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The golden spiral is a logarithmic spiral that grows outward by a factor of the golden ratio for every 90 degrees of rotation (pitch angle about 17.03239 degrees). It can be approximated by a "Fibonacci spiral", made of a sequence of quarter circles with radii proportional to Fibonacci numbers.
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.
logarithmic spiral (also known as equiangular spiral) 1638 [4] = Approximations of this are found in nature Fibonacci spiral: circular arcs connecting the opposite corners of squares in the Fibonacci tiling: approximation of the golden spiral golden spiral
For example, when leaves alternate up a stem, one rotation of the spiral touches two leaves, so the pattern or ratio is 1/2. In hazel the ratio is 1/3; in apricot it is 2/5; in pear it is 3/8; in almond it is 5/13. [56] Animal behaviour can yield spirals; for example, acorn worms leave spiral fecal trails on the sea floor. [57]
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.
This is a different spiral from the golden spiral, which grows by the golden ratio per 90° of turn. [58] Logarithmic spirals are self-similar spirals where distances covered per turn are in geometric progression. A logarithmic spiral whose radius increases by a factor of the golden ratio for each quarter-turn is called the golden spiral.
The chambered nautilus is often used as an example of the golden spiral. While nautiluses show logarithmic spirals, their ratios range from about 1.24 to 1.43, with an average ratio of about 1.33 to 1. The golden spiral's ratio is 1.618. This is visible when the cut nautilus is inspected. [13]
As another example, Carlos Chanfón Olmos states that the sculpture of King Gudea (c. 2350 BC) has golden proportions between all of its secondary elements repeated many times at its base. [3] The Great Pyramid of Giza (constructed c. 2570 BC by Hemiunu) exhibits the golden ratio according to various pyramidologists, including Charles Funck-Hellet.