Spiral Information & Spiral Links at HealthHaven.com

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.

[edit] 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 a Ouroboros symbol and could be mistaken for a torus with a continuously-increasing diameter:

[edit] 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 is hence not a rhumb line.

[edit] 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.

[edit] 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]

[edit] 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.

[edit] References

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

[edit] See also

Seashell surface
Celtic maze (straight-line spiral)
Spirangle

[edit] External links

Wikimedia Commons has media related to: Spiral

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

Geoff Ward (2006). Spirals: the Pattern of Existence. UK: Green Magic. ISBN 0 9547 2309 0.
Spirals by Jürgen Köller

[0] Thompson, D'Arcy (1917,1942), On Growth and Form

[1] Prusinkiewicz, Przemyslaw; Lindenmayer, Aristid (1990). The Algorithmic Beauty of Plants. Springer-Verlag. pp. 101–107. ISBN 978-0387972978. http://algorithmicbotany.org/papers/#webdocs.

[1]

[2]

Contents