Barnsley fern
The Barnsley fern is a fractal named after the British mathematician Michael Barnsley who first described it in his book Fractals Everywhere. He made it to resemble Asplenium adiantum-nigrum; the fern is one of the basic examples of self-similar sets, i.e. it is a mathematically generated pattern that can be reproducible at any magnification or reduction. Like the Sierpinski triangle, the Barnsley fern shows how graphically beautiful structures can be built from repetitive uses of mathematical formulas with computers. Barnsley's 1988 book Fractals Everywhere is based on the course which he taught for undergraduate and graduate students in the School of Mathematics, Georgia Institute of Technology, called Fractal Geometry. After publishing the book, a second course was called Fractal Measure Theory. Barnsley's work has been a source of inspiration to graphic artists attempting to imitate nature with mathematical models; the fern code developed by Barnsley is an example of an iterated function system to create a fractal.
This follows from the collage theorem. He has used fractals to model a diverse range of phenomena in science and technology, but most plant structures. IFSs provide models for certain plants and ferns, by virtue of the self-similarity which occurs in branching structures in nature, but nature exhibits randomness and variation from one level to the next. V-variable fractals allow for such randomness and variability across scales, while at the same time admitting a continuous dependence on parameters which facilitates geometrical modelling; these factors allow us to make the hybrid biological models......we speculate that when a V -variable geometrical fractal model is found that has a good match to the geometry of a given plant there is a specific relationship between these code trees and the information stored in the genes of the plant. —Michael Barnsley et al. Barnsley's fern uses four affine transformations; the formula for one transformation is the following: f = + Barnsley shows the IFS code for his Black Spleenwort fern fractal as a matrix of values shown in a table.
In the table, the columns "a" through "f" are the coefficients of the equation, "p" represents the probability factor. These correspond to the following transformations: f 1 = f 2 = + f 3 = + f 4 = + Though Barnsley's fern could in theory be plotted by hand with a pen and graph paper, the number of iterations necessary runs into the tens of thousands, which makes use of a computer mandatory. Many different computer models of Barnsley's fern are popular with contemporary mathematicians; as long as the math is programmed using Barnsley's matrix of constants, the same fern shape will be produced. The first point drawn is at the origin and the new points are iteratively computed by randomly applying one of the following four coordinate transformations:ƒ1 xn + 1 = 0yn + 1 = 0.16 yn. This coordinate transformation is chosen 1% of the time and just maps any point to a point in the first line segment at the base of the stem; this part of the figure is the first to be completed in during the course of iterations.
Ƒ2 xn + 1 = 0.85 xn + 0.04 ynyn + 1 = −0.04 xn + 0.85 yn + 1.6. This coordinate transformation is chosen 85% of the time and maps any point inside the leaflet represent
Koch snowflake
The Koch snowflake is a mathematical curve and one of the earliest fractals to have been described. It is based on the Koch curve, which appeared in a 1904 paper titled "On a Continuous Curve Without Tangents, Constructible from Elementary Geometry" by the Swedish mathematician Helge von Koch; as the fractal evolves, the area of the snowflake converges to 8/5 the area of the original triangle, while the perimeter of the snowflake diverges to infinity. The snowflake has a finite area bounded by an infinitely long line; the Koch snowflake can be constructed by starting with an equilateral triangle recursively altering each line segment as follows: divide the line segment into three segments of equal length. Draw an equilateral triangle that has the middle segment from step 1 as its base and points outward. Remove the line segment, the base of the triangle from step 2; the first iteration of this process produces the outline of a hexagram. The Koch snowflake is the limit approached; the Koch curve described by Helge von Koch is constructed using only one of the three sides of the original triangle.
In other words, three Koch curves make a Koch snowflake. A Koch curve–based representation of a nominally flat surface can be created by segmenting each line in a sawtooth pattern of segments with a given angle; each iteration multiplies the number of sides in the Koch snowflake by four, so the number of sides after n iterations is given by: N n = N n − 1 ⋅ 4 = 3 ⋅ 4 n. If the original equilateral triangle has sides of length s, the length of each side of the snowflake after n iterations is: S n = S n − 1 3 = s 3 n; the perimeter of the snowflake after n iterations is: P n = N n ⋅ S n = 3 ⋅ s ⋅ n. The Koch curve has an infinite length, because the total length of the curve increases by a factor of 4/3 with each iteration; each iteration creates four times as many line segments as in the previous iteration, with the length of each one being 1/3 the length of the segments in the previous stage. Hence, the length of the curve after n iterations will be n times the original triangle perimeter and is unbounded, as n tends to infinity.
As the number of iterations tends to infinity, the limit of the perimeter is: lim n → ∞ P n = lim n → ∞ 3 ⋅ s ⋅ n = ∞, since |4/3| > 1. An ln 4/ln 3-dimensional measure has not been calculated so far. Only upper and lower bounds have been invented. In each iteration a new triangle is added on each side of the previous iteration, so the number of new triangles added in iteration n is: T n = N n − 1 = 3 ⋅ 4 n − 1 = 3 4 ⋅ 4 n; the area of each new triangle added in an iteration is 1/9 of the area of each triangle added in the previous iteration, so the area of each triangle added in iteration n is: a n = a n − 1 9 = a 0 9 n. where a0 is the area of the original triangle. The total new area added in iteration n is therefore: b n = T n ⋅ a n = 3 4 ⋅ n ⋅ a 0 The total area of the snowflake after n iterations is: A n = a 0 + ∑ k = 1 n b k = a 0 = a 0 ( 1 + 1 3 ∑ k = 0 n − 1
Cantor set
In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of remarkable and deep properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883. Through consideration of this set and others helped lay the foundations of modern point-set topology. Although Cantor himself defined the set in a general, abstract way, the most common modern construction is the Cantor ternary set, built by removing the middle third of a line segment and repeating the process with the remaining shorter segments. Cantor himself mentioned the ternary construction only in passing, as an example of a more general idea, that of a perfect set, nowhere dense; the Cantor ternary set C is created by iteratively deleting the open middle third from a set of line segments. One starts by deleting the open middle third from the interval, leaving two line segments: ∪. Next, the open middle third of each of these remaining segments is deleted, leaving four line segments: ∪ ∪ ∪.
This process is continued ad infinitum, where the nth set is C n = C n − 1 3 ∪ for n ≥ 1, C 0 =. The Cantor ternary set contains all points in the interval that are not deleted at any step in this infinite process: C:= ⋂ n = 1 ∞ C n; the first six steps of this process are illustrated below. Using the idea of self-similar transformations, T L = x / 3, T R = / 3 and C n = T L ∪ T R, the explicit closed formulas for the Cantor set are C = ∖ ⋃ n = 1 ∞ ⋃ k = 0 3 n − 1, where every middle third is removed as the open interval from the closed interval = surrounding it, or C = ⋂ n = 1 ∞ ⋃ k = 0 3 n − 1 − 1 ( ∪ [ 3 k + 2
Fractal
In mathematics, a fractal is a subset of a Euclidean space for which the Hausdorff dimension exceeds the topological dimension. Fractals tend 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 small scales called self similarity known as expanding symmetry or unfolding symmetry. One way that fractals are different from finite geometric figures is the way. Doubling the edge lengths of a polygon multiplies its area by four, two raised to the power of two. If the radius of a sphere is doubled, its volume scales by eight, two to the power of three. However, if a fractal's one-dimensional lengths are all doubled, the spatial content of the fractal scales by a power, not an integer; this power is called the fractal dimension of the fractal, it exceeds the fractal's topological dimension. Analytically, fractals are nowhere differentiable. An infinite fractal curve can be conceived of as winding through space differently from an ordinary line – although it is still 1-dimensional, its fractal dimension indicates that it resembles a surface.
Starting in the 17th century with notions of recursion, fractals have moved through rigorous mathematical treatment of the concept to the study of continuous but not differentiable functions in the 19th century by the seminal work of Bernard Bolzano, Bernhard Riemann, Karl Weierstrass, on to the coining of the word fractal in the 20th century with a subsequent burgeoning of interest in fractals and computer-based modelling in the 20th century. The term "fractal" was first used by mathematician Benoit Mandelbrot in 1975. Mandelbrot based it on the Latin frāctus, meaning "broken" or "fractured", used it to extend the concept of theoretical fractional dimensions to geometric patterns in nature. There is some disagreement among mathematicians about how the concept of a fractal should be formally defined. Mandelbrot himself summarized it as "beautiful, damn hard useful. That's fractals." More formally, in 1982 Mandelbrot stated that "A fractal is by definition a set for which the Hausdorff–Besicovitch dimension exceeds the topological dimension."
Seeing this as too restrictive, he simplified and expanded the definition to: "A fractal is a shape made of parts similar to the whole in some way." Still Mandelbrot settled on this use of the language: "...to use fractal without a pedantic definition, to use fractal dimension as a generic term applicable to all the variants". The consensus is that theoretical fractals are infinitely self-similar and detailed mathematical constructs having fractal dimensions, of which many examples have been formulated and studied in great depth. Fractals are not limited to geometric patterns, but can describe processes in time. Fractal patterns with various degrees of self-similarity have been rendered or studied in images and sounds and found in nature, art and law. Fractals are of particular relevance in the field of chaos theory, since the graphs of most chaotic processes are fractals; the word "fractal" has different connotations for laymen as opposed to mathematicians, where the layman is more to be familiar with fractal art than the mathematical concept.
The mathematical concept is difficult to define formally for mathematicians, but key features can be understood with little mathematical background. The feature of "self-similarity", for instance, is understood by analogy to zooming in with a lens or other device that zooms in on digital images to uncover finer invisible, new structure. If this is done on fractals, however, no new detail appears. Self-similarity itself is not counter-intuitive; the difference for fractals is. This idea of being detailed relates to another feature that can be understood without mathematical background: Having a fractal dimension greater than its topological dimension, for instance, refers to how a fractal scales compared to how geometric shapes are perceived. A regular line, for instance, is conventionally understood to be one-dimensional. A solid square is understood to be two-dimensional. We see that for ordinary self-similar objects, being n-dimensional means that when it is rep-tiled into pieces each scaled down by a scale-factor of 1/r, there are a total of rn pieces.
Now, consider the Koch curve. It can be rep-tiled into four sub-copies, each scaled down by a scale-factor of 1/3. So by analogy, we can consider the "dimension" of the Koch curve as being the unique real number D that satisfies 3D = 4, which by no means is an integer! This number is; the fact th
Binary tree
In computer science, a binary tree is a tree data structure in which each node has at most two children, which are referred to as the left child and the right child. A recursive definition using just set theory notions is that a binary tree is a tuple, where L and R are binary trees or the empty set and S is a singleton set; some authors allow the binary tree to be the empty set as well. From a graph theory perspective, binary trees as defined here are arborescences. A binary tree may thus be called a bifurcating arborescence—a term which appears in some old programming books, before the modern computer science terminology prevailed, it is possible to interpret a binary tree as an undirected, rather than a directed graph, in which case a binary tree is an ordered, rooted tree. Some authors use rooted binary tree instead of binary tree to emphasize the fact that the tree is rooted, but as defined above, a binary tree is always rooted. A binary tree is a special case of an ordered K-ary tree, where k is 2.
In mathematics, what is termed binary tree can vary from author to author. Some use the definition used in computer science, but others define it as every non-leaf having two children and don't order the children either. In computing, binary trees are used in two different ways: First, as a means of accessing nodes based on some value or label associated with each node. Binary trees labelled this way are used to implement binary search trees and binary heaps, are used for efficient searching and sorting; the designation of non-root nodes as left or right child when there is only one child present matters in some of these applications, in particular it is significant in binary search trees. However, the arrangement of particular nodes into the tree is not part of the conceptual information. For example, in a normal binary search tree the placement of nodes depends entirely on the order in which they were added, can be re-arranged without changing the meaning. Second, as a representation of data with a relevant bifurcating structure.
In such cases the particular arrangement of nodes under and/or to the left or right of other nodes is part of the information. Common examples occur with Huffman coding and cladograms; the everyday division of documents into chapters, paragraphs, so on is an analogous example with n-ary rather than binary trees. To define a binary tree in general, we must allow for the possibility that only one of the children may be empty. An artifact, which in some textbooks is called an extended binary tree is needed for that purpose. An extended binary tree is thus recursively defined as: the empty set is an extended binary tree if T1 and T2 are extended binary trees denote by T1 • T2 the extended binary tree obtained by adding a root r connected to the left to T1 and to the right to T2 by adding edges when these sub-trees are non-empty. Another way of imagining this construction is to consider instead of the empty set a different type of node—for instance square nodes if the regular ones are circles. A binary tree is a rooted tree, an ordered tree in which every node has at most two children.
A rooted tree imparts a notion of levels, thus for every node a notion of children may be defined as the nodes connected to it a level below. Ordering of these children makes possible to distinguish left child from right child, but this still doesn't distinguish between a node with left but not a right child from a one with right but no left child. The necessary distinction can be made by first partitioning the edges, i.e. defining the binary tree as triplet, where is a rooted tree and E1 ∩ E2 is empty, requiring that for all j ∈ every node has at most one Ej child. A more informal way of making the distinction is to say, quoting the Encyclopedia of Mathematics, that "every node has a left child, a right child, neither, or both" and to specify that these "are all different" binary trees. Tree terminology so varies in the literature. A rooted binary tree has a root node and every node has at most two children. A full binary tree is a tree in which every node has 2 children. Another way of defining a full binary tree is a recursive definition.
A full binary tree is either:A single vertex. A tree whose root note has two subtrees. In a complete binary tree every level, except the last, is filled, all nodes in the last level are as far left as possible, it can have between 2h nodes at the last level h. An alternative definition is a perfect tree; some authors use the term complete to refer instead to a perfect binary tree as defined above, in which case they call this type of tree an complete binary tree or nearly complete binary tree. A complete binary tree can be efficiently represented using an array. A perfect binary tree is a binary tree in which all interior nodes have two children and all leaves have the same depth or same level. An example of a perfect binary tree is the ancestry chart of a person to a given depth, as each person has two biological parents. In the infinite complete binary tree, every node has two children; the set of all nodes is countably infinite, but the set of all infinite paths from the root
Similarity (geometry)
Two geometrical objects are called similar if they both have the same shape, or one has the same shape as the mirror image of the other. More one can be obtained from the other by uniformly scaling with additional translation and reflection; this means that either object can be rescaled and reflected, so as to coincide with the other object. If two objects are similar, each is congruent to the result of a particular uniform scaling of the other. A modern and novel perspective of similarity is to consider geometrical objects similar if one appears congruent to the other when zoomed in or out at some level. For example, all circles are similar to each other, all squares are similar to each other, all equilateral triangles are similar to each other. On the other hand, ellipses are not all similar to each other, rectangles are not all similar to each other, isosceles triangles are not all similar to each other. If two angles of a triangle have measures equal to the measures of two angles of another triangle the triangles are similar.
Corresponding sides of similar polygons are in proportion, corresponding angles of similar polygons have the same measure. This article assumes that a scaling can have a scale factor of 1, so that all congruent shapes are similar, but some school textbooks exclude congruent triangles from their definition of similar triangles by insisting that the sizes must be different if the triangles are to qualify as similar. In geometry two triangles, △ABC and △A′B′C′, are similar if and only if corresponding angles have the same measure: this implies that they are similar if and only if the lengths of corresponding sides are proportional, it can be shown that two triangles having congruent angles are similar, that is, the corresponding sides can be proved to be proportional. This is known as the AAA similarity theorem. Note that the "AAA" is a mnemonic: each one of the three A's refers to an "angle". Due to this theorem, several authors simplify the definition of similar triangles to only require that the corresponding three angles are congruent.
There are several statements each of, necessary and sufficient for two triangles to be similar: The triangles have two congruent angles, which in Euclidean geometry implies that all their angles are congruent. That is:If ∠BAC is equal in measure to ∠B′A′C′, ∠ABC is equal in measure to ∠A′B′C′ this implies that ∠ACB is equal in measure to ∠A′C′B′ and the triangles are similar. All the corresponding sides have lengths in the same ratio:AB/A′B′ = BC/B′C′ = AC/A′C′; this is equivalent to saying. Two sides have lengths in the same ratio, the angles included between these sides have the same measure. For instance:AB/A′B′ = BC/B′C′ and ∠ABC is equal in measure to ∠A′B′C′; this is known as the SAS similarity criterion. The "SAS" is a mnemonic: each one of the two S's refers to a "side"; when two triangles △ABC and △A′B′C′ are similar, one writes △ABC ∼ △A′B′C′. There are several elementary results concerning similar triangles in Euclidean geometry: Any two equilateral triangles are similar. Two triangles, both similar to a third triangle, are similar to each other.
Corresponding altitudes of similar triangles have the same ratio as the corresponding sides. Two right triangles are similar if one other side have lengths in the same ratio. Given a triangle △ABC and a line segment DE one can, with ruler and compass, find a point F such that △ABC ∼ △DEF; the statement that the point F satisfying this condition exists is Wallis's postulate and is logically equivalent to Euclid's parallel postulate. In hyperbolic geometry similar triangles are congruent. In the axiomatic treatment of Euclidean geometry given by G. D. Birkhoff the SAS similarity criterion given above was used to replace both Euclid's Parallel Postulate and the SAS axiom which enabled the dramatic shortening of Hilbert's axioms. Similar triangles provide the basis for many synthetic proofs in Euclidean geometry. Among the elementary results that can be proved this way are: the angle bisector theorem, the geometric mean theorem, Ceva's theorem, Menelaus's theorem and the Pythagorean theorem. Similar triangles provide the foundations for right triangle trigonometry.
The concept of similarity extends to polygons with more than three sides. Given any two similar polygons, corresponding sides taken in the same sequence are proportional and corresponding angles taken in the same sequence are equal in measure. However, proportionality of corresponding sides is not by itself sufficient to prove similarity for polygons beyond triangles. Equality of all angles in sequence is not sufficient to guarantee similarity. A sufficient condition for similarity of polygons is that corresponding sides and diagonals are proportional. For given n, all regular n-gons are similar. Several types of curves have the property; these include: Circles Parabolas Hyperbolas of a specific eccentricity Ellipses of a specific eccentricity Catenaries Graphs of the logarithm function for different bases Graphs of the exponential function for different bases Logarithmic spirals are self-similar A similarity of a Euclidean space is a bijection f from the space onto itself that multiplies all distances
Viable system model
The viable system model is a model of the organisational structure of any autonomous system capable of producing itself. A viable system is any system organised in such a way as to meet the demands of surviving in the changing environment. One of the prime features of systems that survive is; the VSM expresses a model for a viable system, an abstracted cybernetic description, claimed to be applicable to any organisation, a viable system and capable of autonomy. The model was developed by operations research theorist and cybernetician Stafford Beer in his book Brain of the Firm. Together with Beer's earlier works on cybernetics applied to management, this book founded management cybernetics; the first thing to note about the cybernetic theory of organizations encapsulated in the VSM is that viable systems are recursive. A development of this model has originated the theoretical proposal called viable systems approach. Here we give a brief introduction to the cybernetic description of the organization encapsulated in a single level of the VSM.
A viable system is composed of five interacting subsystems which may be mapped onto aspects of organizational structure. In broad terms Systems 1–3. are concerned with the'here and now' of the organization's operations, System 4 is concerned with the'there and then' – strategical responses to the effects of external and future demands on the organization. System 5 is concerned with balancing the'here and now' and the'there and then' to give policy directives which maintain the organization as a viable entity. System 1 in a viable system contains several primary activities; each System 1 primary activity is itself a viable system due to the recursive nature of systems as described above. These are concerned with performing a function that implements at least part of the key transformation of the organization. System 2 represents the information channels and bodies that allow the primary activities in System 1 to communicate between each other and which allow System 3 to monitor and co-ordinate the activities within System 1.
Represents the scheduling function of shared resources to be used by System 1. System 3 represents the structures and controls that are put into place to establish the rules, resources and responsibilities of System 1 and to provide an interface with Systems 4/5. Represents the big picture view of the processes inside of System 1. System 4 is made up of bodies that are responsible for looking outwards to the environment to monitor how the organization needs to adapt to remain viable. System 5 is responsible for policy decisions within the organization as a whole to balance demands from different parts of the organization and steer the organization as a whole. In addition to the subsystems that make up the first level of recursion, the environment is represented in the model; the presence of the environment in the model is necessary as the domain of action of the system and without it there is no way in the model to contextualize or ground the internal interactions of the organization. Algedonic alerts are alarms and rewards that escalate through the levels of recursion when actual performance fails or exceeds capability after a timeout.
The model is derived from the architecture of nervous system. Systems 3-2-1 are identified with autonomic nervous system. System 4 embodies conversation. System 5, the higher brain functions, include decision making. In "Heart of Enterprise" a companion volume to "Brain...", Beer applies Ashby's concept of Variety: the number of possible states of a system or of an element of the system. There are two aphorisms; these rules ensure the Requisite Variety condition is satisfied, in effect that resources are matched to requirement. These aphorisms are: It is not necessary to enter the black box to understand the nature of the function it performs, it is not necessary to enter the black box to calculate the variety that it may generate. These principles are: Managerial and environmental varieties diffusing through an institutional system, tend to equate; the four directional channels carrying information between the management unit, the operation, the environment must each have a higher capacity to transmit a given amount of information relevant to variety selection in a given time than the originating subsystem has to generate it in that time.
Wherever the information carried on a channel capable of distinguishing a given variety crosses a boundary, it undergoes transduction. The operation of the first three principles must be cyclically maintained without delays; this theorem states: In a recursive organizational structure any viable system contains, is contained in, a viable system. These axioms are: The sum of horizontal variety disposed by n operational elements equals the sum of the vertical variety disposed by the six vertical components of corporate cohesion. (The six are from Environment, System Three*
