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Topic: Cantor dust

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	Cantor set Summary
		Since the Cantor set is the complement of a union of open sets, it itself is a closed subset of the reals, and therefore a complete metric space.
		The Cantor set is a homogeneous space in the sense that for any two points x and y in the Cantor set C, there exists a homeomorphism f : C → C with f(x) = y.
		Cantor himself was led to it by practical concerns about the set of points where a trigonometric series might fail to converge.
	www.bookrags.com /Cantor_set (2571 words)

	Cantor, Georg - Famous mathematicians pictures, posters, gifts items, note cards, greeting cards, and prints
		Cantor's image is flanked by the "Aleph", the first letter of the Hebrew alphabet, which Cantor used (accompanied by subscripts) in his descriptions of transfinite numbers -- quite simply numbers which were not finite.
		The graphic set which backs Cantor's image began with an algorithm to generate the Cantor set, to which color was applied, and then universal operators related to color transition and magnification, ultimately resulting in a unique image whose essence was the Cantor set.
		Cantor came came to the conclusion that the Absolute was beyond man's reach, and identified this concept with God.
	www.mathematicianspictures.com /Mathematicians/Cantor.htm (509 words)

	Georg Cantor Summary
		Cantor, however, was poorly paid by the university, and he strove to obtain a better, more prestigious, teaching appointment in Berlin but was blocked by jealous professional rivals.
		Cantor's chief mathematical pursuit was a deeper understanding of the concept of infinity.
		Cantor died in a mental hospital in 1918.
	www.bookrags.com /Georg_Cantor (6336 words)

	Famous Fractals - Cantor Set (Site not responding. Last check: 2007-10-03)
		Cantor Set is one of the most famous fractal of all, yet it is the most simple one.
		The Cantor Set is composed of 2 identical shapes, each of which is 1/3 the size of the entire figure.
		The pattern of the Cantor Set was found in the rings of Saturn and in the spectra of some molecules.
	library.thinkquest.org /26242/full/fm/fm3.html (242 words)

	Georg Cantor biography .ms (Site not responding. Last check: 2007-10-03)
		Georg Ferdinand Ludwig Philipp Cantor (March 3, 1845 – January 6, 1918) was a mathematician who was born in Russia and lived in Germany for most of his life.
		Cantor is also known for his work on the unique representations of functions by means of trigonometric series (a generalized version of a Fourier series).
		Cantor recognized that infinite sets can have different sizes, distinguished between countable and uncountable sets and proved that the set of all rational numbers Q is countable while the set of all real numbers R is uncountable and hence strictly bigger.
	georg-cantor.biography.ms (507 words)

	Cantor set (Site not responding. Last check: 2007-10-03)
		Therefore, the numbers in the Cantor set can be mapped onto the numbers in [0, 1] by replacing every 2 in the ternary expansion with a 1, and treating the result as a binary expansion.
		It is worth noting that as a topological space, the Cantor set is homeomorphic to the product of countably many copies of the space {0, 1}, where each copy carries the discrete topology, as can easily be shown using the binary expansion used to prove its uncountability.
		This can be used to show that the Cantor set is homogeneous in the sense that for any two points x and y in the Cantor set C, there exists a homeomorphism f : C → C with f(''x'') = y.
	cantor-set.kiwiki.homeip.net (1391 words)

	How the program turns your numbers into a melody
		Cantor's dust is what you are left with if you start with a line, remove the middle third of that, the middle third of each one left, and so on.
		Cantor's dust has the paradoxical property of having no total length, yet having as many points in it as the complete line (see the Maths Encyclopedia article for details).
		The higher notes show the result of doing another Cantor's dust construction on each of the middle thirds that was removed at every stage, then another one on each one of those, and so on.
	www.tunesmithy.connectfree.co.uk /howthe.htm (1279 words)

	cantor dust n dimensions - SciForums.com
		for those who dunno cantor dust can be created by takin a line segment n removing the middle 3rd n continuing the process infinitely.
		The cantor set is a geometrical phenomenon that seems to duplicate the same error distribution for noise in a transmission line.
		The cantor set is fractal in nature and has an infinite number of points but has zero length.
	www.sciforums.com /showthread.php?t=37817 (999 words)

	Georg Cantor (Site not responding. Last check: 2007-10-03)
		The hostile attitude of many contemporaries is believed to have severely aggravated Cantor's emotional ailments and to have caused several nervous breakdowns.
		Today, the vast majority of mathematicians accept Cantor's work on transfinite sets and recognize it as a paradigm shift of major importance.
		Cantor was born in Saint Petersburg Russia, the son of a Danish merchant, Georg Waldemar Cantor, and a Russian musician, Maria Anna Böhm.
	georg-cantor.iqnaut.net (540 words)

	Sierpinski Gasket
		The set of points described here has been attributed to Cantor because of his attempts to imagine what happens when an infinite number of line segments are removed from an initial line interval.
		True Cantor dust is hard to illustrate because of its point nature.
		For the Cantor set described earlier, tau = 1/3 and therefore the dimension = log 2 / log 3 = 0.6309, ie: the dimension is somewhere between a point (dimension = 0) and a line (dimension = 1) Cantor dust can readily be created using L-Systems by using the following axiom and generator.
	local.wasp.uwa.edu.au /~pbourke/fractals/gasket/index.html (2172 words)

	Fractional Dementia
		So in a lower dimension Cantor's Dust (as it came to be known) was infinite when measured, but in the next possible step up it had a one-dimensional measure of zero.
		Let's take one of those dust segments of length n/3 (keep in mind its middle third would also be missing, and the middle thirds of what's left over, and so on in an actual representation of the dust).
		This is another property of dimensionality, defined as how "space-filling" an object is. If our Cantor dust fills.63 of a 1-dimensional plane, we can assume that something of a greater closeness to one would fill the space a little better, getting us closer to 1.0.
	www.imho.com /grae/chaos/fraction.html (2694 words)

	Cantor dust (Site not responding. Last check: 2007-10-03)
		A variant on this curve is given by the box fractal: divide the square into 9 equal parts and let the five diagonal parts remain.
		An alternative Cantor dust has as motif to divide the square into 16 equal parts and let only (arbitrary) four remain.
		Sometimes the Cantor dust name is given to the Cantor set.
	www.2dcurves.com /fractal/fractald.html (199 words)

	J.P. Louvet : Fractals - A History
		Cantor's dust is probably the most ancient known fractal figure (1872 ?)
		With a little imagination, this linear phenomenon made of points (the faulty bits) reminds us of Cantor dust, the elements of witch would have been mixed up.
		Mandelbrot studied the mathematical process that enables us to create random Cantor dust describing perfectly well the fractal structure of the batches of errors on computer lines.
	fractals.iut.u-bordeaux1.fr /jpl/history.html (2730 words)

	Georg Cantor - Wikipedia, the free encyclopedia
		Wangerin was eventually appointed, but he was never close to Cantor.
		Cantor retired in 1913, and suffered from poverty, even hunger, during World War I.
		Many sources say that Cantor was Jewish, and we find such references in, most prominently, the Encyclopedia Judaica (art.
	en.wikipedia.org /wiki/Georg_Cantor (3609 words)

	Cantor dust - Wikipedia, the free encyclopedia
		Cantor dust is a multi-dimensional version of the Cantor set.
		It can be formed by taking a finite cartesian product of the Cantor set with itself, making it a Cantor space.
		Like the Cantor set, Cantor dust has zero measure.
	en.wikipedia.org /wiki/Cantor_dust (72 words)

	Chaos Theory
		Cantor began to wonder what would happen when an infinite number of line segments were removed from an initial line interval.
		In order to understand Cantor Dust, start with a line; remove the middle third; then remove the middle third of the remaining segments; and so on.
		As the transmissions were analyzed to smaller and smaller degrees, it was determined that such dusts, as in the Cantor Dust, were indispensable in modeling intermittency.
	www.fractalfinance.com /chaostheory.html (3538 words)

	The Cantor Set - Examples of Chaos - IMO (Site not responding. Last check: 2007-10-03)
		The Cantor Middle-Thirds Set is an example of a fractal on the real number line.
		This set is seen on the interval between 0 and 1 on the number line.
		The construction of the Cantor Set is pretty simple: (In this example, a line represents a set of numbers, and removing a section is analogous to taking out that part of the set):
	library.thinkquest.org /2647/chaos/cantor.htm (325 words)

	Cantor Set (PRIME)
		We can extend this to a function, called the Cantor function, from the entire unit interval onto itself, by simply agreeing to let its value on the missing intervals be the constant values which equal the values of the original function on the endpoints of those intervals.
		Moreover, every point of the Cantor set is an accumulation point, since within any neighborhood of a number whose ternary expansion consists entirely of 0’s and 2’s one may find other such numbers.
		The Cantor set is an instructively simple example of a fractal, demonstrating that our geometrical intuitions about space (even such simple spaces as the unit interval) can fail to capture much of the deep structure inherent in those very intuitions.
	www.mathacademy.com /pr/prime/articles/cantset/index.asp (1244 words)

	Cantor's Dust and Transmission Errors
		Mandelbrot and the mathematical use of fractals in the form of Cantor’s dust.
		In essence a Cantor set is an interval of numbers from zero to one as represented by a line segment (Gleick, 1987: 92).
		From this Mandelbrot was able to help the engineers by concluding that instead of increasing signal strength to drown out more noise, the engineers should settle for a modest signal and accept the inevitablitity of errors and make use of a strategy of redundancy to find and correct them (Gleick, 1987: 92).
	www.authorhouse.com /BookStore/ItemDetail~bookid~18854.aspx (592 words)

	Fractal History
		The first is a "connected" set which covers a finite area on the X-Y plane; the second is a "not-connected", or NC, set which consists merely of a cloud of discrete points in a region of the X-Y plane.
		This cloud of points was originally called a "Fatou dust" by a few European authors.
		It is now called a "Cantor dust set" by almost everyone.
	home.att.net /~Fractalia/history.htm (661 words)

	[No title] (Site not responding. Last check: 2007-10-03)
		> > From another point of view, the number of steps to produce Cantor dust is > enumerable, but the number of points in the dust is Aleph-one (not > one-to-one with the rationals).
		Suppose I try to make Cantor dust by a different process: instead of subdividing every interval on each step, suppose I only subdivide the largest interval, by removing a subinterval from the middle of it, dividing the remainder into two subintervals.
		This is true of Cantor dust -- for any point in the set, you can find a neighborhood whose boundaries are outside the set, and for any such neighborhood, you can find a smaller one.
	www.canonical.org /~kragen/named-msgs/aleph-null-and-aleph-one (470 words)

	Rossler Attractor (Site not responding. Last check: 2007-10-03)
		The Cantor set is simple to create; take a line, and trisect it; then cut out the middle third.
		One will then have the "Cantor set," or "Cantor dust." This is an infinite number of infinitely small points arranged in a definite pattern.
		Rossler's attractor displays a type of banding, which suggests that perhaps it is related to the Cantor set.
	www.zeuscat.com /andrew/chaos/rossler.html (317 words)

	Untitled
		Thus if the middle of each line of the square was removed and the same process repeated on the remaining lines, what is remaining is a Cantor "dust" [named after the brilliant mathematician] which approaches a collection of zero dimensional points.
		This fractal dust starts off as a solid block of matter, which is divided into stacks of smaller blocks.
		Cantor sets seem to describe cars on a crowded highway, cotton price fluctuations since the nineteenth century, and the rising and falling of the River Nile over more than 2,000 years.
	www.music-mind.com /resour3.htm (13450 words)

	Julia Sets
		The Cantor set of points is a totally disconnected set produced by successively dividing the line segment [0,1] in thirds and discarding the center segment yielding [0,1/3] and [2/3, 1], then repeating for each remaining line segment ad infinitum (Peitgen el al 68).
		The distribution of points in a disconnected Julia set qualitatively resembles the appearance of the more easily envisioned Cantor dust in that they are totally disconnected.
		However, the fractal (non-topological) dimension of fractals (such as the Cantor dust of points or the Sierpinski gasket) incorporates the concept that their infinite ramifications in effect cover more of their Euclidean space than their topological dimension would suggest.
	www.mcgoodwin.net /julia/juliajewels.html (4935 words)

	The Nature of Fractal Music by Solomon
		As an example, one of the first and simplest fractals is called the Cantor fractal, which is based on the Cantor set or function.
		Thus, each of the steps is a binary division of the result of the previous step, and the whole process generates the number series: 1, 2, 4, 8, 16, etc. Looking at the process in reverse, the smaller units are added in binary groups to form the larger units.
		The Cantor fractal has a fractional dimension of 0.63, i.e., lying between a point and a line.
	solomonsmusic.net /fracmus.htm (2502 words)

	College of the Holy Cross \| Iris and B. Gerald Cantor Art Gallery (Site not responding. Last check: 2007-10-03)
		I parallel two days of disaster created by the three groups on August 6th 1945 (Hiroshima) and September 11th 2001 (New York City).
		They are times of horrific destruction and sadness, but the contradiction lies in its own terrific beauty of smoke, fire, and dust.
		The aim of this installation is to allow the viewer to look outside his or her own culture.
	www.holycross.edu /departments/cantor/website/precipicemd.html (198 words)

	CMJ Contents: May 1998
		The Feigenbaum diagram (or bifurcation diagram) of a 1-parameter family of maps on the real line shows the asymptotic states for a typical orbit, for a range of values of the parameter.
		However, it often happens that above a certain parameter value almost all orbits diverge to infinity; only a Cantor set of points remains invariant under the map and this Cantor set cannot be shown realistically on a computer screen.
		Examples are given for the family of tent maps, the logistic maps, and the quadratic maps associated with the Mandelbrot set.
	www.maa.org /pubs/cmj_may98.html (1158 words)

	Introduction to Fractal Images
		It is true that you could see a single fl pixel if one of your grid points coincided with a point of a Cantor dust cloud.
		Since the point z=(0,0) is a member of all connected Julia sets, all that is necessary is to test that one point for the given C. If the following sequence does not diverge, then the corresponding Jset for C is connected.
		This sequence is clearly just a function of C. Treating "C" as a variable, select a grid of points for C and determine which ones have an iterative series, starting from zero, which does not diverge.
	home.att.net /~Fractalia/math.htm (1541 words)

	Chaos and Fractals in Financial Markets, Part 2, by J. Orlin Grabbe
		What George Cantor created was an object whose dimension was more than 0 but less than 1.
		Cantor dust is a fractal with a Hausdorff dimension of.6309 and a topological dimension of 0.
		This is partially clarified in the discussion of Cantor dust, and further discussed in Part 3.
	www.aci.net /Kalliste/chaos2.htm (3503 words)

	The Nature of Fractals
		That depends on the type of fractal we are examining and in the equation that produced the figure in the first place.
		Possibly the first pure fractal object in history, the Cantor dust was described by the German mathematician Georg Cantor-inventor of set theory-around 1872.
		For the Cantor dust example, we start with a large segment (the initiator), divide it in three equal smaller segments, and take out the middle one.
	www.fractovia.org /art/what/what_ing1.shtml (560 words)

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