a curve on a plane that winds around a fixed center point at a continuously increasing or decreasing distance from the point.
a three-dimensional curve that turns around an axis at a constant or continuously varying distance while moving parallel to the axis; a helix.
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 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. In another example, the "center lines" of the arms of a spiral galaxy trace logarithmic spirals.
The second definition includes two kinds of 3-dimensional relatives of spirals:
a conical or volute spring (including the spring used to hold and make contact with the negative terminals of AA or AAA batteries in a battery box), and the vortex that is created when water is draining in a sink is often described as a spiral, or as a conical helix.
quite explicitly, definition 2 also includes a cylindrical coil spring and a strand of DNA, both of which are quite helical, so that "helix" is a more useful description than "spiral" for each of them; in general, "spiral" is seldom applied if successive "loops" of a curve have the same diameter.
In the side picture, the black curve at the bottom is an Archimedean spiral, while the green curve is a helix. The curve shown in red is a conic helix.
The Spiral of Theodorus: an approximation of the Archimedean spiral composed of contiguous right triangles
The involute of a circle, used twice on each tooth of almost every modern gear
spiral of Theodorus
Fibonacci Spiral (golden spiral)
The involute of a circle (black) is not identical to the Archimedean spiral (red).
Hyperbolic spiral as central projection of a helix
An Archimedian spiral is, for example, generated while coiling a carpet.
A hyperbolic spiral apears as image of a helix with a special central projection (see diagram). A hyperbolic spiral is some times called reciproke spiral, because it is the image of an Archimedian spiral with an circle-inversion (see below).
The name logarithmic spiral is due to the equation . Approximations of this are found in nature.
Spirals which do not fit into this scheme of the first 5 examples:
A Cornu spiral has two asymptotic points.
The spiral of Theodorus is a polygon.
The Fibonacci Spiral consists of a sequence of circle arcs.
The involute of a circle looks like an Archimedian, but is not: see Involute#Examples.
The following considerations are dealing with spirals, which can be described by a polar equation , especially for the cases (Archimedian, hyperbolic, Fermat's, lituus spirals) and the logarithmic spiral .
Definition of sector (light blue) and polar slope angle
Polar slope angle
The angle between the spiral tangent and the corresponding polar circle (see diagram) is called angle of the polar slope and the polar slope.
The image of a spiral under the inversion at the unit circle is the spiral with polar equation . For example: The inverse of an Archimedian spiral is a hyperbolic spiral.
A logarithmic spiral is mapped onto the logarithmic spiral
Bounded spirals: (left), (right)
Function of a spiral is usually strictly monotnic, continuous
and unbounded. For the standard spirals is either a power function or an exponential function. If one chooses for a bounded function the spiral is bounded, too. A suitable bounded function is function arcus tangens:
Setting and the choice gives a spiral, that starts at the origin (like an Archimedean spiral) and approaches the circle with radius (diagram, left).
For and one gets a spiral, that approaches the origin (like a hyperbolic spiral) and approaches the circle with radius (diagram, right).
Conic spiral with Archimedian spiral as floor plan
If in the --plane a spiral with parametric representation
is given, then there can be added a third coordinate , such that the now space curve lies on the cone with equation :
Spirals based on this procedure are called conical spirals.
Starting with an archimedean spiral one gets the conical spiral (see diagram)
Spherical spiral with
If one represents a sphere of radius by:
and sets the linear dependency for the angle coordinates, one gets a spherical spiral with the parametric representation (with equal to twice the number of turns)
Spherical spirals were known to Pappus, too.
Remark: a rhumb line is not a spherical spiral in this sense.
A rhumb line (also known as a loxodrome or "spherical spiral") is the curve on a sphere traced by a ship with constant bearing (e.g., travelling from one pole to the other while keeping a fixed angle with respect to the meridians). The loxodrome has an infinite number of revolutions, with the separation between them decreasing as the curve approaches either of the poles, unlike an Archimedean spiral which maintains uniform line-spacing regardless of radius.
A model for the pattern of florets in the head of a sunflower was proposed by H. Vogel. This has the form
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 the golden angle which is related to the golden ratio and gives a close packing of florets.
Spirals in plants and animals are frequently described as whorls. This is also the name given to spiral shaped fingerprints.
An artist's rendering of a spiral galaxy.
Sunflower head displaying florets in spirals of 34 and 55 around the outside.
In the laboratory
When potassium sulfate is heated in water and subjected to swirling in a beaker, the crystals form a multi-arm spiral structure when allowed to settle
Potassium sulfate forms a spiral structure in solution.
The spiral and triple spiral motif is a Neolithic symbol in Europe (Megalithic Temples of Malta). The Celtic symbol the triple spiral is in fact a pre-Celtic symbol. It is carved into the rock of a stone lozenge near the main entrance of the prehistoric Newgrange monument in County Meath, Ireland. Newgrange was built around 3200 BCE predating the Celts and the triple spirals were carved at least 2,500 years before the Celts reached Ireland but has long since been incorporated into Celtic culture. The triskelion symbol, consisting of three interlocked spirals or three bent human legs, appears in many early cultures, including Mycenaean vessels, on coinage in Lycia, on staters of Pamphylia (at Aspendos, 370–333 BC) and Pisidia, as well as on the heraldic emblem on warriors' shields depicted on Greek pottery.
Spirals can be found throughout pre-Columbian art in Latin and Central America. The more than 1,400 petroglyphs (rock engravings) in Las Plazuelas, GuanajuatoMexico, dating 750-1200 AD, predominantly depict spirals, dot figures and scale models. In Colombia monkeys, frog and lizard like figures depicted in petroglyphs or as gold offering figures frequently includes spirals, for example on the palms of hands. In Lower Central America spirals along with circles, wavy lines, crosses and points are universal petroglyphs characters. Spirals can also be found 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.
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 dialectic process and Dialectical monism.
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