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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 = special case of the logarithmic spiral Spiral of Theodorus (also known as Pythagorean spiral)
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.
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.
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]
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]
The Spiralia are a morphologically diverse clade of protostome animals, including within their number the molluscs, annelids, platyhelminths and other taxa. [4] The term Spiralia is applied to those phyla that exhibit canonical spiral cleavage, a pattern of early development found in most members of the Lophotrochozoa.
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The golden spiral (red) and its approximation by quarter-circles (green), with overlaps shown in yellow A logarithmic spiral whose radius grows by the golden ratio per 108° of turn, surrounding nested golden isosceles triangles. This is a different spiral from the golden spiral, which grows by the golden ratio per 90° of turn. [58]