Yin & Yang Logarithmic Spirals

yin_yang

The figure depicts the well-known black and white symbol of Yin and Yang. The dots of different color in the area delimited by each force symbolize the fact that each force bears the seed of its counterpart within itself. According to the principle of Yin and Yang outlined above, neither Yin nor Yang can be observed directly. Both Yin and Yang are intertwined forces always occurring in pairs, rather than being isolated forces independent from each other. In Chinese philosophy, Yin and Yang assume the form of spirals. Let us now show that the net force in

K = −p(K ) ∗ ln(1 − p(K )/p(K))

the performance of a given confidence value K always matches exactly the expectation, i.e. in other words E = p(K), is a spiral too. In order to do so, let us introduce the general definition of the logarithmic spiral before illustrating the similarity to the famous Yin/Yang symbol.

A logarithmic spiral is a special type of spiral curve, which plays an important role in nature. It occurs in all different kinds of objects and processes, such as mollusk shells, hurricanes, galaxies, etc. In polar coordinates (r, θ), the general definition of a logarithmic spiral is

r = ae

Parameter a is a scale factor determining the size of the spiral, while parameter b con- trols the direction and tightness of the wrapping. For a logarithmic spiral, the distances between the turnings increase. This distinguishes the logarithmic spiral from the Archimedean spiral, which features constant distances between turnings. The figure below  depicts a typical example of a logarithmic spiral.

l2_636

Resolving r = ae for θ leads to the following general form of logarithmic spirals:

θ = 1/b ln (r/a)

In order to show that the net force in

K = −p(K ) ∗ ln(1 − p(K )/p(K))

defines a logarithmic spiral, and for the sake of easier illustration, let us look at the negative version of the net force in the above equation for net force and look at the polar coordinates (r, θ) it defines, namely:

θ = −p(K) ∗ ln(p(K)/(1−p(K)))

and

r = (1−p(K)) ∗ e−θ/p(K)

A comparison of θ = −p(K) ∗ ln(p(K)/(1−p(K))), r = (1−p(K)) ∗ e−θ/p(K) with the general form of logarithmic spirals in θ = 1/b ln (r/a) shows that the net force does indeed describe a spiral. Both of these equations match when we set the parameters a and b to the following values:

a = 1−p(K)

and

b = − 1/p(K)

In particular, we can check that a and b are identical when p(K) equals the golden ratio, which happens to be

φ ≈ 1.618 ∨ −0.618

If we let p(K) run from 0 to 1, and mirror the resulting spiral along both axes, we receive two spirals. Figure 8 shows both spirals plotted in a Cartesian coordinate system. Both spirals are, of course, symmetrical and their turnings approach the unit circle. A comparison of the Yin/Yang symbol with the spirals in the figure below shows the strong similarities between both figures.

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A simple mirror operation transforms the spirals in the above figure into the Yin/Yang symbol. The addition of a time dimension to the above figure generates a three-dimensional object.

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The above is an informational universe. Note that the use of performance as time is reasonable because the exponential distribution is typically used to model dynamic time processes and the expectation value is thus typically associated with time.

Yin and Yang with deep roots in Chinese philosophy, stand for two principles that are opposites of each other, and which are constantly trying to gain the upper hand over each other. However, neither one will ever succeed in doing so, though one principle may temporarily dominate the other one. Both principles cannot exist without each other. It is rather the constant struggle between both principles that defines our world and produces the rhythm of life. According to Chinese philosophy, Yin and Yang are the foundation of our entire universe. They flow through, and thus affect, every being. Typical examples of Yin/Yang opposites are, for example, night/day, cold/hot, rest/activity, etc. Chinese philosophy does not confine itself to a mere description of Yin and Yang. It also provides guidelines on how to live in accordance with Yin and Yang. The central statement is that Yin and Yang need to be in harmony. Any imbalance of an economical, biological, physical, or chemical system can be directly attributed to a distorted equilibrium between Yin and Yang.

 

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