Spiral


In mathematics, a spiral is a curve which emanates from a point, moving further away as it revolves around the point. It is a subtype of whorled patterns, a broad group that also includes concentric objects.

Two-dimensional

A two-dimensional, or plane, spiral may be easily described using polar coordinates, where the radius is a monotonic continuous function of angle :
The circle would be regarded as a degenerate case.
In --coordinates the curve has the parametric representation:

  • Examples

Some of the most important sorts of two-dimensional spirals include:
An Archimedean spiral is, for example, generated while coiling a carpet.
A hyperbolic spiral appears as image of a helix with a special central projection. A hyperbolic spiral is some times called reciproke spiral, because it is the image of an Archimedean spiral with a circle-inversion.
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 Archimedean, but is not: see Involute#Examples.

Geometric properties

The following considerations are dealing with spirals, which can be described by a polar equation, especially for the cases and the logarithmic spiral.
;Polar slope angle
The angle between the spiral tangent and the corresponding polar circle is called angle of the polar slope and the polar slope.
From vector calculus in polar coordinates one gets the formula
Hence the slope of the spiral is
In case of an Archimedean spiral the polar slope is
In a logarithmic spiral, is constant.
;Curvature
The curvature of a curve with polar equation is
For a spiral with one gets
In case of
.

Only for the spiral has an inflection point.
The curvature of a logarithmic spiral is
;Sector area
The area of a sector of a curve with polar equation is
For a spiral with equation one gets
The formula for a logarithmic spiral is
;Arc length
The length of an arc of a curve with polar equation is
For the spiral the length is
Not all these integrals can be solved by a suitable table. In case of a Fermat's spiral, the integral can be expressed by elliptic integrals only.
The arc length of a logarithmic spiral is
;Circle inversion
The inversion at the unit circle has in polar coordinates the simple description:.
  • The image of a spiral under the inversion at the unit circle is the spiral with polar equation. For example: The inverse of an Archimedean spiral is a hyperbolic spiral.

    Bounded spirals

The function of a spiral is usually strictly monotonic, 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 the arctan function:
;Example 1
Setting and the choice gives a spiral, that starts at the origin and approaches the circle with radius .
;Example 2
For and one gets a spiral, that approaches the origin and approaches the circle with radius .

Three-dimensional

Helices

Two major definitions of "spiral" in the American Heritage Dictionary are:
  1. a curve on a plane that winds around a fixed center point at a continuously increasing or decreasing distance from the point.
  2. 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 gramophone record closely approximates a plane spiral ; 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, 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 fairly 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 conical spiral.
Two well-known spiral space curves are conical spirals and spherical spirals, defined below.
Another instance of space spirals is the toroidal spiral. A spiral wound around a helix, also known as double-twisted helix, represents objects such as coiled coil filaments.

Conical spirals

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.
;Example
Starting with an archimedean spiral one gets the conical spiral

Spherical spirals

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.
One of the most basic families of spherical spirals is the Clelia curves, which project to straight lines on an equirectangular projection. 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 for the angle coordinates gives a parametric curve in terms of parameter,
Another family of spherical spirals is the rhumb lines or loxodromes, that project to straight lines on the Mercator projection. These are the trajectories traced by a ship traveling with constant bearing. Any loxodrome spirals infinitely around either pole, closer and closer each time, unlike a Clelia curve which maintains uniform spacing in colatitude. Under stereographic projection, a loxodrome projects to a logarithmic spiral in the plane.

In nature

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 shells, 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.
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.

As a symbol

The Celtic 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; triple spirals were carved at least 2,500 years before the Celts reached Ireland, but have long since become part of Celtic culture. 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 and Pisidia, as well as the heraldic emblem on warriors' shields depicted on Greek pottery.
Spirals occur commonly in pre-Columbian art in Latin and Central America. The more than 1,400 petroglyphs in Las Plazuelas, Guanajuato Mexico, 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 include spirals, for example on the palms of hands. In Lower Central America, spirals along with circles, wavy lines, crosses and points are universal petroglyph characters. 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.
Spirals are also a symbol of hypnosis, stemming from the cliché of people and cartoon characters being hypnotized by staring into a spinning spiral. 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 suggest that 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. Due to 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 of Dialectical monism.

The spiral is a frequent symbol for spiritual purification, both within Christianity and beyond. while a helix is repetitive, a spiral expands and thus epitomizes growth – conceptually ad infinitum.