This is a quick visual tour of the probability associated with the 2,400 years old Golden ratio φ (1.618034...) We've covered φ in a couple earlier articles (both related to Fibonacci's confidence intervals for real-world applications, and the probability distribution of fractals). We've also simulated probability distributions with fractals in the past, to predict the difficulty in knowing where flight MH370 impacted Earth. There is some evidence of φ existing in ancient Mediterranean colonial structures. And numerical buffs are eager to paint an association between the arts and science. There have been a number of academics, for example, stating that a number of pieces from 16th century artist, Leonardo da Vinci, contain φ (in the form of flourishing rectangles and spirals). But as we'll see in the movie below, that this is not as easy of a case for many named art -other than by chance alone- given the variety of anchoring points on possibly Mona Lisa's face and the rectangle center (also known as Eye of God) randomly pivoting somewhere on her cheek. Not to mention errors in the rotation of the fractal shape to fit against the art. Some of these basic points are also highlighted in the science research book The Golden Ratio (recently ranked #3), which Da Vinci Code's author Dan Brown endorsed. Note that the constant expression phi is represented by the Greek letter φ.
So in the quick movie here (you must see this on a normal computer, not a hand-held device), we show the alteration of the probability density function to style the location of the undoubtedly enchanting, Golden spiral. This spiral -still- has some symmetrical flaws in that the neighboring quadrants have a discrete difference in radius, even though the spiral arcs are always a continuous shape. So the movie quickly converts the formation of the spiral by introducing the rectangle's neighboring squares, as if they were equally-sized.
The Fibonacci ratio, derived a few hundred years before Da Vinci (also by an individual sharing the same first name: Leonardo), is simply φ-1, φ^2-2, or 1/φ. We can see this development by decomposing the Golden ratio formulae:
(φ + 1)/φ = φ
φ + 1 = φ^2
φ - 1 = φ^2 - 2
Further,
(φ + 1)/φ = φ
1 + 1/φ = φ
1/φ = φ - 1
The Fibonacci ratio shows up in financial markets, namely trading theory. Though in the first blog link above, we show the sensitivity of focusing too much on this ratio (both logically, as well as computationally) in many applied environments. Unlike with any imitating art, there is a wealth of quantifiable financial market data. So it's easier to announce Fibonacci matches that apply (see the professional absurdity on these currency charts: here and here) and keep quiet cases where it didn't apply. Note that 1/φ of roughly 0.618 can be expressed in percent terms: 61.8%. We also see the importance here, in addition to seeing the same in the second blog above, of complementing closed-form probability models with open simulations. Particularly considering the practicality of forcibly extending numerically-irrational fractal shapes beyond science, and on to visual art.
So in the quick movie here (you must see this on a normal computer, not a hand-held device), we show the alteration of the probability density function to style the location of the undoubtedly enchanting, Golden spiral. This spiral -still- has some symmetrical flaws in that the neighboring quadrants have a discrete difference in radius, even though the spiral arcs are always a continuous shape. So the movie quickly converts the formation of the spiral by introducing the rectangle's neighboring squares, as if they were equally-sized.
The Fibonacci ratio, derived a few hundred years before Da Vinci (also by an individual sharing the same first name: Leonardo), is simply φ-1, φ^2-2, or 1/φ. We can see this development by decomposing the Golden ratio formulae:
(φ + 1)/φ = φ
φ + 1 = φ^2
φ - 1 = φ^2 - 2
Further,
(φ + 1)/φ = φ
1 + 1/φ = φ
1/φ = φ - 1
The Fibonacci ratio shows up in financial markets, namely trading theory. Though in the first blog link above, we show the sensitivity of focusing too much on this ratio (both logically, as well as computationally) in many applied environments. Unlike with any imitating art, there is a wealth of quantifiable financial market data. So it's easier to announce Fibonacci matches that apply (see the professional absurdity on these currency charts: here and here) and keep quiet cases where it didn't apply. Note that 1/φ of roughly 0.618 can be expressed in percent terms: 61.8%. We also see the importance here, in addition to seeing the same in the second blog above, of complementing closed-form probability models with open simulations. Particularly considering the practicality of forcibly extending numerically-irrational fractal shapes beyond science, and on to visual art.
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