Phi  x  Pi
Fundamentals of Nature

Dr. Scott Gonzales, MD

Fig.1. Relationship between Phi and Pi in a decagon with radius = Phi.

The relationship between Phi and Pi shown in figure 1 is more than a coincidence.  It reveals something fundamental about nature.  Specifically, three-dimensional objects and space result from the complete integration of similar two-dimensional systems; system A based on a quantum =1 and system B based on a quantum = square root 2.  The purpose of this paper is to consider these relationships.

The unit square shown in figure 2 below has an outside consisting of 4 lines that measure 1 unit each.  The inside, on the other hand, consists of two diagonals that measure square root 2 each.  It is convenient to consider each diagonal as 1/2 of square root 2 folded back on itself as seen in the model picture shown to the right.

Fig 2. Unit square: diagram (left) and model(right).

The folded diagonals of the unit square can be moved to the outside to create an irregular octagon that fits into a circle with radius square root 2.5. When the square root 2 sides of the octagon are folded back into their original position inside the unit square, a new object is seen; a pyramid with base equal to the unit square (figure 3).  With this new perspective, one can consider every unit square on a piece of graph paper as the base of a pyramid, with the structure of the pyramid hidden in 3-dimensions below the graph paper.  The pyramid structure allows the square root 2 part (inside) of the unit square to be managed separately from the outside of the unit square which is based on the unit measurement of one.  In fact, if the inside of the pyramid has structure that requires the base segments (1/2 square root 2) to be squared at all times, then the diagonals viewed from the perspective of the inside of the pyramid would measure one, rather than square root 2.

Figure 3: Irregular octagon (left) and pyramid with base equal to the unit square (right)

It is important to note that the pyramid shown in figure 3 has an apex angle of approximately 36.8 degrees.  Therefore, when consecutive pyramids are placed together, the apices do not meet at the center (figure 4).  Rather, a circle of space is trapped within the pyramid boundary.  The inner circle has a radius Phi - square root 2.5.  The combined pyramids, and captured space, are equivalent to figure 1 at the top of this paper showing the relationships between Phi and Pi.

Figure 4:Decagon with circle.

A question can be asked about the decagon in figure 4.  Is the figure a decagon, a circle, or both at the same time?  Identifying the decagon is self-evident.  But, the trapped circle is legitimately a circle.  It should not be classified as a decagon since the apices of the pyramids are points in space, not lines.  Additionally, when considering only the space component, the inner circle is connected to the outer circle by a radius of Phi.  It may be of interest that the ratio of small circle area to large circle area is equal to (radius small circle/Phi)^2.

In effect, I have taken the internal and external properties of a unit square to create a pyramid which in turn offers a relationship between an object and space summed up by the decagon with radius equal to Phi.

Fig 5: Internal and External properties of pyramid.

It is useful to consider the properties of the pyramid a bit closer. The internal and external parts of the pyramid share a common side edge measuring square root 2.5.  The conversion of square root 2.5 to Phi occurs when 10 pyramids combine to form a decagon with inner circle measuring Phi - square root 2.5.  From the perspective of the outer circle, the conversion of the pyramid is the same as having a radius of Phi extending over the entire base of the pyramid (fig 5, right side image).  This is the first evidence we see of an equality between one and square root 2.  It does not matter if the radius of phi moves along the edge of the pyramid or across the diagonal, both are treated the same.  It is interesting to note that if you were to look at the pyramid from the base perspective, as you would do on graph paper, you would not be able to see the arc caused by Phi.  The base would look flat until you observed it as some angle to the base.

The term Phi is usually considered as a length equal to aproximately 1.618..., created from a specific manipulation of the unit square.  I would argue that Phi is actually a process, not a number.  Any length, rational or irrational, can be converted to Phi  All that is necessary is to have an isoceles triangle with height equal to base; similar to the square root 2 section of the irregular octagon seen in figure 3.  The necessary criteria is that the opposite side of a triangle is exactly 1/2 the length of the adjacent side.  If you add the hypotenuse to the opposite side and then divide by the adjacent side, you always get Phi.  It is possible that this process of Phi is the reason the opposite side of the triangle is always squared when observed from the internal environment of the pyramid, thus causing the diagonal square root 2 to be measured as 1.

Figure 6: 3-D Sphere and Ball

The model of a decagon created from pyramids can be expanded to add an additional decagon at 90 degrees to make a 3-dimensional ball and sphere (figure 6).  The image towards the left shows one side of this model.  The ball has a definite top and bottom that are identical and connected by 4 bridges (red pyramids).  The top seen in the middle image is part of a sphere reproducing the irregular octagon used to create the pyramids.  In this case, though, the octagon curves with a radius of Phi.  Note in the middle image that the curve around the ball is identical regardless of direction, which again, gives an example of equality between 1 on the outside of the pyramid and square root 2 on the inside of the pyramid.

Figure 7: Integration of one and square root 2.

In figure 7, the image to the left is an attempt to show the equality of 1 and square root 2 more clearly.  The decagon and sphere are managed mathematically using the external structure with a quantum of 1.  The inside, or ball, is managed mathematically using the internal structure with a quantum of square root 2.  While it looks as if the outer square with side Phi x Square root 2 extends through only 2 and 1/2 pyramid bases we need to keep in mind that the internal components that serve as an opposite side of a triangle are split in half, squared and finally doubled, just like we did earlier with 1/2 square root 2 to make the diagonal measure 1 in the unit square.  If you use this mathematical manipulation on Phi x Square root 2, the result is Phi squared, the same distance that lies beneath three segments of the decagon or the top and bottom of the ball (figure 7).

The image to the right in figure 7 is a simplified version of the image to the left showing the fundamental structure of the sphere and ball when square root 2 and 1 are equalized.  There appears to be a top and bottom supported by a pyramid with side length equal to Phi and hypotenuse equal to Phi squared.  The remaining volume of the ball is occupied by the 4 bridges and space, including the center circle.

Fig 8: Conversion of ball into 2-Dimensions.

In two dimensions, the ball occupies a 10 x 5 rectangular area.  From this perspective, you would not be able to appreciate the arc created by the radius of Phi.  The lines drawn in red can equally describe the circumference around the ball or the decagon.  What is important is not the actual volume of the ball, but rather, if the ball represents a fundamental unit of mass in the physical world, then accounting for the mass and associated space is what is important.

There are some interesting features of this rectangle that should be pointed out.  The circumference of a great circle around any ball is equal no matter what direction one chooses to follow.  Specific to the ball shown in Figure 8, it does not matter if one follows a great circe along the lines measuring one or square root 2, the circumference must be the same.

It may be coincidence, but it takes 18 pyramids to make a ball and the rectangle has an area of 50 square units.  18/50 is equal to 0.36; the smallest angle of a Phi triangle.  If one takes the hypotenuse of the rectangle split in half and ads the opposite side equal to 5; the result is 10 x Phi, or the total length of all spikes occupying space in figure 4.  You can then take the ten segments measuring 1 from the adjacent side to complete the decagon also seen in figure 4.

Finally, I would like the reader to consider the reason why nature uses three-dimensional space and objects to integrate 1 and square root 2 into a single unified system.  I believe this is a precondition to using two-dimensional space such as graph paper to mathematically model a Higgsless theory of everything based on information theory, a q-bit, and source code identifying the irregular octagon as a fundamental object in space.  Here is a link to such a model that I have previously published on the internet.  Theory of Everything: The Quantum Fundamental Unit and Marginal Analysis.

Information theory, used in this manner, provides a link between quantum mechanics and classical physics.  On the quantum side, a q-bit exists as one of several possibilities at one time in an un-collapsed state.  For example, I use left, right, up, down, open and closed in my Theory of Everything as a q-bit.  Only collapsed closed units in the irregular octagon arrangement shown in figure 3 will define a fundamental particle, pyramid, in classical physics.  The statistical likelihood of obtaining such an outcome is (1/6)^8 or around 1:1.7 x 10^6.  Similarly, the likelihood of obtaining pyramids from irregular octagons arranged next to each other as in figure 8 would be 1.7 x 10^61 if only considering the pyramids needed (10) to accomodate the right sided image in figure 7.  In an information theory model that uses q-bits, with a side equal to the Planck length, expanding at the speed of light in two-dimensions, the total entropy of the system (multiverse) in the Real Space occupying 1 square meter would be 10^70.  These values approximate the known density of protons in the universe (collapsed state).  The channel code available to act on fundamental units, pyramids, is provided by a specific frame of reference that I call the contracting view (as opposed to the expanding view), which is analogous to gravity.  Information theory, thus provides a statistical system defining total entropy of the multiverse that behaves as a wave function while the collapsed fundamental particals (pyramids) behave in a classical manner under the influence of gravity.  In such a model, the standard model of physics would be considered a subgroup of information theory where total entropy of the system and specific definitions of source code and channel code are of primary importance.

Contact the author: scott.gonzales@gmail.com

Cite this article: Gonzales, Scott, MD. "Phi X Pi." www.PhiXPi.com Nov 18, 2012. (Date of access), www.PhiXPi.com

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