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Topic: Logarithmic spiral

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	Cams
		The logarithmic spiral is a mathematical curve which has the unique property of maintaining a constant angle between the radius and the tangent to the curve at any point on the curve (figure 1).
		A logarithmic spiral cam (a "constant angle cam") ensures that the line between the axle and the point of contact (the "line of force") is at a constant angle to the abutting surface, independent of how the cam is oriented.
		The mathematical equation for a logarithmic spiral is R=beaØ.
	www.bigwalls.net /climb/camf (1137 words)

	Logarithmic spiral - Wikipedia, the free encyclopedia
		The logarithmic spiral can be distinguished from the Archimedean spiral by the fact that the distances between the turnings of a logarithmic spiral increase in geometric progression, while in an Archimedean spiral these distances are constant.
		Logarithmic spirals are self-similar in that they are self-congruent under all similarity transformations (scaling them gives the same result as rotating them).
		Hawks approach their prey in a logarithmic spiral: their sharpest view is at an angle to their direction of flight; this angle is the same as the spiral's pitch.
	en.wikipedia.org /wiki/Logarithmic_spiral (847 words)

	Archimedean spiral - Wikipedia, the free encyclopedia
		This Archimedean spiral is distinguished from the logarithmic spiral by the fact that successive turnings of the spiral have a constant separation distance (equal to 2πb if θ is measured in radians), while in a logarithmic spiral these distances form a geometric progression.
		Virtually all static spirals appearing in nature are logarithmic spirals, not Archimedean ones.
		Many dynamic spirals (such as the Parker spiral of the solar wind, or the pattern made by a St.
	en.wikipedia.org /wiki/Archimedean_spiral (291 words)

	logarithmic spiral
		Hawks approach their prey in the form of a logarithmic spiral and their sharpest view is at an angle to their flight direction that is the same as the spiral's pitch.
		In polar coordinates (r, theta) the equation of the logarithmic spiral is
		It can be distinguished from the Archimedean spiral by the fact that the distance between the arms of a logarithmic spiral increase in a geometric sequence while in an Archimedean spiral this distance is constant.
	www.daviddarling.info /encyclopedia/L/logarithmic_spiral.html (405 words)

	Equiangular (Site not responding. Last check: 2007-11-04)
		The equiangular spiral was invented by Descartes in 1638.
		The evolute and the involute of an equiangular spiral is an identical equiangular spiral.
		The caustic of the equiangular spiral, where the pole is taken as the radiant, is an equal equiangular spiral.
	www-groups.dcs.st-and.ac.uk /~history/Curves/Equiangular.html (317 words)

	logarithmic spiral
		The logarithmic relation between radius and angle leads to the name of logarithmic spiral or logistique (in French).
		The logarithmic spiral is the curve for which the angle between the tangent and the radius (the polar tangent) is a constant.
		For that proportionality the curve bears the name of the growth spiral: a growth that is proportional to its size.
	www.2dcurves.com /spiral/spirallo.html (652 words)

	The Logarithmic Spiral (Site not responding. Last check: 2007-11-04)
		The logarithmic spiral was dubbed the "Spira Mirabilis" by Jakob Bernouilli.
		The logarithmic spiral occurs in nature more than any other curve, be it in the form of the seeds on a Sunflower or the shape of the Nautilus shell.
		Probably the most important aspect of the logarithmic spiral is that if we increase the angle ø by equal amounts, the distance r from the poles increases in equal ratios.
	www.bath.ac.uk /~ma2mrm/logarithmicspiral.html (313 words)

	Spirals and the Golden Section by John Sharp for the Nexus Network Journal vol.4 no.1 (Winter 2002)
		Therefore the spiral is a spiral of Archimedes.
		Spirals of shells, particularly the nautilus are logarithmic spirals.
		In order to approximate the multiplication factor for the nautilus logarithmic spiral, measurements were taken for four different 360° rotations of the spiral and the ratio of the radial vectors calculated for each rotation.
	www.maths.tcd.ie /EMIS/journals/NNJ/Sharp_v4n1-pt04.html (1243 words)

	The Nautilus and the Human Embryo and the Golden Ratio (Site not responding. Last check: 2007-11-04)
		Logarithmic spirals are simply spirals that increase at a logarithmic rate.
		The inversion of an equiangular spiral with respect to its pole is an equal spiral.
		The pedal of an equiangular spiral with respect to its pole is an equal spiral.
	www.geocities.com /CapeCanaveral/Station/8228/spiral.htm (737 words)

	Spiral Page
		Long before the discovery of spiral galaxies, the ancients have used the spiral image to represent the universe, the earth's rotation, the moon's orbit, and as a symbol for growth.
		The spiral may be found in three forms: expanding (as in the nebula galaxies), contracting (like whirlwinds or whirlpools), or ossified (like the snail's shell).
		In yoga, the spiral is associated with the Kundalini force dormant at the base of the spine.
	www.wisdomportal.com /Geometry/SpiralPage.html (496 words)

	Part 1 of Spirals and the Golden Section by John Sharp for the Nexus Network Journal vol.4 no.1 (Winter 2002)
		spiral is a plane curve that arises as a result of the movement of a point away from (or towards) a centre combined with a rotation about the centre.
		This spiral can be thought of as being based on multiplication or division since the spiral is formed from the rule that for a given rotation angle (like one revolution) the distance from the pole is multiplied by a fixed amount.
		This is one point of contrast with the Archimedean spiral, the other being that it is not possible to draw the curve all the way to the pole since you need infinite division to reach it.
	www.maths.tcd.ie /EMIS/journals/NNJ/Sharp_v4n1-pt01.html (1233 words)

	Logarithmic spiral - (Site not responding. Last check: 2007-11-04)
		Image:Logarithmic spiral.png Image:NautilusCutawayLogarithmicSpiral.jpg Image:Low pressure system over Iceland.jpg Image:Messier51.jpg A logarithmic spiral, equiangular spiral or growth spiral is a special kind of spiral curve which often appears in nature.
		For b >1 the spiral expands with increasing θ, and for b <1 it contracts.
		One can construct approximate logarithmic spirals with pitch about 17.03239 degrees using Fibonacci numbers or the golden mean as is explained in those articles.
	psychcentral.com /psypsych/Logarithmic_spiral (851 words)

	Notes on the Logarithmic Spiral
		In the second proposition Newton showed that the logarithmic spiral would also be described by a particle attracted to the pole by a force proportional to the square of the density of the medium in which it moves, while this density is at each point inversely proportional to its distance from the pole.
		That the logarithmic spiral is a projection of a certain "elliptic logarithmic spiral" was shown in W. Hamilton, Elements of Quaternions, London, 1866, pp.
		The xylonite logarithmic spiral curve (eight inches in width) sold by Keuffel and Esser Co., New York, furnishes the means for accurately and rapidly drawing the curve.
	www.spirasolaris.ca /rcarchibald.html (7069 words)

	SPACE.com -- Spirals in Nature: The Magical Number behind Hurricanes and Galaxies
		The Golden Ratio also describes the ever-expanding nature of what is termed a logarithmic spiral, not to be confused with the boring spiral created by a roll of toilet paper.
		You've probably seen the logarithmic spiral in a familiar seashell belonging to a creature called the chambered nautilus.
		Livio said the logarithmic spiral is a key shape for anything that grows, because with growth the ratio does not change.
	www.space.com /scienceastronomy/perfect_spirals_030917.html (1213 words)

	Logarithmic Spiral Tilings
		A logarithmic spiral has the polar equation r=exp(ka) where r is the radius and a is the azimuth.
		The radial cells formed by the spiral and its radii correspond to the unit cells or period parallelograms of the plane tesselation.
		This is the logarithmic spiral equivalent of the Greek Cross tiling.
	www.uwgb.edu /dutchs/symmetry/log-spir.htm (587 words)

	Home
		A spiral is a curve that winds around a center of rotation at an ever-increasing distance from it.
		To better understand the logarithmic spiral, look first at the linear spiral, a spiral drawn on a linear chart, a price-time graph in which both dimensions are linear.
		The key is that each point of the spiral has a price component and a time component, and each component is 1.618 times the distance from the center to the previous point at the given angle in the spiral.
	home1.gte.net /simres/e1-logsp.htm (1175 words)

	APOD: 2003 September 25 - Logarithmic Spirals Isabel and M51 (Site not responding. Last check: 2007-11-04)
		For starters, Isabel was hundreds of miles across, while M51 (the Whirlpool Galaxy) spans about 50,000 light-years making them vastly dissimilar in scale, not to mention the extremely different physical interactions which control their formation and evolution.
		But they do look amazingly alike, both exhibiting the shape of a simple and beautiful mathematical curve known as a logarithmic spiral, a spiral whose separation grows in a geometric way with increasing distance from the center.
		Logarithmic spirals also describe, for example, the arrangement of sunflower seeds, the shapes of nautilus shells, and...
	antwrp.gsfc.nasa.gov /apod/ap030925.html (202 words)

	Ivars Peterson's MathTrek - Sea Shell Spirals
		A logarithmic spiral follows the rule that, for a given rotation angle (such as one revolution), the distance from the pole (spiral origin) is multiplied by a fixed amount.
		Such a logarithmic spiral can be inscribed in a rectangle whose sides have lengths defined by the golden ratio.
		Starting with the observation that shell spirals are logarithmic spirals, many people automatically assume that, because the golden ratio can be used to draw a logarithmic spiral, all shell spirals are related to the golden ratio, when, in fact, they are not.
	www.maa.org /mathland/mathtrek_04_04_05.html (584 words)

	Part 3 of Spirals and the Golden Section by John Sharp for the Nexus Network Journal vol.4 no.1 (Winter 2002) (Site not responding. Last check: 2007-11-04)
		The spiral drawn using quarter circles in the set of whirling squares is a like a logarithmic spiral since each rotation of 90º means the radius of the circle is multiplied by the Golden Section.
		If the spiral were to touch the sides of the rectangle, the line from the pole would need to make a tangent angle of 72.9676° with the side of the rectangle.
		This means that the spiral that touches the four sides of the rectangle is the same one as the one in Figure 32, except that it is rotated slightly, so that it touches a little way along the side and not at the point where the vertex square sits.
	www.nexusjournal.com /Sharp_v4n1-pt03.html (1327 words)

	Spiral Synthesis
		Spiral formations are known to occur in the structure of proteins and in the great arms of the Andromeda nebula.
		Indeed, the spiral chamber of the inner ear (cochlea) is of essentially the same structure, within a scale factor, in cats, humans, and elephants (Greenwood 1962).
		In spiral synthesis, however, the imaginary signal is produced automatically, and may optionally be used or discarded.
	staff.washington.edu /bradleyb/spiralsynth/spiral.html (2436 words)

	The logarithmic spiral: a counterexample to the = 2 conjecture, D. B. A. Epstein, V. Markovic
		The logarithmic spiral: a counterexample to the K = 2 conjecture
		We show that if $X$ is a certain logarithmic spiral, then we obtain a counterexample to the conjecture of Thurston and Sullivan that there is a 2-quasiconformal homeomorphism $\Omega\to\dome{\Omega}$ which extends to the identity map on their common boundary in $\Sph^2$.
		Another result is that the average long range bending of the convex hull boundary associated to a certain logarithmic spiral is approximately $.98\pi/2$, which is substantially larger than that of any previously known example.
	projecteuclid.org /getRecord?id=euclid.annm/1115669293 (178 words)

	Math Forum: Lee: Lesson 4 - Where Is the Golden Ratio, part 4 (Site not responding. Last check: 2007-11-04)
		Perhaps you have seen a pinwheel with a logarithmic spiral painted on it in flower garden.
		Mathematically, the radial expansion and contraction are governed by the logarithmic function of increasing and decreasing rotation angle.
		A cutaway view of a chambered nautilus shell shown in figure 7 is a familiar example of logarithmic spirals found in nature.
	mathforum.org /~lisab/fourth_lesson/part_4.htm (279 words)

	Pixel Magic - Spirals (Site not responding. Last check: 2007-11-04)
		A logarithmic spiral, created by feeding the log of the distance to the sin() function.
		An interference between a radial gradient (sin(a8)) and a logarithmic* spiral creates a cluster of bumps, exhibiting a double 9/7 spiral effect.
		A basic spiral effect, made by offseting the distance (d) with the angle (a).
	www.jbum.com /jbum/pixmagic/galspirals.html (185 words)

	Sacred Geometry Exercise-Nautilus Shell Spiral
		The nautilus shell spiral is a logarithmic spiral similar to other spirals such as the Golden Mean or phi spiral, but with slightly different proportions.
		The spiral of the chambered nautilus as well as other logarithmic spirals can be found throughout the human body and nature.
		The drawing of the inner ear and the photo image of a star cluster nebula shown to the right are just two examples which can be seen with a microscope or a telescope.
	www.sacredarch.com /sacred_geo_exer_shell.htm (377 words)

	The Geometry Junkyard: Spirals
		Spirals, fractals, and Escher-like reentrant scenes created by replacing a portion of an image by a copy of the same image.
		This shape, constructed by inscribing circular arcs in a spiral tiling of squares, resembles but is not quite the same as a logarithmic spiral.
		A similar spiral is used as the Sybase Inc. logo.
	www.ics.uci.edu /~eppstein/junkyard/spiral.html (546 words)

	Spiral Tool (Site not responding. Last check: 2007-11-04)
		The spiral of Archimedes is an arithmetic spiral, which increases in radius at a fixed amount for each winding-like a roll of paper.
		The logarithmic spiral increases in radius at an exponential rate-the familiar shape of a Chambered Nautilus.
		To draw a spiral, pace the cursor in the drawing area at a location that will represent the outside end of the spiral.
	www.seqair.com /WildTools/Spirals/Spirals.html (186 words)

	Elliott Wave International - Expert Financial Market Forecasting
		The Golden Spiral, which is a type of logarithmic or equiangular spiral, has no boundaries and is a constant shape.
		The core of a logarithmic spiral seen through a microscope would have the same look as its widest viewable reach from light years away.
		Thus, the Golden Spiral spreads before us in symbolic form as one of nature's grand designs, the image of life in endless expansion and contraction, a static law governing a dynamic process, the within and the without sustained by the 1.618 ratio, the Golden Mean.
	www.elliottwave.com /club/members/tutorial/lesson8/8-7.htm (308 words)

	Logarithmic Spirals
		This elegant spiral pattern is called phyllotaxis and it has a mathematics that is equally lovely.
		Piecewise logarithmic curves are, of course, also an aesthetic choice: Logarithmic spirals are visually dynamic curves.
		Euler proposed that tracks curved in a logarithmic spiral were optimal for slowing and turning trains in railyards.
	xenia.media.mit.edu /~brand/logspiral.html (530 words)

	Logarithmic Spiral (Site not responding. Last check: 2007-11-04)
		]) is identically the length of the spiral in the third quadrant (a
		the center of the spiral located horizontally at 1.170820393 times the length of the side of the main square is at the point on the diagonal 1.175570505
		6.6261 multiplied by the length of the first four-quadrant turn of the spiral (8.640563268) equals, with a difference less than one in 1000, 1 radian, and multiplied by the length of the spiral from the fourth quadrant to -
	dgleahy.com /dgl/p14.html (798 words)

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