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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.
approximation of the golden spiral golden spiral = special case of the logarithmic spiral Spiral of Theodorus (also known as Pythagorean spiral) c. 500 BC: contiguous right triangles composed of one leg with unit length and the other leg being the hypotenuse of the prior triangle: approximates the Archimedean spiral
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]
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 triangles inscribed in a logarithmic spiral. The golden triangle is used to form some points of a logarithmic spiral. By bisecting one of the base angles, a new point is created that in turn, makes another golden triangle. [4] The bisection process can be continued indefinitely, creating an infinite number of golden triangles.
where n is the index number of the floret and c is a constant scaling factor; the florets thus lie on Fermat's spiral. The divergence angle, approximately 137.51°, is the golden angle, dividing the circle in the golden ratio. Because this ratio is irrational, no floret has a neighbor at exactly the same angle from the center, so the florets ...
An Archimedean spiral (black), a helix (green), and a conical spiral (red) Two major definitions of "spiral" in the American Heritage Dictionary are: [5]. a curve on a plane that winds around a fixed center point at a continuously increasing or decreasing distance from the point.
The Lucas spiral, made with quarter-arcs, is a good approximation of the golden spiral when its terms are large.However, when its terms become very small, the arc's radius decreases rapidly from 3 to 1 then increases from 1 to 2.