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Home > Category Index for Science Quotations > Category Index F > Category: Fractal

Fractal Quotes (9 quotes)


A fractal is a mathematical set or concrete object that is irregular or fragmented at all scales.
Cited as from Fractals: Form, Chance, and Dimension (1977), by J.W. Cannon, in review of The Fractal Geometry of Nature (1982) in The American Mathematical Monthly (Nov 1984), 91, No. 9, 594.
Science quotes on:  |  Concrete (21)  |  Definition (152)  |  Fragment (24)  |  Irregular (4)  |  Mathematics (587)  |  Object (110)  |  Scale (49)  |  Set (56)

Fractal geometry will make you see everything differently. There is a danger in reading further. You risk the loss of your childhood vision of clouds, forests, flowers, galaxies, leaves, feathers, rocks, mountains, torrents of water, carpet, bricks, and much else besides. Never again will your interpretation of these things be quite the same.
Fractals Everywhere (2000), 1.
Science quotes on:  |  Brick (12)  |  Carpet (3)  |  Cloud (44)  |  Feather (10)  |  Flower (65)  |  Forest (88)  |  Galaxy (38)  |  Geometry (99)  |  Interpretation (61)  |  Leaf (43)  |  Mountain (111)  |  River (68)  |  Rock (107)  |  Understanding (317)

Fractal is a word invented by Mandelbrot to bring together under one heading a large class of objects that have [played] ... an historical role ... in the development of pure mathematics. A great revolution of ideas separates the classical mathematics of the 19th century from the modern mathematics of the 20th. Classical mathematics had its roots in the regular geometric structures of Euclid and the continuously evolving dynamics of Newton.? Modern mathematics began with Cantor's set theory and Peano's space-filling curve. Historically, the revolution was forced by the discovery of mathematical structures that did not fit the patterns of Euclid and Newton. These new structures were regarded ... as 'pathological,' ... as a 'gallery of monsters,' akin to the cubist paintings and atonal music that were upsetting established standards of taste in the arts at about the same time. The mathematicians who created the monsters regarded them as important in showing that the world of pure mathematics contains a richness of possibilities going far beyond the simple structures that they saw in Nature. Twentieth-century mathematics flowered in the belief that it had transcended completely the limitations imposed by its natural origins.
Now, as Mandelbrot points out, ... Nature has played a joke on the mathematicians. The 19th-century mathematicians may not have been lacking in imagination, but Nature was not. The same pathological structures that the mathematicians invented to break loose from 19th-century naturalism turn out to be inherent in familiar objects all around us.
From 'Characterizing Irregularity', Science (12 May 1978), 200, No. 4342, 677-678. Quoted in Benoit Mandelbrot, The Fractal Geometry of Nature (1977, 1983), 3-4.
Science quotes on:  |  Euclid (28)  |  Idea (440)  |  Imagination (209)  |  Benoit Mandelbrot (13)  |  Mathematician (177)  |  Monster (21)  |  Nature (1029)  |  Sir Isaac Newton (258)  |  Nomenclature (129)  |  Painting (24)  |  Pathological (5)  |  Pure Mathematics (27)  |  Revolution (56)  |  Structure (191)

Fractals are patterns which occur on many levels. This concept can be applied to any musical parameter. I make melodic fractals, where the pitches of a theme I dream up are used to determine a melodic shape on several levels, in space and time. I make rhythmic fractals, where a set of durations associated with a motive get stretched and compressed and maybe layered on top of each other. I make loudness fractals, where the characteristic loudness of a sound, its envelope shape, is found on several time scales. I even make fractals with the form of a piece, its instrumentation, density, range, and so on. Here I’ve separated the parameters of music, but in a real piece, all of these things are combined, so you might call it a fractal of fractals.
Interview (1999) on The Discovery Channel. As quoted by Benoit B. Manelbrot and Richard Hudson in The (Mis)Behaviour of Markets: A Fractal View of Risk, Ruin and Reward (2010), 133.
Science quotes on:  |  Applied (15)  |  Associated (2)  |  Call (68)  |  Characteristic (66)  |  Combined (3)  |  Compressed (3)  |  Concept (102)  |  Density (11)  |  Determine (45)  |  Dream (92)  |  Duration (9)  |  Envelope (5)  |  Form (210)  |  Instrumentation (3)  |  Level (51)  |  Loudness (3)  |  Motive (26)  |  Music (66)  |  Musical (3)  |  Occur (26)  |  Parameter (2)  |  Pattern (56)  |  Piece (32)  |  Pitch (7)  |  Range (38)  |  Real (95)  |  Rhythmic (2)  |  Scale (49)  |  Science And Art (157)  |  Separate (46)  |  Set (56)  |  Shape (52)  |  Sound (59)  |  Theme (8)  |  Thing (37)  |  Time And Space (30)  |  Top (20)

I conceived and developed a new geometry of nature and implemented its use in a number of diverse fields. It describes many of the irregular and fragmented patterns around us, and leads to full-fledged theories, by identifying a family of shapes I call fractals.
The Fractal Geometry of Nature (1977, 1983), Introduction, xiii.
Science quotes on:  |  Nomenclature (129)  |  Pattern (56)  |  Shape (52)  |  Theory (582)

In the mind's eye, a fractal is a way of seeing infinity.
From Chaos (1987), 98.
Science quotes on:  |  Eye (159)  |  Infinity (59)  |  Mind (544)  |  Seeing (48)

The existence of these patterns [fractals] challenges us to study forms that Euclid leaves aside as being formless, to investigate the morphology of the amorphous. Mathematicians have disdained this challenge, however, and have increasingly chosen to flee from nature by devising theories unrelated to anything we can see or feel.
The Fractal Geometry of Nature (1977, 1983), Introduction, xiii.
Science quotes on:  |  Challenge (37)  |  Euclid (28)  |  Mathematician (177)  |  Nature (1029)  |  Pattern (56)  |  Sense (240)  |  Study (331)  |  Theory (582)

The most complex object in mathematics, the Mandelbrot Set … is so complex as to be uncontrollable by mankind and describable as “chaos.”
Science quotes on:  |  Chaos (63)  |  Complexity (80)

Why is geometry often described as “cold” and “dry?” One reason lies in its inability to describe the shape of a cloud, a mountain, a coastline, or a tree. Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line… Nature exhibits not simply a higher degree but an altogether different level of complexity.
From The Fractal Geometry of Nature (1977, 1983), Introduction, xiii.
Science quotes on:  |  Bark (4)  |  Circle (28)  |  Cloud (44)  |  Coast (11)  |  Complexity (80)  |  Cone (5)  |  Geometry (99)  |  Lightning (28)  |  Line (44)  |  Mountain (111)  |  Nature (1029)  |  Shape (52)  |  Smooth (13)  |  Sphere (40)  |  Tree (143)


Carl Sagan Thumbnail In science it often happens that scientists say, 'You know that's a really good argument; my position is mistaken,' and then they would actually change their minds and you never hear that old view from them again. They really do it. It doesn't happen as often as it should, because scientists are human and change is sometimes painful. But it happens every day. I cannot recall the last time something like that happened in politics or religion. (1987) -- Carl Sagan
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