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Topic: Equiangular spiral

	Logarithmic spiral - Wikipedia, the free encyclopedia
		A logarithmic spiral, equiangular spiral or growth spiral is a special kind of spiral curve which often appears in nature.
		The logarithmic spiral can be distinguished from the Archimedean spiral by the fact that the distances between the arms of a logarithmic spiral increase in geometric progression, while in an Archimedean spiral these distances are constant.
		A logarithmic spiral with pitch 0 degrees (b = 1) is a circle; the limiting case of a logarithmic spiral with pitch 90 degrees (b = 0 or b = ∞) is a straight line starting at the origin.
	en.wikipedia.org /wiki/Logarithmic_spiral (734 words)

	Equiangular
		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)

	spiral.htm
		A spiral is plane curve that, in general, unwinds around a point while moving ever farther from the point.
		The famous equiangular spiral was discovered by Rene Descartes, and its properties of self-reproduction by Jacob Bernoulli (1654-1705) who requested that the curve be engraved upon his tomb with the phrase "Eadem mutata resurgo" ("I shall arise the same, though changed.")
		The familiar chambered Nautilus shell is in the form of an equiangular spiral.
	www.math.tamu.edu /~dallen/digitalcam/spiral/spiral.htm (566 words)

	Ultra-wideband high power photon triggered frequency independent radiator with equiangular spiral antenna - Patent ...
		In a preferred embodiment, the device is comprised of a dielectric storage medium, an equiangular spiral antenna composed of two metalized electrodes (spiral arms) mounted opposite one another on a top surface of the dielectric storage medium and a metalized electrode mounted on a bottom surface of the dielectric medium (essentially forming parallel capacitors).
		As such, the equiangular spiral arms on the top surface as well as the electrode on the bottom surface are separated by predetermined gap distance that allows for high power charging without degrading radiation bandwidth.
		Spiral arms 56 and 57 and bottom plates 61 and 62 are positioned such that they form an energy storage device capable of producing wideband radiation.
	www.freepatentsonline.com /5351063.html (1937 words)

	logarithmic spiral (Site not responding. Last check: 2007-10-12)
		This is the spiral for which the radius grows exponentially with the angle.
		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 (651 words)

	[No title]
		Interestingly, an equiangular spiral derived from corresponding points in a system of hexagons (Figure 9) is based on hexagons which are four times as large as the preceding one.
		In the case of the formula for the equiangular spiral, a variable could be added or the angle of tangency adjusted.
		The formula would no longer be that of an equiangular spiral and it would no longer have a slope of -1 on a log log graph, but this was the desired result.
	www-personal.umich.edu /~copyrght/image/FONSECA/CHAPTR10/fonseca10.html (2558 words)

	The Nautilus and the Human Embryo and the Golden Ratio (Site not responding. Last check: 2007-10-12)
		The Fibonacci rectangles spiral increases in size by a factor of Phi (1.618..) in a quarter of a turn, the nautilus spiral curve takes a whole turn before points move a factor of 1.618...
		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)

	[No title]
		the equable spiral, or spiral of Archimedes (Fig 2-A) and the equiangular or logarithmic spiral (Fig 2-B).
		A nice instance of the equiangular spiral (5) is the route which certain insects follow towards a candle (Fig 3-A) "owing to the structure of their compound eyes, the insects do not look straight ahead but make for a light which they see abeam, at a certain angle.
		It is not surprising that the Swiss scientist Jacob Bernoulli called the equiangular spiral the 'spira mirabilis' and asked for it to be engraved on his tombstone.
	www.dchaos.com /portfolio/dchaos1/new_spiral_article.html (1448 words)

	[No title]
		This is because an equiangular spiral is a mathematical plot of a particular Fibonacci sequence.
		The structure of the curves corresponding to these sequences is an equiangular spiral because the length of the radii forming the curve at successive increments of 90 degrees in the angle of the radius are Fibonacci numbers.
		The term equiangular refers to the constant angle (a in equation 1) maintained between an extension of any radius of the curve and a line tangent to the curve at the point where the two meet.
	www-personal.umich.edu /~copyrght/image/FONSECA/CHAPTER1/fonseca1.html (819 words)

	Spirals
		A spiral is a curve in the plane or in the space, which runs around a centre.
		Spirals in their diverse art forms were intended as objects of expressing spirituality for contemplation and meditation.
		Spherical Spirals, Logarithmic Spiral, Logarithmic Spiral Evolute, Loxodrome, Curlicue Fractal, Cornu Spiral,...
	www.mathematische-basteleien.de /spiral.htm (1708 words)

	"Golden" spirals (Site not responding. Last check: 2007-10-12)
		The spiral is a plane line derivated by a driving point, which moves away according to the definite law from the beginning of the ray uniformly rotated around of the beginning.
		Any equiangular spiral represents the scheme of growth or ascending and can be expressed by geometrical progression.
		In this spiral the terms of geometrical progression corresponding to the spiral are the degrees of the golden proportion {
	www.goldenmuseum.com /0212Spirals_engl.html (359 words)

	spirals.nb (Site not responding. Last check: 2007-10-12)
		One particular type of spiral which constantly reappears is the equiangular spiral.
		Notice that the equiangular spiral is the path of an insect which moves towards a light source, keeping the light at a constant angle to its path.
		We can derive the equation for the equiangular spiral from the basic principle that the tangent to the curve is at a constant angle to the center point of the spiral.
	www.bio.miami.edu /tom/bil358/spirals.html (487 words)

	Equiangular Spiral
		The equation for the equiangular spiral was developed by Rene Descartes (1596-1650) in 1638.
		This spiral occurs naturally in many places like sea-shells where the growth of an organism is proportional to the size of the organism.
		It is called an "equiangular" spiral because any radius vector makes the same angle with the curve.
	www.intmath.com /PlnAnalGeom/Equiangular_spiral.php (133 words)

	96 Equiangular or Logarithmic Spiral (Site not responding. Last check: 2007-10-12)
		Its evolute is an equal equiangular spiral, and so is its inverse with respect to the origin.
		If the spiral is rolled along a straight line, then the path of the origin of the spiral, called its pole, is another straight line.
		For example, the whorls of the nautilus shell are equiangular spirals.
	www.bluefish.org /eqspiral.htm (257 words)

	[No title] (Site not responding. Last check: 2007-10-12)
		Following the theory of Rumsey, he fabricated and measured several two-arm balanced equiangular spiral slot antennas by cutting slots in a metal sheet, then bolting the sheet onto a larger ground plane.
		More recently, results were published showing the differences between equiangular spiral slot antennas with and without ground planes, and the effects of changing the spiral arm width to gap ratio5.
		The effects of changing the spiral arm width to gap ratio and dimensions and/or type of feed point on the radiation pattern and impedance characteristics will be simulated.
	ece-www.colorado.edu /~ecen5134/downloads/Stutzke_Prop.doc (537 words)

	Folium of Descartes (Site not responding. Last check: 2007-10-12)
		The radius of the circle is perpendicular to the tangent of both the circle and the curve at point P. The Folium of Descartes and his Equiangular Spiral.
		The investigation of spirals is known to date from the ancient Greeks.
		Jacob Bernoulli (1654 - 1705) was so fascinated by the Equiangular Spiral that he requested it be carved on his tombstone with the phrase "Eadem mutata resurgo" ("I shall arise the same, though changed").
	curvebank.calstatela.edu /descartes/descartes.htm (656 words)

	spiral.html
		You see a ray from the origin to a point P=(x(t),y(t)) on the spiral and a line tangent to the spiral at P as point P moves along the spiral.
		You will investigate the spiral for which the distance from the origin to the point (x(t),y(t)) is r(t)=exp(-k*t) for this value of k.
		Calculate the slope of the line T that is tangent to the spiral at point P on the spiral and the slope of the ray R from the origin to point P. e)
	www.math.uga.edu /calclab/calclab2210/spiral1.html (676 words)

	Stacy Getteson (Site not responding. Last check: 2007-10-12)
		The seashells that contain this spiral are usually on the nautilus shells and on the shell of a snail.
		On the Equiangular Spiral page at http://www-groups.dcs.st-and.ac.uk/history/Curves/Equiangular.html, there is an in depth mathematical evaluation of the equiangular spiral.
		The spiral that Descartes discovered is the equiangular spiral that is found on seashells.
	math.arizona.edu /~lega/195/Fall99/projects/Stacy_Getteson (497 words)

	[No title] (Site not responding. Last check: 2007-10-12)
		To facilitate student construction of an equiangular or logarithmic spiral.
		To be able to recognize the Fibonacci sequence in an equiangular spiral.
		Students could explore the Archimedian spiral, look for the differences between it and an equiangular spiral and report back to the class.
	www.mste.uiuc.edu /courses/ci431kt/modules/burrus/lpequiang.html (354 words)

	Fibonacci (Site not responding. Last check: 2007-10-12)
		At each concentric layer of a Fibonacci spiral, the ratio of a single component piece to one from the next inner layer is a constant, which varies according to how many radially symmetric parts of the spiral there are.
		These spirals occur frequently in nature, for example in a Nautilus shell or the center of a sunflower.
		The individual components for the spirals were designed using a special version of the Symmetry program, and then run through a color progression as they move outwards.
	www.moonstar.com /~nedmay/chromat/fibonaci.htm (236 words)

	The CTK Exchange Forums
		I'm looking for the name and information of the phenomenon the describes a spiral that can get smaller and smaller without ending-like a nautilus shell.
		You are talking of what is known as the logarithmic, or equiangular, spiral.
		I have seen this spiral illustrated in squares within squares getting infinitely smaller.
	www.cut-the-knot.com /htdocs/dcforum/DCForumID4/97.shtml (193 words)

	[No title] (Site not responding. Last check: 2007-10-12)
		This spiral is referred to by some as the "Fibonacci spiral".
		EQUIANGULAR SPIRALS are not just interesting geometric constructions.
		They also can be found in many places in nature, like the spiral in the shell of the "chambered nautilus":
	www.mste.uiuc.edu /courses/ci431kt/modules/hightshoe/stuspi.html (258 words)

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