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Nielsen's spiral, also called the sici spiral (von Seggern 1993) is the spiralwith parametric equations(1)(2)where is the cosine integral and is the sine integral.The curvature is given by(3)and the arc length measured from by(4)

Fermat's spiral, also known as the parabolic spiral, is an Archimedean spiral with having polar equation(1)This curve was discussed by Fermat in 1636 (MacTutor Archive). For any given positive value of , there are two corresponding values of of opposite signs. The left plot above shows(2)only, while the right plot shows equation (1) in red and(3)in blue. Taking both signs, the resulting spiral is symmetrical about the origin.The curvature and arc lengthof the positive branch of Fermat's spiral are(4)(5)(6)where is a hypergeometric function and is an incomplete beta function.

The inverse curve of the epispiralwith inversion center at the origin and inversion radius is the rose

The epispiral is a plane curve with polar equationThere are sections if is odd and if is even.A slightly more symmetric version considers instead

The Theodorus spiral is a discrete spiral formed by connecting the ends of radial spokes corresponding to the hypotenuses of a sequence of adjoining right triangles. The initial spoke is of length , the next spoke is of length , etc., and each segment of the spiral (corresponding to the outer leg of a triangle) has unit length. It is also known as the square root spiral, Einstein spiral, Pythagorean spiral, or--to contrast it with certain continuous analogs--the discrete spiral of Theodorus.The slope of a continuous analog of the discrete Theodorus spiral due to Davis (1993) at the point is sometimes known as Theodorus's constant.

The pedal curve of a logarithmicspiral with parametric equation(1)(2)for a pedal point at the pole is an identical logarithmicspiral(3)(4)so(5)

The inverse curve of the logarithmicspiralwith inversion center at the origin and inversion radius is the logarithmic spiral

For a logarithmic spiral given parametricallyas(1)(2)evolute is given by(3)(4)As first shown by Johann Bernoulli, the evolute of a logarithmic spiral is therefore another logarithmic spiral, having and ,In some cases, the evolute is identical to the original,as can be demonstrated by making the substitution to the new variable(5)Then the above equations become(6)(7)(8)(9)which are equivalent to the form of the original equation if(10)(11)(12)where only solutions with the minus sign in exist. Solving gives the values summarized in the following table.10.2744106319...20.1642700512...30.1218322508...40.0984064967...50.0832810611...60.0725974881...70.0645958183...80.0583494073...90.0533203211...100.0491732529...

A spiral that gives the solution to the central orbitproblem under a radial force law(1)where is a positive constant. There are three solution regimes,(2)where and are constants,(3)(4)and is the specific angular momentum (Whittaker 1944, p. 83). The case gives an epispiral, while leads to a hyperbolic spiral.

The catacaustic of a logarithmic spiral, where the origin is taken as the radiant point, is another logarithmic spiral. For an original spiral with parametric equations(1)(2)the catacaustic with radiant point at the originis(3)(4)

The logarithmic spiral is a spiral whose polarequation is given by(1)where is the distance from the origin, is the angle from the x-axis, and and are arbitrary constants. The logarithmic spiral is also known as the growth spiral, equiangular spiral, and spira mirabilis. It can be expressed parametrically as(2)(3)This spiral is related to Fibonacci numbers, the golden ratio, and the golden rectangle, and is sometimes called the golden spiral.The logarithmic spiral can be constructed from equally spaced rays by starting at a point along one ray, and drawing the perpendicular to a neighboring ray. As the number of rays approaches infinity, the sequence of segments approaches the smooth logarithmic spiral (Hilton et al. 1997, pp. 2-3).The logarithmic spiral was first studied by Descartes in 1638 and Jakob Bernoulli. Bernoulli was so fascinated by the spiral that he had one engraved on his tombstone (although the engraver did not draw..

The pedal curve of a sinusoidalspiralwith pedal point at the center is another sinusoidalspiral with polar equationA few examples are illustrated above.

The inverse curve of a sinusoidalspiralwith inversion center at the origin and inversion radius is another sinusoidal spiral

Whirls are figures constructed by nesting a sequence of polygons (each having the same number of sides), each slightly smaller and rotated relative to the previous one. The vertices give the path of the mice in the mice problem, and form logarithmic spirals.The square whirl appears on the cover of Freund (1993).

The lituus is an Archimedean spiral with , having polar equation(1)Lituus means a "crook," in the sense of a bishop's crosier. The lituus curve originated with Cotes in 1722. Maclaurin used the term lituus in his book Harmonia Mensurarum in 1722 (MacTutor Archive). The lituus is the locus of the point moving such that the area of a circular sector remains constant.The arc length, curvature,and tangential angle are given by(2)(3)(4)where the arc length is measured from .

A sinusoidal spiral is a curve of the form(1)with rational, which is not a true spiral. Sinusoidal spirals were first studied by Maclaurin. Special cases are given in the following table.curvehyperbolalineparabolaTschirnhausen cubicCayley's sexticcardioid1circle2lemniscateThe curvature and tangentialangle are(2)(3)

An Archimedean spiral with polarequation(1)The hyperbolic spiral, also called the inverse spiral (Whittaker 1944, p. 83), originated with Pierre Varignon in 1704 and was studied by Johann Bernoulli between 1710 and 1713, as well as by Cotes in 1722 (MacTutor Archive).It is also a special case of a Cotes' spiral, i.e.,the path followed by a particle in a central orbit with power law(2)when is a constant and is the specific angular momentum.The curvature and tangentialangle are given by(3)(4)

The prime spiral, also known as Ulam's spiral, is a plot in which the positive integers are arranged in a spiral (left figure), with primes indicated in some way along the spiral. In the right plot above, primes are indicated in red and composites are indicated in yellow.The plot above shows a larger part of the spiral in which the primes are shown as dots.Unexpected patterns of diagonal lines are apparent in such a plot, as illustrated in the above grid. This construction was first made by Polish-American mathematician Stanislaw Ulam (1909-1986) in 1963 while doodling during a boring talk at a scientific meeting. While drawing a grid of lines, he decided to number the intersections according to a spiral pattern, and then began circling the numbers in the spiral that were primes. Surprisingly, the circled primes appeared to fall along a number of diagonal straight lines or, in Ulam's slightly more formal prose, it "appears to exhibit a strongly..

The length of the polygonal spiral is found by noting that the ratio of inradius to circumradius of a regular polygon of sides is(1)The total length of the spiral for an -gon with side length is therefore(2)(3)Consider the solid region obtained by filling in subsequent triangles which the spiral encloses. The area of this region, illustrated above for -gons of side length , is(4)The shaded triangular polygonal spiral is a rep-4-tile.

Archimedes' spiral is an Archimedean spiralwith polar equation(1)This spiral was studied by Conon, and later by Archimedes in On Spirals about225 BC. Archimedes was able to work out the lengths of various tangents to the spiral.The curvature of Archimedes' spiral is(2)and the arc length is(3)(4)This has the series expansion(5)(6)(OEIS A091154 and A002595), where is a Legendre polynomial.Archimedes' spiral can be used for compass and straightedge division of an angle into parts (including angle trisection) and can also be used for circle squaring. In addition, the curve can be used as a cam to convert uniform circular motion into uniform linear motion (Brown 1923; Steinhaus 1999, p. 137). The cam consists of one arch of the spiral above the x-axis together with its reflection in the x-axis. Rotating this with uniform angular velocity about its center will result in uniform linear motion of the point where it crosses the y-axis...

The inverse curve of the Archimedeanspiralwith inversion center at the origin and inversion radius is the Archimedean spiral

The beautiful arrangement of leaves in some plants, called phyllotaxis, obeys a number of subtle mathematical relationships. For instance, the florets in the head of a sunflower form two oppositely directed spirals: 55 of them clockwise and 34 counterclockwise. Surprisingly, these numbers are consecutive Fibonacci numbers. The ratios of alternate Fibonacci numbers are given by the convergents to , where is the golden ratio, and are said to measure the fraction of a turn between successive leaves on the stalk of a plant: 1/2 for elm and linden, 1/3 for beech and hazel, 2/5 for oak and apple, 3/8 for poplar and rose, 5/13 for willow and almond, etc. (Coxeter 1969, Ball and Coxeter 1987). A similar phenomenon occurs for daisies, pineapples, pinecones, cauliflowers, and so on.Lilies, irises, and the trillium have three petals; columbines, buttercups, larkspur, and wild rose have five petals; delphiniums, bloodroot, and cosmos have eight petals;..

An Archimedean spiral is a spiral with polarequation(1)where is the radial distance, is the polar angle, and is a constant which determines how tightly the spiral is "wrapped."Values of corresponding to particular special named spirals are summarized in the following table, together with the colors with which they are depicted in the plot above.spiralcolorlituusredhyperbolic spiralorangeArchimedes' spiralgreen1Fermat's spiralblue2The curvature of an Archimedean spiral is given by(2)and the arc length for by(3)where is a hypergeometric function.If a fly crawls radially outward along a uniformly spinning disk, the curve it traces with respect to a reference frame in which the disk is at rest is an Archimedean spiral (Steinhaus 1999, p. 137). Furthermore, a heart-shaped frame composed of two arcs of an Archimedean spiral which is fixed to a rotating disk converts uniform rotational motion to uniform back-and-forth..

A space curve consisting of a spiral wound arounda helix. It has parametric equations(1)(2)(3)

The spherical curve obtained when moving along the surface of a sphere with constant speed, while maintaining a constant angular velocity with respect to a fixed diameter (Erdős 2000). This curve is given in cylindrical coordinates by the parametric equations(1)(2)(3)where is a positive constant and and are Jacobi elliptic functions (Whittaker and Watson 1990, pp. 527-528).Erdős (2000) provides a derivation of the equations of this curve, as well as an analysis of its properties, including conditions for obtaining periodic orbits.

The conical spiral with angular frequency on a cone of height and radius is a space curve given by the parametric equations(1)(2)(3)The general form has parametric equations(4)(5)(6)This curve has arc length function, curvature,and torsion given by(7)(8)(9)

The spherical curve taken by a ship which travels from the south pole to the north pole of a sphere while keeping a fixed (but not right) angle with respect to the meridians. The curve has an infinite number of loops since the separation of consecutive revolutions gets smaller and smaller near the poles.It is given by the parametric equations(1)(2)(3)where(4)and is a constant. Plugging in therefore gives(5)(6)(7)It is a special case of a loxodrome.The arc length, curvature,and torsion are all slightly complicated expressions.A series of spherical spirals are illustrated in Escher's woodcuts "Sphere Surface with Fish" (Bool et al. 1982, pp. 96 and 318) and "Sphere Spirals" (Bool et al. 1982, p. 319; Forty 2003, Plate 67).

The tangent indicatrix of a curve of constant precession is a spherical helix. The equation of a spherical helix on a sphere with radius making an angle with the z-axis is(1)(2)(3)The projection on the -plane is an epicycloid with radii(4)(5)

The involute of the circle was first studied by Huygens when he was considering clocks without pendula for use on ships at sea. He used the circle involute in his first pendulum clock in an attempt to force the pendulum to swing in the path of a cycloid. For a circle of radius ,(1)(2)the parametric equation of the involute is given by(3)(4)The arc length, curvature,and tangential angle are(5)(6)(7)The Cesàro equation is(8)

Successive points dividing a golden rectangle into squares lie on a logarithmic spiral (Wells 1991, p. 39; Livio 2002, p. 119) which is sometimes known as the golden spiral.In the Season 4 episode "Masterpiece" (2008) of the CBS-TV crime drama "Criminal Minds," the agents of the FBI Behavioral Analysis Unit are confronted by a serial killer who uses the Fibonacci number sequence to determine the number of victims for each of his killing episodes. In this episode, character Dr. Reid also notices that locations of the killings lie on the graph of a golden spiral, and going to the center of the spiral allows Reid to determine the location of the killer's base of operations.

Given a rectangle having sides in the ratio , the golden ratio is defined such that partitioning the original rectangle into a square and new rectangle results in a new rectangle having sides with a ratio . Such a rectangle is called a golden rectangle. Euclid used the following construction to construct them. Draw the square , call the midpoint of , so that . Now draw the segment , which has length(1)and construct with this length. Now complete the rectangle , which is golden since(2)Successive points dividing a golden rectangle into squares lie on a logarithmic spiral (Wells 1991, p. 39; Livio 2002, p. 119) which is sometimes known as the golden spiral.The spiral is not actually tangent at these points, however, but passes through them and intersects the adjacent side, as illustrated above.If the top left corner of the original square is positioned at (0, 0), the center of the spiral occurs at the position(3)(4)(5)(6)(7)(8)(9)(10)(11)and..

A fast boat is overtaking a slower one when fog suddenly sets in. At this point, the boat being pursued changes course, but not speed, and proceeds straight in a new direction which is not known to the fast boat. How should the pursuing vessel proceed in order to be sure of catching the other boat?The amazing answer is that the pursuing boat should continue to the point where the slow boat would be if it had set its course directly for the pursuing boat when the fog set in. If the boat is not there, it should proceed in a spiral whose origin is the point where the slow boat was when the fog set in. The spiral must be constructed in such a way that, while circling the origin, the fast boat's distance from it increases at the same rate as the boat being pursued. The two courses must therefore intersect before the fast boat has completed one circuit. In order to make the problem reasonably practical, the fast boat should be capable of maintaining a speed four or five times..

In the mice problem, also called the beetle problem, mice start at the corners of a regular -gon of unit side length, each heading towards its closest neighboring mouse in a counterclockwise direction at constant speed. The mice each trace out a logarithmic spiral, meet in the center of the polygon, and travel a distanceThe first few values for , 3, ..., aregiving the numerical values 0.5, 0.666667, 1, 1.44721, 2, 2.65597, 3.41421, 4.27432, 5.23607, .... The curve formed by connecting the mice at regular intervals of time is an attractive figure called a whirl.The problem is also variously known as the (three, four, etc.) (bug, dog, etc.) problem. It can be generalized to irregular polygons and mice traveling at differing speeds (Bernhart 1959). Miller (1871) considered three mice in general positions with speeds adjusted to keep paths similar and the triangle similar to the original...

A plot in the complex plane of the points(1)where and are the Fresnel integrals (von Seggern 2007, p. 210; Gray 1997, p. 65). The Cornu spiral is also known as the clothoid or Euler's spiral. It was probably first studied by Johann Bernoulli around 1696 (Bernoulli 1967, pp. 1084-1086). A Cornu spiral describes diffraction from the edge of a half-plane.The quantities and are plotted above. The slope of the curve's tangentvector (above right figure) is(2)plotted below. The Cesàro equation for a Cornu spiral is , where is the radius of curvature and the arc length. The torsion is .Gray (1997) defines a generalization of the Cornu spiral given by parametricequations(3)(4)(5)(6)where is a generalized hypergeometric function.The arc length, curvature,and tangential angle of this curve are(7)(8)(9)The Cesàro equation is(10)Dillen (1990) describes a class of "polynomial spirals" for which the..

The golden angle is the angle that divides a full angle in a golden ratio (but measured in the opposite direction so that it measures less than ), i.e.,(1)(2)(3)(4)(5)(6)(7)(OEIS A131988 and A096627;Livio 2002, p. 112).It is implemented in the Wolfram Languageas GoldenAngle.van Iterson showed in 1907 that points separated by on a tightly bound spiral tends to produce interlocked spirals winding in opposite directions, and that the number of spirals in these two families tend to be consecutive Fibonacci numbers (Livio 2002, p. 112).Another angle related to the golden ratio is theangle(8)or twice this angle(9)the later of which is the smaller interior angle in the goldenrhombus.

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