Roots Of Unity

            Scientists, especially those associated with Physics, have been engaged for a long in finding a particle that forms the basis of every material object in the world. The search has taken them from molecules to atoms to subatomic particles. Further development is expected to lead to discovery of an ultimate particle that is at the root of all existence.

            At the same time, Philosophers have been busy in formulating a theory of everything. Such a theory is expected to explain all existing theories and accept them as making part contributions to an understanding of existence.

            The question still remains: What is it that makes a person a Scientist or a Philosopher? There is, it seems, no study that examines the Spirit of eminent personalities engaged in finding Ultimate particle or Ultimate theory. A discipline dedicated to such study may be called Spiritual Philosophy of Science, or Spiritual Science of Philosophy.

            The world presents a diverse picture. Spiritual Scientists (Seers or Kavi in Vedic tradition) talk  of absolute unitary principle in existence. In brief, they are talking about ‘unity in diversty’.  What lies at the root of unity that displays diversity in manifestation? An attempt is made to describe this point through an example of a circle of unit radius. It involves Algebra, Trigonometry, Geometry, Vectors and both real and complex numbers. Consider first an application of algebra and numbers.

            An algebraic equation has an unknown variable symbol on one side whose value must be found to satisfy a given known number on the other side. For example, algebraic equation x=1 is a statement. The statement is true if the unknown variable ‘x’ is assigned a value of 1. In that case, x is assumed solution of the equation. In this case, x=1 represents both, equation as well as solution. Equation 

                                               x²=1

is also a statement. The statement is true if x is assigned values of either +1 or -1. There are thus two solutions of this equation. The solutions are commonly referred to as square roots of unity. A symbolic form of the solution, x=(1)^1/2, has number 2 in its superscript to emphasize the existence of two solutions.  A glimpse of diversity in roots of unity is apparent here. The square root of unity as +1 or -1 presents a pair of opposites in relation to 0 or zero. For geometric representaion, turn to common      human observation. People in general know their location. If they are looking at an object, they know the direction of their attention. Consider the location as origin O and use it as a geometric symbol for zero. The direction of observation is used as a reference line. Choose a point A along the line as a geometric representation of number 1. Therefore, OA is a geometric symbol for unity. The symbol used for it in vector analysis is the bold face letter i.

            Suppose a person standing at O and looking towards A makes a turn to his left or in a counter clockwise direction. A complete turn brings his gaze back to the original direction OA. In trigonometry. a complete turn is represented by a rotation of 2π radians. Half a turn is therefore π and a quarter turn is π/2. A complete turn is also achieved by combining two half turns of π each. This statement has a mathematical representation in algebraic equation: 2π=π+π.

            Consider R( ) as a mathematical symbol for rotation. In terms of R, R(0), R(π) and R(2π) are symbolic statements for no rotation, half rotation and complete rotation. A complete turn R(2π) rotates OA back to its original position and so does two half turns of π each. The original position implies no rotation. A mathematical form of the statement is R(2π)=R(π)R(π) =R(0)=1. A comparison with x²=1 suggests that R(π) is a (square)root of unity; that is R(π)=(1)^1/2 , but it cannot have the value of R(0)=1, hence its value must be -1. Following similar logic, it can be said that two successive quarter rotations of π/2 each equals half rotation π. That is, R(π)=R(π/2)R(π/)=-1. Mathematical form of quarter rotation is therefore square root of -1. That is R(π/2)= (-1)^1/2 = (1)^1/4 can be considered a symbol for quarter turn in counterclockwise direction. It is not necessary to treat it as an imaginary. 

            Suppose a complete rotation of OA is achieved in ‘n’ successive turns in steps of θ=2 π/n each.  Thus R(2π)=R(nθ)=(R(θ))ⁿ=1. Since the equation xⁿ =1 has solution of the form x=(1)^1/n, hence R(θ)=(1)^1/n  is a symbolic form for the roots of unity. In the special case when θ=0 or 2π, it has value 1, which of course is always a root of unity. Note that during the rotation of OA about the point O, the point A traces a circle of radius ‘1’. A paradigm shift in the definition of a unit circle follows immediately. Every point on the circumference of a circle of unit radius is in fact geometric representaion of roots of unity. Further, every point on the circumference of a circle of radius ‘a’ geometrically represents roots of ‘a’. This definition can be extended easily to apply to a sphere. All that need to be done is to use the circle obtained due to rotation of OA about O and rotate it about any of its diameter. The logic that has been used to show that points on the circumference of a circle of radius ‘1’ geometrically represent roots of unity can also be used to show that each point on a sphere of radius one is a geometric representation of roots of unity.

            The framework of Vector Analysis and Complex Variables can be used to construct a mathametical formula for R(θ). The ideas from Vector Analysis are taken first. Symbol for unit vector along OA is i. Since R is a symbol for rotation and R(0) stands for no rotation, i =R(0) i . Suppose counter clockwise rotation R(θ) turns vector i into another unit vector i’=R(θ) i . Use the vector symbol j when the angle of rotation is θ=π/2. Thus, j =R(π/2) i .  Suppose cosθ is the value of projection of or component of i’ in the direction i (along θ=0) and sinθ is its projection on j (on θ=π/2), then 

            i’= R(θ) i  = (cosθ) i + (sinθ) j = R(0)  (cosθ) i + R(π/2) (sinθ) i

An expression for the symbol of rotation 

            R(θ) = R(0) (cosθ) + R(π/2) (sinθ)

follows immediately. It is interesting to compare it with a formula in complex variables. For this purpose, use R(0)=1 and the symbol √(-1) for R(π/2) to obtain the formula of complex variables expressed in exponential form:

            e^{√(-1) θ} =R(θ)=(cosθ) + √(-1) (sinθ)

In complex variables, R(-θ) is considered conjugate to R(θ) and their product yields its magnitude. In the present case, R() is a symbol for numerous positions occupied by the line OA during its rotation.  Thus, R(θ)R(-θ)=R(0)=1: Or, in terms of sine and cosine functions,

            (sin(θ))² +(cos(θ))²=R(θ ) R(-θ)=R(0)=1

which is a restatement of Pythagoras Theorem in a different form.

            The idea that the roots of unity can be given various mathematical forms has another interesting outcome. In terms of 2π=nθ or θ=2π/n , the relation R(θ))ⁿ=R(nθ) can be solved for

            R(θ)=R(0) cos(θ) + R( π/2) sin(θ) = (R(0) cos(2π)+R(π/2) sin(2π))^1/n=(1)^1/n

for any value of 1. Solution for n=1 is θ=0. One solution for n=2 is θ=0. For the other solution, divide the circle in two equal parts marked by angles θ=0 and θ=π. The two solutions of the square root of unity that correspond to n=2 are therefore R(0)=1 and R(π)=-1. For n=3, divide the circle in three equal parts marked by angles θ=0, θ=2π/3 and θ=4 π/3. Hence R(0), R(2 π/3) and R(4 π/3) are the three cube roots of unity. The four roots of unity for n=4 are R(0), R(π/2), R(π) and R(3π/2) because θ=0, θ=π /2, θ=π  and θ=3π/2 divide the circle in four equal parts. It seems possible that roots of unity can be found for any integral value of n. There is also a possibility that the concept of  roots of unity can be extended  to Differential Calculus as well. Instead, focus is shifted to a discussion of cocept like Brahman, Jagat and Sansar (often written Samsara) of ancient Indian Philosophy.

            Jagat and Sansar are the two words often used in Sanskrit to refer to the world and its objects.  In Sanskrit, complex words are often created by a combination of roots. The meaning of the roots is incorporated in the combined word. For example, when root ‘ja’ implying birth or creation is combined with ‘gati’ (a variation of ‘ga’ that implies movement) to form Jagati or Jagat as a symbolic reference for the world and its objects, the implication is clear. For ancient Indian Philosophers, Jagat (the world and its objects) is born or created in movement and is mired in movement. The world is called sansar because everything it contains undergoes cycles of appearance and disppearance; it exhibits a tendency to repeat.

            But the ultimate source of all is called Brahman by ancient Indian Philosophers. The meaning of the word Brahman is to grow. The meaning suggests that the world can be interpreted as symbolic representation of the growth of Brahman. Scientists feel that the world is made out of, or has its origin in, sub-atomic particles. They have been trying to find this origin. Philosophers are looking for a theory that describes the path of growth. Indian Philosopher of the Vedic era, called Kavi or Seer-Poet, suggest that existence can be explained in terms of Sat and Asat. Sat is that element of existence which always remains the same. More recent term used for Sat is eternal (sanatan). On other hand, Asat is always in flux relative to Sat. Asat in flux can exhibit multiplicity.     

            Reconsider the technical part of this presentation. What is a circle or a sphere? Both can be shown as geometric figures. Or, they can be thought of as having a center, multiple radial lines emanating from the center and terminating in multiple points on a surrounding curve or surface. If the radius has value one, then starting point of each radius is zero and at a distance of one is its terminating point. But for a rotating radius (radius in flux), the terminal point has different mathematical representation when expressed relative to a fixed radius. The fact that each terminal point is a root of unity explains the apparent multiplicity or diversity in what is essentially one. If the radius grows to ‘a’ (recall the meaning of Brahman), each point on the sphere is root of unity multiplied by the value of ‘a’. In this context, the center seems to be a symbolic representation for Sat while radius and its other terminal points indicate the presence of Asat.