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A true meaning of univocity is that the number one has a purely geometric expression. The system of a circle transforming into and through a cone yields a pure geometry that can be accurately described by algebraic functions. The term Univocity was first advanced by the work of John Duns Scotus. For him, I would think that Univocity meant, “that which speaks with one voice.” His idea was called the univocity of being, and we can see that when analyzing the pure geometry of one, we can visualize the being of oneness. In my thinking, the meaning of univocity has two interpretations. First, the number one has a purely geometric expression. Secondly, when we apply the Lorentz transformation to the height of the cone, in such a way that it should cancel out with itself (equal one and thus have no effect on the function to which it was applied), when using the exact speed of light in scientific notation, we find a solution to the innate velocity in the Lorentz transformation. This is further evidence that oneness speaks (univocity). The idea and philosophy of univocity was later developed by Edmund Husserl in his masterful phenomenological treatises. The following paper about the phenomenon of the square root of negative one (the imaginary number) was composed by Parker Emmerson with the assistance of Wolfram Alpha and attempts to visualize, perceive, and further the understanding of univocity. The use of the imaginary number, i (the square root of negative one) is the field of complex analysis. We have thus unified mathematics and philosophy.

Please refer to the following diagram to understand the meaning of the symbols involved in the transformation.

System of a Circle Transforming into a cone by Parker Emmerson 2009

The first meaning of univocity (the number one has a purely geometric, algebraic expression):

It can be proven that the height of the cone can be calculated in terms of only the initial radius (the radius of the larger circle in the diagram) and Alpha (the angle taken out of the initial circle), thus Beta is a function of Alpha alone. From that statement, it is implied that:

Proof: \[Alpha] r = 2 \[Pi] r – 2 \[Pi] Sqrt[(r^2 - \[Eta]^2)]
\[Eta] := r Sin[\[Beta]]
From (2 \[Pi] \[Eta])/Sqrt[4 \[Pi] \[Alpha] – \[Alpha]^2] = r,
we note that : r = (2 \[Pi] r Sin[\[Beta]])/Sqrt[
4 \[Pi] \[Alpha] – \[Alpha]^2]

1=(2 pi sin(beta))/sqrt(4 pi alpha-alpha^2), and the function is periodic in beta with period 2 pi.

Elaboration on the first meaning of univocity (one has a purely geometric, algebraically described expression):

We can elaborate on the first meaning of univocity by taking the square root of negative one and performing an exposition of the different configurations and implications of the algebraic expression of the square root of negative one.

The following was computed with with Wolfram Alpha by Parker Emmerson from his original ideas.

Since both variables in the expression for the square root of negative one are angular, we can visualize a 3D Spherical Plot of the equation. The purely geometric, algebraically described expression of the square root of negative one (Sqrt[-((2 pi sin(beta))/sqrt(4 pi alpha-alpha^2))] when plotted spherically looks like this:

Imaginary Number (square root of negative one) by Parker Emmerson ©2009

and also,

The imaginary number i the square root of negative one by parker emmerson ©2009

The following is an exposition using wolfram alpha of the algebraic and geometric expression of the square root of negative one.

For the following description, the variable referred to as “alpha” will now be referred to as “theta”

i=Sqrt[-((2 Pi Sin[β])/Sqrt[4 Pi θ - θ^2])]=Sqrt[2 Pi] Sqrt[-(Sin[β]/Sqrt[4 Pi θ - θ^2])]

The square root of negative one by Parker Emmerson computed by Wolfram Alpha

Contour plots of the square root of negative one by Parker Emmerson computed with Mathematica

Alternate forms of the equation, i=Sqrt[-((2 Pi Sin[β])/Sqrt[4 Pi θ - θ^2])]=Sqrt[2 Pi] Sqrt[-(Sin[β]/Sqrt[4 Pi θ - θ^2])] are:

i=Sqrt[-((2 Pi Sin[β])/Sqrt[4 Pi θ - θ^2])]=Sqrt[2 Pi] Sqrt[-(Sin[β]/Sqrt[4 Pi θ - θ^2])]=
Sqrt[2 Pi] Sqrt[-(Sin[β]/(Sqrt[4 Pi - θ] Sqrt[θ]))]=
Sqrt[Pi] Sqrt[((-I) (E^((-I) β) - E^(I β)))/Sqrt[4 Pi θ - θ^2]]=
(I E^(I Pi Floor[Arg[4 Pi θ - θ^2]/(4 Pi) – Arg[Sin[β]]/(2 Pi)]) Sqrt[2 Pi] Sqrt[Sin[β]])/(4 Pi θ – θ^2)^(1/4)

If theta and beta are positive, then, i=(I Sqrt[2 Pi] Sqrt[Sin[β]])/((4 Pi – θ) θ)^(1/4)

The roots of the equation are: {{4 Pi θ – θ^2 != 0, β == Pi C[1], Element[C[1], Integers]}}

The series expansion is: i=SeriesData[β, 0, {Sqrt[2 Pi] Sqrt[-(1/Sqrt[4 Pi θ - θ^2])], 0, 0, 0, -(Sqrt[Pi/2] Sqrt[-(1/Sqrt[4 Pi θ - θ^2])])/6, 0, 0, 0, (Sqrt[Pi/2] Sqrt[-(1/Sqrt[4 Pi θ - θ^2])])/720, 0, 0, 0, -(Sqrt[Pi/2] Sqrt[-(1/Sqrt[4 Pi θ - θ^2])])/12096, 0, 0, 0, (-67 Sqrt[Pi/2] Sqrt[-(1/Sqrt[4 Pi θ - θ^2])])/14515200, 0, 0, 0, -(Sqrt[Pi/2] Sqrt[-(1/Sqrt[4 Pi θ - θ^2])])/2838528, 0, 0, 0, (-64397 Sqrt[Pi/2] Sqrt[-(1/Sqrt[4 Pi θ - θ^2])])/2324754432000, 0, 0, 0, (-113249 Sqrt[Pi/2] Sqrt[-(1/Sqrt[4 Pi θ - θ^2])])/50214695731200, 0, 0, 0, (-3679787 Sqrt[Pi/2] Sqrt[-(1/Sqrt[4 Pi θ - θ^2])])/19511996055552000, 0, 0, 0, (-810304169 Sqrt[Pi/2] Sqrt[-(1/Sqrt[4 Pi θ - θ^2])])/50304927676760064000, 0, 0, 0, (-6040635661561 Sqrt[Pi/2] Sqrt[-(1/Sqrt[4 Pi θ - θ^2])])/4316162794666013491200000}, 1, 43, 2]

The derivative of the algebraic (geometric) expression of i (the square root of negative one) is:

Possible derivation:
d/dbeta(sqrt(2 pi) sqrt(-(sin(beta))/sqrt(4 pi theta-theta^2)))
| Factor out constants:
= | sqrt(2 pi) (d/dbeta(sqrt(-(sin(beta))/sqrt(4 pi theta-theta^2))))
| Use the chain rule, d/dbeta(sqrt(-(sin(beta))/sqrt(4 pi theta-theta^2))) = ( dsqrt(u))/( du) ( du)/( dbeta), where u = -(sin(beta))/sqrt(4 pi theta-theta^2) and ( dsqrt(u))/( du) = 1/(2 sqrt(u)):
= | sqrt(2 pi) (d/dbeta(-(sin(beta))/sqrt(4 pi theta-theta^2)))/(2 sqrt(-(sin(beta))/sqrt(4 pi theta-theta^2)))
| Factor out constants:
= | (sqrt(pi/2) (-(d/dbeta((sin(beta))/sqrt(4 pi theta-theta^2)))))/sqrt(-(sin(beta))/sqrt(4 pi theta-theta^2))
| Factor out constants:
= | -(sqrt(pi/2) (d/dbeta(sin(beta)))/sqrt(4 pi theta-theta^2))/sqrt(-(sin(beta))/sqrt(4 pi theta-theta^2))
| The derivative of sin(beta) is cos(beta):
= | -(sqrt(pi/2) cos(beta))/(sqrt(4 pi theta-theta^2) sqrt(-(sin(beta))/sqrt(4 pi theta-theta^2)))

-((Sqrt[Pi/2] Cos[β])/(Sqrt[4 Pi θ - θ^2] Sqrt[-(Sin[β]/Sqrt[4 Pi θ - θ^2])]))

The following are alternative representations:

Sqrt[-((2 Pi Sin[β])/Sqrt[4 Pi θ - θ^2])] == Sqrt[(-2 Pi)/(Csc[β] Sqrt[4 Pi θ - θ^2])]
Sqrt[-((2 Pi Sin[β])/Sqrt[4 Pi θ - θ^2])] == Sqrt[(2 Pi Cos[Pi/2 + β])/Sqrt[4 Pi θ - θ^2]]
Sqrt[-((2 Pi Sin[β])/Sqrt[4 Pi θ - θ^2])] == Sqrt[(2 I Pi)/(Csch[I β] Sqrt[4 Pi θ - θ^2])]

The following are series representations:

Sqrt[-((2 Pi Sin[β])/Sqrt[4 Pi θ - θ^2])] == 2 Sqrt[Pi] Sqrt[-(Sum[(-1)^k BesselJ[1 + 2 k, β], {k, 0, Infinity}]/Sqrt[(4 Pi - θ) θ])]

Sqrt[-((2 Pi Sin[β])/Sqrt[4 Pi θ - θ^2])] == Sqrt[2 Pi] Sqrt[-(Sum[((-1)^k β^(1 + 2 k))/(1 + 2 k)!, {k, 0, Infinity}]/Sqrt[(4 Pi - θ) θ])]

Sqrt[-((2 Pi Sin[β])/Sqrt[4 Pi θ - θ^2])] == Sqrt[2 Pi] Sqrt[-(Sum[((-1)^k (-Pi/2 + β)^(2 k))/(2 k)!, {k, 0, Infinity}]/Sqrt[(4 Pi - θ) θ])]

The following are integral representations of the pure geometry and algebra of the square root of negative one:

Sqrt[-((2 Pi Sin[β])/Sqrt[4 Pi θ - θ^2])] == Sqrt[2 Pi] Sqrt[-((β Integrate[Cos[t β], {t, 0, 1}])/Sqrt[(4 Pi - θ) θ])]

Sqrt[-((2 Pi Sin[β])/Sqrt[4 Pi θ - θ^2])] == (Pi^(1/4) Sqrt[(I β Integrate[E^(s - β^2/(4 s))/s^(3/2), {s, (-I) Infinity + γ, I Infinity + γ}])/Sqrt[(4 Pi - θ) θ]])/Sqrt[2] /; γ > 0

Sqrt[-((2 Pi Sin[β])/Sqrt[4 Pi θ - θ^2])] == Pi^(1/4) Sqrt[(I Integrate[(2^(-1 + 2 s) β^(1 - 2 s) Gamma[s])/Gamma[3/2 - s], {s, (-I) Infinity + γ, I Infinity + γ}])/Sqrt[(4 Pi - θ) θ]] /; 0 < γ < 1 && β > 0

Elaboration on the second meaning of univocity (the Lorentz transformation applied to the height of the cone in such a way that it should cancel out with itself actually yields a solution to the velocity within the Lorentz transformation when using the exact speed of light in scientific notation):

We set up the equation and solve it:

“Solve[(Sqrt[r Sqrt[1 - (v)^2/c^2]] Sqrt[\[Theta]/Sqrt[1 - (v)^2/c^2]]
Sqrt[4 \[Pi] r – r \[Theta]])/(2 \[Pi]) == r Sin[\[Beta]], v]” ©Parker Emmerson 2009

Solution to the innate velocity in the lorentz transformation (phenomenological velocity) © Parker Emmerson 2009