In geometry, a golden spiral is a logarithmic spiral whose growth factor is φ , the golden ratio. That is, a golden spiral gets wider (or further from its origin) by a factor of φ for every quarter turn it makes.
There are several comparable spirals that approximate, but do not exactly equal, a golden spiral.
For example, a golden spiral can be approximated by first starting with a rectangle for which the ratio between its length and width is the golden ratio. This rectangle can then be partitioned into a square and a similar rectangle and this rectangle can then be split in the same way. After continuing this process for an arbitrary number of steps, the result will be an almost complete partitioning of the rectangle into squares. The corners of these squares can be connected by quarter-circles. The result, though not a true logarithmic spiral, closely approximates a golden spiral.
Another approximation is a Fibonacci spiral, which is constructed slightly differently. A Fibonacci spiral starts with a rectangle partitioned into 2 squares. In each step, a square the length of the rectangle's longest side is added to the rectangle. Since the ratio between consecutive Fibonacci numbers approaches the golden ratio as the Fibonacci numbers approach infinity, so too does this spiral get more similar to the previous approximation the more squares are added, as illustrated by the image.
Approximate logarithmic spirals can occur in nature, for example the arms of spiral galaxies φ and the Fibonacci series. In truth, spiral galaxies and nautilus shells (and many mollusk shells) exhibit logarithmic spiral growth, but at a variety of angles usually distinctly different from that of the golden spiral. This pattern allows the organism to grow without changing shape.[ citation needed ]- golden spirals are one special case of these logarithmic spirals, although there is no evidence that there is any general tendency towards this case appearing. Phyllotaxis is connected with the golden ratio because it involves successive leaves or petals being separated by the golden angle; it also results in the emergence of spirals, although again none of them are (necessarily) golden spirals. It is sometimes stated that spiral galaxies and nautilus shells get wider in the pattern of a golden spiral, and hence are related to both
A golden spiral with initial radius 1 is the locus of points of polar coordinates satisfying
The polar equation for a golden spiral is the same as for other logarithmic spirals, but with a special value of the growth factor b:
with e being the base of natural logarithms, a being the initial radius of the spiral, and b such that when θ is a right angle (a quarter turn in either direction):
Therefore, b is given by
The numerical value of b depends on whether the right angle is measured as 90 degrees or as radians; and since the angle can be in either direction, it is easiest to write the formula for the absolute value of (that is, b can also be the negative of this value):
for θ in degrees, or
for θ in radians.
An alternate formula for a logarithmic and golden spiral is:
where the constant c is given by:
which for the golden spiral gives c values of:
if θ is measured in degrees, and
if θ is measured in radians.
With respect to logarithmic spirals the golden spiral has the distinguishing property that for four collinear spiral points A, B, C, D belonging to arguments θ, θ + π, θ + 2π, θ + 3π the point C is the projective harmonic conjugate of B with respect to A, D, i.e. the cross ratio (A,D;B,C) has the singular value −1. The golden spiral is the only logarithmic spiral with (A,D;B,C) = (A,D;C,B).
In the polar equation for a logarithmic spiral:
the parameter b is related to the polar slope angle :
In a golden spiral, being constant and equal to (for θ in radians, as defined above), the slope angle is:
if measured in degrees, or
if measured in radians.
Its complementary angle
in radians, or
in degrees, is the angle the golden spiral arms make with a line from the center of the spiral.
In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals. Originally, they arose in connection with the problem of finding the arc length of an ellipse and were first studied by Giulio Fagnano and Leonhard Euler. Modern mathematics defines an "elliptic integral" as any function f which can be expressed in the form
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,
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 is called the pole, and the ray from the pole in the reference direction is the polar axis. The distance from the pole is called the radial coordinate, radial distance or simply radius, and the angle is called the angular coordinate, polar angle, or azimuth. The radial coordinate is often denoted by r or ρ, and the angular coordinate by φ, θ, or t. Angles in polar notation are generally expressed in either degrees or radians.
A logarithmic spiral, equiangular spiral, or growth spiral is a self-similar spiral curve that often appears in nature. The first to describe a logarithmic spiral was Albrecht Dürer (1525) who called it an "eternal line". More than a century later, the curve was discussed by Descartes (1638), and later extensively investigated by Jacob Bernoulli, who called it Spira mirabilis, "the marvelous spiral".
The Archimedean 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 constant speed along a line that rotates with constant angular velocity. Equivalently, in polar coordinates (r, θ) it can be described by the equation
In mathematics, a spiral is a curve which emanates from a point, moving farther away as it revolves around the point.
In geometry, a solid angle is a measure of the amount of the field of view from some particular point that a given object covers. That is, it is a measure of how large the object appears to an observer looking from that point. The point from which the object is viewed is called the apex of the solid angle, and the object is said to subtend its solid angle from that point.
A Fermat's spiral or parabolic spiral is a plane curve named after Pierre de Fermat. Its polar coordinate representation is given by
In mathematics, the inverse trigonometric functions are the inverse functions of the trigonometric functions. Specifically, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions, and are used to obtain an angle from any of the angle's trigonometric ratios. Inverse trigonometric functions are widely used in engineering, navigation, physics, and geometry.
In trigonometry, tangent half-angle formulas relate the tangent of half of an angle to trigonometric functions of the entire angle. The tangent of half an angle is the stereographic projection of the circle onto a line. Among these formulas are the following:
In Euclidean geometry, Ptolemy's theorem is a relation between the four sides and two diagonals of a cyclic quadrilateral. The theorem is named after the Greek astronomer and mathematician Ptolemy. Ptolemy used the theorem as an aid to creating his table of chords, a trigonometric table that he applied to astronomy.
The Havriliak–Negami relaxation is an empirical modification of the Debye relaxation model in electromagnetism. Unlike the Debye model, the Havriliak–Negami relaxation accounts for the asymmetry and broadness of the dielectric dispersion curve. The model was first used to describe the dielectric relaxation of some polymers, by adding two exponential parameters to the Debye equation:
In calculus, the Leibniz integral rule for differentiation under the integral sign, named after Gottfried Leibniz, states that for an integral of the form
A golden triangle, also called a sublime triangle, is an isosceles triangle in which the duplicated side is in the golden ratio to the base side:
The main trigonometric identities between trigonometric functions are proved, using mainly the geometry of the right triangle. For greater and negative angles, see Trigonometric functions.
A ratio distribution is a probability distribution constructed as the distribution of the ratio of random variables having two other known distributions. Given two random variables X and Y, the distribution of the random variable Z that is formed as the ratio Z = X/Y is a ratio distribution.
In geometry, the tangential angle of a curve in the Cartesian plane, at a specific point, is the angle between the tangent line to the curve at the given point and the x-axis.
In geometry, a golden rhombus is a rhombus whose diagonals are in the golden ratio:
Landen's transformation is a mapping of the parameters of an elliptic integral, useful for the efficient numerical evaluation of elliptic functions. It was originally due to John Landen and independently rediscovered by Carl Friedrich Gauss.