How To Use Fractal In A Sentence

Related, but not directly, I have spent much time getting to know fractals, 4d geometry, and other ‘strange’ mathematical phenomena.

With the help of catastrophe criterion, the anomalism criterions of the crossfeed between dam body and foundation are acquired from the real contact area or fractal dimension curve vs.

A fractal object is self  affine instead of self  similar if it has scaling property in different direction.

This is an example of what is known as a fractal set since its dimension is not a whole number.

In particular we show that pointlessness and dimensional function implies fractality.
Backreaction

A structure may be called fractal if it presents the same irregularity at all scales.

This is a very deep zoom into a fractal called the Mandelbrot set. the boundary of the Mandelbrot set is infinite in length and detail. the deeper you go the more complex the image gets.
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Fractals, using simple mathematical algorithm to produce complex objects typical of those which occur naturally.

The word that comes to mind is exothermic, the first nanoseconds of an explosion splintering into whitehot fractal fissures.
Lambo's Latest Rambo Has a Heart

This paper applies wavelet and fractal theories to analyze multipath fading signal in mobile radio communications for the first time.

Ancient Celts believed in a giant man, the Greeks observed some basic shapes, the word "fractal" was coined in the Seventies.
Telegraph.co.uk  Telegraph online, Daily Telegraph and Sunday Telegraph

When carbonized, they have fractal properties that allow the hydrogen to be pushed into them at relatively low pressure.
Chicken Feathers May Fuel Hydrogen Cars in the Future  Inhabitat

Benoît Mandelbrot was that IBM employee who successfully measured the coastline of England in 1967, coined the term fractal in 1975 and opened new worlds in mathematics and now, I believe, human emotional understanding.
Fractal Geometry, Human Emotion and Social Marketing  Tom Troja  MediaBizBloggers

Out of interest, here are the Ultra Fractal files  limulus. ufr and limulus. ufm  for a 'horseshoe crab' biomorph I designed in Fractint a while back.
Archive 20031101

The researchers discovered that Pollock's patterns could be characterized as fractals  shapes that repeat themselves on different scales within the same object.

Some of the early investigators of chaos were the American physicist Mitchell Feigenbaum; the Polishborn mathematician and inventor of fractals see fractal geometry Benoit Mandelbrot; the American mathematician James Yorke, who popularized the term chaos; and the American meteorologist Edward Lorenz.
Exponential Growth in Physical Systems #2 « Climate Audit

Many natural chaotic systems form fractals in the patterns that record the process.

Fractals are used in the study of things like forked lightning and to produce some types of computer graphics.

Would the present obsession with fractal geometry have come about anyway as part of the ever changing taste of architectural stylists, or is it an inevitable result of computer power?

The influence of Pablo Picasso, Thomas Hart Benton and the Mexican Muralist David Alfaro Siqueiros is still apparent, but giving way to Native American sandpainting and "all over" painting techniques that would eventually lead Pollock to abandon traditional notions of perspective and geometric shape for fractallike patterns.
A Pollock Saved From the Flood

Benoit Mandelbrot pioneered fractal geometry

Euclidean geometry, Fibonacci numbers, the digits of pi, the notion of algorithms, concepts of infinity, fractals, and other ideas furnished the mathematical underpinnings.

Broadly expressed cell markers, such as the neuronal nuclei antigen NeuN, the glial fibrillary acidic protein (GFAP), myelin proteolipid protein (PLP) and the fractalkine receptor CX3CR1 (which identify neurons, astrocytes, oligodendrocytes and microglia, respectively), are commonly used in immunohistochemical analysis of brain development or disease.
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With these branching patterns, roots can be characterized by fractal analysis over a range of scales from about a quarter of a millimetre to several centimetres.

You see infinitely detailed multicoloured fractal alphabets, they see growing paisley patterns.

His research focuses on the interplay between fractal geometry and stochastic processes.

Mathematicians have long been taken with the concept of fractals, which posits that patterns are made up of smaller, selfcontained patterns.

It should be remembered that in his era there was little thought or knowledge of fractal geometry, curved space or different degrees of infinity.

Strange attractors have detailed structure on all scales of magnification and were one of the early fractals to be studied.

The result is a division of the initial planar shape into several pieces with ragged fractal edges.

Each level of expression contributed to the next, and over the course of millennia we created mathematical theory, from basic arithmetic to algebra, from calculus to fractal geometry.

It reports the quantitative analysis of Purkinje cell fractality in a number of placentates, one marsupial and two bird species, showing that fractal complexity is essentially identical in these species.
Another unintelligent move  The Panda's Thumb

Fractal dimension is an important parameter to characterize roughness in an image.

Then we study the technique formally, and then apply it to images of fractal art.

Fractals are used in the study of things like forked lightning and to produce some types of computer graphics.

The Fractal Brownian Motion ( FBM ) provides effective mathematics model, which has important ...

Some of the complexity has been captured in modern pictures of fractals, many of which are created by iterating functions of this kind.

But then I could actually feel my unconscious brain straining to generate fractal landscapes as I flew, and start to realize that it was just a dream.
Writer's Block: Dream On

We equation in fractal porous media , and present reiterated two  scale probabilistic model and numerical method.

Computer programs help diagram elaborate formations of tiny spirals that form massive, spiraled fractals to describe, say, the economic tendencies of your new, island society.

A fractal is a geometric shape with a pattern that repeats itself at different scales of magnitude.
The Fractal Murders: The Story Behind the Story

The implications of map theory, game theory, topology, the fractals of chaos theory, have all lurked in ornament, awaiting their elevation to science.

What this installation actually consists of is somewhat obscured by its tendency to catalyse an onset of multicoloured fractal hallucinations.
This week's new exhibitions

In his view, "Science and religion do not glower at each other … [but] interdigitate in patterns of complex fingering, and at every fractal scale of selfsimilarity.
Metacentricities

The appearance of fractal geometry extended our former understanding about dimension and scale of architecture.

Fractal scientists and chaologists, in contrast, operate from the assumption that natural systems are necessarily nonlinear, dynamic, and disordered.

Fractals are used in the study of things like forked lightning and to produce some types of computer graphics.

Serves me right: apparently (and this was explained to me very carefully before I moved here) there's a ring of fractalized subspace around the entire block.

Mathematicians can use similar algorithms to generate fractals and other forms.

I like some the choppy rhythm, and the alternating difficult rhymes  endless/tortoise, fractal/awful, commitments/balance.

He finds in the fractal, and its "hologram"  like selfsimilarityacross scales, a kind of "secret order" to the apparent complexity of the worlda confirmation of an overall design which we discover stamped on the most infinitesimal components of that design.
Complexity and Order.

What is the mathematical, fractal relationship between shapes found in beehive honeycombs, a pineapple, tesselating hexagons, and AfricanAmerican hair braiding?
Boing Boing: April 25, 2004  May 1, 2004 Archives

The researchers also found that the greater the length scales at which the pattern persisted  or the more complete its "fractality"  the higher the maximum temperature at which the crystal could superconduct.
New Scientist  Online News

Fractal dimension does have some correspondence with the type of failure.

The column introduced the public to puzzles and concepts such as fractals and Chinese tangram puzzles, as well as the work of artist M.C. Escher.
Martin Gardner, Layman's Mathematician And Fierce Foe Of Pseudoscience, Dies At 95

Outlines may be captured as rectangular coordinates or Fourier coefficients and, even when nonfractal, analysed in terms of fractal dimensions.

Like a mountain range whose profile appears equally craggy when observed from both far and near, fractals are used to define curves and surfaces, independent of their scale.

Classically, a dissipative chaotic system tends towards a strange attractor with a fractal structure; that is, the attractor exhibits substructure at all length scales.
Quantum Hyperion

The way that the trunk of a tree divides into smaller and smaller branches and twigs is an approximate fractal pattern.

Blake would appear to find the latter conceptually and aesthetically interesting and productive up to a point, but boring in infinity, however intricate fractality may be, such as that of the famous Mandelbrot set.
Chaosmic Orders: Nonclassical Physics, Allegory, and the Epistemology of Blake's Minute Particulars.

Mathematicians can use similar algorithms to generate fractals and other forms.

Ronell avoids the trap by proceeding in everdecreasing circles  or fractal geometry, as it is now known.

But he spent 10 years consciously refining his pouring technique to increase the visual complexity of these fractal patterns.
Smithsonian Mag

For example, we observe the presence of coexisting lipid phases (one phase less ordered than the other phase) and unstable domain growth that results in patterns such as fractals.

They've talked about Mondo 2000, virtual raves, about smart drugs and new psychedelics, about chaos theory and fractals.

Bone radiograph texture measurement based on power spectrum and fractal analysis have been proposed.

We want to lift the restrictions imposed on this old concept by the recent mathematical chaology, identifying only a family of dynamic processes transcending linearity, or the geometry of fractals, transcending scales and the integerdimensions.

An artist can look at a mathematical shape and envision unlimited possibilities, even from a shape as seemingly simple as a tetrahedron, a trefoil knot, a Möbius strip, or a fractal surface.

Fractals, using simple mathematical algorithm to produce complex objects typical of those which occur naturally.

Chaos theory is everywhere now: landscapes and starscapes in scifi movies, image compression software, even your laser printer uses fractal algorithms.

The lines strictly demarcating (and hence, in contrast to fractals, terminating) the process or nonfractal changes in the shapes of the bodies themselves may give one pause, but, I admit, this is a matter of interpretation, since such figures may be read as indicating potential fractality of the type one finds in the Mandelbrot set.
Chaosmic Orders: Nonclassical Physics, Allegory, and the Epistemology of Blake's Minute Particulars.

Fractals, quarks and chaos theory share space with morphic resonance, channelling and UFOlore.

The way that the trunk of a tree divides into smaller and smaller branches and twigs is an approximate fractal pattern.

Fractals are used in the study of things like forked lightning and to produce some types of computer graphics.

Critical of science's objectification of nature, Smith constructs mechanized assemblages whose structural workings derive from natural laws and principles, such as chaos theory, fractal geometry and biomathematics.

This variant form of cauliflower is the ultimate fractal vegetable.

Many reports have been made about the applications of fractal theory in the fields of natural image simulation, image texture analysis, pattern recognition etc.

Another instance of fractal selfsimilarity: pomps are pageantries, but 'of mist' maps them onto themselves with a ratio of slightly more than one.
_Queen Mab_ as Topological Repertoire

These include such topics as histogram information, suggested palettes for lowcolor displays, fractalimage support and pixel calibration.

Topics ranged from the golden mean, spirals, and tessellations to fractals, origami, and impressionist art.

Fractals, using simple mathematical algorithm to produce complex objects typical of those which occur naturally.

With the help of catastrophe criterion, the anomalism criterions of the crossfeed between dam body and foundation are acquired from the real contact area or fractal dimension curve vs.

A number of fractal dimension algorithms were developed and evaluated using a field data set of 100 incidents.

Benoit Mandelbrot, the father of fractal theory and a pioneer in the study of market swings, argues that finance is prone to a "wild" randomness not usually seen in nature.

The metamorphosis of lion into lamb in canto viii is also fractal: we are unsure whether he is now exactly like a lamb.
_Queen Mab_ as Topological Repertoire

The reward for the reader is polysemous meaning, subtlety, fractal flavor bursts of emotion and idea.
BOOK VIEW CAFE BLOG » Writers on Writing: Sentence Structure Oopsies

But as the technology and computing power advanced, those working on the project would test more and more complex patterns, such as the most obvious candidates  fractals.

When the game starts, your selected object is presented in the center of a spherical space bounded by fractal walls.

The Von Koch snowflake is a fractal which is constructed from an equilateral triangle as follows.

More precisely, the recursive nature, which is essential for fractal models, is lacking in the procedural approach.

Each level of expression contributed to the next, and over the course of millennia we created mathematical theory, from basic arithmetic to algebra, from calculus to fractal geometry.

The appearance of fractal geometry extended our former understanding about dimension and scale of architecture.

The concept of fractal dimension offers a new way to image texture analysis, because it accords image texture a statistical description and exhibits complexity and roughness of texture.

In one of them, researcher Jonathan Mackenzie is using the ideas behind fractal geometry to synthesise sounds.

His unique knowledge of subjects such as fractal dynamics, subspace theory, a trivector system and others, has made this energetic feedback system possible.

(a) Vertebrate research; identify PC with excessive fractality, requiring excessive recursion (and thus, “junk DNA”).
Another unintelligent move  The Panda's Thumb

The approach of this paper was originally developed in to study the Hausdorff measure of certain parameterized families of fractals.

Urban is the congeries of population and economy, the urban system of an area has the trait of fractal at different timespace preconditions.

A close examination of our surroundings, as has been done by chaologists, has revealed a whole lot of fractal shapes every way they have looked: from seashores and clouds, even to the stock market.

But he spent 10 years consciously refining his pouring technique to increase the visual complexity of these fractal patterns.
Smithsonian Mag

Furthermore, could we explore the idea of the body as a dynamic nonlinear hologram existing within a fractal space?

Plus I'm using fractal imaging software (called apophysis) and hubble images to make some cool flying through space videos using Magix video software.
Lowell Sun Forum

The most famous computergenerated fractal is called the Mandelbrot set  a swirling, feathery, seemingly organic landscape that is reminiscent of the natural world, but is nonetheless completely virtual.
BBC News  Home

Nature builds its fractals using statistical selfsimilarity: the patterns observed at different magnifications, although not identical, are described by the same statistics.

The current visual novelty of abstractions derived from digital technology, fractal geometry and literary theory is usually the product of inventive imagination rather than empathic imagination.

The sharpwitted column, packed with cultural references, humor and accessible logic puzzles instead of academic jargon, featured the mathematical concepts behind fractals, Chinese tangram puzzles, and the art of surrealist M.C. Escher.
Martin Gardner, 95, a journalist, provided indepth analysis of Lewis Carroll's Cheshire Cat

To explain the concept of fractal dimension, it is necessary to understand what we mean by dimension in the first place.

The concept of fractal dimension offers a new way to image texture analysis, because it accords image texture a statistical description and exhibits complexity and roughness of texture.

Finally, the thesis does analysis on generative images of holo  embryo architecture from fractal dimension aspect.

Water tree of power cable is a fractal structure, which is attribute to diffusion  limited aggregation model.

Topics ranged from the golden mean, spirals, and tessellations to fractals, origami, and impressionist art.

While classical geometry assumes that objects exist in integer dimensions, fractal geometry deals with objects that have noninteger dimensions.

You are not claiming that DNA fractality is generally associated with metamorphosis, right?
Another unintelligent move  The Panda's Thumb

With these branching patterns, roots can be characterized by fractal analysis over a range of scales from about a quarter of a millimetre to several centimetres.

Many natural chaotic systems form fractals in the patterns that record the process.

Engineering superconductors to increase their fractality could yield materials that work at yet higher temperatures, he adds, making it much easier to harness them for practical applications. and you'll get:
New Scientist  Online News

The new sciences of complexity  fractals, nonlinear dynamics, the new cosmology, selforganizing systems  have brought about the change in perspective.

Importantly, such objects, known as fractals, turn out to be very useful for describing the shapes of physical objects such as mountains, coast lines, fractures, cracks and so on.

The relationship be  tween grain size and fractal dimension is investigated.

As such, I was not aware of the startling developments in fractalfirework technology.
July « 2008 « Morgan Dempsey

Divide as above and voila, you have your fractal dimension.

They've talked about Mondo 2000, virtual raves, about smart drugs and new psychedelics, about chaos theory and fractals.

He likes problems in diverse fields (harmonic analysis, fractals, and tilings, to name a few) that involve algebraic integers, such as Pisot and Salem numbers.

A fractal geometry of the most commonly used recursive algorithm painting cockscomb , just a few lines of code.

Storm goes to Egypt to help out on old friend, taking Jubilee, and getting involved in fractal magic.
Superhero Prose Fiction: XMen  The Ultimate XMen 09 Order From Chaos

During this time the writer also became increasingly curious about the link between science and spiritual potential, developing a keen interest in fractal geometry, quantum mechanics and chaos theory.

For example, a snowflake is a highlystructured, nearly fractal, characteristically hexagonal lattice of ice.
Demarcation, Credentials, and Science Education

Fractal geometry offered the quantitative description method for this kind of phenomenon.

These are fractal poems, insinuating themselves in the angles and nooks of every last lochan in sight.
The Times Literary Supplement

This result also solved a longstanding open problem concerning the existence of certain fractal sets in the plane.

A particle  cluster model for fractal aggregation that have been presented.

The lateral 'skeletonized' evolution of this fractal sequence suggests the final lower very lower high is close at hand.
Did Gladwell Libel Economists?, Arnold Kling  EconLog  Library of Economics and Liberty

Another part of us recognizes instinctively that nature designs in fractals, with intriguing detail on every scale from the microscopic to the macroscopic.

The first condition that must be fulfilled to apply scaling analysis techniques is that the growth behavior of a process lie in the fractal nature of the interface.

Qubits, foglets, gigaflops, haptic interfaces, probabilistic fractals: Kurzweil is not writing science for sissies.

Images are represented by measurements of their fractal dimension, fractal model fit, anisotropy, and rectilinearity.
Posthuman Blues

Of course, no class on maths and the visual arts would be complete without fractals.

Debt load becomes a primary factor in the fractal decay process, where assets are liquated in an attempt to pay down debt.
A Framing Puzzle, Bryan Caplan  EconLog  Library of Economics and Liberty

He showed how fractals can occur in many different places in both mathematics and elsewhere in nature.

The article analysis mine exploitation sink and resume infection feature to land use types, It utilize fractal dimension to analysis diggings sight pattern's direction of evolvement.

He argued that the combination of the personal computer, the Internet, and interest in fractals, chaos, and related mathematical topics has in recent years greatly facilitated the intertwining of art and math.

Definitely concur on the entheogen connection: at the height of every such experience, I've 'seen' great vistas of fractal patterns and structures.
Mandelbrot Fractals

Applying the fractal mathematics model, we can estimate the dimension of various kinds of cotton fiber surface, then, make the sample consulting the quantitative relation.

The relationship be  tween grain size and fractal dimension is investigated.

All of these elements are combined into seemingly aleatory compositions that, like fractals, operate on several scales at once.