Logarithmic spiral (pitch 10°) Cutaway of a nautilus shell showing the chambers arranged in an approximately logarithmic spiral A low pressure area over Iceland shows an approximately logarithmic spiral pattern The arms of spiral galaxies often have the shape of a logarithmic spiral, here the Whirlpool Galaxy Romanesco broccoli, showing fractal forms

A logarithmic spiral, equiangular spiral or growth spiral is a special kind of spiral curve which often appears in nature. Nautilus (from Greek ναυτίλος, 'sailor' is the common name of any marine creatures of the Cephalopod family Nautilidae, the sole A low pressure area, or " low " is a region where the Atmospheric pressure is lower in relation to the surrounding area Iceland, officially the Republic of Iceland ( ( Ísland or Lýðveldið Ísland ( A spiral galaxy is a Galaxy belonging to one of the three main classes of galaxy originally described by Edwin Hubble in his 1936 work “The Realm of the The Whirlpool Galaxy (also known as Messier 51a, M51a, or NGC 5194) is an interacting Spiral galaxy located at a distance of Romanesco broccoli is an edible flower of the species Brassica oleracea and a variant form of Cauliflower. A fractal is generally "a rough or fragmented geometric shape that can be split into parts each of which is (at least approximately a reduced-size copy of the whole" In Mathematics, a spiral is a Curve which emanates from a central point getting progressively farther away as it revolves around the point In Mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object The logarithmic spiral was first described by Descartes and later extensively investigated by Jakob Bernoulli, who called it Spira mirabilis, "the marvelous spiral". For other family members named Jacob see Bernoulli family. Jacob Bernoulli (also known as James or Jacques) ( Basel

Definition

In polar coordinates (r, θ) the curve can be written as[1]

$r = ae^{b\theta}\,$

or

$\theta = \frac{1}{b} \ln(r/a),$

with e being the base of natural logarithms, and a and b being arbitrary positive real constants. In Mathematics, the polar coordinate system is a two-dimensional Coordinate system in which each point on a plane is determined by

In parametric form, the curve is

$x(t) = r \cos(t) = ae^{bt} \cos(t)\,$
$y(t) = r \sin(t) = ae^{bt} \sin(t)\,$

with real numbers a and b. In Mathematics, the real numbers may be described informally in several different ways

The spiral has the property that the angle ɸ between the tangent and radial line at the point (r,θ) is constant. For the tangent function see Trigonometric functions. For other uses see Tangent (disambiguation. This article is about the mathematical concept for the medical meaning see Arterial line. This property can be expressed in differential geometric terms as

$\arccos \frac{\langle \mathbf{r}(\theta), \mathbf{r}'(\theta) \rangle}{\|\mathbf{r}(\theta)\|\|\mathbf{r}'(\theta)\|} = \arctan \frac{1}{b} = \phi.$

The derivative r'(θ) is proportional to the parameter b. This article only considers curves in Euclidean space Most of the notions presented here have analogues for curves in Riemannian and pseudo-Riemannian In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change In other words, it controls how "tightly" and in which direction the spiral spirals. In the extreme case that b = 0 (ɸ = π/2) the spiral becomes a circle of radius a. Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the Conversely, in the limit that b approaches infinity (ɸ → 0) the spiral tends toward a straight line. In Mathematics, the concept of a " limit " is used to describe the Behavior of a function as its argument either "gets close" In Mathematics, the affinely extended real number system is obtained from the Real number system R by adding two elements +∞ and &minus∞ (pronounced The complement of ɸ is called the pitch. A pair of Angles is complementary if the sum of their measures add up to 90 degrees.

Spira mirabilis and Jakob Bernoulli

Spira mirabilis, Latin for "miraculous spiral", is another name for the logarithmic spiral. Latin ( lingua Latīna, laˈtiːna is an Italic language, historically spoken in Latium and Ancient Rome. Although this curve had already been named by other mathematicians, the specific name ("miraculous" or "marvelous" spiral) was given to this curve by Jakob Bernoulli, because he was fascinated by one of its unique mathematical properties: the size of the spiral increases but its shape is unaltered with each successive curve. In Mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object For other family members named Jacob see Bernoulli family. Jacob Bernoulli (also known as James or Jacques) ( Basel In Mathematics, a spiral is a Curve which emanates from a central point getting progressively farther away as it revolves around the point Possibly as a result of this unique property, the spira mirabilis has evolved in nature, appearing in certain growing forms such as nautilus shells and sunflower heads. Nautilus (from Greek ναυτίλος, 'sailor' is the common name of any marine creatures of the Cephalopod family Nautilidae, the sole The sunflower ( Helianthus annuus) is an Annual plant in the family Asteraceae and native to the Americas, with a large flowering Jakob Bernoulli wanted such a spiral engraved on his headstone, but, by error, an Archimedean spiral was placed there instead. A headstone, tombstone or gravestone is a marker normally carved from stone, placed over or next to the site of a Burial The Archimedean spiral (also known as the arithmetic spiral) is a Spiral named after the 3rd century BC Greek Mathematician

Properties

The logarithmic spiral can be distinguished from the Archimedean spiral by the fact that the distances between the turnings of a logarithmic spiral increase in geometric progression, while in an Archimedean spiral these distances are constant. The Archimedean spiral (also known as the arithmetic spiral) is a Spiral named after the 3rd century BC Greek Mathematician In Mathematics, a geometric progression, also known as a geometric sequence, is a Sequence of Numbers where each term after the first is found

Logarithmic spirals are self-similar in that they are self-congruent under all similarity transformations (scaling them gives the same result as rotating them). In Linear algebra, two n -by- n matrices A and B over the field K are called similar if there exists Scaling by a factor eb gives the same as the original, without rotation. They are also congruent to their own involutes, evolutes, and the pedal curves based on their centers. In the Differential geometry of curves, an involute of a smooth Curve is another curve obtained by attaching an imaginary taut string to the given curve and tracing In the Differential geometry of curves, the evolute of a Curve is the locus of all its centers of curvature. In the Differential geometry of curves, a pedal curve is a Curve derived by construction from a given Curve (as is for example the Involute

Starting at a point P and moving inward along the spiral, one can circle the origin an unbounded number of times without reaching it; yet, the total distance covered on this path is finite; that is, the limit as θ goes toward -∞ is finite. In Mathematics, the concept of a " limit " is used to describe the Behavior of a function as its argument either "gets close" This property was first realized by Evangelista Torricelli even before calculus had been invented. Evangelista Torricelli ( ( October 15, 1608 &ndash October 25, 1647) was an Italian physicist and mathematician Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives The total distance covered is r/cos(ɸ), where r is the straight-line distance from P to the origin.

The exponential function exactly maps all lines not parallel with the real or imaginary axis in the complex plane, to all logarithmic spirals in the complex plane with centre at 0. The exponential function is a function in Mathematics. The application of this function to a value x is written as exp( x) (Up to adding integer multiples of 2πi to the lines, the mapping of all lines to all logarithmic spirals is onto. In Mathematics, the phrase " up to xxxx" indicates that members of an Equivalence class are to be regarded as a single entity for some purpose In Mathematics, a function f is said to be surjective or onto, if its values span its whole Codomain; that is for every ) The pitch angle of the logarithmic spiral is the angle between the line and the imaginary axis.

The function $x \mapsto x^k$, where the constant k is a complex number with non-zero imaginary part, maps the real line to a logarithmic spiral in the complex plane. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Mathematics, the imaginary part of a Complex number z is the second element of the ordered pair of Real numbers representing z In Mathematics, the real line is simply the set R of singleton Real numbers However this term is usually used when R is to be treated as a

One can construct a golden spiral, a logarithmic spiral that grows outward by a factor of the golden ratio for every 90 degrees of rotation (pitch about 17. In Geometry, a golden spiral is a Logarithmic spiral whose growth factor b is related to &phi the Golden ratio. In Mathematics and the Arts two quantities are in the Golden ratio if the Ratio between the sum of those quantities and the larger one is the 03239 degrees), or approximate it using Fibonacci numbers. In Mathematics, the Fibonacci numbers are a Sequence of numbers named after Leonardo of Pisa, known as Fibonacci

Logarithmic spirals in nature

In several natural phenomena one may find curves that are close to being logarithmic spirals. Here follows some examples and reasons:

• The approach of a hawk to its prey. The term hawk can be used in several ways In strict usage in Europe and Asia, to mean any of the Species in the Subfamily Their sharpest view is at an angle to their direction of flight; this angle is the same as the spiral's pitch. [2]
• The approach of an insect to a light source. They are used to having the light source at a constant angle to their flight path. Usually the sun (or moon for nocturnal species) is the only light source and flying that way will result in a practically straight line.
• The arms of spiral galaxies. A galaxy is a massive gravitationally bound system consisting of Stars an Interstellar medium of gas and dust, and Dark matter Our own galaxy, the Milky Way, is believed to have four major spiral arms, each of which is roughly a logarithmic spiral with pitch of about 12 degrees, an unusually small pitch angle for a galaxy such as the Milky Way. The Milky Way (a translation of the Latin Via Lactea, in turn derived from the Greek Γαλαξίας (Galaxias sometimes referred to simply In general, arms in spiral galaxies have pitch angles ranging from about 10 to 40 degrees.
• The nerves of the cornea. The cornea is the transparent front part of the Eye that covers the iris, Pupil, and Anterior chamber.
• The arms of tropical cyclones, such as hurricanes. A tropical cyclone is a storm system characterized by a low pressure center and numerous Thunderstorms that produce strong winds and Flooding
• Many biological structures including the shells of mollusks. Foundations of modern biology There are five unifying principles Molluscs are animals belonging to the phylum Mollusca. There are around 250000 extant Species within the phylum with an estimated 70000 In these cases, the reason is the following: Start with any irregularly shaped two-dimensional figure F0. Expand F0 by a certain factor to get F1, and place F1 next to F0, so that two sides touch. Now expand F1 by the same factor to get F2, and place it next to F1 as before. Repeating this will produce an approximate logarithmic spiral whose pitch is determined by the expansion factor and the angle with which the figures were placed next to each other. This is shown for polygonal figures in the accompanying graphic. In Geometry a polygon (ˈpɒlɨɡɒn ˈpɒliɡɒn is traditionally a plane figure that is bounded by a closed path or circuit

References

1. ^ Priya Hemenway (2005). In Geometry, a golden spiral is a Logarithmic spiral whose growth factor b is related to &phi the Golden ratio. Divine Proportion: Φ Phi in Art, Nature, and Science. Sterling Publishing Co. ISBN 1402735227.
2. ^ Organismal Biology: Flying Along a Logarithmic Spiral.