Spiral

For other uses of this word, see spiral (disambiguation).

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.

Spiral or helix

An Archimedean spiral, a helix, and a conic spiral

While a "spiral" and a "helix" are distinct as technical terms, a helix is sometimes described as a spiral in non-technical usage. The two primary definitions of a spiral are provided by the American Heritage Dictionary:

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 varying distance while moving parallel to the axis.

The first definition is for a 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 (a logarithmic spiral) are examples of a spiral.

The second definition is for the 3-dimensional variant of a spiral, for example a conical spring can be described as a spiral whereas a cylindrical spring or a strand of 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 conic helices.

Two-dimensional spirals

A two-dimensional spiral may be described most easily using polar coordinates, where the radius r is a monotonic continuous 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:

• The Archimedean spiral: $r=a+b \cdot \theta$ (see also:Involute)
• The Euler spiral, Cornu spiral or clothoid
• Fermat's spiral: $r= \theta^{1/2}$
• The hyperbolic spiral: $r = a/ \theta$
• The lituus: $r = \theta^{-1/2}$
• The logarithmic spiral: $r=a\cdot e^{b\theta}$; approximations of this are found in nature
• The Fibonacci spiral and golden spiral: special cases of the logarithmic spiral
• The Spiral of Theodorus: an approximation of the Archimedean spiral composed of contiguous right triangles

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  Archimedean spiral

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  Cornu spiral

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  Fermat's spiral

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  hyperbolic spiral

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  lituus

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  logarithmic spiral

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  spiral of Theodorus

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:

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

Spherical spiral

Rhumb line

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 hence is 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, for example, the triple spiral.

The World Pantheist Movement's spiral-based logo

The most recurring motive in the Nazca Lines is the spiral. There is more than 100 spirals in the desert, in all places and sizes, sometimes even being included in a figure (e.g. the monkey's tail).

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 found in structures as small as the double helix of DNA and as large as a galaxy. Because of this frequent natural occurrence, the spiral is the official symbol of the World Pantheist Movement.

The spiral is also a symbol of the process of dialectic.

In nature

The 53rd plate from Ernst Haeckel's Kunstformen der Natur (1904), depicting organisms classified as Prosobranchia

The study of spirals in nature has 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.

Spirals in plants and animals are frequently described as whorls. This is also the name given to spiral shaped fingerprints.

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.

In art

The spiral has inspired artists throughout the ages. Among the most famous of spiral-inspired art is Robert Smithson's earthwork, "Spiral Jetty", at the Great Salt Lake in Utah. The spiral theme is also present in David Wood's Spiral Resonance Field at the Balloon Museum in Albuquerque, as well as in the critically acclaimed Nine Inch Nails 1994 concept album The Downward Spiral. The Spiral is also a prominent theme in the anime Gurren Lagann, where it represents a philosophy and way of life. It also central in Mario Merz and Andy Goldsworthy's work.

See also

• Celtic maze (straight-line spiral)
• Patterns in nature
• Seashell surface
• Spirangle

Related publications

• Cook, T., 1903. Spirals in nature and art. Nature 68 (1761), 296.
• Cook, T., 1979. The curves of life. Dover, New York.
• Habib, Z., Sakai, M., 2005. Spiral transition curves and their applications. Scientiae Mathematicae Japonicae 61 (2), 195 – 206.
• Dimulyo, S., Habib, Z., Sakai, M., 2009. Fair cubic transition between two circles with one circle inside or tangent to the other. Numerical Algorithms 51, 461–476 [1].
• Harary, G., Tal, A., 2011. The natural 3D spiral. Computer Graphics Forum 30 (2), 237 – 246 [2].
• Xu, L., Mould, D., 2009. Magnetic curves: curvature-controlled aesthetic curves using magnetic fields. In: Deussen, O., Hall, P. (Eds.), Computational Aesthetics in Graphics, Visualization, and Imaging. The Eurographics Association [3].
• Wang, Y., Zhao, B., Zhang, L., Xu, J., Wang, K., Wang, S., 2004. Designing fair curves using monotone curvature pieces. Computer Aided Geometric Design 21 (5), 515–527 [4].
• A. Kurnosenko. Applying inversion to construct planar, rational spirals that satisfy two-point G2 Hermite data. Computer Aided Geometric Design, 27(3), 262–280, 2010 [5].
• A. Kurnosenko. Two-point G2 Hermite interpolation with spirals by inversion of hyperbola. Computer Aided Geometric Design, 27(6), 474–481, 2010.
• Miura, K.T., 2006. A general equation of aesthetic curves and its self-affinity. Computer-Aided Design and Applications 3 (1–4), 457–464 [6].
• Miura, K., Sone, J., Yamashita, A., Kaneko, T., 2005. Derivation of a general formula of aesthetic curves. In: 8th International Conference on Humans and Computers (HC2005). Aizu-Wakamutsu, Japan, pp. 166 – 171 [7].
• Meek, D., Walton, D., 1989. The use of Cornu spirals in drawing planar curves of controlled curvature. Journal of Computational and Applied Mathematics 25 (1), 69–78 [8].
• Farin, G., 2006. Class A Bézier curves. Computer Aided Geometric Design 23 (7), 573–581 [9].
• Farouki, R.T., 1997. Pythagorean-hodograph quintic transition curves of monotone curvature. Computer-Aided Design 29 (9), 601–606.
• Yoshida, N., Saito, T., 2006. Interactive aesthetic curve segments. The Visual Computer 22 (9), 896–905 [10].
• Yoshida, N., Saito, T., 2007. Quasi-aesthetic curves in rational cubic Bézier forms. Computer-Aided Design and Applications 4 (9–10), 477–486 [11].
• Ziatdinov, R., Yoshida, N., Kim, T., 2012. Analytic parametric equations of log-aesthetic curves in terms of incomplete gamma functions. Computer Aided Geometric Design 29 (2), 129 – 140 [12].
• Ziatdinov, R., Yoshida, N., Kim, T., 2012. Fitting G2 multispiral transition curve joining two straight lines, Computer-Aided Design 44(6), 591–596 [13].
• Ziatdinov, R., 2012. Family of superspirals with completely monotonic curvature given in terms of Gauss hypergeometric function. Computer Aided Geometric Design 29(7): 510–518 [14].
• Ziatdinov, R., Miura K.T., 2012. On the Variety of Planar Spirals and Their Applications in Computer Aided Design. European Researcher 27(8–2), 1227-–1232 [15].

External links

• SpiralZoom.com, an educational website about the science of pattern formation, spirals in nature, and spirals in the mythic imagination.
• Spirals by Jürgen Köller
• Spirals – an Encyclopedia of Life collection with examples of spirals in nature.
• Archimedes' spiral transforms into Galileo's spiral. Mikhail Gaichenkov, OEIS