Over the past four years, I have been doing in-depth research into the system of a circle’s transforming through a cone into a line that is orthogonal (at ninety degrees) to the initial circle. I have illustrated this transformation below. It is not drawn to scale. I have also found that this system is a seemingly simple one that holds, “folded” up within it, an immense wealth of beauty, complexity, and application.
Transformation of a Circle into a Cone by Parker Emmerson
If you imagine removing an arc length from a circle’s circumference, you could use the remaining length to construct a smaller circle. It just so happens that when you remove half of the circumference of a circle and use the remaining length to construct the circumference of a smaller circle, the resulting circle has a radius that is half the size of the radius of the initial circle.
The function is expressed as Circumference1-Circumference2 = π(Diameter1) – π(Diameter2) = 2πr – 2πx = θr = s = arc length. In essence, the difference in the circumferences of two circles equals an arc length of the initial circle. The initial circle is the circle whose radius is r. After more introspection, one can say that the difference in the circumference of the two circles is equal to an arc length of the “changed” circle (that circle whose radius is x), but for now, we will stick with the premise that the difference in the circumferences of two circles equals an arc length of the initial circle, which can be proven through Euclidean geometry.
We can calculate the height of the cone in terms of the Pythagorean theorem. This is written as:
Initial Radius of Circle = Hypotenuse of Right Triangle within Right Cone
Therefore,
r^2 = (base of cone)^2 + (height of cone)^2 = x^2 + h^2 = r^2
I calculate the base of the cone to be: x = sqrt(r^2 – h^2)
I then substitute the solution for the base of the cone, x into the equation:
Circumference1-Circumference2 = π(Diameter1) – π(Diameter2) = 2πr – 2πx = θr = s = arc length, which yields,
2πr – 2πx = 2πr – 2π(sqrt(r^2 – h^2)) = θr
The Difference in Circumferences of Two Circles Applied to the Pythagorean Theorem
Solving that equation for h, we get:
h = (sqrt(4(Pi)(r^2)θ – (r^2)(θ^2)))/(2(Pi))
We also know that the height of the cone can be calculated using trigonometry as
h = rSin(β), where β is the angle opposite the hypotenuse.
Therefore,
h = (sqrt(4(Pi)(r^2)θ – (r^2)(θ^2)))/(2(Pi)) = rSin(β)
The height of a cone in terms of the system of a difference in circumferences of two circles. The height of a cone in terms of the trigonometric function, sine.
We can then apply something called the Lorentz Coefficient (Lorentz Factor) in such a way that it cancels out with itself. However, when we use the exact speed of light in scientific notation, and only in scientific notation, we can compute a solution to the velocity variable, v, within the Lorentz Factor, which should cancel out with itself.
Application of the Lorentz Factor to the Height of a Cone in Such a Way that it Cancels out with Itself (Solution of Phenomenological-Transcendental-Computational Velocity by Parker Emmerson)
What does this imply for the nature of our reality? The fact that something ought not be, but is, is sort of akin to the idea of something from nothing. This system has spiritual implications, and I will let you ponder them, while moving onto one of the other prominent mysteries within the system for the next article.
The “Transcendentally-Computational-Phenomenological” velocity solution yields beautiful diagrams like:
"Tulip Shell" Black Background, "Phenomenological, computed velocity" by Parker Emmerson from A Geometric Pattern of Perception Theorems
As well as a free wealth of these available at:
http://parkeremmerson.com/portfolio-of-art-from-equations/
and
http://www.scribd.com/mathangle
Please see the “Perception and Geometry” collection under the collections section of that user’s profile.