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Topic: Koch snowflake

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Fractal

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	CS212 S98 Problem Set 1
		For the von Koch snowflake, the initiator is an equilateral triangle.
		In the case of the von Koch snowflake, the zero'th figure would be an equilateral triangle, the first figure would be a a six-pointed star, the second figure would be a six-pointed star with little bumps on each point, etc.
		The snowflake figure is just one of a large family of figures that can be drawn using the same general method.
	www.cs.cornell.edu /courses/cs212/1998sp/psets/ps1.html (1561 words)

	efg's Fractals And Chaos -- von Koch Curve Lab Report
		The purpose of this project is to show how to create a von Koch curve, including a von Koch snowflake.
		Swedish mathematician Helge von Koch introduced the "Koch curve" in 1904.
		For the von Koch "snowflake," start with an equilateral triangle, and apply the "rule" above to each of the line segments.
	www.efg2.com /Lab/FractalsAndChaos/vonKochCurve.htm (696 words)

	Reference.com/Encyclopedia/Koch snowflake
		The Koch snowflake (or Koch star) is a mathematical curve and one of the earliest fractal curves to have been described.
		The lesser known Koch curve is the same as the snowflake, except it starts with a line segment instead of an equilateral triangle.
		The Koch curve is a special case of the de Rham curve.
	www.reference.com /browse/wiki/Koch_snowflake (623 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.
		While this may seem odd at first, doing this sort of thing is not at all uncommon.
		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 Snowflake -- Mudd Math Fun Facts
		One way to model a snowflake is to use a fractal which is any mathematical object showing "self-similarity" at all levels.
		Amazingly, the Koch snowflake is a curve of infinite length!
		You can see that the boundary of the snowflake has infinite length by looking at the lengths at each stage of the process, which grows by 4/3 each time the process is repeated.
	www.math.hmc.edu /funfacts/ffiles/20006.3.shtml (272 words)

	Department of Computing: Duncan White's Fractal Koch Curves Page
		I have written koch.cgi, a Perl CGI script which draws Koch curves using a turtle graphics implementation.
		One of the simplest type of fractals are those called Koch curves, or Koch Snowflakes.
		You can download the Koch CGI script and it's support modules here.
	www.doc.ic.ac.uk /~dcw/webprogs/koch.html (289 words)

	A+ Snowflakes - Make Paper Snowflake ornaments, Free Snow flake Patterns, Virtual snowflake designs
		Our easy paper snowflake instruction directions are simple to locate, just click on the grinning snow man for Pattern Instruction for Paper Snowflakes.
		A miniature masterpiece of symmetry, the snowflake pattern is known throughout the world.
		And because snowflake patterns are so symmetrical and beautiful they are frequently used in weaving, knitting, crochet, tatting, wood work, paper weaving and other fine arts and craft projects.
	www.papersnowflakes.com (411 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/KochSnowflake.html (258 words)

	Koch Snowflake
		The Koch Snowflake is the limiting image of the construction.
		The boundary of the snowflake consists of three copies of the Koch curve placed around the three sides of the initial equilateral triangle.
		The Koch snowflake is also known as the Koch island.
	ecademy.agnesscott.edu /~lriddle/ifs/ksnow/ksnow.htm (378 words)

	Koch Snowflake (Site not responding. Last check: )
		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 snowflake Summary
		Hence the area of the Koch snowflake is eight fifths of the area of the original triangle.
		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 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)

	Famous Fractals - Koch Snowflake (Site not responding. Last check: )
		Koch Snowflake is one of the most famous fractals.
		That is the area of the entire snowflake.
		The Koch Snowflake can be very beautifully filled in by any of the series of fractals called the Snowflake Sweeps.
	library.thinkquest.org /26242/full/fm/fm16.html (299 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.
		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-groups.dcs.st-and.ac.uk /~history/Biographies/Koch.html (742 words)

	Koch's Snowflake (Site not responding. Last check: )
		In his 1904 paper entitled "Sur une courbe continue sans tangente, obtenue par une construction géométrique élémentaire" he used the Koch Snowflake to show that it is possible to have figures that are continuous everywhere but differentiable nowhere.
		The Koch Snowflake is an example of a figure that is self-similar, meaning it looks the same on any scale.
		Therefore, while the perimeter of the snowflake, which is an infinite series, is continuous because there are no breaks in the perimeter, it is not differentiable since there are no smooth lines.
	www.math.ubc.ca /~cass/courses/m308-05b/projects/fung/page.html (444 words)

	Koch's Snowflake, Mandelbrot's Coastline, Alaska Science Forum
		Koch had shown that a finite area can be enclosed by an infinite line.
		In the 1960s, Koch's snowflake resurfaced in the hands of another mathematician given to upsetting his colleagues.
		Benoit Mandelbrot looked at the snowflake's edge and saw "a rough but vigorous model of a coastline," as he later said.
	www.gi.alaska.edu /ScienceForum/ASF9/920.html (780 words)

	Applications of Series
		The Koch snowflake is a geometric shape created by a repeated set of steps.
		Thus, we found that the area inside the Koch snowflake is finite, and that it sums up to eight fifths of the area in the original triangle.
		Thus, the perimeter of the Koch snowflake is infinite, even though its area is finite.
	www.ugrad.math.ubc.ca /coursedoc/math101/notes/series/serapp.html (549 words)

	The Koch Snowflake (Site not responding. Last check: )
		Similarly, once we have reached the nth iteration of the snowflake curve, the (n+1)st iteration is obtained by removing the middle third of each line segment and bridging the gap with two sides of a smaller equilateral triangle.
		The snowflake is the limiting figure obtained by this procedure.
		To see that the length of the boundary is infinite, just notice that the length of the curve at iteration n is 4/3 times the length of the curve at the previous iteration, as is apparent from the "replacement" sketch.
	csunix1.lvc.edu /~lyons/problems/snowflake.html (231 words)

	EROSION CONTROL \| FEATURE ARTICLE - The Fractal Nature of Erosion
		Discovered by mathematician Helge von Koch in 1904, this snowflake is a prime example of the fractal geometry related to the problem of measuring Great Britain's coastline.
		The snowflake is beginning to resemble the complexity of an eroded coastline.
		The von Koch snowflake and Sierpinski triangle are certainly bizarre shapes, but it is easier to harness these figures with mathematics than it is with the random coastline of Great Britain.
	www.forester.net /ecm_0205_fractal.html (3248 words)

	Chaos & Fractals: Fractals
		As an example, let us compute the dimension of the famous curve of Von Koch, which is sometimes referred to as the "Koch Snowflake." The Koch Snowflake is generated by a simple recursive geometric procedure:
		Its fractal dimension is given from the definition of the curve: N = 4 and r = 1/3 (remember 4 segments each 1/3 size of the original line segment).
		Another interesting property of the Koch Snowflake is that it encloses a finite area with an infinite perimeter.
	www.pha.jhu.edu /~ldb/seminar/fractals.html (472 words)

	Michael Levitin - triadic von Koch snowflake (Site not responding. Last check: )
		Heat Equation on the Triadic Von Koch Snowflake
		The figure shows the distribution of temperature inside the region (von Koch snowflake) with fractal boundary.
		At the initial moment the region is kept at temperature 1, and at all the times the boundary is kept at temperature 0.
	www.ma.hw.ac.uk /~levitin/snowflake.html (82 words)

	The Philosopher’s Stone ‘A Stone Most Precious’ by Kris Weber Sherwood
		The spectacular ‘Snowflakes’ of 1997 had multiple influencing factors instrumental in their creation, and there were at least three major events of Collective Conscious focus that, in my view, contributed directly to the genesis of their particular designs.
		Snowflakes and six-pointed stars were in the forefront of consciousness as the new Crop Circle season began; the genuine ‘Circlemaker’ had answered back stunningly, but it was just one layer of the fantastic ‘glass onion’ in store for us in 1997.
		Browsing in the Avebury bookshop after visiting the fresh Koch I Snowflake formation in July of 1997, I was drawn to a book of Runes that I bought and tucked away for the long flight back to California.
	www.cropcircleconnector.com /Millennium/Philosopher.html (2450 words)

	Snowflake Curve (Site not responding. Last check: )
		The snowflake curve has some interesting properties that may seem paradoxical.
		The snowflake curve is connected in the sense that it does not have any breaks or gaps in it.
		The snowflake never escapes the dashed square you see in figures 1-4, so it encloses a finite amount of area no larger than a credit card.
	scidiv.bcc.ctc.edu /Math/Snowflake.html (315 words)

	Koch Snowflake
		Snowflakes can be seen as one of nature's beautiful works of art.
		The Koch Snowflake is not found in nature, but is an interesting shape that looks somewhat like a real snowflake.
		The Koch Snowflake is made of many different sized equilateral triangles.
	educ.queensu.ca /~fmc/january2003/KochSnowflake.html (482 words)

	Koch snowflake
		The Koch curve has infinite length because each iteration increases the length of a line segment one third, and the iterations go on forever.
		The cube has side length t/√2, where t is the length of one of the edges of the regular tetrahedron you started with.
		Variations on the flat Koch snowflake include the so-called exterior snowflake, the Koch antisnowflake, and the flowsnake curves.
	www.daviddarling.info /encyclopedia/K/Koch_snowflake.html (272 words)

	Koch Snowflake (Site not responding. Last check: )
		This "snowflake" was created by Niels Fabian Helge von Koch, to demonstrate a curve which is continuous but NOWHERE differentiable (i.e.
		The Koch snowflake has infinite perimeter (since it becomes 4/3 as long after each iteration), but finite area (if you circumscribed a circle about the original triangle, the end snowflake would fit in the same circle).
		This is similar to the Koch Quadric Island and to many other fractals.
	wso.williams.edu /~nyates/fractal/kochsnowflake.html (163 words)

	IFS Attractor: Koch Snowflake (Site not responding. Last check: )
		A single size of the Koch snowflake does not tile the plane, but the plane may be tiled with a unit consisting of 1 larger copy and 2 smaller copies.
		An alternative construction for the Koch Snowflake uses 7 large and 6 small copies.
		This construction allows fractals to be derived from the Koch Snowflake, in addition to those that can derived from the 1
	www.meden.demon.co.uk /Fractals/kochsnowflake.html (91 words)

	Make a math Paper Koch Snowflake \| math instruction for koch snowflake
		Koch Snowflakes are made out of triangle shapes
		A pre-colored version is available in the Koch Snowflake Drag and Drop game.
		To create a giant Koch snowflake in the classroom, print these triangle shapes and have the children assemble them just like this picture.
	www.papersnowflakes.com /pages/pattern/koch_snowflake.htm (157 words)

	Shodor Interactivate: Koch's Snowflake
		Step through the generation of the Koch Snowflake -- a fractal made from deforming the sides of a triangle, and explore number patterns in sequences and geometric properties of fractals.
		This activity allows the user to step through the process of building the Koch Snowflake.
		This activity would work well in groups of two to four for about thirty to thirty-five minutes if you use the exploration questions and five minutes otherwise.
	www.shodor.org /interactivate/activities/KochSnowflake (168 words)

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