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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.
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.
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]
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.
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In geometry, a golden rectangle is a rectangle with side lengths in golden ratio +:, or :, with approximately equal to 1.618 or 89/55. Golden rectangles exhibit a special form of self-similarity : if a square is added to the long side, or removed from the short side, the result is a golden rectangle as well.
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