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The respective lengths a, b, and c of the sides of these three polygons satisfy the equation a 2 + b 2 = c 2, so line segments with these lengths form a right triangle (by the converse of the Pythagorean theorem). 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]
Euclid's Elements (c. 300 BC) provides several propositions and their proofs employing the golden ratio, [15] [c] and contains its first known definition which proceeds as follows: [16] A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser. [17] [d]
Its true value is the real solution of the equation x 3 = 2x 2 + 1. The name supersilver ratio results from analogy with the silver ratio , the positive solution of the equation x 2 = 2 x + 1 , and the supergolden ratio .
A supergolden rectangle is a rectangle whose side lengths are in a : ratio. Compared to the golden rectangle , the supergolden rectangle has one more degree of self-similarity . Given a rectangle of height 1 , length ψ {\displaystyle \psi } and diagonal length ψ 3 {\displaystyle {\sqrt {\psi ^{3}}}} (according to 1 + ψ 2 = ψ ...
For example, start with a 1-by-Φ rectangle, where Φ is the golden ratio. Add an adjacent Φ-by-Φ square and get another golden rectangle. Add an adjacent (1+Φ)-by-(1+Φ) square and get a larger golden rectangle, and so on. Now, in order to separate more than 1/3 of the shapes, the separator must separate O(N) shapes from two different vertices.
The diagonal of a half square forms the basis for the geometrical construction of a golden rectangle.. The golden ratio φ is the arithmetic mean of 1 and . [4] The algebraic relationship between , the golden ratio and the conjugate of the golden ratio (Φ = − 1 / φ = 1 − φ) is expressed in the following formulae:
Consider a rectangle such that the ratio of its length L to its width W is the n th metallic ratio. If one remove from this rectangle n squares of side length W, one gets a rectangle similar to the original rectangle; that is, a rectangle with the same ratio of the length to the width (see figures).
Lis (Library of Iterative Solvers for linear systems; pronounced lis]) is a scalable parallel software library to solve discretized linear equations and eigenvalue problems that mainly arise from the numerical solution of partial differential equations using iterative methods.
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