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The golden-section search is a technique for finding an extremum (minimum or maximum) of a function inside a specified interval. For a strictly unimodal function with an extremum inside the interval, it will find that extremum, while for an interval containing multiple extrema (possibly including the interval boundaries), it will converge to one of them.
A root rectangle is a rectangle in which the ratio of the longer side to the shorter is the square root of an integer, such as √ 2, √ 3, etc. [2] The root-2 rectangle (ACDK in Fig. 10) is constructed by extending two opposite sides of a square to the length of the square's diagonal. The root-3 rectangle is constructed by extending the two ...
The 2D:4D ratio is calculated by dividing the length of the index finger by the length of the ring finger of the same hand. Other digit ratios are also calculated similarly in the same hand. The digit length is typically measured on the palmar (ventral, "palm-side") hand, from the midpoint of the bottom crease to the tip of the finger. [8]
Geometrically, the square root of 2 is the length of a diagonal across a square with sides of one unit of length; this follows from the Pythagorean theorem. It was probably the first number known to be irrational. [1] The fraction 99 / 70 (≈ 1.4142857) is sometimes used as a good rational approximation with a reasonably small denominator.
Thales measured the length of the pyramid's base and the height of his pole. Then at the same time of the day he measured the length of the pyramid's shadow and the length of the pole's shadow. This yielded the following data: height of the pole (A): 1.63 m; shadow of the pole (B): 2 m; length of the pyramid base: 230 m; shadow of the pyramid: 65 m
The length of the side of a larger square to the next smaller square is in the golden ratio. For a square with side length 1, the next smaller square is 1/φ wide. The next width is 1/φ², then 1/φ³, and so on. There are several comparable spirals that approximate, but do not exactly equal, a golden spiral. [2]
The ratio of the side length of the hexagon to the decagon is the golden ratio, so this triangle forms half of a golden rectangle. [8] The convex hull of two opposite edges of a regular icosahedron forms a golden rectangle.