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Patterns
A pattern is a regularity in the world, in human-made design, or in abstract ideas. As such, the elements of a pattern repeat in a predictable manner. A geometric pattern is a kind of pattern formed of geometric shapes and typically repeated like a wallpaper design. Any of the senses may directly observe patterns. Conversely, abstract patterns in science, mathematics, or language may be observable only by analysis. Direct observation in practice means seeing visual patterns, which are widespread in nature and in art. Visual patterns in nature are often chaotic, rarely exactly repeating, and often involve fractals. Natural patterns include spirals, meanders, waves, foams, tilings, cracks, and those created by symmetries of rotation and reflection. Patterns have an underlying mathematical structure; indeed, mathematics can be seen as the search for regularities, and the output of any function is a mathematical pattern. Similarly in the sciences, theories explain and predict re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Patterns In Nature
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, the Belgian physicist Joseph Plateau examined soap films, leading him to formulate the concept of a minimal surface. The 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, the British mathematician Alan Turing predicted mechanisms of morp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tessellation
A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation 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 (an aperiodic set of prototiles). A '' tessellation of space'', also known as a space filling or honeycomb, can be defined in the geometry of higher dimensions. A real physical tessellation is a tiling made of materials such as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fractal
In mathematics, a fractal is a Shape, geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illustrated in successive magnifications of the Mandelbrot set. This exhibition of similar patterns at increasingly smaller scales is called self-similarity, also known as expanding symmetry or unfolding symmetry; if this replication is exactly the same at every scale, as in the Menger sponge, the shape is called affine geometry, affine self-similar. Fractal geometry lies within the mathematical branch of measure theory. One way that fractals are different from finite geometric figures is how they Scaling (geometry), scale. Doubling the edge lengths of a filled 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 conventional dimension of the filled polygon). ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetry
Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is Invariant (mathematics), invariant under some Transformation (function), transformations, such as Translation (geometry), translation, Reflection (mathematics), reflection, Rotation (mathematics), rotation, or Scaling (geometry), scaling. Although these two meanings of the word can sometimes be told apart, they are intricately related, and hence are discussed together in this article. Mathematical symmetry may be observed with respect to the passage of time; as a space, spatial relationship; through geometric transformations; through other kinds of functional transformations; and as an aspect of abstract objects, including scientific model, theoretic models, language, and music. This article describes symmetry from three perspectives: in mathematics, including geometry, the m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chaos Theory
Chaos theory is an interdisciplinary area of Scientific method, scientific study and branch of mathematics. It focuses on underlying patterns and Deterministic system, deterministic Scientific law, laws of dynamical systems that are highly sensitive to initial conditions. These were once thought to have completely random states of disorder and irregularities. Chaos theory states that within the apparent randomness of chaotic complex systems, there are underlying patterns, interconnection, constant feedback loops, repetition, self-similarity, fractals and self-organization. The butterfly effect, an underlying principle of chaos, describes how a small change in one state of a deterministic nonlinear system can result in large differences in a later state (meaning there is sensitive dependence on initial conditions). A metaphor for this behavior is that a butterfly flapping its wings in Brazil can cause or prevent a tornado in Texas. Text was copied from this source, which is avai ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spirals
In mathematics, a spiral is a curve which emanates from a point, moving further away as it revolves around the point. It is a subtype of whorled patterns, a broad group that also includes concentric objects. Two-dimensional A two-dimensional, or plane, spiral may be easily described using polar coordinates, where the radius r is a monotonic continuous function of angle \varphi: * r=r(\varphi)\; . The circle would be regarded as a degenerate case (the function not being strictly monotonic, but rather constant). In ''x-y-coordinates'' the curve has the parametric representation: * x=r(\varphi)\cos\varphi \ ,\qquad y=r(\varphi)\sin\varphi\; . Examples Some of the most important sorts of two-dimensional spirals include: * The Archimedean spiral: r=a \varphi * The hyperbolic spiral: r = a/ \varphi * Fermat's spiral: r= a\varphi^ * The lituus: r = a\varphi^ * The logarithmic spiral: r=ae^ * The Cornu spiral or ''clothoid'' * The Fibonacci spiral and golden spiral * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Meander
A meander is one of a series of regular sinuous curves in the Channel (geography), channel of a river or other watercourse. It is produced as a watercourse erosion, erodes the sediments of an outer, concave bank (cut bank, cut bank or river cliff) and deposits sediments on an inner, convex bank which is typically a point bar. The result of this coupled erosion and sedimentation is the formation of a sinuous course as the channel migrates back and forth across the axis of a floodplain. The zone within which a meandering stream periodically shifts its channel 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 challenges for local municipalities attempting to maintain stable roads and bridges.Neuendorf, K.K.E., J.P. Mehl Jr., and J.A. Jackson, J.A., eds. (2005) ''Glossary of Geology'' (5th ed.). Alexandria, Virginia, American Geological I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abstraction
Abstraction is a process where general rules and concepts are derived from the use and classifying of specific examples, literal (reality, real or Abstract and concrete, concrete) signifiers, first principles, or other methods. "An abstraction" is the outcome of this process — a concept that acts as a common noun for all subordinate concepts and connects any related concepts as a ''group'', ''field'', or ''category''.Suzanne K. Langer (1953), ''Feeling and Form: A Theory of Art Developed from Philosophy in a New Key'', p. 90: "Sculpture, Sculptural form is a powerful abstraction from actual objects and the three-dimensional space which we construe ... through sensory system, touch and sight." Conceptual abstractions may be made by filtering the information content of a concept or an observable phenomenon, selecting only those aspects which are relevant for a particular purpose. For example, abstracting a leather soccer ball to the more general idea of a ball selects only the in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
