Spiral

[1][2][3][4] It is a subtype of whorled patterns, a broad group that also includes concentric objects.

A two-dimensional, or plane, spiral may be easily described using polar coordinates, where the radius

: The circle would be regarded as a degenerate case (the function not being strictly monotonic, but rather constant).

[5] A hyperbolic spiral appears as image of a helix with a special central projection (see diagram).

From vector calculus in polar coordinates one gets the formula Hence the slope of the spiral

Two major definitions of "spiral" in the American Heritage Dictionary are:[7] The first definition describes a planar curve, that extends in both of the perpendicular directions within its plane; the groove on one side of a gramophone record closely approximates a plane spiral (and it is by the finite width and depth of the groove, but not by the wider spacing between than within tracks, that it falls short of being a perfect example); note that successive loops differ in diameter.

one gets the conical spiral (see diagram) Any cylindrical map projection can be used as the basis for a spherical spiral: draw a straight line on the map and find its inverse projection on the sphere, a kind of spherical curve.

These are curves for which longitude and colatitude are in a linear relationship, analogous to Archimedean spirals in the plane; under the azimuthal equidistant projection a Clelia curve projects to a planar Archimedean spiral.

If one represents a unit sphere by spherical coordinates then setting the linear dependency

D’Arcy Wentworth Thompson's On Growth and Form gives extensive treatment to these spirals.

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

A spiral like form has been found in Mezine, Ukraine, as part of a decorative object dated to 10,000 BCE.

[15] It is carved into the rock of a stone lozenge near the main entrance of the prehistoric Newgrange monument in County Meath, Ireland.

[16] The triskelion symbol, consisting of three interlocked spirals or three bent human legs, appears in many early cultures: examples include Mycenaean vessels, coinage from Lycia, staters of Pamphylia (at Aspendos, 370–333 BC) and Pisidia, as well as the heraldic emblem on warriors' shields depicted on Greek pottery.

[17] Spirals occur commonly in pre-Columbian art in Latin and Central America.

The more than 1,400 petroglyphs (rock engravings) in Las Plazuelas, Guanajuato Mexico, dating 750-1200 AD, predominantly depict spirals, dot figures and scale models.

[18] In Colombia, monkeys, frog and lizard-like figures depicted in petroglyphs or as gold offering-figures frequently include spirals, for example on the palms of hands.

[19] In Lower Central America, spirals along with circles, wavy lines, crosses and points are universal petroglyph characters.

[20] Spirals also appear among the Nazca Lines in the coastal desert of Peru, dating from 200 BC to 500 AD.

The geoglyphs number in the thousands and depict animals, plants and geometric motifs, including spirals.

[21] Spiral shapes, including the swastika, triskele, etc., have often been interpreted as solar symbols.

[citation needed] Roof tiles dating back to the Tang dynasty with this symbol have been found west of the ancient city of Chang'an (modern-day Xi'an).

[citation needed][year needed] 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).

The spiral is also found in structures as small as the double helix of DNA and as large as a galaxy.

Due to this frequent natural occurrence, the spiral is the official symbol of the World Pantheist Movement.

[...] while a helix is repetitive, a spiral expands and thus epitomizes growth - conceptually ad infinitum.

Among the most famous of spiral-inspired art is Robert Smithson's earthwork, "Spiral Jetty", at the Great Salt Lake in Utah.

The Spiral is also a prominent theme in the anime Gurren Lagann, where it represents a philosophy and way of life.

2012 A Piece of Mind By Wayne A Beale also depicts a large spiral in this book of dreams and images.

[25][full citation needed][26][verification needed] The coiled spiral is a central image in Australian artist Tanja Stark's Suburban Gothic iconography, that incorporates spiral electric stove top elements as symbols of domestic alchemy and spirituality.