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The main use for the golden ratio is its aesthetic appeal - in art and architecture. Rectangles with the golden ratio as their aspect appeal to the human mind (for some reason). So various aspects of the Parthenon in Athens, for example, have dimensions whose ratio is phi.
Phi is closely related to the Fibonacci sequence: the ratio of successive terms of the sequence approaches phi and so, just like the Fibonacci sequence, phi appears in many natural situations.
However, there is no particular application based on phi.
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What is divine proportion in math?
The divine proportion, also known as the Golden Ratio and symbolised by the Greek letter, phi, is [1+sqrt(5)]/2 = 1.6180, approx.
How is the golden ratio worked out?
(a+b)/a=a/b=phi (the golden ratio, as defined) (a+b)/a=phi (we'll solve this equation) 1+b/a=phi (just changing the form of the left side a little) 1+1/phi=phi (a/b=phi so b/a=1/phi) phi+1=phi2 (multiply both sides by phi) phi2-phi-1=0 (rearrange) From here, we can use the quadratic equation to find the positive solution: phi=(-b+√(b2-4ac))/(2a) phi=(1+√(1+4))/2 phi=(1+√5)/2≈1.618
What number is phi?
In mathematics, phi represents the Golden Ratio. Two numbers are in the Golden Ratio if the ratio of the smaller to the larger number is the same as the ratio of the larger number to the sum of the two.Thus, is a and b are the two numbers and a < b, thenif a/b = b/(a+b), the two ratios equal phi.phi is irrational and = [1 + sqrt(5) ]/2 = 1.618034 approx.Alternatively, phi is the positive root of the quadratic x^2 - x - 1 = 0The ratio has aesthetically pleasing properties and has been used extensively by artists and architects. Also, the A4 family of paper as well as the B series, have sides in the phi ratio.
How does the golden ratio relate to platonic solids?
The golden ratio, phi occurs many places in the platonic solids. The dihedral angle on the dodecahedron is 2*atan(phi), and the dihedral angle on the icosahedron is 2*atan(phi2) or 2*atan(phi + 1). The mid radius of the dodecahedron is similarly phi2/2 or (phi + 1)/2, and the mid radius on the icosahedron is phi/2. There are several other measures within Platonic solids which involve phi.
Which ratio is known as the golden ratio?
The Golden Ratio denoted by the Greek letter phi, usually lower case (φ) states that the division of a line segment into two creates a ratio of the shorter part to the longer equal to that of the longer to the whole. This can be solved, using algebra, to give φ = (1 + √5)/2= 1.61803 approx.
