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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" ("ewige Linie").
Golden spirals are self-similar. The shape is infinitely repeated when magnified. In geometry, a golden spiral is a logarithmic spiral whose growth factor is φ, the golden ratio. [1] That is, a golden spiral gets wider (or further from its origin) by a factor of φ for every quarter turn it makes.
For <, spiral-ring pattern; =, regular spiral; >, loose spiral. R is the distance of spiral starting point (0, R) to the center. R is the distance of spiral starting point (0, R) to the center. The calculated x and y have to be rotated backward by ( − θ {\displaystyle -\theta } ) for plotting.
Conical spiral with an archimedean spiral as floor projection Floor projection: Fermat's spiral Floor projection: logarithmic spiral Floor projection: hyperbolic spiral. In mathematics, a conical spiral, also known as a conical helix, [1] is a space curve on a right circular cone, whose floor projection is a plane spiral.
# Output to svg file: set terminal svg size 1024 768 set output "logarithmic_spiral.svg" # Same scale for both axes, half-size output: set size ratio -1 0.5, 0.5 # More sample points to produce smoother picture: set samples 480 # Axes in the center, no tick marks: set zeroaxis unset xtics unset ytics unset border set polar plot [-4*pi:4*pi] [-8:10] [-8:6] 1.19**t notitle
In mathematics, a conchospiral a specific type of space spiral on the surface of a cone (a conical spiral), whose floor projection is a logarithmic spiral. Conchospirals are used in biology for modelling snail shells, and flight paths of insects [1] [2] and in electrical engineering for the construction of antennas. [3] [4]
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 logarithmic spiral through the vertices of adjacent triangles has polar slope = (). The parallelogram between the pair of upright grey triangles has perpendicular diagonals in ratio φ {\displaystyle \varphi } , hence is a golden rhombus .