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Topic: Koch curve

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	Koch Curve (Site not responding. Last check: 2007-09-18)
		Koch constructed his curve in 1904 as an example of a non-differentiable curve, that is, a continuous curve that does not have a tangent at any of its points.
		Therefore the length of the Koch curve is infinite.
		Three copies of the Koch curve placed around the three sides of an equilateral triangle, form a simple closed curve that form the boundary of the Koch snowflake.
	ecademy.agnesscott.edu /~lriddle/ifs/kcurve/kcurve.htm (563 words)

	PlanetMath: Koch curve
		A Koch curve is a fractal generated by a replacement rule.
		A Koch snowflake is the figure generated by applying the Koch replacement rule to an equilateral triangle indefinitely.
		This is version 4 of Koch curve, born on 2002-01-03, modified 2005-02-28.
	planetmath.org /encyclopedia/KochCurve.html (258 words)

	Koch Curves and Snowflakes
		The Koch curve was, unsurprisingly, constructed by the mathematician Niels Koch in 1904 as a demonstration of a non-differentiable continuous curve, i.e.
		As each iteration increases the length of the curve by a factor of 4/3 it is clear that after n generations the curve is of length (4/3)^n, assuming that the initial 'curve' is of length 1.
		The Koch snowflake is an attractive visualisation of the Koch curve.
	people.bath.ac.uk /ma3gkwc/koch.html (719 words)

	Curve
		This definition of curve captures our intuitive notion of a curve as a connected, smooth figure that is "like" a line; although it also includes figures that are not called curves in common usage.
		For example, a mathematical curve can have any number of "kinks" or corners, since these have no effect on whether or not the resulting figure can be "stretched" and smoothed into a straight line; this is roughly the meaning of "can be continuously mapped".
		Some extreme examples of "kinky" curves are the fractal Koch curve and the dragon curve[?].
	www.ebroadcast.com.au /lookup/encyclopedia/cu/Curve.html (238 words)

	Koch snowflake
		The Koch curve is one of the earliest fractal curves to have been described, appearing in a 1906 paper entitled "Une méthode géométrique élémentaire pour l'étude de certaines questions de la théorie des courbes plane" by the Swedish mathematician Helge von Koch (1870 - 1924) [1].
		The Koch curve is the limit which you approach as you follow the above steps over and over again.
		The Koch snowflake is the same as the above except you start with an equilateral triangle instead of a line segment.
	www.ebroadcast.com.au /lookup/encyclopedia/ko/Koch_curve.html (205 words)

	Koch Curves, general discussion
		Clearly, the generation of a Koch curve is a prime candidate for recursion.
		Notice as well that when the curve "grows" from one iteration to the next, that all the straight line segments transition from the base case Koch curve (straight line) to the non-base case Koch curve (a pattern of Koch curves) which is the series of straight lines.
		A Koch snowflake is a Koch curve whose base case consists of a set of Koch curves all in their base cases.
	www.owlnet.rice.edu /~comp212/02-fall/projects/kochCurve/koch.html (463 words)

	Koch snowflake Summary
		Incidentally, the self-similarity dimension of the Koch curve is equal to its Hausdorff dimension.
		The Koch snowflake (or Koch star) is a mathematical curve, and one of the earliest fractal curves to have been described.
		The Koch curve is a special case of the de Rham curve.
	www.bookrags.com /Koch_snowflake (1195 words)

	Koch curve - ExampleProblems.com
		The Koch curve is a mathematical curve, and one of the earliest fractal curves to have been described.
		The better known Koch snowflake (or Koch star) is the same as the curve, except it starts with an equilateral triangle instead of a line segment.
		The Koch curve is continuous, but not differentiable anywhere.
	www.exampleproblems.com /wiki/index.php?title=Koch_curve&printable=yes (459 words)

	Koch Snowflake
		Like the Sierpinski curve, the Koch curve is a closed limit curve of infinite length that bounds a region of finite area.
		But unlike the Hilbert and Sierpinski curves, the Koch snowflake is a fractal curve that is not plane-filling.
		Thus the Koch snowflake is an infinitely long curve that encloses a finite area.
	home.comcast.net /~davebowser/fractals/koch.html (457 words)

	Koch curve (Site not responding. Last check: 2007-09-18)
		Koch constructed the Koch curve in 1904 as an example of a continuous, non-differentiable curve.
		The curve is a base motif fractal which uses a line segment as base.
		Three copies of the Koch curve placed at the the sides of an equilateral triangle, form a Koch snowflake:
	www.2dcurves.com /fractal/fractalk.html (121 words)

	Von Koch Curves
		The genuine Von Koch curve, also called snowflake curve, is derived as the limit of a polygonal contour.
		Ideally the process should go on indefinitely, but, in practice, the curve displayed on the screen no longer changes when the elementary side becomes less than the pitch, and then the iterations can be stopped.
		For a while, it was thought that such a curve was a possible starting point for the design of tools for drawing complex natural curves (such as the rocky coast of the celebrated example of Benoît Mandelbrot), with a little number of control parameters.
	perso.orange.fr /charles.vassallo/en/art/von_koch.html (632 words)

	Koch biography
		In July 1911 von Koch succeeded Mittag-Leffler as professor of mathematics at Stockholm University.
		Von Koch is famous for the Koch curve which appears in his paper Une méthode géométrique élémentaire pour l'étude de certaines questions de la théorie des courbes plane published in 1906.
		The von Koch snowflake is a continuous curve which does not have a tangent at any point.
	www-history.mcs.st-andrews.ac.uk /Biographies/Koch.html (742 words)

	De Rham curve - Wikipedia, the free encyclopedia
		The Cantor function, Minkowski's question mark function, the Lévy C curve, the blancmange curve and the Koch curve are all special cases of the general de Rham curve.
		The self-symmetries of all of the de Rham curves are given by the monoid that describes the symmetries of the infinite binary tree or Cantor set.
		The Cesaro-Faber and Peano-Koch curves are both special cases of the general case of a pair of affine linear transformations on the complex plane.
	en.wikipedia.org /wiki/De_Rham_curve (415 words)

	Fractals: Von Koch Curve
		Note: the figures are valid for 'classical' Von Koch curves, for which the similarity ratio is standard.
		The area of the curve on the next iteration continues to increase but to a smalller extent: 16 small triangles are added but their area are now 81 times smaller than the very first triangle of the Von Koch curve.
		Still, this very interesting property of the Von Koch curve: its area converges rapidly to a finite limit while the total length of the segments that composed that curve have no limit.
	users.skynet.be /TGMDev/curvevonkoch.htm (1208 words)

	The Koch Curve and Visual Resolution at Nonoscience (Site not responding. Last check: 2007-09-18)
		The Koch Curve or the von Koch snowflake was discovered by Helge von Koch (1870-1924) in 1904.
		It is a closed fractal curve of infinite length within a finite region of space, enclosing a finite area.
		This is the “zeroth” iteration of the Koch curve.
	www.arunn.net /scienceblog/2006/08/28/the-koch-curve-and-visual-resolution (1916 words)

	sam.ufm Helpfile (Site not responding. Last check: 2007-09-18)
		As it is composed of circles and uses an algorithm similar to the one for the Koch Curve, I gave it this name...
		Depending on which default gradient you have, the Koch Curve may appear all fl when you load the formula.
		Maybe the two transform Koch Curve Scissor and Rotating Koch Curve Scissor will be more useful to create interesting images.
	www.p-gallery.net /help/kochcurves.htm (283 words)

	Fractal dimension
		See this figure for an example of a curve approximated by a sequence of line segments, which is usually called a polygonal curve.
		The generations produced by the L-system associated to the Koch curve (see this figure) are in fact polygonal approximations to the limit curve.
		This constant is the fractal dimension of the Koch curve.
	www.math.okstate.edu /mathdept/dynamics/lecnotes/node37.html (767 words)

	NetLogo Models Library: Koch Curve
		The Koch curve is as difficult to understand as the Cantor set or the Sierpinski tree.
		Actually Koch's motivation for finding this curve was to provide another example for the discovery made by the German mathematician Karl Weierstrass, who in 1872 had precipitated a minor crisis in mathematics.
		It therefore might come as a surprise that the area enclosed by the Koch curve is finite; the proof of this we leave as an exercise for the reader.
	ccl.northwestern.edu /netlogo/models/KochCurve (767 words)

	The Koch Snowflake
		However, the inner area of the Koch curve remains less than the area of a circle drawn around the original triangle.
		The Koch curve is rougher than a smooth curve or line, which has one dimension.
		So it makes sense that the dimension of the Koch curve is somewhere in between the two.
	home.iprimus.com.au /ajwalker/mh/koch.htm (210 words)

	Recursion and making a Koch Snowflake with Maple
		In the previous section, we didn't give the specifics of exactly how we generated the Koch curve of level 5 which we showed.
		Since the turtle command for the curve has 2388 letters, certainly it isn't something you would expect someone to type in directly.
		This is an infinite length ``curve'' which bounds a finite area, and resembles a snowflake.
	www.math.sunysb.edu /~scott/Book331/Recursion_making_Koch.html (839 words)

	Koch biography
		The second of von Koch's papers was published in 1892, the year in which von Koch was awarded a doctorate for his thesis which contained the results of the two papers.
		In July 1911 von Koch succeeded Mittag-Leffler as professor of mathematics at Stockholm University.
		Von Koch is famous for the Koch curve which appears in his paper Une méthode géométrique élémentaire pour l'étude de certaines questions de la théorie des courbes plane published in 1906.
	www-groups.dcs.st-and.ac.uk /~history/Biographies/Koch.html (742 words)

	Koch and Sierpinski Fractals
		The Koch Snowflake is one of the oldest known fractals that was designed and studied.
		The Koch Curve was actually studied first by von Koch and the Snowflake came from that.
		To be more specific, if you took a piece from one section of the curve and another piece from another section they will be practically identical.
	www.people.iup.edu /plxk/final/koch_home.htm (208 words)

	Turtle graphics and L-systems
		If we scale by a factor of three so that the starting and ending points of the curve remain the same, we may describe this production by saying that the middle third of the original line segment was replaced by the top of the equilateral triangle spanned by that middle third.
		The limit is a fractal curve of the type that caused some anxiety among the mathematicians (Weierstrass and others) of the nineteenth century who were at work building the rigorous foundations of the calculus.
		This curve is an example of a one-to-one continuous mapping of the unit interval into the two-dimensional plane which does not have a well-defined tangent line anywhere.
	www.math.okstate.edu /mathdept/dynamics/lecnotes/node18.html (734 words)

	Koch
		The second of von Koch's papers was published in 1892, the year in which von Koch was awarded a doctorate for his thesis which contained the results of the two papers.
		Von Koch was then appointed to the chair of pure mathematics at the KTH.
		At the end of his paper, von Koch gives a geometric construction, based on the von Koch curve, of such a function which he also expresses analytically.
	www.educ.fc.ul.pt /icm/icm2003/icm14/Koch.htm (634 words)

	L-System Based Fractals
		The Koch Curve fractal, one of the classic ones, is also one of the easiest ones to implement.
		The Hilbert Curve, proposed by David Hilbert, is called a space-filling fractal, because as you can see, it continuously attempts to fill in the empty area within.
		It is the square version of the Koch Curve which as it grows, becomes an interesting pyramid with lots of tiny steps and interesting decorations or tunnels into the pyramid.
	ejad.best.vwh.net /java/fractals/lsystems.shtml (1020 words)

	L-System Based Fractals
		The Koch Curve fractal, one of the classic ones, is also one of the easiest ones to implement.
		The Hilbert Curve, proposed by David Hilbert, is called a space-filling fractal, because as you can see, it continuously attempts to fill in the empty area within.
		It is the square version of the Koch Curve which as it grows, becomes an interesting pyramid with lots of tiny steps and interesting decorations or tunnels into the pyramid.
	arcytech.org /java/fractals/lsystems.shtml (1020 words)

	Koch curve exercise (Site not responding. Last check: 2007-09-18)
		The Koch curve is made by starting a line segment.
		Keep the first and last thirds, but replace the middle by a path that goes up 60 degrees for a distance length/3 and then comes back down to the original segment (angle -60 degrees).Continue this replacement process until lengths are small enough to make everyone happy, then finally draw a segment.
		It would be nice to rename everything from "ccurve" to "koch" so we are clear that this is a different program.
	www.csm.astate.edu /~rossa/cs3363/koch.html (130 words)

	Comp200 lecture -- Koch curve practice
		A level two Koch curve is like level one, except that each straight-line segment is replaced with a copy of the original level-one Koch curve (of appropriate size).
		To draw a level-three Koch curve, draw a small level-two curve, turn, draw another small level-two curve, turn again, draw a third level-two curve, turn, and finally draw the last level-two curve.
		A level three Koch curve is like level two, except that each of the level two's segments is replaced with a smaller copy of the (level-one) Koch curve.
	www.owlnet.rice.edu /~comp200/01spring/Lecture-notes/lect11-snowflake.html (638 words)

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