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Cutaway of a nautilus shell showing the chambers arranged in an approximately logarithmic spiral.

In mathematics, a spiral is a curve which emanates from a central point, getting progressively farther away as it revolves around the point.

Contents

Spiral or helix

An Archimedean spiral, a helix, and a conic spiral.

A "spiral" and a "helix" are both technically spirals even though they each represent a different object. The two primary definitions of a spiral are as follows:

a. A curve on a plane that winds around a fixed center point at a continuously increasing or decreasing distance from the point.
b. A three-dimensional curve that turns around an axis at a constant or continuously varying distance while moving parallel to the axis.

The first definition is for a flat, often 2-Dimensional, planar curve that extends primarily in length and width, but not in height. A groove on a record or the arms of a spiral galaxy are examples of a spiral. The second definition is for the 3-Dimensional cylindrical variant of a spiral, called a Helix, that extends primarily in height. A spring (device) or a strand of a DNA are examples of a helix.

The length and width of a Helix typically remain static and do not grow like on a planar spiral. If they do, then the helix becomes a Conic Helix. You can make a conic helix with an Archimedean or equiangular spiral by giving height to the center point, thereby creating a cone-shape from the spiral.

In the side picture, the black curve at the bottom is an Archimedean spiral, while the green curve is a helix. A cross between a spiral and a helix, such as the curve shown in red, is known as a conic helix. The spring used to hold and make contact with the negative terminals of AA or AAA batteries in remote controls and the vortex that is created when water is draining in a sink are examples of a conic helix.

Two-dimensional spirals

A two-dimensional spiral may be described most easily using polar coordinates, where the radius r is a continuous monotonic function of angle θ. The circle would be regarded as a degenerate case (the function not being strictly monotonic, but rather constant).

Some of the more important sorts of two-dimensional spirals include:

Three-dimensional spirals

For simple 3-d spirals, a third variable, h (height), is also a continuous, monotonic function of θ. For example, a conic helix may be defined as a spiral on a conic surface, with the distance to the apex an exponential function of θ.

The helix and vortex can be viewed as a kind of three-dimensional spiral.

For a helix with thickness, see spring (math).

Another kind of spiral is a conic spiral along a circle. This spiral is formed along the surface of a cone whose axis is bent and restricted to a circle:

TORUSA-4 Konische Spirale entlang eines Kreises.PNG

This image is reminiscent of a Ouroboros symbol and could be mistaken for a torus with a continuously-increasing diameter:

TORUSA-1 Torus mit variablem Ringdurchmesser.PNG

Spherical spiral

Archimedean Spherical Spiral

A spherical spiral (rhumb line or loxodrome, left picture) is the curve on a sphere traced by a ship traveling from one pole to the other while keeping a fixed angle (unequal to 0° and to 90°) with respect to the meridians of longitude, i.e. keeping the same bearing. The curve has an infinite number of revolutions, with the distance between them decreasing as the curve approaches either of the poles.

The gap between the curves of an Archimedean spiral (right picture) remains constant as the radius changes and is hence not a rhumb line.

As a symbol

The Newgrange entrance slab

The spiral plays a specific role in symbolism, and appears in megalithic art, notably in the Newgrange tomb or in many Galician petroglyphs such as the one in Mogor. See also triple spiral.

While scholars are still debating the subject, there is a growing acceptance that the simple spiral, when found in Chinese art, is an early symbol for the sun. Roof tiles dating back to the Tang Dynasty with this symbol have been found west of the ancient city of Chang'an (modern-day Xian).

Spirals are also a symbol of hypnosis, stemming from the cliché of people and cartoon characters being hypnotized by staring into a spinning spiral (One example being Kaa in Disney's The Jungle Book). They are also used as a symbol of dizziness, where the eyes of a cartoon character, especially in anime and manga, will turn into spirals to show they are dizzy or dazed. The spiral is also a prominent symbol in the anime Gurren Lagann, where it symbolizes the double helix structure of DNA, representing biological evolution, and the spiral structure of a galaxy, representing universal evolution.

In nature

The 53rd plate from Ernst Haeckel's Kunstformen der Natur (1904), depicting organisms classified as Prosobranchia (now known to be polyphyletic).

The study of spirals in nature have a long history, Christopher Wren observed that many shells form a logarithmic spiral. Jan Swammerdam observed the common mathematical characteristics of a wide range of shells from Helix to Spirula and Henry Nottidge Moseley described the mathematics of univalve shells. D’Arcy Wentworth Thompson's On Growth and Form gives extensive treatment to these spirals. He describes how shells are formed by rotating a closed curve around a fixed axis, the shape of the curve remains fixed but its size grows in a geometric progression. In some shell such as Nautilus and ammonites the generating curve revolves in a plane perpendicular to the axis and the shell will form a planar discoid shape. In others it follows a skew path forming a helico-spiral pattern.

Thompson also studied spirals occurring in horns, teeth, claws and plants.[1]

Spirals in plants and animals are frequently described as whorls.

A model for the pattern of florets in the head of a sunflower was proposed by H Vogel. This has the form

\theta = n \times 137.5^{\circ},\ r = c \sqrt{n}

where n is the index number of the floret and c is a constant scaling factor, and is a form of Fermat's spiral. The angle 137.5° is related to the golden ratio and gives a close packing of florets.[2]

In art

The spiral has inspired artists down the ages. The most famous piece of 60s Land Art was Robert Smithson's Spiral Jetty, at the Great Salt Lake in Utah. The theme continues in David Wood's Spiral Resonance Field at the Balloon Museum in Albuquerque.

References

  1. ^ Thompson, D'Arcy (1917,1942), On Growth and Form  
  2. ^ Prusinkiewicz, Przemyslaw; Lindenmayer, Aristid (1990). The Algorithmic Beauty of Plants. Springer-Verlag. pp. 101–107. ISBN 978-0387972978. http://algorithmicbotany.org/papers/#webdocs.  

See also

External links

  • SpiralZoom.com, an educational website about the science of pattern formation, spirals in nature, and spirals in the mythic imagination.

1911 encyclopedia

Up to date as of January 14, 2010

From LoveToKnow 1911

SPIRAL, in mathematics, the locus of the extremity of a line (oi radius vector) which varies in length as it revolves about a fixed point (or origin). Here we consider some of the more important plane spirals. Obviously such curves are conveniently expressed by polar equations, i.e. equations which directly state a relation existing between the radius vector and the vector angle; another form is the "p, r" equation, wherein r is the radius vector of a point, and p the length of the perpendicular from the origin to the tangent at that point.

The equiangular or logarithmic spiral (fig. I) is such that as the vector angle increases arithmetically, the radius vector increases 176' ' 'F/C geometrically; this definition leads to an equation of the form r=AeB S, where e is the base of natural logarithms and A, B are constants. Another definition is that the tangent makes a constant angle (a, say) with the radius vector; this leads to p =r sin a. This curve has the property that its positive pedals, inverse, polar reciprocal and evolutes are all equal equiangular spirals. A group of spirals are included in the "parabolic spirals" given by the equation r=aO n; the more important are the Archimedean spiral, r =a0 (fig. 2); the hyperbolic or reciprocal spiral, r = ae-I (fig. 3); and the lituus, r =a6 1 -- (fig. 4). The first-named was discovered by Conon, whose studies were completed by Archimedes. Its "p, r" equation is p =r2/.J (a 2 -}-r 2), and the angle between the radius vector and the tangent equals the vector angle. The second, called hyperbolic on account of the analogy of its equation (polar) to that (Cartesian) of a hyperbola between the asymptotes, is the inverse of the Archimedean. Its p, r equation is p-2 = r-2 +a-2, and it has an asymptote at the distance a above the initial line. The lituus has the initial line as asymptote. Another group of spirals - termed Cotes's spirals - appear as the path of a particle moving under the influence of a central force varying as the inverse cube of the distance (see Mechanics). Their general equation is p-2 =Are +B, in which A and B can have any values. If B =0, we have p = rl A, and the locus is the equiangular spiral. If A = i we have p-2 = r2+B, which leads to the polar equation rB = I/d B, i.e. the reciprocal spiral. The more general investigation is as follows: Writing u= r--1 we have p- 2 = Au 2 +B, and since p2 = u 2 + (du/de) 2 (see Infinitesimal Calculus), then Au 2 +B=u 2 +(du/de) 2, i.e. (du/dO)2= (A - i)u2+B. The right-hand side may be written as C 2 (u 2+ D2), C2 (// 2 D2), C 2 (D 2 - 14 2) according as A - i and B are both positive, A - I positive and B negative, and as A - i negative and B positive. On integration these three forms yield the polar equations u = C sin hDO, u = C cos hDO, and u =C sin DO. Of interest is the spiral r =a0 2 / (02-1), which has the circle r=a as an asymptote in addition to a linear asymptote.


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Simple English

shell showing the chambers arranged in an approximately logarithmic spiral.]]

A Spiral is a special curve in mathematics. This curve starts at a point, and then goes around the point, but gets farther and farther away from it. This is different from a circle (which is always at the same distance or an ellipse. A spiral is an "open" curve, unlike circles and ellipses which are closed curves.

Contents

Two-dimensional spirals

A two-dimensional spiral may be described most easily using polar coordinates. There the radius r is a continuous monotonic function of angle θ (theta). The circle would be regarded as a degenerate case. With the circle, the function would not be strictly monotonic, but constant.

Some of the more important sorts of two-dimensional spirals include:

  • The Archimedean spiral: r = a + bθ
  • The Euler spiral, Cornu spiral or clothoid
  • Fermat's spiral: r = θ1/2
  • The hyperbolic spiral: r = a
  • The lituus: r = θ-1/2
  • The logarithmic spiral: r = abθ; approximations of this are found in nature
  • The Fibonacci spiral and golden spiral: special cases of the logarithmic spiral
  • The Spiral of Theodorus: an aproximation of the Archimedean spiral composed of contiguous right triangles

Three-dimensional spirals

For simple 3-d spirals, a third variable, h (height), is also a continuous, monotonic function of θ. For example, a conic helix may be defined as a spiral on a conic surface, with the distance to the apex an exponential function of θ.

The helix and vortex can be viewed as a kind of three-dimensional spiral.

For a helix with thickness, see spring (math).

In nature

's Kunstformen der Natur (1904), showing organisms classified as Prosobranchia (now known to be polyphyletic).]] The study of spirals in nature have a long history, Christopher Wren found out that many shells form a logarithmic spiral. Jan Swammerdam observed the common mathematical characteristics of a wide range of shells from Helix to Spirula and Henry Nottidge Moseley described the mathematics of univalve shells. D’Arcy Wentworth Thompson's On Growth and Form gives extensive treatment to these spirals. He describes how shells are formed by rotating a closed curve around a fixed axis, the shape of the curve remains fixed but its size grows in a geometric progression. In some shell such as Nautilus and ammonites the generating curve revolves in a plane pirpendicular to the axis and the shell will form a planer discoid shape. In others it follows a skew path forming a helico-spiral pattern.

Thompson also studied spirals occurring in horns, teeth, claws and plants.[1]

Spirals in plants and animals are frequently described as whorls.

A model for the pattern of florets in the head of a sunflower was proposed by H Vogel. This has the form

\theta = n \times 137.5^{\circ},\ r = c \sqrt{n}

where n is the index number of the floret and c is a constant scaling factor, and is a form of Fermat's spiral. The angle 137.5° is related to the golden ratio and gives a close packing of florets.[2]

The spiral also represents infinance, or 'infinity.' Starting at a single point, and revolving outwardly until the end of the universe. Because of this, some civilizations believe that the Spiral is a pathway to the afterlife.

As a symbol

The spiral plays an important role in symbolism. It appears in megalithic art, notably in the Newgrange tomb or in many Galician petroglyphs such as the one in Mogor. See also triple spiral.

Scholars are still talking about the subject, but many of them now believe that the simple spiral in Chinese art may be a symbol of the sun. Roof tiles from the Tang Dynasty with this symbol have been found west of the ancient city of Chang'an (modern-day Xian).

The spiral is the most ancient symbol found on every civilized continent. Because it appears at burial sites across the globe, the spiral most likely represented the "life-death-rebirth" cycle. Similarly, the spiral symbolized the sun, as ancient people thought the sun was born each morning, died each night, and was reborn the next morning.

Spirals are also a symbol of hypnosis. This most likely comes from the cliché of people and cartoon characters being hypnotized by staring into a spinning spiral (One example being Kaa in Disney's The Jungle Book). They are also used as a symbol of dizziness, where the eyes of a cartoon character, especially in anime and manga, will turn into spirals to show they are dizzy or dazed. The spiral is also a prominent symbol in the anime Gurren Lagann, where it symbolizes the double helix structure of DNA, representing biological evolution, and the spiral structure of a galaxy, representing universal evolution.

References

  1. Thompson, D'Arcy (1917,1942), [Expression error: Unexpected < operator On Growth and Form] 
  2. Prusinkiewicz, Przemyslaw; Lindenmayer, Aristid (1990). The Algorithmic Beauty of Plants. Springer-Verlag. pp. 101–107. ISBN 978-0387972978. http://algorithmicbotany.org/papers/#webdocs. 







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