Definitions for **"Fractal"**

Term coined by Benoit Mandelbrot in 1975, referring to objects built using recursion, where some aspect of the limiting object is infinite and another is finite, and where at any iteration, some piece of the object is a scaled down version of the previous iteration.

A term coined by Benoit Mandelbrot to refer to items with fractional dimensions as opposed to the integer dimensions such as 1, 2 and 3 associated with length, area and volume. Often used to refer to a structure bearing statistically similar details over a wide range of scales.

Object which is self-similar at all scales. Regardless of scale the same level of detail and appearance is present.

an algorithm, or shape, characterized by self-similarity and produced by recursive sub-division; more generally the branch of mathematics named and explored by Benoit Mandelbrot

A nested pattern that shows the same symmetry/geometry at any scale a pattern is looked at. It is an example of an infinite pattern in both larger and smaller dimensions. Fractals are geometrical abstractions and sometimes used to explain complexity in living organisms, also the comparison does not hold up on closer inspection. As abstractions, they are thought to continue into infinity, even the infinitely small, although physicists believe that there is a material limit to the what constitutes the smallest dimension. As for complex biological organisms, their structure is hierarchical with higher levels having emergent structures and properties not found at the lower levels. Thus it is not a true fractal.

A mathematically generated pattern that is reproducible at any magnification or reduction.

A set of data elements, frequently representing the points of a geometric figure, characterized by self affinity (parts resemble the whole) and invariance to scale (having similar attributes at various degrees of analytical precision).

An irregular shape with self-similarity. It has infinite detail, and cannot be differentiated. "Wherever chaos, turbulence, and disorder are found, fractal geometry is at play" (Briggs and Peat, 1989).

Submitted By The Authors Fractals are rough or fragmented geometric shapes which display the following two properties: First, most fractals are self-similiar or indendant of scale. Second, fractals have fractal dimensions. More information about fractals are available in the fractal division of the fractal geometry main page.

(adj.) The term, short for fractional dimensional, used to describe graphics with randomly generated curves and surfaces that exhibit a degree of self-similarity. Fractal design tools provide new opportunities for designers to produce complex patterns with more visual realism than can be output from conventional geometry programs.

irregular and fragmented self-similar shapes; e.g., fractal curves can wriggle so much that they fall into the gap between dimensions.

Short for Fractional Dimension. See also Mandelbrot.

A graphics term, originally defined by mathematician Benoit Mandelbrot, to describe a category of geometric shapes charachterized by irregular shape and design and used by computer software, such as Genuine Fractals, as a mathematical model for resizing and enlarging image files.

(mathematics) a geometric pattern that is repeated at every scale and so cannot be represented by classical geometry

a colored image that mathematically models how well things survive in their environments

a complex empirical object or geometrical figure which appears to have an ordinary "intuitive" dimension (i

a complex geometric figure made up of patterns that repeat themselves at smaller and smaller, or larger and larger, scales

a complex geometric pattern that can be subdivided in parts, each of which is (at least approximately) a smaller copy of the whole

a complex geometric shape that has the property of self-similarity

a complex pattern (or random shape) plotted by coordinates that are generated by repeating an equation

a complex shape in which each part of an image is a smaller version of the whole

a distribution or shape that is in general not homogeneous, but which possesses the property that each part is a simulacrum of the whole, i

a figure where a small part of the figure has the same shape as the whole figure

a figure with a fractional dimension

a form with parts that resemble smaller copies of the whole over a range of scale

a fun geometric shapes whose dimension is hard to pin down

a general price pattern that repeats itself at different scales

a geometrical figure that has the following two properties

a geometrical object with the property that subsections of the object appear identical to (but smaller than) the whole object

a geometrical or physical structure that has an irregular or fragmented shape at all scales at certain mathematical or physical properties of the structure are greater than the spatial dimensions

a geometrical shape or pattern made up of identical parts, which are in turn identical to the overall pattern

a geometrical shape that is repeated at ever smaller (or larger) scales

a geometrical shape which looks the same when you magnify it an arbitrary number of times

a geometrical shape whose structure is such that magnification by a given factor reproduces the original object

a geometric figure in which a single motif is repeated at a continuously decreasing scale ("self-similarity")

a geometric figure that can be separated into parts, each of which is similar to the whole

a geometric figure that consists of an identical motif repeating itself on an ever reducing scale

a geometric figure with two special properties

a geometric form that looks the same (that is, it is self-similar) no matter how much it is enlarged or reduced

a geometric object which can be divided into parts, each of which is similar to the original object

a geometric object which is 'broken up' in aradical way

a geometric object which is 'broken up' in a radical way

a geometric object which is rough or irregular on all scales of length, and so which appears to be broken

a geometric object with a non-integer dimension

a geometric pattern capable of being repeated at any scale

a geometric pattern exhibiting an infinite level of repeating, self-similar detail that can't be described with classical geometry

a geometric pattern repeated at ever-smaller scales so that each is a smaller copy of the whole

a geometric pattern that is repeated at ever smaller scales to produce irregular shapes and surfaces that cannot be represented by classical geometry

a geometric pattern that repeats itself at smaller and smaller scales to produce irregular shapes and surfaces that cannot be represented by classical geometry

a geometric pattern whose smaller parts

a geometric shape, a geometric shape having a special property that as you look closer and closer to it you see it is essentially the same thing

a geometric shape or curve that repeats itself irregularly, like a mountain range or a coastline

a geometric shape that can be divided into parts, each of which is (approximately) a reduced size copy of the original

a geometric shape that can be recursively broken down into smaller parts

a geometric shape that can be separated into parts, each of which is a reduced-scale version of the whole

a geometric shape that is complex and detailed at every level of magnification, as well as self-similar

a geometric shape with a pattern that repeats itself at different scales of magnitude

a geometric shape with patterns that repeat themselves without limit at smaller and smaller scales

a graphic representation of a mathematical equasion that represents and image, pattern, or tendency for objects, concepts, and thier behavior in our world

a great example of a pattern

a kind of repeating structure displaying properties of self similarity

a mathematical construct based on a simple equation repeated infinitely, where the graph of the equation results create a form of infinite complexity and depth, resembling nature itself in its endless variations

a mathematical construct which appears deceptively simple at a glance

a mathematical equation that makes repetative patterns to form an image, and in this equation, the patterns create infinite Triforces, circumscribed within each other

a mathematical figure calculated using complex values (e

a mathematical formula of a pattern that repeats over a wide range of size and time scales

a mathematical function or geometric shape that is self-similar on all scales

a mathematical function that--when represented graphically--produces an often amazingly beautiful geometric shape

a mathematical object that has a detailed structure at any scale

a mathematical object that has detailed structure no matter how closely you look at it, no matter how great the magnification

a mathematical object that is self-similar and chaotic

a mathematical set or object whose form is extremely irregular and/or fragmented at all scales

a natural pattern found in a variety of places in nature

an identical shape or pattern that appears at all different scales

an image formed through advanced mathematical formulas, so each time the image is zoomed in or out, the formulas rebuild the image giving more diversity to the art

an image of a chaos pattern

an image of a pattern in nature

an image or set that has been created by applying a simple, recursive rule to a mathematical set

an infinitely bumpy line

an infinite pattern compressed into a finite space

an irregularly shaped object that is nonrandom in the sense that its discontinuities (i

an object exhibiting self similarity on different scales that can be related mathematically

an object of awkward dimensionality (formally, its Hausdorff-Besicovitch dimension is strictly greater than its topological dimension)

an object or quantity that displays self-similarity, in a somewhat technical sense, on all scales

an object that displays self-similarity at various scales

an object that has the same irregularity or crinkliness at different magnifications

an object that is similarly shaped at different scales

an object that looks like itself at different scales, or levels of magnification

an object whose volume is not a simple product of its dimensions ," explained Arthur Epstein, director of the Center for Materials Research at Ohio State University

an object with a fractional dimension

an object with structure at many different scales or magnifications

a pattern that appears the same no matter what scale it's viewed at

a repeating pattern, whose iterations generate both larger and smaller "self-similar" images or systems

a rough or fragmented shape that can be divided into parts, each of which is a slightly different copy of the whole

a self-similar and infinitely recursive object whose parts somehow resemble the whole

a self-similar graphical form with a fractional dimension

a set of points that are exactly or statistically self similar

a shape, often drawn by a computer, that repeats itself in a pattern

a shape or curve that is self-similar

a shape or sequence or system that is infinite and contains a copy of itself within itself

a shape that has a repeating pattern, but where occurrences of the pattern change in location and size (often times getting smaller)

a shape that is self-similar and has a fractional dimension

a shape that is self similar, that is that repeats the same basic shape at smallers levels within the same structure

a shape that is symmetric in scale, meaning that it looks the same, or similar, magnified as it does unmagnified

a shape which is generated mathematically

a special image that is generated using a very simple mathematical formula

a structure with an infinitely repeating pattern

a surface which is so bumpy that it is not really a line and not really an area -- it is something in between

a system which has the same structure on many measurement scales

a tree-like object composed of subunits that resemble the larger scale structure

a type of curve which has some feature repeated on a different scale, such as the pattern of a coastline

a whole system, which within hold reflections of itself at increasingly lower levels

an object in which the parts are in some way related to the whole. That is, the individual components are "self-similar." An example is the branching network in a tree. While each branch, and each successive smaller branching is different, they are qualitatively similar to the structure of the whole tree

Computer-generated images corresponding to mathematical equations, that repeat self-similar patterns at infinitely receding levels of organization.

Infinitely repeating patterns of colored waves targeted to newageoholics who dropped too much acid in the 60s; usually marketed as computer screen savers.

A System having similar detail at all scales, leading to intricate patterns and unexpected features. Fractal geometry explores systems with non-integer dimensions.

A form of computer generated art making process that creates complex, repetitive, mathematically based geometric shapes and patterns that resemble those found in nature.

a set for which the Hausdorff Besicovitch dimension strictly exceeds the topological dimension; a set (of numbers which may define an object or image) having a non-integer dimension

Any of various extremely irregular curves, shapes, or patterns for which any part is similar to a given larger or smaller part when magnified or reduced to the same size.

An object having a fractional dimension: one which has variation that is self-similar at all scales, in which the final level of detail is never reached and never can be reached by increasing the scale at which observations are made.

A geometric shape that is self-similar and has fractional dimensions. Natural phenomena such as the formation of snowflakes, clouds, mountain ranges, and landscapes involve patterns. Their pictorial representations are fractals and are usually generated by computers.

Of or pertaining to an object with fractional dimension and self-similarity at many or all scales; also, the name of such an object. The most famous fractal is the Mandelbrot set, which is basically a two-dimensional blob with a boundary of dimension greater than one but less than two. Around the boundary are tiny, distorted copies of the main blob, and each copy has its own copies nearby; this attribute is called self-similarity, and it is the basis for fractal compression.

Any function that contains elements of self-similarity (after the work of Benoit Mandelbrot).

Term coined by Benoit Mandelbrot, refers to identical or similar shapes on ever decreasing scales. Branches of trees are natural fractals as are bays and coves of coastlines. Mathematical fractals are closer to identical.

Fractals are patterns within patterns within patterns. 1/2

A curve or shape so irregular that its dimension (according to the technical definition of dimension) is a fraction, rather than an integer. Many interesting examples of fractals have the property of being self-similar, in the sense that portions of the fractal are similar in shape to magnified parts of itself at arbitrarily high rates of magnification.

A pattern which is repeated infinitely; a pattern of consistent and infinite detail; the underlying structure of the Universe; the underlying structure of consciousness; the navigation system to metaphysical abilities (also see www.fractal.org).

An object with a fractal dimension. Fractals are self-similar and may be deterministic or stochastic. See also Cantor Set, Diffusion Limited Aggregation, IFS, Julia Set, L-Systems, MRCM, Mandelbrot Set, and Strange Attractor.

A geometrical structure in which the same pattern is repeated at many different scales; a self-similar object.

Fractals are elements with fractional dimensions between whole integers. Fractals may represent patterns which are infinitely self similar at varying scales. The fractal boundary of a strange attractor can be thought to represent a bounded system that is an evolving complex system recognizable by the self similar patterns being formed from the surrounding disorder.

An algebraically generated complex geometric shape having the property of being endlessly self-similar under magnification. Some computer screen savers utilize fractals.

Fractal Dimension Fractal Distribution

An irregular geometric object that is self-similar to its substructure at any level of refinement. The fractal dimension is a measure of the irregularity of the boundary of the object. Clouds have been shown to exhibit fractal characteristics with cloud perimeters having fractal dimension of about 1.35. The result follows from the observation that over a large size range, cloud area is proportional to cloud perimeter raised to a power slightly less than 1.5 (rather than proportional to the square of the perimeter, as is the case in Euclidean geometry); fractal dimension of the perimeter then is calculated as the ratio of 2 (the exponent in the Euclidean power law) to the exponent in the observed power relation. Lovejoy, S., 1982: Area-perimeter relation for rain and cloud areas. Science, 216, 185â€“187.

In colloquial usage, a fractal is "a rough or fragmented geometric shape that can be subdivided in parts, each of which is (at least approximately) a reduced/size copy of the whole". The term was coined by BenoÃ®t Mandelbrot in 1975 and was derived from the Latin fractus meaning "broken" or "fractured".

a measure of the degree to which a system uses the space available to it

an estimate of the degree to which a system occupies a region in an outcome basin

an independently active scientific domain , whose aims and achievements are clearly definable