In
mathematics, a spiral is a
curve which emanates from a point, moving farther away as it revolves around the point.[1][2][3][4] It is a subtype of
whorled patterns, a broad group that also includes
concentric objects.
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
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. 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.[5]
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.
The involute of a circle (black) is not identical to the Archimedean spiral (red).
An Archimedean spiral is, for example, generated while coiling a carpet.[6]
A hyperbolic spiral appears 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 Archimedean spiral with a circle-inversion (see below).[7]
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 (Archimedean, hyperbolic, Fermat's, lituus spirals) and the logarithmic spiral .
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 Archimedean spiral is a hyperbolic spiral.
A logarithmic spiral is mapped onto the logarithmic spiral
Bounded spirals
The function of a spiral is usually strictly monotonic, continuous
and un
bounded. 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 (like an Archimedean spiral) and approaches the circle with radius (diagram, left).
Example 2
For and one gets a spiral, that approaches the origin (like a hyperbolic spiral) and approaches the circle with radius (diagram, right).
Three-dimensional
"Space spiral" redirects here. For the building, see
Space Spiral.
Two well-known spiral
space curves are conical spirals and spherical spirals, defined below.
Another instance of space spirals is the toroidal spiral.[8] A spiral wound around a helix,[9] also known as double-twisted helix,[10] represents objects such as
coiled coil filaments.
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 (see diagram)
Spherical spirals
If one represents a sphere of radius by:
and sets the linear dependency for the angle coordinates, one gets a
spherical curve called spherical spiral[11] 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.
Spherical spiral
Loxodrome
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[13] 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.[14]
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.
As a symbol
A spiral like form has been found in
Mezine,
Ukraine, as part of a decorative object dated to 10,000 BCE.[citation needed]
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.[15] 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.[16] 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.[17]
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.[18] 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.[19] In Lower Central America spirals along with circles, wavy lines, crosses and points are universal petroglyphs characters.[20] 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.[21]
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.[22]
The spiral is also a symbol of the
dialectic process and
Dialectical monism.
Gothic Revival spiralling bell tower of the Maison des compagnons du tour de France,
Nantes, unknown architect,
c.1910
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.[23] 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. The spiral is the central theme of the horror manga Uzumaki by
Junji Ito, where a small coastal town is afflicted by a curse involving spirals. 2012 A Piece of Mind By Wayne A Beale also depicts a large spiral in this book of dreams and images.[24][full citation needed][25][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.[26][27]
^Anthony Murphy and Richard Moore, Island of the Setting Sun: In Search of Ireland's Ancient Astronomers, 2nd ed., Dublin: The Liffey Press, 2008, pp. 168-169
^For example, the trislele on
Achilles' round shield on an Attic late sixth-century hydria at the
Boston Museum of Fine Arts, illustrated in John Boardman, Jasper Griffin and Oswyn Murray, Greece and the Hellenistic World (Oxford History of the Classical World) vol. I (1988), p. 50.
^Harrison, Paul.
"Pantheist Art"(PDF). World Pantheist Movement. Retrieved 7 June 2012.
^Israel, Nico (2015). Spirals : the whirled image in twentieth-century literature and art. New York Columbia University Press. pp. 161–186.
ISBN978-0-231-15302-7.
Harary, G., Tal, A., 2011. The natural 3D spiral. Computer Graphics Forum 30 (2), 237 – 246
[1]Archived 2015-11-22 at the
Wayback Machine.
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
[2].
Kurnosenko, A. (2010). "Applying inversion to construct planar, rational spirals that satisfy two-point G2 Hermite data". Computer Aided Geometric Design. 27 (3): 262–280.
arXiv:0902.4834.
doi:
10.1016/j.cagd.2009.12.004.
S2CID14476206.
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
[3]Archived 2013-06-28 at the
Wayback Machine.
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
[4]Archived 2013-06-28 at the
Wayback Machine.
Yoshida, N., Saito, T., 2006. Interactive aesthetic curve segments. The Visual Computer 22 (9), 896–905
[5]Archived 2016-03-04 at the
Wayback Machine.
Yoshida, N., Saito, T., 2007. Quasi-aesthetic curves in rational cubic Bézier forms. Computer-Aided Design and Applications 4 (9–10), 477–486
[6]Archived 2016-03-03 at the
Wayback Machine.
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
[7].
Ziatdinov, R., Yoshida, N., Kim, T., 2012. Fitting G2 multispiral transition curve joining two straight lines, Computer-Aided Design 44(6), 591—596
[8].
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, 2012
[9].
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
[10].