artists, musicians, historians, architects, psychologists, and even mystics have pondered and debated the basis of its ubiquity and appeal. In fact, it is fair to say that the Golden Ratio has inspired thinkers of all disciplines like no other number in the history of mathematics. Ancient Greek mathematicians first studied what we now call the golden ratio because of its frequent appearance in geometry, the division of a line into extreme and mean ratio is important in the geometry of regular pentagrams and pentagons. Euclid explains a construction for cutting a line in extreme and mean ratio, throughout the Elements, several propositions and their proofs employ the golden ratio. The golden ratio is explored in Luca Paciolis book De divina proportione, since the 20th century, the golden ratio has been represented by the Greek letter φ or less commonly by τ. Timeline according to Priya Hemenway, Phidias made the Parthenon statues that seem to embody the golden ratio, plato, in his Timaeus, describes five possible regular solids, some of which are related to the golden ratio

3.
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

4.
Angle
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In planar geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. Angles formed by two rays lie in a plane, but this plane does not have to be a Euclidean plane, Angles are also formed by the intersection of two planes in Euclidean and other spaces. Angles formed by the intersection of two curves in a plane are defined as the angle determined by the tangent rays at the point of intersection. Similar statements hold in space, for example, the angle formed by two great circles on a sphere is the dihedral angle between the planes determined by the great circles. Angle is also used to designate the measure of an angle or of a rotation and this measure is the ratio of the length of a circular arc to its radius. In the case of an angle, the arc is centered at the vertex. In the case of a rotation, the arc is centered at the center of the rotation and delimited by any other point and its image by the rotation. The word angle comes from the Latin word angulus, meaning corner, cognate words are the Greek ἀγκύλος, meaning crooked, curved, both are connected with the Proto-Indo-European root *ank-, meaning to bend or bow. Euclid defines a plane angle as the inclination to each other, in a plane, according to Proclus an angle must be either a quality or a quantity, or a relationship. In mathematical expressions, it is common to use Greek letters to serve as variables standing for the size of some angle, lower case Roman letters are also used, as are upper case Roman letters in the context of polygons. See the figures in this article for examples, in geometric figures, angles may also be identified by the labels attached to the three points that define them. For example, the angle at vertex A enclosed by the rays AB, sometimes, where there is no risk of confusion, the angle may be referred to simply by its vertex. However, in geometrical situations it is obvious from context that the positive angle less than or equal to 180 degrees is meant. Otherwise, a convention may be adopted so that ∠BAC always refers to the angle from B to C. Angles smaller than an angle are called acute angles. An angle equal to 1/4 turn is called a right angle, two lines that form a right angle are said to be normal, orthogonal, or perpendicular. Angles larger than an angle and smaller than a straight angle are called obtuse angles. An angle equal to 1/2 turn is called a straight angle, Angles larger than a straight angle but less than 1 turn are called reflex angles

5.
Arc (geometry)
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In Euclidean geometry, an arc is a closed segment of a differentiable curve. A common example in the plane, is a segment of a circle called a circular arc, in space, if the arc is part of a great circle, it is called a great arc. Every pair of points on a circle determines two arcs. The length, L, of an arc of a circle with radius r and this is because L c i r c u m f e r e n c e = θ2 π. Substituting in the circumference L2 π r = θ2 π, and, with α being the angle measured in degrees, since θ = α/180π. For example, if the measure of the angle is 60 degrees and this is so because the circumference of a circle and the degrees of a circle, of which there are always 360, are directly proportional. The area of the sector formed by an arc and the center of a circle is A =12 r 2 θ. The area A has the proportion to the circle area as the angle θ to a full circle. We can cancel π on both sides, A r 2 = θ2, by multiplying both sides by r2, we get the final result, A =12 r 2 θ. Using the conversion described above, we find that the area of the sector for an angle measured in degrees is A = α360 π r 2. The area of the bounded by the arc and the straight line between its two end points is 12 r 2. To get the area of the arc segment, we need to subtract the area of the triangle, determined by the circles center and the two end points of the arc, from the area A. Using the intersecting chords theorem it is possible to calculate the radius r of a circle given the height H and its perpendicular bisector is another chord, which is a diameter of the circle. The length of the first chord is W, and it is divided by the bisector into two halves, each with length W/2. The total length of the diameter is 2r, and it is divided into two parts by the first chord, the length of one part is the sagitta of the arc, H, and the other part is the remainder of the diameter, with length 2r − H. Applying the intersecting chords theorem to these two chords produces H =2, whence 2 r − H = W24 H, so r = W28 H + H2

6.
Circle
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A circle is a simple closed shape in Euclidean geometry. The distance between any of the points and the centre is called the radius, a circle is a simple closed curve which divides the plane into two regions, an interior and an exterior. Annulus, the object, the region bounded by two concentric circles. Arc, any connected part of the circle, centre, the point equidistant from the points on the circle. Chord, a segment whose endpoints lie on the circle. Circumference, the length of one circuit along the circle, or the distance around the circle and it is a special case of a chord, namely the longest chord, and it is twice the radius. Disc, the region of the bounded by a circle. Lens, the intersection of two discs, passant, a coplanar straight line that does not touch the circle. Radius, a line segment joining the centre of the circle to any point on the circle itself, or the length of such a segment, sector, a region bounded by two radii and an arc lying between the radii. Segment, a region, not containing the centre, bounded by a chord, secant, an extended chord, a coplanar straight line cutting the circle at two points. Semicircle, an arc that extends from one of a diameters endpoints to the other, in non-technical common usage it may mean the diameter, arc, and its interior, a two dimensional region, that is technically called a half-disc. A half-disc is a case of a segment, namely the largest one. Tangent, a straight line that touches the circle at a single point. The word circle derives from the Greek κίρκος/κύκλος, itself a metathesis of the Homeric Greek κρίκος, the origins of the words circus and circuit are closely related. The circle has been known since before the beginning of recorded history, natural circles would have been observed, such as the Moon, Sun, and a short plant stalk blowing in the wind on sand, which forms a circle shape in the sand. The circle is the basis for the wheel, which, with related inventions such as gears, in mathematics, the study of the circle has helped inspire the development of geometry, astronomy and calculus. Some highlights in the history of the circle are,1700 BCE – The Rhind papyrus gives a method to find the area of a circular field. The result corresponds to 256/81 as a value of π.300 BCE – Book 3 of Euclids Elements deals with the properties of circles

7.
On-Line Encyclopedia of Integer Sequences
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The On-Line Encyclopedia of Integer Sequences, also cited simply as Sloanes, is an online database of integer sequences. It was created and maintained by Neil Sloane while a researcher at AT&T Labs, Sloane continues to be involved in the OEIS in his role as President of the OEIS Foundation. OEIS records information on integer sequences of interest to professional mathematicians and amateurs, and is widely cited. As of 30 December 2016 it contains nearly 280,000 sequences, the database is searchable by keyword and by subsequence. Neil Sloane started collecting integer sequences as a student in 1965 to support his work in combinatorics. The database was at first stored on punched cards and he published selections from the database in book form twice, A Handbook of Integer Sequences, containing 2,372 sequences in lexicographic order and assigned numbers from 1 to 2372. The Encyclopedia of Integer Sequences with Simon Plouffe, containing 5,488 sequences and these books were well received and, especially after the second publication, mathematicians supplied Sloane with a steady flow of new sequences. The collection became unmanageable in book form, and when the database had reached 16,000 entries Sloane decided to go online—first as an e-mail service, as a spin-off from the database work, Sloane founded the Journal of Integer Sequences in 1998. The database continues to grow at a rate of some 10,000 entries a year, Sloane has personally managed his sequences for almost 40 years, but starting in 2002, a board of associate editors and volunteers has helped maintain the database. In 2004, Sloane celebrated the addition of the 100, 000th sequence to the database, A100000, in 2006, the user interface was overhauled and more advanced search capabilities were added. In 2010 an OEIS wiki at OEIS. org was created to simplify the collaboration of the OEIS editors and contributors, besides integer sequences, the OEIS also catalogs sequences of fractions, the digits of transcendental numbers, complex numbers and so on by transforming them into integer sequences. Sequences of rationals are represented by two sequences, the sequence of numerators and the sequence of denominators, important irrational numbers such as π =3.1415926535897. are catalogued under representative integer sequences such as decimal expansions, binary expansions, or continued fraction expansions. The OEIS was limited to plain ASCII text until 2011, yet it still uses a form of conventional mathematical notation. Greek letters are represented by their full names, e. g. mu for μ. Every sequence is identified by the letter A followed by six digits, sometimes referred to without the leading zeros, individual terms of sequences are separated by commas. Digit groups are not separated by commas, periods, or spaces, a represents the nth term of the sequence. Zero is often used to represent non-existent sequence elements, for example, A104157 enumerates the smallest prime of n² consecutive primes to form an n×n magic square of least magic constant, or 0 if no such magic square exists. The value of a is 2, a is 1480028129, but there is no such 2×2 magic square, so a is 0

8.
Radian
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The radian is the standard unit of angular measure, used in many areas of mathematics. The length of an arc of a circle is numerically equal to the measurement in radians of the angle that it subtends. The unit was formerly an SI supplementary unit, but this category was abolished in 1995, separately, the SI unit of solid angle measurement is the steradian. The radian is represented by the symbol rad, so for example, a value of 1.2 radians could be written as 1.2 rad,1.2 r,1. 2rad, or 1. 2c. Radian describes the angle subtended by a circular arc as the length of the arc divided by the radius of the arc. One radian is the angle subtended at the center of a circle by an arc that is equal in length to the radius of the circle. Conversely, the length of the arc is equal to the radius multiplied by the magnitude of the angle in radians. As the ratio of two lengths, the radian is a number that needs no unit symbol, and in mathematical writing the symbol rad is almost always omitted. When quantifying an angle in the absence of any symbol, radians are assumed, and it follows that the magnitude in radians of one complete revolution is the length of the entire circumference divided by the radius, or 2πr / r, or 2π. Thus 2π radians is equal to 360 degrees, meaning that one radian is equal to 180/π degrees, the concept of radian measure, as opposed to the degree of an angle, is normally credited to Roger Cotes in 1714. He described the radian in everything but name, and he recognized its naturalness as a unit of angular measure, the idea of measuring angles by the length of the arc was already in use by other mathematicians. For example, al-Kashi used so-called diameter parts as units where one part was 1/60 radian. The term radian first appeared in print on 5 June 1873, in examination questions set by James Thomson at Queens College, Belfast. He had used the term as early as 1871, while in 1869, Thomas Muir, then of the University of St Andrews, in 1874, after a consultation with James Thomson, Muir adopted radian. As stated, one radian is equal to 180/π degrees, thus, to convert from radians to degrees, multiply by 180/π. The length of circumference of a circle is given by 2 π r, so, to convert from radians to gradians multiply by 200 / π, and to convert from gradians to radians multiply by π /200. This is because radians have a mathematical naturalness that leads to a more elegant formulation of a number of important results, most notably, results in analysis involving trigonometric functions are simple and elegant when the functions arguments are expressed in radians. Because of these and other properties, the trigonometric functions appear in solutions to problems that are not obviously related to the functions geometrical meanings

9.
Degree (angle)
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A degree, usually denoted by °, is a measurement of a plane angle, defined so that a full rotation is 360 degrees. It is not an SI unit, as the SI unit of measure is the radian. Because a full rotation equals 2π radians, one degree is equivalent to π/180 radians, the original motivation for choosing the degree as a unit of rotations and angles is unknown. One theory states that it is related to the fact that 360 is approximately the number of days in a year. Ancient astronomers noticed that the sun, which follows through the path over the course of the year. Some ancient calendars, such as the Persian calendar, used 360 days for a year, the use of a calendar with 360 days may be related to the use of sexagesimal numbers. The earliest trigonometry, used by the Babylonian astronomers and their Greek successors, was based on chords of a circle, a chord of length equal to the radius made a natural base quantity. One sixtieth of this, using their standard sexagesimal divisions, was a degree, Aristarchus of Samos and Hipparchus seem to have been among the first Greek scientists to exploit Babylonian astronomical knowledge and techniques systematically. Timocharis, Aristarchus, Aristillus, Archimedes, and Hipparchus were the first Greeks known to divide the circle in 360 degrees of 60 arc minutes, eratosthenes used a simpler sexagesimal system dividing a circle into 60 parts. Furthermore, it is divisible by every number from 1 to 10 except 7 and this property has many useful applications, such as dividing the world into 24 time zones, each of which is nominally 15° of longitude, to correlate with the established 24-hour day convention. Finally, it may be the case more than one of these factors has come into play. For many practical purposes, a degree is a small enough angle that whole degrees provide sufficient precision. When this is not the case, as in astronomy or for geographic coordinates, degree measurements may be written using decimal degrees, with the symbol behind the decimals. Alternatively, the sexagesimal unit subdivisions can be used. One degree is divided into 60 minutes, and one minute into 60 seconds, use of degrees-minutes-seconds is also called DMS notation. These subdivisions, also called the arcminute and arcsecond, are represented by a single and double prime. For example,40. 1875° = 40° 11′ 15″, or, using quotation mark characters, additional precision can be provided using decimals for the arcseconds component. The older system of thirds, fourths, etc. which continues the sexagesimal unit subdivision, was used by al-Kashi and other ancient astronomers, but is rarely used today

10.
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

11.
Floret
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This glossary is incomplete, you can help by expanding it, you can also help by adding illustrations that assist an understanding of the terms. Ab- A prefix meaning from, away from, or outside of, abaxial The surface of an organ facing away from the axis, e. g. the lower surface of a lateral organ such as a leaf or petal. Abort To abandon development of a structure or organ, abscission shedding of an organ that is mature or aged, e. g. a ripe fruit or an old leaf. Abscission zone a specialised layer of tissue formed, for example, acaulescent An adjective descriptive of a plant that has no apparent stem, or at least none visible above ground. Examples include some species of Agave, Oxalis and Attalea, accrescent Increasing in size with age, such as a calyx that continues to grow after the corolla has fallen, for example in Physalis peruviana. -aceae The suffix added to the stem of a name to form the name of a family. Achene A dry 1-seeded indehiscent fruit, e. g. in the genus Ranunculus, acropetal Moving from roots to leaves, e. g. of molecular signals in plants. Acrophyll The regular leaves of a plant, produced above the base. Acrostichoid covering the entire surface of the frond, usually densely so. Actino- A prefix that indicates a radial form, actinodromous palmate or radially arranged venation with three or more primary veins arising from at or near the base of the leaf, the primary veins reaching the margin or not. Actinomorphic regular, radially symmetrical, may be bisected into similar halves in at least two planes. Applies e. g. to steles and flowers in which the segments within each whorl are alike in size and shape, compare regular, contrast with asymmetrical, irregular. Aculeate Armed with prickles, e. g. the stem of a rose, acuminate Tapering gradually to a point. Acute Sharply pointed, converging edges making an angle of less than 90°, ad- A prefix meaning near or towards. Adaxial The surface of an organ facing towards the axis, e. g. the upper surface of an organ such as a leaf or petal. Adnate grown or fused to an organ of a different kind, especially along a margin, e. g. a stamen fused to a petal, cf. connate. Adventitious A structure produced in a position, e. g. an adventitious bud produced from a stem rather than from the axil of a leaf. Aerial Of the air, growing or borne above the surface of the ground, aestivation The arrangement of sepals and petals or their lobes in an unexpanded flower bud, cf. vernation, the arrangement of leaves in a bud. aff

12.
Sunflower
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Helianthus or sunflowers L. /ˌhiːliˈænθəs/ is a genus of plants comprising about 70 species in the family Asteraceae. The genus is one of many in the Asteraceae that are known as sunflowers, except for three species in South America, all Helianthus species are native to North America. The common name, sunflower, typically refers to the annual species Helianthus annuus, or the common sunflower. This and other species, notably Jerusalem artichoke, are cultivated in regions as food crops. The largest sunflower field is located in Tuscany, Italy, the domesticated sunflower, H. annuus, is the most familiar species. Perennial sunflower species are not as popular for gardens due to their tendency to spread rapidly and become invasive. Whorled sunflowers, H. verticillatus, were listed as a species in 2014 when the U. S. Fish. The primary threats are industrial forestry and pine plantations in Alabama, Georgia and they grow to 1.8 m and are primarily found in woodlands, adjacent to creeks and moist, prairie-like areas. Sunflowers are usually annual or perennial plants that grow to a height of 300 centimetres or more. They bear one or more wide, terminal capitula, with yellow ray florets at the outside. Several ornamental cultivars of Helianthus annuus have red-colored ray florets, all of them stem from a single original mutant, during growth, sunflowers tilt during the day to face the sun, but stop once they begin blooming. This tracking of the sun in young sunflower heads is called heliotropism, by the time they are mature, sunflowers generally face east. The rough and hairy stem is branched in the part in wild plants but is usually unbranched in domesticated cultivars. The petiolate leaves are dentate and often sticky, the lower leaves are opposite, ovate or often heart-shaped. They are distinguished technically by the fact that the ray florets are sterile, some species also have additional shorter scales in the pappus, and there is one species that lacks a pappus entirely. Another technical feature that distinguishes the genus more reliably, but requires a microscope to see, is the presence of a prominent, sunflowers are especially well known for their symmetry based on Fibonacci numbers and the Golden angle. There is quite a bit of variability among the species that make up the bulk of the species in the genus. Some have most or all of the leaves in a rosette at the base of the plant

13.
Aristid Lindenmayer
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Aristid Lindenmayer was a Hungarian biologist. In 1968 he developed a type of formal languages that is today called L-systems or Lindenmayer Systems, using those systems Lindenmayer modelled the behaviour of cells of plants. L-systems nowadays are used to model whole plants. Lindenmayer worked with yeast and filamentous fungi and studied the patterns of various types of algae. Originally the L-systems were devised to provide a description of the development of such simple multicellular organisms. Later on, this system was extended to higher plants. Lindenmayer studied chemistry and biology at the University of Budapest from 1943 to 1948 and he received his Ph. D. in plant physiology in 1956 at the University of Michigan. In 1968 he became professor in Philosophy of Life Sciences and Biology at the University of Utrecht, from 1972 onward he headed the Theoretical Biology Group at Utrecht University. Aristid Lindenmayer, Mathematical models for interaction in development

14.
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

15.
Metallic mean
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The metallic means or silver means of the successive natural numbers are the continued fractions, n +1 n +1 n +1 n +1 n + ⋱ = = n + n 2 +42. The golden ratio is the mean between 1 and 2, while the silver ratio is the metallic mean between 2 and 3. The term bronze ratio, or terms using other names of metals, are used to name subsequent metallic means. The values of the first ten metallic means are shown at right, notice that each metallic mean is a root of the simple quadratic equation, x 2 − n x =1, where n is any positive natural number. As the golden ratio is connected to the pentagon, the ratio is connected to the octagon. As the golden ratio is connected to the Fibonacci numbers, the ratio is connected to the Pell numbers. Taking successive Fibonacci numbers as ratios, these ratios approach the golden mean, the Pell number ratios approach the silver mean, and these properties are valid only for integers m, for nonintegers the properties are similar but slightly different. The above property for the powers of the ratio is a consequence of a property of the powers of silver means. For the silver mean S of m, the property can be generalized as S m n = K n S m + K where K n = m K + K. Using the initial conditions K0 =1 and K1 = m, the powers of silver means have other interesting properties, If n is a positive even integer, S m n − ⌊ S m n ⌋ =1 − S m − n. In general, S m 2 n +1 = S ∑ k =0 n 2 n +12 k +1 m 2 k +1. The silver mean S of m also has the property that 1 S m = S m − m meaning that the inverse of a silver mean has the decimal part as the corresponding silver mean. S m = a + b where a is the part of S and b is the decimal part of S, then the following property is true. Because, the part of Sm = m, a = m. For m >1, we then have S m 2 = m a + m b +1 S m 2 = m +1 S m 2 = m +1. Therefore, the mean of m is a solution of the equation x 2 − m x −1 =0. It may also be useful to note that the silver mean S of −m is the inverse of the silver mean S of m 1 S m = S = S m − m, another interesting result can be obtained by slightly changing the formula of the silver mean. The silver mean of m is given by the integral S m = ∫0 m d x

16.
Golden ratio base
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Golden ratio base is a non-integer positional numeral system that uses the golden ratio as its base. It is sometimes referred to as base-φ, golden mean base, phi-base, or, colloquially, any non-negative real number can be represented as a base-φ numeral using only the digits 0 and 1, and avoiding the digit sequence 11 – this is called a standard form. A base-φ numeral that includes the digit sequence 11 can always be rewritten in standard form, despite using an irrational number base, when using standard form, all non-negative integers have a unique representation as a terminating base-φ expansion. Other numbers have standard representations in base-φ, with rational numbers having recurring representations and these representations are unique, except that numbers with a terminating expansion also have a non-terminating expansion, as they do in base-10, for example,1 =0. 99999…. In the following example the notation 1 is used to represent −1. 211. 01φ is not a standard base-φ numeral, since it contains a 11 and a 2, which isnt a 0 or 1, and contains a 1 = −1, which isnt a 0 or 1 either. To standardize a numeral, we can use the following substitutions, 011φ = 100φ, 0200φ = 1001φ, 010φ = 101φ and we can apply the substitutions in any order we like, as the result is the same. Below, the applied to the number on the previous line are on the right. Any positive number with a non-standard terminating base-φ representation can be standardized in this manner. If we get to a point where all digits are 0 or 1, except for the first digit being negative and this can be converted to the negative of a base-φ representation by negating every digit, standardizing the result, and then marking it as negative. For example, use a sign, or some other significance to denote negative numbers. If the arithmetic is being performed on a computer, a message may be returned. We can either consider our integer to be the digit of a nonstandard base-φ numeral, therefore, we can compute + =, − = and × =. So, using integer values only, we can add, subtract and multiply numbers of the form, > if and only if 2 − > × √5. If one side is negative, the positive, the comparison is trivial. Otherwise, square sides, to get an integer comparison, reversing the comparison direction if both sides were negative. On squaring both sides, the √5 is replaced with the integer 5, so, using integer values only, we can also compare numbers of the form. To convert an integer x to a number, note that x =

17.
Fibonacci number
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The Fibonacci sequence is named after Italian mathematician Leonardo of Pisa, known as Fibonacci. His 1202 book Liber Abaci introduced the sequence to Western European mathematics, the sequence described in Liber Abaci began with F1 =1. Fibonacci numbers are related to Lucas numbers L n in that they form a complementary pair of Lucas sequences U n = F n and V n = L n. They are intimately connected with the ratio, for example. Fibonacci numbers appear unexpectedly often in mathematics, so much so that there is a journal dedicated to their study. The Fibonacci sequence appears in Indian mathematics, in connection with Sanskrit prosody, in the Sanskrit tradition of prosody, there was interest in enumerating all patterns of long syllables that are 2 units of duration, and short syllables that are 1 unit of duration. Counting the different patterns of L and S of a given duration results in the Fibonacci numbers, susantha Goonatilake writes that the development of the Fibonacci sequence is attributed in part to Pingala, later being associated with Virahanka, Gopāla, and Hemachandra. He dates Pingala before 450 BC, however, the clearest exposition of the sequence arises in the work of Virahanka, whose own work is lost, but is available in a quotation by Gopala, Variations of two earlier meters. For example, for four, variations of meters of two three being mixed, five happens, in this way, the process should be followed in all mātrā-vṛttas. The sequence is also discussed by Gopala and by the Jain scholar Hemachandra, outside India, the Fibonacci sequence first appears in the book Liber Abaci by Fibonacci. The puzzle that Fibonacci posed was, how many pairs will there be in one year, at the end of the first month, they mate, but there is still only 1 pair. At the end of the month the female produces a new pair. At the end of the month, the original female produces a second pair. At the end of the month, the original female has produced yet another new pair. At the end of the nth month, the number of pairs of rabbits is equal to the number of new pairs plus the number of pairs alive last month and this is the nth Fibonacci number. The name Fibonacci sequence was first used by the 19th-century number theorist Édouard Lucas, the most common such problem is that of counting the number of compositions of 1s and 2s that sum to a given total n, there are Fn+1 ways to do this. For example, if n =5, then Fn+1 = F6 =8 counts the eight compositions, 1+1+1+1+1 = 1+1+1+2 = 1+1+2+1 = 1+2+1+1 = 2+1+1+1 = 2+2+1 = 2+1+2 = 1+2+2, all of which sum to 5. The Fibonacci numbers can be found in different ways among the set of strings, or equivalently

18.
Kepler triangle
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A Kepler triangle is a right triangle with edge lengths in geometric progression. The ratio of the edges of a Kepler triangle is linked to the golden ratio φ =1 +52 and can be written,1, φ, φ, the squares of the edges of this triangle are in geometric progression according to the golden ratio. The first we may compare to a mass of gold, the second we may call a precious jewel, some sources claim that a triangle with dimensions closely approximating a Kepler triangle can be recognized in the Great Pyramid of Giza. For positive real numbers a and b, their arithmetic mean, geometric mean, take any Kepler triangle with sides a, a φ, a φ, and consider, the circle that circumscribes it, and a square with side equal to the middle-sized edge of the triangle. Then the perimeters of the square and the circle coincide up to a less than 0. 1%. This is the mathematical coincidence π ≈4 / φ, the square and the circle cannot have exactly the same perimeter, because in that case one would be able to solve the classical problem of the quadrature of the circle. In other words, π ≠4 / φ because π is a transcendental number, according to some sources, Kepler triangles appear in the design of Egyptian pyramids. However, the ancient Egyptians probably did not know the mathematical coincidence involving the number π and the golden ratio φ

19.
Golden rectangle
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In geometry, a golden rectangle is a rectangle whose side lengths are in the golden ratio,1,1 +52, which is 1, φ, where φ is approximately 1.618. A golden rectangle can be constructed with straightedge and compass by four simple steps. Draw a line from the midpoint of one side of the square to an opposite corner, use that line as the radius to draw an arc that defines the height of the rectangle. A distinctive feature of this shape is that when a section is removed, the remainder is another golden rectangle. An alternative construction of the golden rectangle uses three polygons circumscribed by congruent circles, a decagon, hexagon, and pentagon. The respective lengths a, b, and c of the sides of these three polygons satisfy the equation a2 + b2 = c2, so line segments with these lengths form a right triangle. The ratio of the length of the hexagon to the decagon is the golden ratio. The convex hull of two edges of a regular icosahedron forms a golden rectangle. The twelve vertices of the icosahedron can be decomposed in this way into three mutually-perpendicular golden rectangles, whose boundaries are linked in the pattern of the Borromean rings, the proportions of the golden rectangle have been observed in works predating Paciolis publication. Le Corbusiers 1927 Villa Stein in Garches features a ground plan, elevation

20.
Golden rhombus
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In geometry, a golden rhombus is a rhombus whose diagonals are in the ratio p q = φ, where φ is the golden ratio. They include the two golden rhombohedra, the Bilinski dodecahedron, the icosahedron, the rhombic triacontahedron. The first five of these are the only convex polyhedra with golden rhomb faces, Golden rectangle Golden triangle M. Livio, The Golden Ratio, The Story of Phi, the Worlds Most Astonishing Number, New York, Broadway Books, p.206,2002

21.
Golden-section search
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The golden-section search is a technique for finding the extremum of a strictly unimodal function by successively narrowing the range of values inside which the extremum is known to exist. The technique derives its name from the fact that the algorithm maintains the function values for triples of points whose distances form a golden ratio, the algorithm is the limit of Fibonacci search for a large number of function evaluations. Fibonacci search and golden-section search were discovered by Kiefer, the discussion here is posed in terms of searching for a minimum of a unimodal function. Unlike finding a zero, where two function evaluations with opposite sign are sufficient to bracket a root, when searching for a minimum, the golden-section search is an efficient way to reduce progressively the interval locating the minimum. The diagram above illustrates a single step in the technique for finding a minimum, the functional values of f are on the vertical axis, and the horizontal axis is the x parameter. The value of f has already been evaluated at the three points, x 1, x 2, and x 3. Since f 2 is smaller than either f 1 or f 3, the next step in the minimization process is to probe the function by evaluating it at a new value of x, namely x 4. It is most efficient to choose x 4 somewhere inside the largest interval, i. e. between x 2 and x 3. From the diagram, it is clear if the function yields f 4 a, then a minimum lies between x 1 and x 4, and the new triplet of points will be x 1, x 2. However, if the function yields the value f 4 b, then a minimum lies between x 2 and x 3, and the new triplet of points will be x 2, x 4, and x 3. Thus, in case, we can construct a new narrower search interval that is guaranteed to contain the functions minimum. From the diagram above, it is seen that the new search interval will be either between x 1 and x 4 with a length of a + c, or between x 2 and x 3 with a length of b. The golden-section search requires that these intervals be equal, if they are not, a run of bad luck could lead to the wider interval being used many times, thus slowing down the rate of convergence. To ensure that b = a + c, the algorithm should choose x 4 = x 1 +, however, there still remains the question of where x 2 should be placed in relation to x 1 and x 3. The golden-section search chooses the spacing between points in such a way that these points have the same proportion of spacing as the subsequent triple x 1, x 2, x 4 or x 2, x 4, x 3. However, if f is f 4 b and our new triplet of points is x 2, x 4, because smooth functions are flat near a minimum, attention must be paid not to expect too great an accuracy in locating the minimum. The check is based on the size relative to its central value. For that same reason, the Numerical Recipes text recommends that τ = ε, let be interval of current bracket

22.
Golden spiral
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In geometry, a golden spiral is a logarithmic spiral whose growth factor is φ, the golden ratio. That is, a golden spiral gets wider by a factor of φ for every turn it makes.0053468 for θ in degrees. A212224 if θ is measured in radians, there are several similar spirals that approximate, but do not exactly equal, a golden spiral. These are often confused with the golden spiral, for example, a golden spiral can be approximated by first starting with a rectangle for which the ratio between its length and width is the golden ratio. This rectangle can then be partitioned into a square and a similar rectangle, after continuing this process for an arbitrary amount of steps, the result will be an almost complete partitioning of the rectangle into squares. The corners of squares can be connected by quarter-circles. The result, though not a true logarithmic spiral, approximates a golden spiral, another approximation is a Fibonacci spiral, which is constructed slightly differently. A Fibonacci spiral starts with a rectangle partitioned into 2 squares, in each step, a square the length of the rectangles longest side is added to the rectangle. e. The cross ratio has the singular value -1, the golden spiral is the only logarithmic spiral with =. Approximate logarithmic spirals can occur in nature, golden spirals are one special case of these logarithmic spirals, a recent analysis of spirals observed in mouse corneal epithelial cells indicated that some can be characterized by the golden spiral, and some by other spirals. It is sometimes stated that spiral galaxies and nautilus shells get wider in the pattern of a golden spiral, in truth, spiral galaxies and nautilus shells exhibit logarithmic spiral growth, but at a variety of angles usually distinctly different from that of the golden spiral. This pattern allows the organism to grow without changing shape, Golden angle Golden ratio Golden rectangle Logarithmic spiral Fibonacci number

23.
Golden triangle (mathematics)
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A golden triangle, also known as the sublime triangle, is an isosceles triangle in which the duplicated side is in the golden ratio to the distinct side, a b = φ =1 +52. Golden triangles are found in the nets of several stellations of dodecahedrons and icosahedrons, also, it is the shape of the triangles found in the points of pentagrams. The vertex angle is equal to θ = cos −1 = π5 =36 ∘, since the angles of a triangle sum to 180°, base angles are therefore 72° each. The golden triangle can also be found in a decagon, or a ten-sided polygon and this will form a golden triangle. This is because, 180/10=144 degrees is the angle and bisecting it through the vertex to the center. The golden triangle is uniquely identified as the only triangle to have its three angles in 2,2,1 proportions. The golden triangle is used to form a logarithmic spiral, by bisecting the base angles, a new point is created that in turn, makes another golden triangle. The bisection process can be continued infinitely, creating a number of golden triangles. A logarithmic spiral can be drawn through the vertices and this spiral is also known as an equiangular spiral, a term coined by René Descartes. If a straight line is drawn from the pole to any point on the curve, it cuts the curve at precisely the same angle, the golden gnomon is also uniquely identified as a triangle having its three angles in 1,1,3 proportion. The acute angle is 36 degrees, which is the same as the apex of the golden triangle, the distance of AX and BX are both equal to φ, as seen in the figure. The golden triangle has a ratio of length to side length equal to the golden section φ. A golden triangle can be bisected into a triangle and a golden gnomon. The same is true for a golden gnomon, a golden gnomon and a golden triangle with their equal sides matching each other in length, are also referred to as the obtuse and acute Robinson triangles. These isosceles triangles can be used to produce Penrose tilings, Penrose tiles are made from kites and darts. A kite is made from two golden triangles, and a dart is made from two gnomons, Golden rectangle Golden rhombus Kepler triangle Lute of Pythagoras Pentagram Weisstein, Eric W. Golden triangle. Robinson triangles at Tilings Encyclopedia Golden triangle according to Euclid The extraordinary reciprocity of golden triangles at Tartapelago by Giorgio Pietrocola

24.
Silver ratio
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This defines the silver ratio as an irrational mathematical constant, whose value of one plus the square root of 2 is approximately 2.4142135623. The silver ratio is denoted by δS and these fractions provide accurate rational approximations of the silver ratio, analogous to the approximation of the golden ratio by ratios of consecutive Fibonacci numbers. Multiplying by δS and rearranging gives δ S2 −2 δ S −1 =0, using the quadratic formula, two solutions can be obtained. In fact it is the second smallest quadratic PV number after the golden ratio and this means the distance from δ n S to the nearest integer is 1/δ n S ≈0. 41n. Thus, the sequence of parts of δ n S, n =1,2,3. In particular, this sequence is not equidistributed mod 1, the paper sizes under ISO216 are rectangles in the proportion 1, √2, sometimes called A4 rectangles. Removing a largest possible square from a sheet of such paper leaves a rectangle with proportions 1, √2 −1 which is the same as 1 + √2,1, removing a largest square from one of these sheets leaves one again with aspect ratio 1, √2. A rectangle whose aspect ratio is the ratio is sometimes called a silver rectangle by analogy with golden rectangles. Confusingly, silver rectangle can also refer to the paper sizes specified by ISO216, however, only the 1, √2 rectangles have the property that by cutting the rectangle in half across its long side produces two smaller rectangles of the same aspect ratio. The silver rectangle is connected to the regular octagon, if the edge length of a regular octagon is t, then the inradius of the octagon is δSt, and the area of the octagon is 2δSt2. Metallic means Ammann–Beenker tiling Buitrago, Antonia Redondo, polygons, Diagonals, and the Bronze Mean, Nexus Network Journal 9,2, Architecture and Mathematics, p. 321-2. An Introduction to Continued Fractions, The Silver Means, Fibonacci Numbers, Silver rectangle and its sequence at Tartapelago by Giorgio Pietrocola

25.
Pell number
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In mathematics, the Pell numbers are an infinite sequence of integers, known since ancient times, that comprise the denominators of the closest rational approximations to the square root of 2. This sequence of approximations begins 1/1, 3/2, 7/5, 17/12, and 41/29, so the sequence of Pell numbers begins with 1,2,5,12, and 29. The numerators of the sequence of approximations are half the companion Pell numbers or Pell–Lucas numbers, these numbers form a second infinite sequence that begins with 2,6,14,34. As with Pells equation, the name of the Pell numbers stems from Leonhard Eulers mistaken attribution of the equation, the Pell–Lucas numbers are also named after Édouard Lucas, who studied sequences defined by recurrences of this type, the Pell and companion Pell numbers are Lucas sequences. The Pell numbers are defined by the recurrence relation P n = {0 if n =0,1 if n =1,2 P n −1 + P n −2 otherwise. In words, the sequence of Pell numbers starts with 0 and 1, and then each Pell number is the sum of twice the previous Pell number and the Pell number before that. The first few terms of the sequence are 0,1,2,5,12,29,70,169,408,985,2378,5741,13860, …. The Pell numbers can also be expressed by the closed form formula P n = n − n 22, a third definition is possible, from the matrix formula = n. Pell numbers arise historically and most notably in the rational approximation to √2. If two large integers x and y form a solution to the Pell equation x 2 −2 y 2 = ±1 and that is, the solutions have the form P n −1 + P n P n. The approximation 2 ≈577408 of this type was known to Indian mathematicians in the third or fourth century B. C, the Greek mathematicians of the fifth century B. C. also knew of this sequence of approximations, Plato refers to the numerators as rational diameters. In the 2nd century CE Theon of Smyrna used the term the side and these approximations can be derived from the continued fraction expansion of 2,2 =1 +12 +12 +12 +12 +12 + ⋱. As Knuth describes, the fact that Pell numbers approximate √2 allows them to be used for accurate rational approximations to an octagon with vertex coordinates. All vertices are equally distant from the origin, and form uniform angles around the origin. Alternatively, the points, and form approximate octagons in which the vertices are equally distant from the origin. A Pell prime is a Pell number that is prime, the first few Pell primes are 2,5,29,5741, …. The indices of these primes within the sequence of all Pell numbers are 2,3,5,11,13,29,41,53,59,89,97,101,167,181,191, … These indices are all themselves prime. As with the Fibonacci numbers, a Pell number Pn can only be prime if n itself is prime, the only Pell numbers that are squares, cubes, or any higher power of an integer are 0,1, and 169 =132. However, despite having so few squares or other powers, Pell numbers have a connection to square triangular numbers