In
mathematics , a spiral is a
curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.
Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ... which emanates from a point, moving farther away as it revolves around the point.
Helices
Two major definitions of "spiral" in the
American Heritage Dictionary
American(s) may refer to:
* American, something of, from, or related to the United States of America, commonly known as the " United States" or "America"
** Americans, citizens and nationals of the United States of America
** American ancestry, ... are:
Spiral
# 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
A helix () is a shape like a corkscrew or spiral staircase. It is a type of smooth space curve with tangent lines at a constant angle to a fixed axis. Helices are important in biology, as the DNA molecule is formed as two intertwined hel ... .
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
A record, recording or records may refer to:
An item or collection of data Computing
* Record (computer science), a data structure
** Record, or row (database), a set of fields in a database related to one entity
** Boot sector or boot 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
Spiral galaxies form a class of galaxy originally described by Edwin Hubble in his 1936 work ''The Realm of the Nebulae''logarithmic spiral s.
The second definition includes two kinds of 3-dimensional relatives of spirals:
# a conical or volute spring
A volute spring, also known as a conical spring, is a compression spring in the form of a cone (somewhat like the classical volute decorative architectural ornament). Under compression, the coils slide past each other, thus enabling the spring to ... battery box ), and the vortex
In fluid dynamics, a vortex ( : vortices or vortexes) is a region in a fluid in which the flow revolves around an axis line, which may be straight or curved. Vortices form in stirred fluids, and may be observed in smoke rings, whirlpools in t ... 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
The Archimedean spiral (also known as the arithmetic spiral) is a spiral named after the 3rd-century BC Greek mathematician Archimedes. It is the locus corresponding to the locations over time of a point moving away from a fixed point with a cons ... Two-dimensional
A two-dimensional
In mathematics, a plane is a Euclidean ( flat), two-dimensional surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. Planes can arise ... polar coordinates
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to t ... radius
In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ... r is a monotonic
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of ord ... continuous function of angle \varphi :
* r=r(\varphi)\; .
The circle would be regarded as a degenerate
Degeneracy, degenerate, or degeneration may refer to:
Arts and entertainment
* ''Degenerate'' (album), a 2010 album by the British band Trigger the Bloodshed
* Degenerate art, a term adopted in the 1920s by the Nazi Party in Germany to descr ... function not being strictly monotonic, but rather constant ).
In ''x -y -coordinates'' the curve has the parametric representation:
* x=r(\varphi)\cos\varphi \ ,\qquad y=r(\varphi)\sin\varphi\; .
Examples
Some of the most important sorts of two-dimensional spirals include:
* The Archimedean spiral
The Archimedean spiral (also known as the arithmetic spiral) is a spiral named after the 3rd-century BC Greek mathematician Archimedes. It is the locus corresponding to the locations over time of a point moving away from a fixed point with a cons ... r=a \varphi
* The hyperbolic spiral : r = a/ \varphi
* Fermat's spiral : r= a\varphi^
* The lituus : r = a\varphi^
* The logarithmic spiral : r=ae^
* The Cornu spiral or ''clothoid''
* The Fibonacci spiral and golden spiral
* The Spiral of Theodorus : an approximation of the Archimedean spiral composed of contiguous right triangles
* The involute
In mathematics, an involute (also known as an evolvent) is a particular type of curve that is dependent on another shape or curve. An involute of a curve is the locus of a point on a piece of taut string as the string is either unwrapped from o ... gear
A gear is a rotating circular machine part having cut teeth or, in the case of a cogwheel or gearwheel, inserted teeth (called ''cogs''), which mesh with another (compatible) toothed part to transmit (convert) torque and speed. The basic p ...
Image:Archimedean spiral.svg, Archimedean spiral
Image:Hyperspiral.svg, hyperbolic spiral
Image:Fermat's spiral.svg, Fermat's spiral
Image:Lituus.svg, lituus
Image:Logarithmic Spiral Pylab.svg, logarithmic spiral
Image:Cornu Spiral.svg, Cornu spiral
Image:Spiral of Theodorus.svg, spiral of Theodorus
Image:Fibonacci_spiral.svg, Fibonacci Spiral (golden spiral)
Image:Archimedean-involute-circle-spirals-comparison.svg, The involute of a circle (black) is not identical to the Archimedean spiral (red).
\varphi=\tfrac\cdot \ln \tfrac . 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.Involute#Examples .
Geometric properties
The following considerations are dealing with spirals, which can be described by a polar equation r=r(\varphi) , especially for the cases r(\varphi)=a\varphi^n (Archimedean, hyperbolic, Fermat's, lituus spirals) and the logarithmic spiral r=ae^ .
\alpha between the spiral tangent and the corresponding polar circle (see diagram) is called ''angle of the polar slope and \tan \alpha the ''polar slope''.
From vector calculus in polar coordinates one gets the formula
:\tan\alpha=\frac\ .
Hence the slope of the spiral \;r=a\varphi^n \; is
* \tan\alpha=\frac\ .
In case of an ''Archimedean spiral'' (n=1 ) the polar slope is \; \tan\alpha=\tfrac\ .
The ''logarithmic spiral'' is a special case, because of \ \tan\alpha=k\ ''constant'' !
;curvature
The curvature \kappa of a curve with polar equation r=r(\varphi) is
:\kappa = \frac\ .
For a spiral with r=a\varphi^n one gets
* \kappa = \dotsb = \frac\frac\ .
In case of n=1 ''(Archimedean spiral)''
\kappa=\tfrac .-1 the spiral has an ''inflection point''.
The curvature of a ''logarithmic spiral'' \; r=a e^ \; is \; \kappa=\tfrac \; .
;Sector area
The area of a sector of a curve (see diagram) with polar equation r=r(\varphi) is
:A=\frac\int_^ r(\varphi)^2\; d\varphi\ .
For a spiral with equation r=a\varphi^n\; one gets
* A=\frac\int_^ a^2\varphi^\; d\varphi
=\frac\big(\varphi_2^- \varphi_1^\big)\ , \quad \text\quad n\ne-\frac,
:A=\frac\int_^ \frac\; d\varphi
=\frac(\ln\varphi_2-\ln\varphi_1)\ ,\quad \text \quad n=-\frac\ .
The formula for a ''logarithmic spiral'' \; r=a e^ \; is
\ A=\tfrac\ .
;Arc length
The length of an arc of a curve with polar equation r=r(\varphi) is
:L=\int\limits_^\sqrt\,\mathrm\varphi \ .
For the spiral r=a\varphi^n\; the length is
* L=\int_^ \sqrt\; d\varphi
= a\int\limits_^\varphi^\sqrtd\varphi
\ .
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 integral
In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (). Their name originates from their originally arising in ... \; r=a e^ \; is \ L=\tfrac\big(r(\varphi_2)-r(\varphi_1)\big) \ .
;Circle inversion
The inversion at the unit circle has in polar coordinates the simple description: \ (r,\varphi) \mapsto (\tfrac,\varphi)\ .
* The image of a spiral \ r= a\varphi^n\ under the inversion at the unit circle is the spiral with polar equation \ r= \tfrac\varphi^\ . For example: The inverse of an Archimedean spiral is a hyperbolic spiral.
:A logarithmic spiral \; r=a e^ \; is mapped onto the logarithmic spiral \; r=\tfrac e^ \; .
Bounded spirals
r(\varphi) of a spiral is usually strictly monotonic, continuous
and unbounded
Boundedness or bounded may refer to:
Economics
* Bounded rationality, the idea that human rationality in decision-making is bounded by the available information, the cognitive limitations, and the time available to make the decision
* Bounded e ... r(\varphi) is either a power function or an exponential function. If one chooses for r(\varphi) a ''bounded'' function the spiral is bounded, too. A suitable bounded function is the arctan
In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted domains). S ... \;r=a \arctan(k\varphi)\; and the choice \;k=0.1, a=4, \;\varphi\ge 0\; gives a spiral, that starts at the origin (like an Archimedean spiral) and approaches the circle with radius \;r=a\pi/2\; (diagram, left).
;Example 2
For \;r=a (\arctan(k\varphi)+\pi/2)\; and \;k=0.2, a=2,\; -\infty<\varphi<\infty\; one gets a spiral, that approaches the origin (like a hyperbolic spiral) and approaches the circle with radius \;r=a\pi\; (diagram, right).
Three-dimensional
space curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.
Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition tha ... coiled coil filament
An incandescent light bulb, incandescent lamp or incandescent light globe is an electric light with a wire filament heated until it glows. The filament is enclosed in a glass bulb with a vacuum or inert gas to protect the filament from oxidat ... Slinky
The Slinky is a helical spring toy invented by Richard James in the early 1940s. It can perform a number of tricks, including travelling down a flight of steps end-over-end as it stretches and re-forms itself with the aid of gravity and its own ... Conical spirals
If in the x -y -plane a spiral with parametric representation
:x=r(\varphi)\cos\varphi \ ,\qquad y=r(\varphi)\sin\varphi
is given, then there can be added a third coordinate z(\varphi) , such that the now space curve lies on the cone with equation \;m(x^2+y^2)=(z-z_0)^2\ ,\ m>0\; :
* x=r(\varphi)\cos\varphi \ ,\qquad y=r(\varphi)\sin\varphi\ , \qquad \color \ .
Spirals based on this procedure are called conical spirals.
;Example
Starting with an ''archimedean spiral'' \;r(\varphi)=a\varphi\; one gets the conical spiral (see diagram)
:x=a\varphi\cos\varphi \ ,\qquad y=a\varphi\sin\varphi\ , \qquad z=z_0 + ma\varphi \ ,\quad \varphi \ge 0 \ .
Spherical spirals
If one represents a sphere of radius r by:
:
\begin
x &=& r \cdot \sin \theta \cdot \cos \varphi \\
y &=& r \cdot \sin \theta \cdot \sin \varphi \\
z &=& r \cdot \cos \theta
\end
and sets the linear dependency \; \varphi=c\theta , \; c> 2\; , for the angle coordinates, one gets a spherical curve
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is th ... c equal to twice the number of turns)
*
\begin
x &=& r \cdot \sin \theta \cdot \cos \\
y &=& r \cdot \sin \theta \cdot \sin \\
z &=& r \cdot \cos \theta\qquad \qquad 0\le\theta\le \pi \ .
\end
Spherical spirals were known to Pappus, too.
Remark: a rhumb line is ''not'' a spherical spiral in this sense.
KUGSPI-5 Archimedische Kugelspirale.gif, Spherical spiral
KUGSPI-9_Loxodrome.gif, 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
Pole may refer to:
Astronomy
*Celestial pole, the projection of the planet Earth's axis of rotation onto the celestial sphere; also applies to the axis of rotation of other planets
* Pole star, a visible star that is approximately aligned with th ... angle
In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the '' vertex'' of the angle.
Angles formed by two rays lie in the plane that contains the rays. Angles ... meridians ). The loxodrome has an infinite
Infinite may refer to:
Mathematics
*Infinite set, a set that is not a finite set
*Infinity, an abstract concept describing something without any limit
Music
*Infinite (group)
Infinite ( ko, 인피니트; stylized as INFINITE) is a South Ko ... revolution
In political science, a revolution (Latin: ''revolutio'', "a turn around") is a fundamental and relatively sudden change in political power and political organization which occurs when the population revolts against the government, typically due ... Archimedean spiral
The Archimedean spiral (also known as the arithmetic spiral) is a spiral named after the 3rd-century BC Greek mathematician Archimedes. It is the locus corresponding to the locations over time of a point moving away from a fixed point with a cons ... In nature
The study of spirals in nature
Nature, in the broadest sense, is the physical world or universe. "Nature" can refer to the phenomena of the physical world, and also to life in general. The study of nature is a large, if not the only, part of science. Although humans ar ... Christopher Wren
Sir Christopher Wren PRS FRS (; – ) was one of the most highly acclaimed English architects in history, as well as an anatomist, astronomer, geometer, and mathematician-physicist. He was accorded responsibility for rebuilding 52 churc ... shells form a logarithmic spiral ; Jan Swammerdam observed the common mathematical characteristics of a wide range of shells from ''Helix
A helix () is a shape like a corkscrew or spiral staircase. It is a type of smooth space curve with tangent lines at a constant angle to a fixed axis. Helices are important in biology, as the DNA molecule is formed as two intertwined hel ... Spirula
''Spirula spirula'' is a species of deep-water squid-like cephalopod mollusk. It is the only extant member of the genus ''Spirula'', the family Spirulidae, and the order Spirulida. Because of the shape of its internal shell, it is commonly kno ... Henry Nottidge Moseley described the mathematics of univalve shells. D’Arcy Wentworth Thompson 's ''On Growth and Form
''On Growth and Form'' is a book by the Scottish mathematical biology, mathematical biologist D'Arcy Wentworth Thompson (1860–1948). The book is long – 793 pages in the first edition of 1917, 1116 pages in the second edition of 1942.
The ... shape
A shape or figure is a graphical representation of an object or its external boundary, outline, or external surface, as opposed to other properties such as color, texture, or material type.
A plane shape or plane figure is constrained to lie on ... geometric progression
In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the ''common ratio''. For e ... Nautilus
The nautilus (, ) is a pelagic marine mollusc of the cephalopod family Nautilidae. The nautilus is the sole extant family of the superfamily Nautilaceae and of its smaller but near equal suborder, Nautilina.
It comprises six living species ... ammonite
Ammonoids are a group of extinct marine mollusc animals in the subclass Ammonoidea of the class Cephalopoda. These molluscs, commonly referred to as ammonites, are more closely related to living coleoids (i.e., octopuses, squid and cuttle ... helico -spiral pattern. Thompson also studied spirals occurring in horn s, teeth
A tooth ( : teeth) is a hard, calcified structure found in the jaws (or mouths) of many vertebrates and used to break down food. Some animals, particularly carnivores and omnivores, also use teeth to help with capturing or wounding prey, te ... claw
A claw is a curved, pointed appendage found at the end of a toe or finger in most amniotes (mammals, reptiles, birds). Some invertebrates such as beetles and spiders have somewhat similar fine, hooked structures at the end of the leg or tars ... plant
Plants are predominantly Photosynthesis, photosynthetic eukaryotes of the Kingdom (biology), kingdom Plantae. Historically, the plant kingdom encompassed all living things that were not animals, and included algae and fungi; however, all curr ... floret
This glossary of botanical terms is a list of definitions of terms and concepts relevant to botany and plants in general. Terms of plant morphology are included here as well as at the more specific Glossary of plant morphology and Glossary ... sunflower was proposed by H. Vogel. This has the form
:\theta = n \times 137.5^,\ r = c \sqrt
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
In geometry, the golden angle is the smaller of the two angles created by sectioning the circumference of a circle according to the golden ratio; that is, into two arcs such that the ratio of the length of the smaller arc to the length of the l ... golden ratio
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0,
where the Greek letter phi ( ... whorls . This is also the name given to spiral shaped fingerprint
A fingerprint is an impression left by the friction ridges of a human finger. The recovery of partial fingerprints from a crime scene is an important method of forensic science. Moisture and grease on a finger result in fingerprints on surfa ...
The center Galaxy of Cat's Eye.jpg, An artist's rendering of a spiral galaxy.
Helianthus whorl.jpg, 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
Ukraine ( uk, Україна, Ukraïna, ) is a country in Eastern Europe. It is the second-largest European country after Russia, which it borders to the east and northeast. Ukraine covers approximately . Prior to the ongoing Russian invas ... triple spiral motif is a Neolithic
The Neolithic period, or New Stone Age, is an Old World archaeological period and the final division of the Stone Age. It saw the Neolithic Revolution, a wide-ranging set of developments that appear to have arisen independently in several part ... 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
Newgrange ( ga, Sí an Bhrú) is a prehistoric monument in County Meath in Ireland, located on a rise overlooking the River Boyne, west of Drogheda. It is an exceptionally grand passage tomb built during the Neolithic Period, around 3200 ... County Meath
County Meath (; gle, Contae na Mí or simply ) is a county in the Eastern and Midland Region of Ireland, within the province of Leinster. It is bordered by Dublin to the southeast, Louth to the northeast, Kildare to the south, Offaly to the ... Ireland
Ireland ( ; ga, Éire ; Ulster Scots dialect, Ulster-Scots: ) is an island in the Atlantic Ocean, North Atlantic Ocean, in Northwestern Europe, north-western Europe. It is separated from Great Britain to its east by the North Channel (Grea ... triskelion
A triskelion or triskeles is an ancient motif consisting of a triple spiral exhibiting rotational symmetry.
The spiral design can be based on interlocking Archimedean spirals, or represent three bent human legs. It is found in artefacts of t ... Mycenaean vessels, on coinage in Lycia
Lycia ( Lycian: 𐊗𐊕𐊐𐊎𐊆𐊖 ''Trm̃mis''; el, Λυκία, ; tr, Likya) was a state or nationality that flourished in Anatolia from 15–14th centuries BC (as Lukka) to 546 BC. It bordered the Mediterranean Sea in what is t ... stater
The stater (; grc, , , statḗr, weight) was an ancient coin used in various regions of Greece. The term is also used for similar coins, imitating Greek staters, minted elsewhere in ancient Europe.
History
The stater, as a Greek silver curre ... Pamphylia
Pamphylia (; grc, Παμφυλία, ''Pamphylía'') was a region in the south of Asia Minor, between Lycia and Cilicia, extending from the Mediterranean to Mount Taurus (all in modern-day Antalya province, Turkey). It was bounded on the north b ... Aspendos
Aspendos or Aspendus ( Pamphylian: ΕΣΤϜΕΔΥΣ; Attic: Ἄσπενδος) was an ancient Greco-Roman city in Antalya province of Turkey. The site is located 40 km east of the modern city of Antalya.
It was situated on the Eurymedon Ri ... Pisidia
Pisidia (; grc-gre, Πισιδία, ; tr, Pisidya) was a region of ancient Asia Minor located north of Pamphylia, northeast of Lycia, west of Isauria and Cilicia, and south of Phrygia, corresponding roughly to the modern-day province of A ... 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
A petroglyph is an image created by removing part of a rock surface by incising, picking, carving, or abrading, as a form of rock art. Outside North America, scholars often use terms such as "carving", "engraving", or other description ... Las Plazuelas , Guanajuato
Guanajuato (), officially the Free and Sovereign State of Guanajuato ( es, Estado Libre y Soberano de Guanajuato), is one of the 32 states that make up the Federal Entities of Mexico. It is divided into 46 municipalities and its capital city i ... Mexico
Mexico ( Spanish: México), officially the United Mexican States, is a country in the southern portion of North America. It is bordered to the north by the United States; to the south and west by the Pacific Ocean; to the southeast by Guate ... 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.
Spiral shapes, including the swastika
The swastika (卐 or 卍) is an ancient religious and cultural symbol, predominantly in various Eurasian, as well as some African and American cultures, now also widely recognized for its appropriation by the Nazi Party and by neo-Nazis. I ... triskele , etc., have often been interpreted as solar symbol
A solar symbol is a symbol representing the Sun.
Common solar symbols include circles (with or without rays), crosses, and spirals.
In religious iconography, personifications of the Sun or solar attributes are often indicated by means of a hal ... Tang Dynasty
The Tang dynasty (, ; zh, t= ), or Tang Empire, was an imperial dynasty of China that ruled from 618 to 907 AD, with an interregnum between 690 and 705. It was preceded by the Sui dynasty and followed by the Five Dynasties and Ten Kingdo ... Chang'an
Chang'an (; ) is the traditional name of Xi'an. The site had been settled since Neolithic times, during which the Yangshao culture was established in Banpo, in the city's suburbs. Furthermore, in the northern vicinity of modern Xi'an, Qin ... hypnosis
Hypnosis is a human condition involving focused attention (the selective attention/selective inattention hypothesis, SASI), reduced peripheral awareness, and an enhanced capacity to respond to suggestion.In 2015, the American Psychologica ... cliché
A cliché ( or ) is an element of an artistic work, saying, or idea that has become overused to the point of losing its original meaning or effect, even to the point of being weird or irritating, especially when at some earlier time it was consi ... Kaa in Disney's ''The Jungle Book
''The Jungle Book'' (1894) is a collection of stories by the English author Rudyard Kipling. Most of the characters are animals such as Shere Khan the tiger and Baloo the bear, though a principal character is the boy or "man-cub" Mowgli, w ... dizziness
Dizziness is an imprecise term that can refer to a sense of disorientation in space, vertigo, or lightheadedness. It can also refer to disequilibrium or a non-specific feeling, such as giddiness or foolishness.
Dizziness is a common medical ... anime
is hand-drawn and computer-generated animation originating from Japan. Outside of Japan and in English, ''anime'' refers specifically to animation produced in Japan. However, in Japan and in Japanese, (a term derived from a shortening of ... manga
Manga ( Japanese: 漫画 ) are comics or graphic novels originating from Japan. Most manga conform to a style developed in Japan in the late 19th century, and the form has a long prehistory in earlier Japanese art. The term ''manga'' is use ... double helix of DNA and as large as a galaxy
A galaxy is a system of stars, stellar remnants, interstellar gas, dust, dark matter, bound together by gravity. The word is derived from the Greek ' (), literally 'milky', a reference to the Milky Way galaxy that contains the Solar Sys ... dialectic
Dialectic ( grc-gre, διαλεκτική, ''dialektikḗ''; related to dialogue; german: Dialektik), also known as the dialectical method, is a discourse between two or more people holding different points of view about a subject but wishing to ... Dialectical monism
Dialectical monism, also known as dualistic monism or monistic dualism, is an ontological position that holds that reality is ultimately a unified whole, distinguishing itself from monism by asserting that this whole necessarily expresses itself i ... In art
The spiral has inspired artists throughout the ages. Among the most famous of spiral-inspired art is Robert Smithson
Robert Smithson (January 2, 1938 – July 20, 1973) was an American artist known for sculpture and land art who often used drawing and photography in relation to the spatial arts. His work has been internationally exhibited in galleries and mu ... earthwork , " Spiral Jetty ", at the Great Salt Lake in Utah. 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
Nine Inch Nails, commonly abbreviated as NIN and stylized as NIИ, is an American industrial rock band formed in Cleveland in 1988. Singer, songwriter, multi-instrumentalist, and producer Trent Reznor was the only permanent member of the ban ... The Downward Spiral ''. The Spiral is also a prominent theme in the anime Gurren Lagann
''Gurren Lagann'', known in Japan as , is a Japanese mecha anime television series animated by Gainax and co-produced by Aniplex and Konami. It ran for 27 episodes on TV Tokyo between April and September 2007. It was directed by Hiroyuki ... 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. The coiled spiral is a central image in Australian artist Tanja Stark's Suburban Gothic
Suburban Gothic is a subgenre of Gothic fiction, art, film and television, focused on anxieties associated with the creation of suburban communities, particularly in the United States and the West, from the 1950s and 1960s onwards.
Criteria
It o ... electric stove top elements as symbols of domestic alchemy and spirituality. See also
* Celtic maze (straight-line spiral)
*Concentric circles
In geometry, two or more objects are said to be concentric, coaxal, or coaxial when they share the same center or axis. Circles, regular polygons and regular polyhedra, and spheres may be concentric to one another (sharing the same center ... DNA
*Fibonacci number
In mathematics, the Fibonacci numbers, commonly denoted , form a integer sequence, sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start ... Hypogeum of Ħal-Saflieni
* Megalithic Temples of Malta
*Patterns in nature
Patterns in nature are visible regularities of form found in the natural world. These patterns recur in different contexts and can sometimes be modelled mathematically. Natural patterns include symmetries, trees, spirals, meanders, wave ... Seashell surface
* Spirangle
* Spiral vegetable slicer
* Spiral stairs
*Triskelion
A triskelion or triskeles is an ancient motif consisting of a triple spiral exhibiting rotational symmetry.
The spiral design can be based on interlocking Archimedean spirals, or represent three bent human legs. It is found in artefacts of t ... References
Related publications
* Cook, T., 1903. ''Spirals in nature and art''. Nature 68 (1761), 296.
* Cook, T., 1979. ''The curves of life''. Dover, New York.
* Habib, Z., Sakai, M., 2005. ''Spiral transition curves and their applications''. Scientiae Mathematicae Japonicae 61 (2), 195 – 206.
*
* Harary, G., Tal, A., 2011. ''The natural 3D spiral''. Computer Graphics Forum 30 (2), 237 – 24 External links
Jamitzer Jamnitzer, Jamnitzer Archimedes' spiral transforms into Galileo's spiral. Mikhail Gaichenkov, OEIS
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In mathematics, a spiral is a 
Two major definitions of "spiral" in the 
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 (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).
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.
;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''.
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
The ''logarithmic spiral'' is a special case, because of ''constant'' !
;curvature
The curvature of a curve with polar equation is
:
For a spiral with one gets
*
In case of ''(Archimedean spiral)''
.