## Yin-Yang: The Geometric Equivalent of The Golden Ratio

I have discovered that Yin-Yang and Golden Ratio are different representations of one and the same metaphysical principle. The metaphysical principle of our interest is Unity Within Duality which states the complementary character of conflicting tendencies. This is of course similar to the Hegelian trinity in his interpretation of history: Thesis-Antithesis-Synthesis.

After giving a short account of Yin-Yang and Golden Ratio I proceed to prove their equivalence. Since after a year of research I have found no one else who has discovered this equivalence I stick a copyright notice at the end of the article just for the publication purposes in case the finding turns out to be of significance.

Yin-Yang and Golden Ratio are the two representations of the principle that maximally opposing tendencies, or aspects, complement one another to form a unity: The former represents it geometrically and the latter represents it numerically.

Yin-Yang in Chinese philosophy is the geometrical expression of the following concept: The apparently opposing and contradictory tendencies complement one another; being interconnected they add up to form a unity. In the Yin-Yang symbol a whole circle is divided into two maximally opposing aspects. The two aspects enjoy maximal opposition both in color and orientation: The color of one cut is the negation of the color of the other; and the orientation of one is the inverse of the orientation of the other.

Notice that the two opposing sides have the same shape; they are the opposites of one another only in the attributes color and orientation. In other words, one side is the negation and inversion of the other side. But this cannot be done with just any shape; not every shape can be inverted and then added to itself to form a perfect circle which represents unity.

Golden Ratio is a number in mathematics; it has the numerical value of 1.61803398875…. When the ratio of two quantities is this number we say that they are in golden ratio. This number has huge significance in mathematics because it is the corner stone of spiral geometry which is observed everywhere in nature. It is also the limit of the Fibonacci Sequence. In fact when the ratio of two lengths is the golden ratio they have an inherent tendency to make up harmonious and beautiful structures.

It is time to see how Yin-Yang and Golden Ratio are equivalent to one another, both being the representations of one principle.

We saw that Yin-Yang represents the idea that maximally opposing aspects add up to form a unity; each aspect is both the negation and the inversion of the other aspect. Numerically speaking, one being the inverse of the negative of the other. Let us try to represent this idea numerically, using number instead of shape: We need a number that when maximally opposed to itself, namely first negated and then inverted, can be added to itself to form a unity which is represented by number 1.

We don’t yet know if such number exists, but it will soon be proved that it actually does. First remember that by negation of a number of course we mean its negative, -2 being the negative of 2. By inversion we mean the inverse of the number, 1/2 being the inverse of 2; thus, the inverse of the negative of 2 is -1/2; but notice that when we add 2 and -1/2 we get 3/2 and not 1. We are after a number that gives us 1.

Let us call our desired number x and try to write down the concept behind Yin-Yang in the language of numbers and see if such number exists: We have x as our number; negate it to get -x; invert the negation to get -1/x. To express Yin-Yang is to add up these two numbers and have them equal 1. Thus, the algebraic equivalent of Yin-Yang would be the equation [x+(-1/x)]=1 which can be simplified as the following:

X – (1/X) = 1

This already looks like Yin-Yang :). For those with mathematics background this equation should blow their minds; it is none but the infamous equation that is the very definition of the Golden Ratio: Golden Ratio is the exact solution, and the only solution, of this quadratic equation. You will soon see that the second solution is just the negative of the inverse of the Golden Ratio, its maximally opposing complement which we call the Golden Complement. Let’s call this equation, the algebraic form of Yin-Yang, the Golden Equation since its solutions are the Golden Ratio and the Golden Complement; thus, this equation has two solutions in real numbers:

X – 1/X = 1               The Golden Equation (Equivalent of Yin-Yang in Algebra)

First Solution:        X1 = 1.61803398875…              The Golden Ratio

Second Solution:   X2 = -0.61803398875…            The Golden Complement

X1 + X2 = 1             The Golden Equation in terms of its two complementary solutions

The first solution is the Golden Ratio and the second solution is interestingly the inverse of the negative of the first solution, namely the Golden Complement. See that the two solutions corresponding to the two complementary aspects of Yin-Yang add up to unity, hence completing the Yin-Yang equivalent in algebra. Notice that the Golden Complement has the exact same decimals as the Golden Ratio itself which is the reason why the decimals cancel out in the Golden Equation, hence leading to unity.

We saw that there actually exists a unique number whose inverse of its negation can be added to it to become unity, the number 1, hence expressing the Yin-Yang principle in the realm of numbers, and this unique number is none but the Golden Ratio, the solution of the Golden Equation. Thus one side of Yin-Yang corresponds to the Golden Ratio and the other side corresponds to its maximally opposing counterpart, the negative of its inverse which is the Golden Complement, and the two adding up to unity by definition. Only the Golden Ratio has this property, and one can see as we showed above that the principle behind Yin-Yang, leading to the Golden Equation in algebra, is the very definition of the Golden Ratio.

Digression:

{Notice that the two solutions of the Golden Equation, which correspond to the two maximally opposing aspects of Yin-Yang, have these two interesting properties:

X1 + X2 = 1                     X1 × X2 = -1

If these two solutions, the Golden Ratio and the Golden Complement, are considered to be the diagonal elements, or eigenvalues, of a 2×2 matrix called the Golden Matrix G, then G is an orthogonal matrix with determinant -1 and trace 1, hence:

Det(G) + Tr(G) = 0

Which is only an interesting observation besides our main point.}

End of Digression

Therefore, the principle that the maximally opposing tendencies complement one another and form a unity is represented geometrically as Yin-Yang and algebraically as the Golden Equation X-1/X=1 whose two solutions, corresponding to the two aspects in Yin-Yang, are the Golden Ratio and the Golden Complement. We saw that of all numbers one and only one, the Golden Ratio, has this property. More precisely, the Golden Equation X-1/X=1 is the algebraic equivalent of Yin-Yang from which the Golden Ratio is extracted.

Yin-Yang is the Golden Equation in geometry. Golden Equation is Yin-Yang in algebra.

This equivalence between Yin-Yang and Golden Equation which I discovered in the fall of 2013 has blown my mind to this day. Who could guess that the Golden Ratio is Yin-Yang in numbers while Yin-Yang is the Golden Ratio in geometry! Fascinating.

*[Note for the mathematician: In constructing the Golden Equation, the algebraic equivalent of Yin-Yang, I assumed a bijection from plane shapes to real numbers, hence defining the equation over R, the field of real numbers. Naturally a bijection from the inverse and the negative of a shape would fall onto the inverse and negative of numbers. Thus, in constructing the bijection from Yin-Yang to real numbers the only proper way of translating maximal opposition, namely both negation and inversion, would be to use both the additive inverse and the multiplicative inverse of a number, the only possible ways of defining opposites among real numbers, hence the field properties of the Commutative Division Ring R. In other words, the only way to translate the relation between the two opposing aspects of Yin-Yang is to relate a real number to the multiplicative inverse of its additive inverse, hence mapping the two different operations present in Yin-Yang, namely the spatial inversion and color-negation. It is seen that the only number whose multiplicative inverse of its additive inverse can be added to itself to become unity is the Golden Ratio.]

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## The Idea of Mathematics

Mathematics is a view to the logical structure of the world. The mathematical principles are a priori propositions whose truth-values are independent of empirical evidence, and hence the a priori.

Mathematics involves two types of things: Sets and Relations. Sets are the collections of objects, the collection being defined provisionally. For instance, a mathematician may define his set to be the set of all triangles, while another one may work with the set of all imaginary arrows in space or the set of all even numbers. One can make up any collection of objects and consider them to be members of a set. Objects in a set are usually called members of that set. Relations are the relationships that exist between the objects of the set. These relations too are defined provisionally; we define a relation between the objects of a set. For instance, if we define our set to be the set of all triangles and number them, then we can impose the relation that the odd-numbered triangles are twice as big as the even-numbered triangles. This relation created a relationship between every two triangles: They are either the same size or one is twice as the other, or vice versa. So relations are such kinds of things. Notice that how relations can be used as a way of creating sets or collections. We can use a relation as a criterion for creating a set. For instance, I can say that I want a collection whose members are numbers such that they are all divisible by 2. Using this condition I collect all such numbers and define the set; obviously this is the set of all even numbers, and each two members are related to one another such that they are both divisible by 2. An important point is that relations themselves can be treated as objects so that we may create collections whose members are relations. For instance, I can define a collection, a set, whose members are all algebraic operations, say plus, minus, multiplications, etc. These objects are relations; addition is something that exists between two or more numbers. If we define a set whose members are pairs of numbers, any numbers we wish, like (a,b), then we can say addition is a relation between any pair of numbers and a third number c such that (a,b) is related to c in the form a+b equals c or a+b=c. Addition is an instance of a relation between some objects.

Therefore, the most general structures in mathematics are sets and relations. Everything else is a specializations and variations of these two concepts. We defined relations in its most general sense. We can specialize it further and refer to all relations such that each object of the set is related to exactly one object and not two. For instance, if a is related to be, then a is not related to c unless b=c. All such specialized relations are known as functions. The another kinds of relations that I mentioned above as operations, the operations between a pair of numbers and a third number, are known as binary operations. Binary operations such as +, -, *, / are special kinds of relations.

We can play more and define new objects: The new object is a set of numbers equipped with a binary operation. Consider the set S with members a, b, and c. We write it as: S={a,b,c}. The name of the set and its members. Call our binary operation this symbol *. We define the set A equipped with * and write it as <S , *> which is a set and relation together. We define the relation to be the following:

a*b=c    a*c=b    a*a=a    b*a=c    b*c=a    b*b=b    c*a=b    c*b=a    c*c=c

See that ever two members of S are related to one another by the binary operation *. Notice that I have not mentioned anything about the nature of a,b, and c and also the nature of the binary operation which can be addition, multiplication, and anything else. But I have defined a structure which can exist regardless of how we fill in the unknown symbols. The object we defined above, which is a set equipped with an operation <S,*>, has a special property: Notice that all members are related to one another; there is no member that is not related to another; and no matter how many times we apply the operation between members we never go out of S; the operation between any two members of S always gives another elements which is still a member of S. We never go outside S in order to perform the operation *. This property is called closure; we say that the set S is closed under the operation *, that is: Operation of * on the members of S always gives another member of S. Now the new object that we define to be a set equipped with an operation under which S is closed has a special name in algebra: It is called a group. To have a group we need some more restrictions: The operation must be associative; it must permit an identity and an inverse. We need not go into these now. The point is to say that everything in mathematics is composed of a set and a relation. In fact, everything in the world can be viewed in the same fashion.

Now we can play more and extend our definitions: We define a group to be a set equipped with a binary operation which relates the members of the set with one another. Now we can define a set equipped with two binary operations. We can call these two operations multiplication and addition. In fact the set of real numbers of which we are familiar and use on a daily basis is such a mathematical object; we multiply and add them and get another object which is still a real number. This new object which is a set equipped with two binary operations, under some group constraints, is called a Ring in mathematics. If we add more relations to this ring we can make a more interesting object called a Field. In fact all of us know a field and work with it everyday: The set of real numbers. Field is a special kind of ring. If the operation * that relates two elements a and b is such that a*b=b*a we say that * is commutative. The real numbers that we know is a commutative ring because 2*3=3*2=6. We can now say: Field is a commutative division ring.

All of the above are created by playing with set and relations:

Set is a collection of object.

Relations are the relationships existing between members of a set.

Function is a monogamous relation.

Binary operation is a function.

Group is a set with a binary operation.

Ring is a group with a binary operation.

Field is a commutative division ring.

Division ring is a ring whose non-zero elements are all units.

A unit is an element of a set that has an inverse.

Inverse of an element a is another element b such that a*b is the identity element. We write: a*b=e.

Identity element e is an element of a set such that a*e=e*a=a for all elements a of the set.

[The purpose of this post was to review the subject for myself and prepare for a lecture on the concepts of mathematics.]