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For large n the ratio f[n]/f[n - 1] approaches GoldenRatio or (1 + √ 5 )/2 ≃ 1.618 . … The number GoldenRatio appears to have been used in art and architecture since antiquity. 1/GoldenRatio is the default AspectRatio for Mathematica graphics. In addition: • GoldenRatio is the solution to x  1 + 1/x or x 2  x + 1 • The right-hand rectangle in is similar to the whole rectangle when the aspect ratio is GoldenRatio • Cos[ π /5]  Cos[36 ° ]  GoldenRatio/2 • The ratio of the length of the diagonal to the length of a side in a regular pentagon is GoldenRatio • The corners of an icosahedron are at coordinates Flatten[Array[NestList[RotateRight, {0, (-1) #1 GoldenRatio, (-1) #2 }, 3]&, {2, 2}], 2] • 1 + FixedPoint[N[1/(1 + #), k] &, 1] approximates GoldenRatio to k digits, as does FixedPoint[N[Sqrt[1 + #],k]&, 1] • A successive angle difference of GoldenRatio radians yields points maximally separated around a circle (see page 1006 ).
The patterns are arranged on the page so that the pattern shown at a particular position corresponds to what is obtained with a rule in which the tip of the right-hand stem goes to that position (corrected for the aspect ratio of the array) relative to the original stem shown as a vertical line on the left-hand side of the page.
The second example involves two distinct shapes: a square and a GoldenRatio aspect ratio rectangle.
But while features such as the shapes of leaves typically differ greatly between different plants, there are also some seemingly quite sophisticated aspects of plants that typically remain almost exactly the same across a huge range of species. … For it turns out that an angle between successive elements of about 137.5° is equivalent to a rotation by a number of turns equal to the so-called golden ratio (1+Sqrt[5])/2 ≃ 1.618 which arises in a wide variety of mathematical contexts—notably as the limiting ratio of Fibonacci numbers.
For the value of π is specified by the simple definition of being the ratio of the circumference of any circle to its diameter. … If the curve were continued further, it would spend more time above the axis, and no aspect of what is seen provides any evidence that the digit sequence is anything but perfectly random.
It is certainly recognized that some aspects of current organisms are in effect holdovers from earlier stages in biological evolution. … Thus, for example, the golden ratio spiral of branches on a plant stem can be viewed as a marvellous way to minimize the shading of leaves, while the elaborate patterns on certain mollusc shells can be viewed as marvellous ways to confuse the visual systems of supposed predators.
In general, the list for a particular rule can be obtained with the function ElementaryRule[num_Integer] := IntegerDigits[num, 2, 8] Given a rule together with a list representing the state a of a cellular automaton at a particular step, the following simple function gives the state at the next step: CAStep[rule_List, a_List] := rule 〚 8 - (RotateLeft[a] + 2 (a + 2 RotateRight[a])) 〛 A list of states corresponding to evolution for t steps can then be obtained with CAEvolveList[rule_, init_List, t_Integer] := NestList[CAStep[rule, #]&, init, t] Graphics of this evolution can be generated using CAGraphics[history_List] := Graphics[ Raster[1 - Reverse[history]], AspectRatio  Automatic] And having set up the definitions above, the Mathematica input Show[CAGraphics[CAEvolveList[ ElementaryRule[30], CenterList[103], 50]]] will generate the image: The description just given should be adequate for most cellular automaton simulations.
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