Imaginary Space

The Cosmology of Imaginary Space

Each system of belief has had a cosmology describing the relationship of what is considered "here" with wherever their God and Heaven is thought to be. Most have shown reality as a series of layers, with the "here" on the bottom or center and "God on the top. The difference between the various layers is usually described as a difference in vibration or granularity, and travel from one layer to the other is described in terms of "going there," or "being over there" or "the other side." Lacking a better set of terms, such these words are useful for conversation so long as it is understood that they are just metaphors for what the Spiritualists refer to as "different atmospheres and awareneses."

In this essay, I propose that a better way of depicting the etheric cosmology is to model it after the virtual world of the Mandelbrot Set. In that virtual reality, "movement" is accomplished by changing the initial state for iterating the basic equation. One is "attracted" to a "loci of creation" by an orderly change of assumptions, which might relate well to a change in worldview brought through an orderly progression of experiences.

How the "apple man" fractal is derived is explained here and the use of the Mandelbrot set as a parable for the structure of reality is explained.

Please keep in mind that I am not saying that the Mandelbrot Set is a description of reality. I am saying that the principles of fractal geometry and chaos theory expressed in the Set may better describe the nature of reality than any spatially defined cosmology.

Formation of the Apple Man Fractal

Figure 1
The "Apple Man" fractal in the Mandelbrot number set.
The two arrows point to the location of "0x + 0yi."

The New Cabala--The Mandelbrot Set

An important feature of chaos theory is what mathematicians refer to as an attractor. This is a mathematical area or point on a plot that is the focus for a large percentage of the plotted points representing numbers that satisfy some mathematical test. You can think of the attractor as a loci around which the results tend to gather.

A plot is an graphic made by repeatedly testing points in the complex number plane to see how quickly the result will go to infinity (or become very large). The plots in the examples shown here are formed by assigning a color value to the point being tested, depending on how stable it is in the calculation. A degree of order tends to emerge as the range of possible numbers, X and Y are tested and plotted. The most popular example of this is the plot derived from what is commonly called, the Mandelbrot Set, named after the mathematician, Benoit Mandelbrot. The Mandelbrot Set itself, is only one of a group of numbers known as the Julia Set, which was documented by Gaston Julia. Figure 1 is a plot of the Mandelbrot Set made with the Black Saturn Mandelbrot Set Explorer at http://www.users.nac.net/thegangof4/blackSaturn/mandelbrot/msvVeryHigh.htm .

The Mandelbrot Set is represented by the nonlinear mathematical formula: Z_n+1 = Z_n² + C where both Z and C are complex numbers (x + yi). The complex plane is represented in Figure 2 as the horizontal plane bounded by X and Y. The formula is calculated by selecting a value from the complex plane for C so that C = x+yi, to be used as a constant, and letting Z be a variable. After each iteration, the resulting complex number is substituted for Z and the calculation is repeated with C held constant. The usual way that the results are illustrated are shown in Figure 1, in which the changes in color are indicative of the number of times the calculation can be iterated before the resulting answer exceeds a predetermined limit--usually approaching infinity. Thus: Z begins at zero and C is the complex number (x+yi) being tested. So to begin an iterative process for this formula , substitute 0 for Z to get 0² + C = C. Take the result, C, and add it to the original value of Z to get (O + C)² + C = C₂ + C, then repeat by substituting C₂ + C for Z and on until your threshold is reached or until enough iterations have been made to determine that the selected number for C is not likely to go to infinity. Then, the next number for C is selected and the process is repeated to produce the next point in the plot. This sequence is repeated until all of the numbers, x + yi, in the target area of the complex plane have been tested.

In Figure 1, -2.0 X is at the left, +1.0 X is at the right, -1.5Y at the top and +1.5 Y at the bottom. The two arrows in the middle point to 0 + 0i.

For our purposes, the point of the Apple Man fractal--the black area--is that calculations using numbers from those regions can be iterated an infinite number of times and the result will never go to a "large" number. However, beyond the rather definite boundary of the fractal, the numbers will take the calculation to infinity with decreasingly fewer and fewer iterations. Looking at it from the perspective of a very large complex number moving toward the area of stability, you can see why mathematicians have come up with the term, "attractor," meaning that the numbers or points outside of the attractor seem to be attracted to it as an area of stability.

Fractals

Now I must introduce yet another term--fractals. In chaos theory, a fractal is a mathematically determined shape that can be infinitely divided with each part having the same shape as the whole. A simple example of this is illustrated in Figure 2. To derive the triangle, three points numbered on through three are selected to form an equilateral triangle if they were connected. Then, a number from one to three is randomly selected and a dot is place at the appropriate point of the triangle. After that, repeatedly select a random number from one to three and calculate half the distance from the last dot toward the point of the triangle associated with the selected number. For example, the first number was a three and a dot was placed at the third point of the triangle. The next number was a two and a dot was placed half the distance between the third point of the triangle and the second point. The next number was a one and a dot was placed half the distance from the last dot toward the first point of the triangle. Using a computer, this process was repeated thousands of times resulting in the plot shown in Figure 2. What you see are triangles within triangles--by definition, fractals. A low resolution process was used for the example, but had a higher resolution been used, and/or had the display been of a higher resolution, the plotted would have shown more levels of smaller triangles since each triangle is actually shaped from three smaller triangles which are, in turn, shaped from three triangles and on, and on. The three triangle fractal is called a Sierpinski Triangle after the mathematician Waclaws Sierpinski who first defined it in 1916.

Figure 2
The Sierpinski Triangle

The objective here is to introduce the concept of boundaries, fractals, and the use of simple instructions to convey complex systems of information. All of these features are present in nonlinear equations, but mathematicians are finding that simple linear processes can also produce fractals and very complicated systems of information, as is illustrated in the Sierpinski Triangles.

The Mandelbrot Set also contains fractals, but on a much more complicated scale. Figure 3 illustrates how the virtual world of the Mandelbrot Set can be viewed, as if with a microscope. It is possible to select a specific area of the complex plane and expand the image to see more detail. What you see is part of the “V” shaped curving section at the top of the large cardioid shape in Figure 1. The large black area is the stable region of numbers that can be used in the formula, Z_n+1 = Z_n² + C and that will not take the result out of range even after a very large number of calculations (200 for each point in this case). The bands of color represents groups of complex numbers that take the calculation to infinity, faster and faster, as you move away from the black area. In a bigger computer, this would be shown as a gradual shading, from left to right, and not as abrupt divisions as is shown here. The fringe between the large black area and the color pattern is a range of complex numbers that approach infinity very slowly and represent the boundary between stability and chaos (infinity). It is in this thin boundary that the wonders of the Mandelbrot Set unfolds.

Figure 3
Magnification of the area located above and between the main cardioid and the larger circle in Figure 1

Traversing Imaginary Space

The boundary area can be magnified even more as is shown in Figure 4. The area shown is from the bottom-left of the larger circle of black. You can see that the same Apple Man shape is beginning to appear all along the boundary. The apple man shape is usually the same but not always identical with that seen in Figure 1. You can see an example of a highly distorted apple man shape in Figure 5, found at the far right of Figure 1 and considerably magnified. Some of the black areas are of other but often repeated shapes, but I think they all qualify as fractals.

Figure 4
"Magnification" of the bottom of the large circle in Figure 1

Figure 5
Far right of the cardioid in Figure 1 at even greater magnification. As a reference, for the point where the spine intersects the left side of the picture, X = 0.4242606570763012 and Y = 0.3412283517941925
That is a very extreme amount of magnification.

If these plots are considered contour maps with the vertical axis representing number of iterations of the equation for each point, then the large black area of stability might represent a very high plateau, and pattern of colors might represent some distance down a very long slope. As the boundary is approached, the lines would begin to get closer and closer together. The numbers represented here would eventually take the calculation to infinity, but only after more and more iterations. As is illustrated in Figure 6, the contour lines continue to get closer to the black in a fashion known as "asymptotic," meaning that the distance between them approaches zero, but that it will not become zero before they reach infinity--they never actually reach zero. Here and there in this well of infinity are other islands of stable numbers that might look like sharp, flat-topped spires reaching up from the sides of the descending curves. They are flat toped only because of our method of representing them. In reality, they will apparently continue to infinity.

Figure 6
The Mandelbrot Set unstable (goes to infinity) and stable (never goes to infinity) boundary

The top of the black area on the left represents the Apple Man. The curve represents how many iterations can be applied to the Mandelbrot Set before the result will go to infinity. This figure is an approximate illustration of what is occurring near the boundary between the stable set of Mandelbrot numbers and the unstable numbers outside of the set.

The Apple Man Fractal as a Parable for the etheric cosmology.

his is just one example of the work being done in chaos theory. The true meaning of these forms in complex space is still being debated. For one thing, we can see that a seemingly endless amount of information can be derived from a simple formula, Z_n+1 = Z_n² + C. Fortunately, my objective here is not to derive specific meaning from the Mandelbrot Set. My objective is to illustrate the structure of reality in a way that better agrees with our observations than do cosmologies based on such concepts as vibration and distance.

The characteristics of fractal geometry we are interested in include:

The complex numbers of imaginary space are not allowed in basic mathematics. "i" is equal to the square root of minus one, which is a disallowed state since a number times itself is always a positive number. We see a similar constraint in physical science, in that an etheric aspect of reality is not allowed.
A relatively simple rule can produce very complex results. In the imaginary space cosmology, the simple formula, in the case of the illustration above, Z_n+1 = Z_n² + C.
Imaginary space is not bounded, in that it extends to infinity in all directions. This includes apparently an infinite range in granularity or scale. The above illustrations are constrained by the ability of the computer and display system and is far more complex than can be shown.
Figure 1 is the projection of a two dimensional plane into a third dimension based on the number of iterations that are allowed. In the colored regions, the test is how fast the results of the iterated equation exceeds some very large number. In the black regions, the test is whether or no the results are still less than some predetermined, small number after a set number of iterations--in the case of Figure 1, 200 iterations. All of the computer routines I have seen for this set are about the same, but as shown in the Wikipedia, the feature found in the projected structure will be different if different limiting numbers are used. For instance, the black area in Figure 5 would probably better agree with the Apple Man shape if more iterations were allowed before the point was entered into the plot. As such, Figure 1 is not a simple projection of a two dimensional number set, but is actually a three dimensional structure determined by the parameters of X, Yi and the number of iterations allowed in the calculation.
If the real number plane represents the surface on which you stand, then the structures within imaginary space would be all around you, but would be invisible and the only way that you would become aware of the structures would be if you could somehow align yourself with imaginary space.

This is not the same as the the "multiverse" space imagined in quantum-holographics, nor is it the same as parallel universes sometimes described in fiction. In this model the imaginary space represents the whole of reality, of which the X and Y number set is an aspect representing the physical universe.

A important point to remember is that the familiar references of the physical too easily lead us to formulate models for the greater reality that can keep us from seeing the truer nature of reality. Concepts like, "above or below," "higher or lower vibration" and " "the other side," are useful as placeholders until we can agree on more accurate terms--we generally know what a person means when the terms are used--but the terms misdirect our thinking and cause us to construct images in our mind that only keep us from further growth. Of course, the same can be said by my effort here, but the difference in imagery I am proposing should awaken new curiosity in other researchers who might bring new ideas to the field of study.

With that said, it is clear that at least limited understanding of the greater reality can be extrapolated by what is experienced in the physical. Reality is a continuum, but as suggested by item 4, above, the basic building blocks of reality are the same everywhere. However, it may be that the rules by which those blocks may be assembled to form objects of reality are local to where the formation might occur. If so, then it should be reasonable to include what is known of the physical with what we are told of the greater realty and what we are able to experimentally determine, and formulate reasonable hypotheses that can lead us in further study.

I will simplify my explanation of this model as a list of elements.

There are two views to be taken of this hypothesis. The energy view is about the fundamental makeup of reality and the processes of formation. I will call this the "Formative view." The second view is concerned with the way in which we experience reality. I will cal this the "Developmental view."

The Formative View:

Formation is the result of attention on imagination with the intention of forming an object of reality. An "object of reality" is anything, including a person (Self), an idea and a thought.
The simple formula, Z_n+1 = Z_n² + C, might be considered the basic element of energy in imaginary reality, but the point determined by x + yi holds the rules determining how the formula will manifest a point in imaginary space--whether or not the results of the calculation will go to infinity and if so, how fast.

One of the tenets of the concept of Natural Law is that there is a fundamental form of energy that is universally present in reality. In the Spiritualist view, Infinite Intelligence is the source of this energy, and as such, is the source of the rules by which the energy must behave. Those rules are described as the Principles of Natural Law. I have always assumed that the rules governing this fundamental energy must necessarily be embedded in the energy; however, the model I am proposing here offers an alternative view. The rules by which the results manifest from the formula depend on the initial conditions of each iterative round of calculations. The structure of imaginary space resulting from the equation, Z_n+1 = Z_n² + C, is the result of the differences in embedded rules found in each combination of X and Y.

This initial condition dependency is evident in what we know of reality. For instance, all points in the physical are apparently governed by the same principles, but we know that intention more quickly acts on imagination in other aspects of reality. As explained below, the physical universe would be bounded by a single fractal.

In this cosmology, the fractals represent regions of stability toward which the regions of instability are attracted. The number of iterations of the calculation represents a more energetic fundamental unit of energy because it determines how stable the point is and because it is determined by an initial condition that supports more complex processing. The fractal itself, is very energetic, and in this model, the relative size of a fractal and its relationship to the primary fractal determines how energetic it is. All initial points within a fractal are equally energetic.
The fractals, then, are loci of creation and regions of imaginary space in which the rules of formation permit more complex and longer lasting objects of reality than do the s of reality separating. They represent regions such as the physical aspect of reality that contain many opportunities for experience operating under the same rules of formation. Smaller loci may be as simple as a thought form or even an idea.

Developmental View

Expanding on the idea that an individual point on the real number plane can be compared to a unit of energy, the point can also be compared to a set of initial assumptions, which would be a person's worldview as he or she embarks on an experience. In this case, the number of iterations that can be applied to the basic formula would be compared to understanding gained from the experience.

In other words, any point x + yi could be compared to my lifetime and the initial understanding and inclinations I came into this lifetime with. It can also be compared to the worldview I have as I enter into a lifetime experience, such as marriage or a work experience. The number of iteration the calculation can be taken to before it reaches some limit would represent the understanding I gain in these experiences.
Movement in imaginary space is not movement from "here" to "there," in the same sense of changing locations. Movement is accomplished by selecting experiences (initial assumptions or x + yi) that provide the greatest opportunity for increased understanding. These experiences increase in complexity and value to our learning experience as we improve our ability to select helpful initial conditions for experiences because of our increasing understanding. In that way, we are "attracted" to the fractals, which represent some region of stability, such as the physical or the astral universe but not necessarily the final objective.
The set of initial conditions represented by a fractal has only energy and a common set of rules governing how that energy can be used in formation. In order to manifest as an object of reality, the rules of formation must be applied to an image held in the mind of a creator entity. Here, it should be noted that imaginary space is an environment in which Self must exist. Imaginary space constrains how Self may move about and behave, but it is also managed by Self. As an individual Self gains sufficient maturity through gaining understanding about the operation of reality, it is more able to move about.
The Mandelbrot Set does not suggest a mechanism by which a Self might initially come into existence. It is just an illustration suggesting the basic nature of the cosmology, but it does suggest behavior once the journey has begun. If you assume that a Self is an aspect of a creator entity sent into reality to gain understanding about the operation of Natural Law by experiencing its operation in certain venues, then you must also accept that Self is preoccupied by gaining that understanding and returning with it to its creator entity. So, imagine that Self is cast out to the farthest reaches of reality, which in this model, are the regions in which an initial assumption will take the calculation to infinity very quickly.

Self has the urge to gain in understanding and can only do so by testing different initial assumptions. Once tested, the resulting experience must manifest as understanding, and so, the initial assumptions may be tested many times, or all of the adjacent assumptions may be required before understanding does occur. Once understanding is reached, I think of this as gaining spiritual maturity, Self is able to move on to the next set of initial assumptions, which are venues for experiences. Self cannot go further than understanding allows. In effect, Self is energetically in agreement with the set of initial assumptions it is testing and cannot move on until it changes the nature of its energy--its understanding or worldview. As such, Self cannot return to its creator entity until it has achieved sufficient spiritual maturity to be energetically in agreement with the aspect of reality its creator entity inhabits.
A creator entity may exist in one of the more substantial fractals, and itself be an aspect of another creator entity occupying an even more substantial fractal. As such, Self may be one of many Selfs working their way back to the same creator entity, and the creator entity may be one of many working their way back to yet another creator entity. In this cosmology, their might be many rounds of aspectation Self must deal with before it is finally reunited with its prime creator.

Closing Comments

A cosmology is just a hypothesis, and as such, is a working model that requires continued testing and reformulating based on current evidence. In effect, I am proposing the "Imaginary Space" cosmology as a refinement of the layer cake models one sees in some of the systems of belief. Terms such as "plane" and "vibration" remain useful metaphors, but they are too often taken literally--as I am sure some people take Imaginary Space literally despite my cautions to the contrary. I am not sufficiently insightful to know what descriptive terms might come from this cosmology. I like "loci of reality" rather than "plane," "aspect" and "Self" rather than "Soul" and "energetic agreement" rather than "same vibration."

It will be interesting if you contemplate this cosmology for a time and see how it fits your understanding of things etheric. Perhaps after you have tried it on for a while, you can offer an alternative or modifications to this one. One of the things I should warn you, though, is that I will apply a litmus test to any theory that is intended to describe the relationship of Self to reality. There is way more evidence that Self is an etheric being that is in a symbiotic relationship with the physical body. There is no real proof that Self does has a biological origin, and any theory that proposes otherwise has probably ignored the evidence.

Tom Butler


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