My 2001 paper "Parallel Universes In Math".

Here is my 2001 paper entitled "Parallel Universes In Math". I finished it in June of 2001 and put it on my website(nolonger exists) at that point in time. The website stayed up until the Fall of 2001. The primary concern of the website was to publish webpages I had created, for a course "final" in an HTML college course I had taken. That "final" had a bunch of poems I had written on it and links to the Poetry Webring. I added my parallel universe paper because I didn't want to publish it in the same place as my 1997 paper. By the way, the status on my 2006 paper is less then what I alluded to in my previous post. I think I said I was writing it. A more accurate statement might be, I'm writing it - in my head. I hope you enjoy my 2001 paper.

Parallel Universes In Math.
By Jim Akerlund


This paper will try to show that there are parallel universes in math. In order to do this, we will need to explain to you what we mean by the statement; "there are parallel universes in math". The concept of parallel universes as brought down to us from history basically says; given the same set of circumstances, a different set of events may have happened in a different universe. That previous statement divides the set of events in our universe into two different types; events that are "universe invariant" and events that are "universe specific". We will define universe invariant as; events, ideas, memes, rules, and laws that are the same in all parallel universes. And we will define universe specific as; events, ideas, memes, rules, and laws that are specific to exactly one universe. Our current thinking of math is that all of math is universe invariant. This paper will try to show that some of math is universe specific and therefore there are parallel universes in math.

Summary
A model of our universes expansion will be developed. It will be shown that this model is a derivative of Hubble's model of universal expansion. We then go onto create a curved spacetime diagram from this model. From the curved spacetime diagram we calculate an expansion rate and postulate the existance of parallel universes based on different values of π. We then split math into two seperate groups: "universe invariant" and "universe specific" math. We use the universe specific number of π to show that our number system is also universe specific. This leads to another way to define universe specific and universe invariant. In the discussion, we discuss if the paper itself is universe invariant and finally we look for what parts of math that mankind already has developed that maybe universe invariant. One of the goals of this paper is to give you enough information about parallel universes and math so that hopefully you won't be looking into the same mathematical abyss that we were looking into nine years ago when we first conceived of parallel universes based on different values of π.

The Retroverse
There is more then one way to describe how the universe expands. The Hubble model (Peebles 1993) of universal expansion is what we will describe as an event-to-event model of expansion. Meaning that the model is not referenced to any particular event in the universe. A different way of describing the universes expansion would be event-to-specific-event. Meaning that all expansion in the universe is referenced to one specific event in it's history. For this type of model to work effectively, we would need an event that has no preceeding events before it and the specific event is in the histories of all subsequent events. The only event in the universe that fulfills this requirement is the Big Bang event. This model of the universes expansion is unique, because it requires no observation. We imagine a photon emitted at the Big Bang and arriving at some event in the universe. We then ask the question, how far did the event expand from the Big Bang in the time that it took the photon to reach the event. This can be put down into an equation,

d*q = a. (1)

a is the length of the geodesic for the photon and it is also the age of the universe when the photon reached the event. d is the distance the event expanded from the Big Bang and q is the factor of the difference between a and d.

The Hubble model of the universes expansion is based on a scaling factor. This scaling factor, ignoring local irregularities, is thought to be the same for all of spacetime. Recent research that suggests an accelerating expansion(Hogan, Kirshner, and Suntzeff 1999), calls for this scaling to be in question. We will call the Hubble scaling "flat" scaling. It describes a situation where an object twice as far as another object will be expanding at twice the expansion rate. The accelerating expansion model, suggests a "hyperbolic" scaling. Hyperbolic scaling occurs when an object twice as far away will be expanding at greater then twice the expansion rate. The model described in this paper will display "elliptic" scaling. Elliptic scaling occurs when an object twice as far away will be expanding at less then twice the expansion rate.

Research into Einstein Rings(Wambsganss 2001) may resolve whether the universe has "flat", "hyperbolic", or "elliptic" scaling. Einstein Rings provide two seperate ways too evaluate distance and expansion. The expansion rate of the two objects can be determined by redshift. The distance of the two objects can be determined by the geometery of the gravitational lens. If the distance determined by the lens geometery agrees with the expansion via redshift, then this is evidence for flat scaling. If the distance via lens geometery is less then the expected expansion via redshift, then this is evidence of hyperbolic scaling. If the distance via lens geometery is greater then the expected expansion via redshift, then this is evidence of elliptic scaling

If we set q = 1, we get the strange situation where the event is expanding from the Big Bang at the speed of light. To set q > 1, we get the situation where the geodesic of light(a) is longer then the path of the event(d) from the Big Bang. This leads to a conclusion that some how the geodesic of light is curved. General Relativity (Einstein 1956) would tell us that gravity is causing the curvature of the light. In this case, because the light has traversed the universe to get to the event, then the mass of the universe is causing the curvature of the light. Other examples of the curvature of light due to mass include; light bending around the sun and gravitational lensing due to intervening mass between event and observer(Ciufolini and Wheeler 1995). We believe that this is another manifestation of light bending in our observable universe.

One of the unique things about q is that each value of q determines a seperate mass of the
universe. Because matter can neither be created nor destroyed then the universe has exactly one
mass that is the same from it's creation. In other words, once the value of q is established at the Big Bang it can never change for all events in the same universe.

Of all the possible curvatures for the values q > 1, q = π/2 defines the simplest curve. It also implies a universe that spatially homogeneous and isotropic on the very large scale. When q = π/2, we can
then draw a curved spacetime diagram.



Diagram #1, above simulates two photons traveling from the Big Bang to the event, each from
opposite directions. Distances measured in spacetime are measured along geodesics of light, since d isn't measured along a geodesic of light then it is an imaginary distance (not the mathematical term, imaginary). The diagram shows that the event is at the antipodal from the Big Bang in spacetime. All events are at the antipodal of spacetime from the Big Bang

This diagram is very similiar to a diagram presented by (Osserman 1995) in his book. He calls the
diagram the "retroverse". Mr. Osserman doesn't apply variables to his diagram. This paper will take
the retroverse far beyond what is described in his book.

Also from the above diagram we can eliminate other values of q > 1, because when drawn(Diagrams #2 and #3) they will reveal a universe that isn't spatially homogeneous and isotropic on the very large scale, or they don't satisfy the cosmological principle(Peebles 1993).






The curved spacetime diagram also shows a time relation to all events in the universe relative to the event illustrated. Here is a diagram showing the time relations.



Diagram #4 shows that everything outside the circle is composed of both future events and
"Unknowable events" and everything inside the circle is composed of past events. The area on the
diagram called "Unknowable Events" is a manifestation of the two dimensional diagram that is drawn. We don't know what this would look like in the four dimensional spacetime we live in, nor do we know if the "Unknowable Events" area actually exists in our four dimensional universe.

Parallel Universes
For any value of q we can determine the expansion rate of that event from the Big Bang. Here is
equation 1 again,

d*q = a. (1)

To get a rate of expansion r of the event from the Big Bang, we need to divide the distance the
event expanded d, by the age of the universe a, and we get this equation,

d/a = r. (2)

Substituting equation 1 into equation 2 we get

1/q = r. (3)

Substituting π/2 for q we get,

2/π = r = 0.6366197724... lightyears. (3.1)

When the speed of light is not unity, equation 3.1 becomes,

2c/π = r, (3.2)

where c is the speed of light.

If every event in the universe is expanding at exactly 0.6366197724... lightyears from the Big Bang,
then what about the other expansion rates from the Big Bang? We are faced with two seperate
solutions here. The first solution is that the Big Bang expanded with exactly one expansion rate and
that all other expansion rates are not possible. Or the Big Bang expanded with many expansion
rates and each expansion rate defines it's own separate and complete universe.

The "exactly one expansion rate" solution, suggests an Anthropic principle(Barrow and Tipler 1983) is at work in the Big Bang. This paper will not explore that solution. This paper will explore the seperate expansion rates that define seperate universes, or a parallel universes solution. This solution is based off of an idea originally proposed by Hugh Everett III(Everett 1957) and followed up by(DeWitt and Graham 1973).

Here is equation 3.1 again,

2/π = r. (3.1)

If we change the expansion rate r, then some variable on the left hand side of the equation will also
have to change. That varaible is π. This immediately implies that each seperate universe has it's own different value of π. Universes where π > 3.14159... will be expanding at r < 0.6366197724... lightyears, and universes where π < 3.14159... will be expanding at r > 0.6366197724... lightyears. In equation 3.1, the speed of light is unity, it will also be unity in the parallel universes, so it can not be a variable that determines the existance of parallel universes.

"Universe Invariant" and "Universe Specific" Math
Now we need to ask the question; what do we mean π changes in the parallel universes? π is the
ratio between a circles diameter and it's circumference in Euclidian spacetime. In a parallel universe, the ratio between a circles diameter and it's circumference will be different in that universes Euclidian spacetime.

From the above paragraph we can show you two concepts of parallel universes. These two concepts are that some ideas are specific to each individual universe or "universe specific" and some ideas are applicable to all universes or "universe invariant". We will define universe invariant as; events, ideas, memes, rules, and laws that are the same in all parallel universes. We will define universe specific as; events, ideas, memes, rules, and laws that are specific to one universe. The value of π is universe specific as shown above. Circle, circumference, diameter, ratio, and Euclidian spacetime are universe invariant. If Euclidian spacetime were universe specific then that would lead to the absurd proposition that somehow our universe exhibits Euclidian spacetime and the parallel universes don't or are Non-euclidian spacetimes. There is a "Copernican" view in physics that says that the section of the universe we reside in isn't special from the other sections in the universe. We now extend that "Copernican" view to the relationship between parallel universes, in that, no specific universe is more special then any other universe. This translates to, if this universe exhibits Euclidian spactime, then the parallel universes will also exhibit Euclidian spactimes, or Euclidian Spacetime is universe invariant. Euclidian spacetime becomes a feature to the "flatness" of the spacetime you are measuring, irrespective of the universe you are measuring it in. This same type of arguement can also be applied to circle, circumference, diameter, and ratio.

Circle, circumference, diameter, ratio, π, and Euclidian geometery are all concepts of math, but in the previous paragraph we seperated them into universe specific and universe invariant concepts. That means that somehow math is divided into two seperate groups; math that is universe specific and math that is universe invariant.

That previous paragraph is contrary to our current thinking in math. Our current thinking in math is that all of math is the same in all parallel universes, or that all of math is universe invariant. The only math that discusses parallel universes is math used by Quantum Physics to discuss possible interpretations of quantum theory(Everett 1957) and(DeWitt and Graham 1973). The math used in quantum theory was formulated by John von Neuman(von Neuman 1955). This math was formulated without consideration that there are might be parallel universes and that math may change in these parallel universes.

As we write this, we don't know how deep and extensive the splitting of math into universe invariant and universe specific groups goes, but using the universe invariant concept of ratio we can show you another universe specific concept in math. Suppose we are in a universe where the value of π is not equal to 3.14159... and we decide to use equation 4 to calculate the value of π. We will run into a paradox.

π/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9.... (4)

We can not have two different values of π in the same universe. One value based on the spacetime we are in and another value based on a calculation we did on paper, or computer, or in our heads. Equation 4 is known as Liebniz' equation(Blatner 1997).

Equation 4 shows the very special relationship π has with the number system. We know from above that π is universe specific. In order for equation 4 to calculate a value of π not equal to 3.14159..., then some part of the equation needs to change in some way. We have come up with three possible solutions to this problem.

1. The whole equation is universe specific.
2. The numbers only are universe specific and the equation form is universe invariant.
3. Some, unknown aspect of the equation is universe specific and another, unknown aspect of the equation is universe invariant

Of the three possiblities, #2 the situation where the numbers are universe specific and the equation form is universe invariant, is the only solution where ratio remains a universe invariant concept. Solution #2 also allows for this hypothetical situation to occur. Where a parallel universe equaliant of Gottfried Liebniz formulates an equation to calculate π in his universe and his equation looks like this,

<π>/<4> = <1> - <1>/<3> + <1>/<5> - <1>/<7>... (5)

The symbols inside the < > are this universes intrepretation of the parallel universes Gottfried Liebniz' symbols. The < > symbols do you a disservice. They help us in explaining the thinking, but when actually applied to mathematical equations they will preduce lots of confusion. This universes number system is specific to this universe. Three apples in this universe is in no way equivalent to three apples in another universe. Sure the concept of "three" does exist in the parallel universes, but "three" is universe specific. The best that we have been able to do is to turn Liebniz' equation into a kind of parallel universe number system translator. Here is that equation,

u*π/4 = u/1 - u/3 + u/5 - u/7 + u/9.... (6)

u is the value of 1, relative to this universe, of the universe you are looking at. When u = 1 that defines this universe for equation 6. u*π is the value of π in the parallel universe relative to this universe.

The above demonstration of π showing that the number system is universe specific leads us to another definition of universe invariant and universe specific. A universe invariant concept will not change when π changes. A universe specific concept will change when π changes. This can be generalized too; given an unknown concept(unknown as to whether it is universe invariant or universe specific) and a known universe specific concept. The unknown concept will be universe invariant if when the two concepts are brought together and the universe specific concept is changed, the unknown will not change. If the unknown is universe specific, then it will change under the above conditions. We are not quite sure what we mean when we say "two concepts are brought together".

Ultimately, all concepts of math are concepts that are invented by mankind to deal with problems that he has encountered. Math could be classified as the set of solutions and methods, mankind has invented to solve a certain class of problems(Davis and Hersh 1981), (Skemp 1971). Numbers, are a solution mankind has come up with to solve a problem of counting(Barrow 1992). So, when we say that numbers are universe specific; we are saying that should mankind had been in a parallel universe and once again created the "number" solution to his counting problem and we were to compare that parallel universe solution to the same solution in this universe, then equation 6 would show how they are different. We don't know if there is a universe invariant "number" solution. To reiterate our definition of math again and this time include the concepts of universe invaraint and universe specific, we can say: Math is a set of solutions and methods, mankind has come up with to solve a certain class of problems, some of those solutions and methods are universe invariant and some of those solutions and methods are universe specific.

Discussion

The idea where each different value of π from infinity to zero is stacked like the layers of an onion as a result of the Big Bang is one of the most compelling ideas we have ever run across. The concepts of universe invariant and universe specific are byproducts of the different values of π. Throughout this paper we have struggled with defining, describing, and manipulating the concepts of universe invariant and universe specific. One of the key aspects of universe specific is that universe specific concepts can not be used to describe parallel universes. You can only use universe invariant concepts to describe parallel universes. That begs the question then; is this paper itself universe invariant? The answer to that question we do not have, but if it is it may explain why the math is so simple in it. You can not use universe specific math to define parallel universes, it has to be universe invariant. We believe the ideas in this paper are also simple for the same reason. We don't think this means that all universe invariant concepts are simple. We don't know. The Friedman cosmological models(Ciufolini and Wheeler 1995) all appear to be universe specific and therefore can not define parallel universes. One other thing about universe invariant and universe specific. In our definitions of these two concepts, we said they applied to one universe or all universes. We don't know if there are concepts that may apply to more the one universe but less then all. In any case we have called these types of concepts semi-universe invariant.

One of the questions that was continually raised while working on this model is, has mankind formulated any math where unbeknownst to the mathematician, he has created a universe invariant math. From the conditions outlined above, we would be looking for any maths that don't use; a specific value of π, number, or quantity. The best example of universe invariant math we have run across so far, is a book written in 1854 by George Boole. Here is a quote from "The Laws Of Thought" by George Boole, page 12(Boole 1854) "It is not the essence of mathematics to be conversant with the ideas of number and quantity." We have found threads of universe invariant math in many of the fields of math; Ring theory(Rowen 1991), all sorts of Algebras(Birkhoff and Mac Lane 1977), Euclidian and Non-euclidian geometries(Gans 1973), and even Number theory(Ore 1948) has universe invariant threads running through it. We would need to pull all these universe invariant concepts of math together, before we could go on to create a universe invariant physics. With a universe invariant physics in our hands we could then start building the methods that will take us to all these parallel universes that are described in this paper. We have dubbed the act of crossing into a parallel universe, pi-crossing.

References

Barrow, John, D. and Tipler, F.J., "The Anthropic Cosmological Principle". (1986). Oxford University Press.

Barrow, John, D., "Pi in the Sky: Counting, Thinking, and Being". (1992) Oxford University Press.

Birkhoff, Garrett, and Mac Lane, Saunders., "A Survey of Modern Algebra". (1977). Macmillian.

Blatner, David., "The Joy Of p". (1997). Walker Publishing.

Boole, George., "An Investigation Of The Laws Of Thought On Which Are Founded The Mathematical Theories Of Logic And Probabilities". (1958). Dover. Originally published (1854). Macmillan.

Ciufolini, Ignazio, and Wheeler, John, A., "Gravitation and Inertia". (1995). Princeton University Press.

Davis, Philip, J. and Hersh, Reuben., "The Mathematical Experience". (1981). Birkhäuser.

DeWitt, Bryce, S. and Graham, Neill., "The Many-Worlds Interpretation Of Quantum Mechanics". (1973). Princeton University Press.

Einstein, Albert., "The Meaning Of Relativity", 5th ed. (1956). Princeton University Press.

Everett, Hugh, III., "Relative State, Formulation Of Quantum Mechanics". Rev. Mod. Phys. 29, 454, (1957)

Gans, David., "An Introduction To Non-euclidian Geometery". (1973). Academic Press.

Hogan, Craig, J., Kirshner, Robert, P., and Suntzeff, Nicholas, B., "Surveying Space-time with Supernova". Sci. Am. January (1999).

Ore, Oystein., "Number Theory and It's History". (1948). McGraw-Hill.

Osserman, Robert., "Poetry Of The Universe: A Mathematical Exploration Of The Cosmos". (1995). Anchor books.

Peebles, P. J. E., "Principles Of Physical Cosmology". (1993). Princeton University Press.

Rowen, Louis, H., "Ring Theory". (1991). Academic Press.

Skemp, Richard, R., "The Psychology of Learning Mathematics". (1971). Penguin Books.

von Neuman, John., "Mathematical foundations of Quantum Mechanics". (1955). Princeton University Press.

Wambsganss, Joachim., "Gravity's Kaleidoscope". Sci. Am. November. (2001).

The ideas in this paper that were originated by Jim Akerlund were originally published in 1997 in the Newsgroup Alt.sci.physics.new-theories in the July/August time frame under the title "Foundations of Parallel Universe Math".

My 1997 paper "Foundations Of Parallel Universe Math"

I have written several papers about Parallel Universes; one in 1997, one 2001, and one I am currently in the process of writing for 2006. The paper I wrote in 2001, I posted to my own webpage which now onlonger exists. I'm guessing also that nobody ever saw it. The paper I wrote in 1997 I posted to the newsgroup Alt.Sci.Physics.New-Theories several times in the July/August time frame of 1997. I do know that some people actually read it there, because two people sent me emails concerning it. The emails were questions concerning early ideas in the paper and not part of the meat and potatos of the paper. This year (2006) I ran across a conversation in a different NG that seems to be related to one of the meat and potatos ideas of the paper, but I have no idea as to whether the conversation was based on my 1997 paper. Anyway I tell you all of this in preperation of the 1997 paper itself. Here it is for your enjoyment.

Foundations Of Parallel Universe Math
by
Jim Akerlund

Abstract

A simple mathematical relationship is developed between an event and the Big Bang. This relationship is then compared with known rules of universe expansion rates to extrapolate parallel universes. A curvature of spacetime based on the events relationship to the Big Bang is shown, and an explanation of the "matter in the universe seems older then the universe it resides in" is presented, also the curvature of spacetime that is derived suggests an "arrow of time". The curvature of spacetime is then shown to require an "advanced" light particle to complete the transaction. Finally, a connection is developed between math in this universe, and math in the Parallel universes.


The Geodesic Of Light

What, you may ask, is a simple relationship between an event and the Big Bang? It is the thought that there are only three things that occured in the universe. The Big Bang, the event(it doesn't matter when or what, as long as it occured after the Big Bang), and one photon of light traveling between the two(from the Big Bang to the event). We now ask the question, how long is the length of the path that the photon traveled(called a Geodesic)? The answer is, the age of the universe at the time of the event. But if we also ask, how far has the universe expanded, when the length of the geodesic is questioned. We get an answer that will suggest a shape to the universe. If we let A be the length of the geodesic, and D be the distance the universe has expanded, and we set these equal to each other, then we will get

D = A. (1)

This equation shows a very flat universe, with the expansion rate being equal to the speed of light. Our universe is not like this, so that means that the light that traveled from the Big Bang to an event had to get curved in some way. This curvature could be represented in this equation

DZ = A. (2)

In Eq. (2), Z is some number greater then 1. In the case where 1>Z>0, this would describe a universe where the expansion rate exceeds the speed of light. For the case where Z < or = 0, I do not know what to make of it. For Z < or = 0, these numbers would describe expansion rates faster then infinite.
I stumbled across these equations because I came to the conclusion that the universe on the whole was curved.
Here is what I did. Draw two dots. Label one dot "Us" (the event), and the other dot "BB" (the Big Bang). Draw a half circle connecting the two. The half circle represents the geodesic of light traveling from the Big Bang to the event. I started out with the half circle, because it was the easiest curve I could think of. I was expecting to graduate to other curves once I understood what was happening with the simplist. Draw a straight line connecting the two dots. The straight line represents the distance the universe has expanded when the event occured, it is an imaginary distance(not to be confused with imaginary numbers), and is not something that can be measured physically.
If we were to extend the curve from "BB" to "Us", so that it meets "BB" again, then we would have a circle. The completed circle shows a special relationship between the event and the Big Bang, the Big Bang is at the antipodal from the event. An antipodal is the point opposite another point on a circle, this can also be extended to a sphere, an example on the earth is the antipodal of the North Pole, is the South Pole. We could also draw other events, all at different distances and different angles from the Big Bang, with there corresponding circles. All of these events would also be at their own antipodal from the Big Bang
The areas of this drawing can be labeled, and show a time relationship between other events in the universe and the "Us" event. An event that happens inside the curve from "Us" to "BB", is an event that occured in the past for "Us". An event that happens along the curve between "Us" and "BB", is a simultaneous event as the "Us" event. An event that happens outside the curve, will be in "Us"'s future. Because this is drawn on paper, there is an area on the paper where if events happen there, they can not be in "Us"'s future, past, or present (It is the area defined as: everything on the opposite side of "BB" from "Us", including a line passing through "BB" that is perpendicular to the line from "Us" to "BB".). The author does not know if this type of area also exists in reality.
If we were to rotate the circle through the third dimension, then we would get a sphere, with the "Us" event at the antipodal from the Big Bang. If we were to then rotate the sphere through the fourth dimension, then we would get a hypersphere, with, once again, the "Us" event at the antipodal from the Big Bang. And that is the complete picture of the relationship between an event and the Big Bang. We are rotating the geodesic through these dimensions to show that the geodesic can be rotated. The results from Eq. (2) will not change when rotated into other dimensions. The shape of the geodesic doesn't have to be a smooth curve either. It can be any convoluted shape, just as long as the difference between A and D remain the same. But, we are then faced with the question of why the convoluted shape when a smooth curve satisfies the same equation? As a general rule, if the universe can be precieved to be doing something a complex way, or a simple way, the universe will choose the simple way. But this in no way eliminates the convoluted shape.
This is not the first time the universe has been diagramed in this fashion. Robert Osserman in his book "Poetry of the Universe: a mathematical exploration of the cosmos". (1995). Anchor Books., pages 114-120, describes essentially the same thing. He even gives a name to the curved universe he describes, he calls it the "retroverse", we will also use the same name. Mr. Osserman arrives at the retroverse from a different prespective, and does not derive an equation from his model, nor does he label the parts of the model other then the event and the Big Bang. This model of the retroverse will be different then what is explained in Mr. Osserman's book.


The Hubble model of Universe expansion versus the retroverse model.


The equation for the circumference of a circle is: (diameter) x (Pi) = (circumference). Our diagram is half of a circle, so the equation becomes: (diameter) x (Pi)/2 = (circumference)/2. In the diagram, (circumference)/2, is distance between "Us" and "BB" as measured along the curve, and diameter, is the distance between "Us" and "BB" as measured along the straight line. Substituting A and D for circumference/2 and diameter, we get this equation

D*(Pi)/2 = A. (3)

Since this model uses light as the measuring unit, we can set the speed of light to equal 1. That is the same as saying, the speed of light is unity. So if light were to travel for one light year, we would get two pieces of information from that; the distance it has traveled, and the time it traveled in. We get the same type of information when light has traveled one light second. When that is applied to this model, "A" becomes two different values at the same time; a distance, and a time. With that in mind, we can proceed.
The equation for the rate of expansion of an event from the Big Bang, in this model is: (distance the universe has expanded at the time of the event) / (age of the universe at the time of the event) = (rate of expansion). In the diagram, the age of the universe, is "A", and the distance the universe has expanded, is "D". We will set "R" to be the, rate of expansion, and we get this equation

D/A = R. (4)

Substituting Eq. (3) into Eq. (4), we get this equation

D/D*Pi/2 = R. (5)

Eq. (5) reduces to this equation

2/Pi = R. (6)

According to this model, every event that has and will occur in this universe, since the Big Bang, is expanding at .6366197724... lightyears from the Big Bang. When the speed of light is not unity, Eq. (6) becomes

2*C/Pi = R (6.1)

where C is the speed of light.
Edwin P. Hubble showed by observation that the velocity of recession is proportional to the distance of a galaxy("The Expansion Rate And Size Of The Universe". Wendy L. Freedman. (Nov. 92). Sci. Am.). In other words, galaxies at different distances have different recession rates. This seems to be at odds with the retroverse model. Well, actually they are not at odds with each other. The Hubble model determines "event to event" expansion rates. The Retroverse model determines "event to Big Bang" expansion rates. If we were to turn the Hubble model into an "event to Big Bang" type of model, the model would produce the same type of results as the retroverse model. Mainly, all events are expanding at one rate from the Big Bang.
Here is the Hubble model of universe expansion when the Big Bang is the other event to be measured. We shall use "Us" and "BB" again and this time a third set of events, "X". It is observed from "Us" that an event "X1" is moving away form "Us" some rate Q1. "X2" is observed to be further away then "X1", and it's rate of moving away from "Us" is proportional to the distance. There is some "X" at the Big Bang where it's moving away from "Us" is the absolute limit. Meaning, there is no event further back in time then the Big Bang, so nothing can expand faster then an event at the Big Bang. This puts an upper limit on the expansion rate one event can expand from another event. What that upper limit is, the Hubble model does not say.
The retroverse model, gives an expansion rate from the Big Bang that is time invariant. Is the upper limit, from the Hubble model, time invariant? Meaning, is the upper limit, the same value at one minute after the Big Bang as a trillion years after the Big Bang? The Hubble model is not designed to answer that question. We shall now see that the Hubble model is very time specific. One of the things the Hubble model comes up with, is the Hubble constant. It is a measure of the recession velocity of a galaxy divided by it's distance. It is measured in kilometers per second per megaparsec. The value of the Hubble constant is not part of the scope of this paper, so we will set the value to Q per second per megaparsec. We continue with this question; for an observer, when she was exactly one megaparsec from the Big Bang, what was the value of the Hubble constant then? If the observer says Q per second per megaparsec, then the universe is speeding up it's expansion rate over time. This is not what we expect of the expanding universe. The other solution is, the Hubble constant is not constant, and is dependent upon when it is measured, or time dependent.
This is what can be concluded with both the Hubble model of universe expansion, and the retroverse model of universe expansion. The Hubble model does not contradict the retroverse model. The retroverse model reveals some limitations of the Hubble model. The Hubble model can not confirm nor deny that the value of Z from Eq. (2) is equal to Pi/2. The Hubble model and the retroverse model are two different perspectives of an expanding universe from a single Big Bang. The Hubble model is based on "event to event" expansion rates, and the retroverse model is based on "event to Big Bang" expansion rates.


The Wow Section

If every event that has and will occur in this universe is expanding at .6366197724... lightyears, then what about the other expansion rates? We are faced with two solutions here. The first one, is that the Big Bang exploded at exactly one expansion rate, and all other expansion rates are not possible. The second solution is, the Big Bang exploded with many expansion rates, and each seperate expansion rate defines it's own seperate and complete universe.
If the Big Bang created one expansion rate, then there has to be a physical reason why this occured, or an explanation has to be available why the other expansion rates are not possible. An Anthropic principle will not suffice here("The Anthropic Cosmological Principle", Barrow, J.D. and Tipler, F.J. (1986). Oxford University Press.). For these reasons, this paper will not explore the exacly one expansion rate solution. This paper will explore the seperate expansion rates that define seperate universes, or parallel universes solution. This solution is based off of an idea originally proposed by Hugh Everett III, in his paper ("Relative State" Formulation Of Quantum Mechanics In Quantum Theory And Measurement (1957) Rev. Of Mod. Phys. 29, 454-462).
The equation that gave us the expansion rate is,

2/Pi = R. (6)

If we change the expansion rate (R), then some variable on the left hand side of the equation will also have to change. the only variable available to us is Pi. This immediately implies that each seperate universe has it's own different value of Pi. The values of Pi approaching infinite as the expansion rate approaches 0, from Eq. (6), and Pi approaching 0 as the expansion rate approaches infinite.
A few things about Pi before we go on. Pi is the ratio between a circles circumference and it's diameter. If we take the circumference of a circle and divide it by the diameter, we will get Pi. Pi's expansion is infinite in length, and the numbers do not repeat. Pi is a transendental number. Pi can be calculated both mathematically and physically. A math equation to calculate the value of Pi is this

Pi/4 = 1 - 1/3 + 1/5 - 1/7 + ... (7)

This equation is called Liebniz' series. There are many other equations to calculate Pi. Pi can be calculated by droping sticks on Parallel lines, first observed in the eighteenth century by Count Buffon("One Two Three...Infinity" by George Gamov, Bantom books 1965). In 1995, it was discovered that any number in the expansion of Pi can be extracted without knowledge of the perceeding numbers in the expansion (On The Rapid Computation Of Various Polylogrithmic Constants by Bailey, Bowein, and Plouffe, http://www.mathsoft.com/asolve/plouffe/plouffe.html).
Different values of Pi are not new to math. In both the Geometeries of Riemann(eliptic) and Lobachevsky(hyperbolic), different values of Pi can be derived. But there is a catch, in both of the Geometeries, the smaller the space that is measured, the closer to "Pi" the different values of Pi get. This is the equation for the circumference of a circle in Hyperbolic Geometery

C = 2*Pi*sinh*r. (8)

C is the circumference of the circle, r is the radius of the circle, and sinh is measure of the hyperbolic surface. As r approaches 0, the difference between C and r gets closer to 2*Pi.
This is the equation for the circumference of a circle in Double Eliptic Geometery, sin(a/r) is a measure of the eliptic surface,

C = 2*Pi*r*sin(a/r). (9)

Once again, as r approaches 0, the difference between C and r gets closer to 2*Pi. Equations (8) and (9), were taken from ("An Introduction To Non-Euclidian Geometery". David Gans. (1973). Academic Press.).
We will use Pi-bar to set the values of Pi in different universes. This will eliminate the confusion between Pi in this universe and Pi in another universe. When talking about a specific universe, this notation will show up, Pi-bar = 10, this would indicate that wherever the variable Pi-bar shows up, the value of Pi in that universe is 10. We can also set Pi-bar = Pi, and that would describe situations in this universe.
For reasons that will be explained later, we can not use 2*r in the circle equation for parallel universes, we will use d for the diameter, instead. This is the equation for the circumference of a circle in a parallel universe

C = d*Pi-bar. (10)

As d approaches 0, the difference between d and C does not change, so the changes in Pi, in a parallel universe, is not a change to Non-Euclidian Geometery. To the best of my knowledge, no other math we use to describe spacetime, suggests different values of Pi. I am 95% sure of this about math that was produced before the 20th century, and I am 51% sure of this, for the 20th century itself.
We will now sum up what has just been presented. The Big Bang "exploded" with many different rates of expansion, and each seperate expansion rate is it's own universe with it's own different value of Pi. Humankind has created a math that has different Pi's, but that math is not the same as what is presented here, and as far as this author knows, no other math uses different values of Pi to describe spacetime.


The Curvature of Spacetime

The equation for the curvature of a circle is

k = 1/r. (11)

Where k is the curvature of the circle, and r is the circles radius. Knowing that the retroverse is a half circle, we can apply Eq. (11) to this model, and we get this

k = 2/D. (12)

This is an equation for the curvature of the geodesic when we assume a non-convoluted shape to the geodesic. The geodesic is a part of spacetime, so we will say the curvature of spacetime. We can also set k equal to other variables from Eq. (3)

k = 2/D = Pi/A. (13)

This shows an important thing about the curvature of spactime, remembering that A is the age of the universe, the curvature of spacetime is getting smaller as the universe gets older, and an event determines the curvature of spacetime. Eq. (13) also gives a reason why events are not time reversible, the so called, arrow of time. In order for a series of events to reverse process, the universe would also have to reverse it's expansion to get the curvature of spacetime to get larger, so that the geodesics can return to their original paths.
If the event is an observer, then the observer is faced with a time illusion about the universe. The observer will observe an event in the past from her curvature, and will falsely assume that the event in the past is at her same curvature. The observer will then falsely give an age after the Big Bang when the event occured, when in fact, the event in the past has it's own different curvature, which will determine a different age after the Big Bang. The Hubble Space Telescope seems to be reaching the distances where this effect is most noticable. It is the, "matter in the universe seems to be older then the universe it resides in" problem ("Hubble Space Telescope measures precise distance to the most remote Galaxy yet". Press release No. STScI-PR94-49. (10/26/94). http://oposite.stsci.edu/pubinfo/press-release/94-49.txt).
Here are three equations to determine the actual age of the event, versus the preceived age of the event. A is the age of the universe for the observer. B is the preceived age of the universe, relative to the Big Bang, of the observed event. F is the actual age of the event, relative to the Big Bang. D = 2*A/Pi. This is the equation when B is older then A/2, but younger then A,

F = Pi((90(2B-1)sin*D+D)^2+(90(2B-1)cos*D)^2)^-1/2
_ _ _ _ _ _
2 A 2 2 A 2. (14)

This is the equation, when B is exactly A/2

F = Pi/2*((D/2)^2+(D/2)^2)^-1/2. (15)

This is the equation, when B is younger then A/2

F = Pi/2((D-90(2B)cos(D))^2+(90(2B)sin(D))^2)^-1/2
_ _ _ _ _
2 A 2 A 2. (16)

Equations (14, 15, & 16) are corrected in a post on this blog called Corrections to Equations 14, 15, and 16 for My 1997 paper "Foundations Of Parallel Universe Math".

This time illusion is a feature of curved spacetime. The difference between observed time of the event, and actual time of the event is a measure of the curvature of spacetime. The time illusion discrepancey is governed by the value of Z from Eq. (2). When Z = 1, there will be no time illusion. The three equations (14), (15), and (16) are for Z = Pi/2 only.
The author suspects that there are shorter equations for (14), (15), and (16), but he is unable to derive them.


Maxwell's advanced light and the Wheeler-Feynman absorber theory

When James Clerk Maxwell produced his wave equation for light, it had two solutions; the "retarded solution" for light that travels forward in time, and the "advanced solution" for light that travels backward in time ("Faster Than Light: Superluminal Loopholes In Physics". Nick Herbert. (1988). Plume.). The "advanced solution" will be the one we are talking about when we say advanced light.
For an observer(receiver), all light that arrives to her, arrives with the curvature k = Pi/A (Eq. (13)). What about light that is emitted by the observer, what is the curvature of the emitted light? This model is past based, so the only way to determine the curvature of an emitted light, is to look at the receiver of the light, and the receiver always receives her light with the curvature k = Pi/A where the value of A is based on the when the receiving event occured, relative to the Big Bang. Relative to the emitter, the emitted light has the curvature of

k = Pi/(A + t), (17)

where t is the time between the emitter and the receiver. Some how the emitter has to "know" what curvature to emit the light for it to reach the receiver. But that is only for the photons that reach that event, there are other photons emitted, by the same emitter, that will have a curvature of k = Pi/(A + ?), where ? is the time between the emitter and any other future receiver. We are presented with two possible solutions here; a "non-local" model, or Maxwell's advanced light/retarded light. The author does not believe in "non-local" models, suggesting some magical transferance of information, so that leaves Maxwell's solution as the only solution that fits, where "advanced light" being emitted by the receiver travels backward in time and "tell" the emitter what curvature to emit at. The curvature for the advanced light is

k = -Pi/A (18)

relative to the receiver.
This transaction is very similiar to an advanced light and retarded light transaction that was originally proposed in the paper ("Interaction with the Absorber as the Mechanism of Radiation." Wheeler, J.A. and Feynman, R.P. (1945) Reviews Of Modern Physics 17, 157). The only thing this paper adds to the transaction, is the curvature of the geodesic.
Mr. Herbert, in his book "Faster Then Light: Superluminal Loopholes In Physics", also mentions two other Absorber theories using advanced and retarded light ("Advanced Effects In Particle Physics." Csonka, Paul L. (1969) Physical Review 180, 1266), and ("The Transactional Interpretation Of Quantum Mechanics." Cramer, John G. (1986) Reviews Of Modern Physics 58, 647). Mr. Cramer's paper can also be found at http://mist.npl.washington.edu/npl/int_rep/tiqm/ti_toc.html.
The advanced light solution of how a photon "knows" what curvature to follow also might lead to a solution to a question raised from Eq. (2). When this model was first conceived, we drew a half circle, because it was "the easiest curve I could think of",and after we understood what was happening with the easiest, we could advance on to other values of Z. This is a conjecture, but advanced and retarded light geodesics create a closed "circuit", it is believed that there are only two values of Z(Z = 1 or Pi/2) that allow this closed "circuit" to be completed geometerically, for all events since the Big Bang. It is also believed that this can be proved mathmatically, but the author does not know how to go about it. If this can be proved, then the Z = Pi/2 is the actual shape of the universe we reside in; because, due to the Time Illusion mentioned earlier, we know that we do not live in Z = 1.


Parallel Universe Math

We stated earlier that each seperate universe has it's own seperate value of Pi. In this universe, the number system and Pi are intimately connected, and that is most vividly shown in Eq. (7).

Pi/4 = 1 - 1/3 + 1/5 - 1/7 + ... (7)

This universe does not have a special mathematical distinction over any of the Parallel universes, so the number system in the Parallel universes must change in some way so that an Eq. (7) in the Parallel universe will produce the given value of Pi in that universe. This equation extends the Liebniz' series to Parallel universes

U*Pi/4 = U/1 - U/3 + U/5 - ... (19)

The U*Pi is the value of Pi in the Parallel universe, and the U is the unit value in the parallel universe relative to this universe. This is the reason why we could not use 2*r in Eq. (10), the equation for a circle in a parallel universe; the "value", "quantity" of all numbers is specific to this universe only.
The dividing line that we seem to be looking at when we consider as to weather the math in the Parallel Universe is different from this universe, is if the math is somehow derived or connected to Pi. The Axioms of Euclid do not give a specific value of Pi, and are not dependent on what the value of Pi is. That suggests that the Axioms of Euclid are valid in the Parallel Universes along with the field of Non-Euclidian Geometery with very little alteration required. The author is not sure about other Fields in Math.

I close this Paper with a Poem I wrote.

You're Nuts
by
Jim Akerlund

You're not playing with a full deck.
Your elevator doesn't go all the way to the top.
You're a few pancakes short of a stack.
Your antenna isn't receiving all stations.
You're out of your tree.
Your cart isn't rolling on all wheels.
You're one brain short of a brainstorm.
Your train isn't pulling as much weight anymore.
You don't have a clue.
You're a few pieces short of a puzzle.
You're a few degrees short of a summer day.
If sanity were a holiday, yours would be April first.
You're a few notes short of a song.
You're a few gallions short of an ocean.
It seems to me your toilet doesn't flush anymore.
You're a few birds short of a flock.
You're a few cars short of a traffic jam.
You've come to a gun fight with a knife.
Your legs don't reach all the way to the ground.
But all of this is a matter of opinion,
by a guy who thinks he know where Einstein went wrong.

In the book "Relativity, The Special And The General Theory" by Albert Einstein, 1961 Crown Trade Paperbacks, Mr. Einstein describes a rotating disc and how it is effected by relativity. He says that a measuring rod used to measure the the circumference of the rotating disc will be shortened by the rotation of the disc, and this will effect the total value of Pi on the disc, arriving at a value of Pi larger then 3.1415... If this model is correct, then that is wrong. This model is based one the idea that there is exactly one value of Pi in this universe no matter how relativity may effect a rotating disc. The invariance of Pi in the frame of reference.

This was monkey # e * 10^googool typing paper # Pi * 10^googool.

This paper was posted to Alt.Sci.Physics.New-Theories several times in July/August of 1997.