Abstract

A mathematical treatment of the idea that within you is a reflection of everything in the holographic universe...

First, the unit circle is taken to represent an individual object or being, having within, without, and a boundary.

Second, a relationship is defined corresponding each point without to a point within the unit circle. Essentially,  this relationship is the reciprical along the line to the origin.

In this model, information approaches infinite density near the origin center of the unit circle, reciprical of points more and more distant, when it appears to be equally distributed across the without. A mapping is created by which points within the unit circle take up an equal region as those from the without of the unit circle. This distorts the information density of the points scattered without, but makes the information density within reflect the density without.

Several features visible in the mapping are demonstrated, with open-ended philosophical interpretation.

Tic-Tac-Toe Transformation

The Inspiration of Religion

Rumi, the Muslim mystic writes:

Let yours eyes see God everywhere. Give up fears
and expectations.  The Friend, the Beloved, your Soul,
is a River with the trees and buds of the world
reflected in it.  And it’s no illusion!

The reflections are real, real images, through which
God is made real to you.  This River Water
is an orchard that fills your basket.  Be splashed!
            -We Are Three pages 54-55.

The Christian declares, on Scriptural authority,

God that made the world and all things therein... giveth to all life, and breath, and all things;
And hath made of one blood all nations of men for to dwell on all the face of the earth...
that they should seek [God], if haply they might feel after him, and find him, though he
be not far from every one of us:
For in him we live, and move, and have our being; as certain of your own poets have said,
For we are also his offspring.
            -Acts 17, 24-28.

Finally, the Hindu teaches:

See, the whole evolution of man  is from being somebody -
being somebody is ego - "I am great." "I am very evolved." -
and you are not evolved. That is an ego. See. Recognize it.

That from being somebody to being nobody and being nobody to being everybody,
these are two steps in evolution.  to An enlightened person everyone is form of God,
everyone is form of divinity.  An enlightened person, when he speaks, he doesn't speak from the position
"oh, you're all ignorant I'm very enlightened i'm going to tell you something." No. He knows the Divine
has provided this beautiful knowledge and it is taking back, coming in another form.
            -Sri Sri Ravi Shankar

There are many implications to these words holy in several cultures.  The idea I am focusing on in this mathematical philosophical dialectic is this:  Within each individual is a reflection of the entire universe.  It may be, as the religionists say, that God fills the universe, and God, as Life, fills us.

From a physical standpoint, it is noted that each individual is influenced, in a causal way, by ALL the universe. This is true in the moment, and even more true if ALL the universe eventuated in one quantum event at the Big Bang, by which we are entangled throughout history, even as a pair of photons, quantumly entangled, emitted in opposite directions, at the speed of photons, light, yet correlated instantaneously across space in experiment. I expect that this entanglement is the method by which patterns and life arise, springing from small to large.

ALL reflected within One: Posing the Question in Mathematics

Can the idea that All the universe is reflected within each arbitrarily chosen object or being, holographically, be modeled in mathematical terms?  Here's how I explored the issue:                              

First, I took the Unit Circle as a model for an arbitrary individual. There is within the circle (bounded) and without the circle (boundless) and a well-defined boundary to the circle. The unit circle is taken to be relative, that is, any individual, object or being, may be represented by the unit circle.  Outside the circle, points represent information, objects and beings, in the universe. Inside the circle, the reciprical of those points represent the internalized reflection of the all in one.

There are two considerations to note already: First, in this model the individual has a well defined boundary; this is likely to be a false assumption. It simplifies the model to use a clearly bounded model of an individual. Second, there is no mapping defined between objects and beings and points on the plane. We assume that there is a distance function definable relative to the individual represented by the circle, by which all outside objects can be put in correspondance with points outside the unit circle.

Philosophical Ideas  <-> Mathematical Definitions
so far...

ASSUME: The within reflects the without... relative to ONE.

1. The Inside Reflection  <-> Points WITHIN the UNIT  CIRCLE:  or X and Y such that X^2 + Y^2 < 1; or Distance from Origin < 1.
2. The Outside Reality <-> Points WITHOUT the UNIT CIRCLE: or X and Y such that X^2 + Y^2 >1;  or Distance from Origin > 1.
3. A Distinct Boundary <-> The UNIT CIRCLE; or X and Y such that X^2 + Y^2 = 1; or Distance from Origin EQUALS 1.

4. A relationship: The Within reflects the Without, modeled by a relationship connecting points within the unit circle with points without in a one-to-one fashion.
For a Point Without, X and Y such that X^2 + Y^2 >1, the Point Within, A and B such that A = X/(X^2 + Y^2) and B = Y/(X^2 + Y^2) corresponds.

Defined in Mathematica, the above statements are:

1. WithinUnitCircle[{X_,Y_}] := If[X^2 + Y^2 < 1, True, False];
2. WithoutUnitCircle[{X_,Y_}] := If[X^2 + Y^2 > 1, True, False];
3. CircleBoundary[{X_,Y_}] := If[X^2 + Y^2 == 1, True, False];

4. CircleRecipricalPoint[{X_,Y_}] := {X/(X^2 + Y^2), Y/(X^2 + Y^2)};

Mathematica Code...

Features visible in the model.

Figure 1: shows the features described so far.

Figure 2: shows a set of points scattered randomly, except within the area of the unit circle.  
Within the unit circle, in a matching color to the point outside, is the reciprical point on 
the line between the outside point and the origin.

Notice in figure 2 that while points are scattered randomly across the without, they are dense near the center of the unit circle, approaching infinite density at the origin (0,0) center. When the point without is farther away, its reciprical representation within the unit circle is nearer the origin center.

Finding a mapping of this unit circle model in which the information within is equally dense as the information without was the motivation to develop the tic-tac-toe transformation.

I noticed that the line y = 1 * x had a slope of one and divided evenly Quadrants I and III and the line orthoganal, with a negative reciprical slope, y = (-1) * x, bisected Quadrants II and IV. The quadrants are bounded with an asyomptotically approached vertical slope of the Y-axis, with plus and minus infinity. The horizontal boundary, the X-axis is the constant function y = 0, with a slope of zero. If somehow the inside of the unit circle could be mapped onto lines with slopes of zero to one, and the negatives, negative one to zero, the boundary to the lines y = 1 * x and y = (-1) * x, evenly dividing the Cartesian plane into equal parts, so that the without of the unit circle, mapped to lines with slopes greater than one and less than negative one takes the area of the inside of the unit circle.

I wondered if there was a way to map the slope of one to a circle with radius of one. Looking at the problem in polar coordinates, I saw that essentially, what was required was taking Point (R, Theta) and reversing the R and Theta, appropriately scaled, to make Point' (R' = Scale[Theta], Theta' = Scale[R]).

Mixing the Cartesian and Polar coordinate systems to take any LinearPoint (X,Y) on the linear map and making CircularPoint' (R, Theta)  in the circular map showed me the scale function. The radius in CircularPoint' should just be the slope of the line between LinearPoint (X,Y) and the origin, (0,0): R = Y/X. To check: any point on the line y = 1 * x, has an R of y/x = 1. But: the line y = (-1) * x should also be mapped from the linear to the circular on the unit circle or R = 1. To include this case, for any LinearPoint (X,Y) the R in the CircularPoint' (R, Theta) is the absolute value of the slope Y/X or AbsoluteValue[Y/X].

Finding an appropriate scale for Theta in the CircularPoint' (R, Theta) is a little more difficult. The idea of reversing the polar coordinates had been in mind. To convert AnyPoint (X,Y) in Cartesian coordinates to Polar coordinates AnyPoint (R, Theta) the R is found by the distance of AnyPoint (X,Y) from the origin center (0,0). This is, by the Distance Formula: Square Root of X^2 + Y^2. Next I want to change this radius, or distance from the origin, to a corresponding Theta to make CircularPoint' (R = AbsoluteValue[Y/X], Theta). In the first part of the transformation, the slope became a radius, now how to make an angle or slope from a radius?  Well, the Tangent function returns Y/X ( = Sin/Cos) when given an angle Theta.  The ArcTangent or inverse of the Tangent takes Y/X and gives the angle back, but is restricted to map negative infinity towards - Pi/2 and positive infinity towards Pi/2. The Distance Formula only returns values from zero towards positive infinity, so the ArcTangent of Distance ranges from zero towards Pi/2.  The function must be divided into Quadrants.

After some trial and error, a balanced formula was found.

For the first Quadrant, distances from the origin on the linear mapping are mapped to angles, Theta in CircularPoint' (R = AbsoluteValue[Y/X], Theta), whose angle Theta equals the value of the distance on the linear map (X) being used as a slope on the circular (O) map, which means Theta is the ArcTangent of the distance, so we know:

Quadrant I: Points (X, Y) such that X and Y are positive;
    maps to CircularPoint'
    R = Y/X and
    Theta = ArcTangent[Distance] = ArcTangent[SquareRoot[X^2 + Y^2] ].
    
following a pattern,

Quadrant IV: Points (X, Y) such that X is positive and Y is negative
    maps to CircularPoint'
    R = negative Y/X = AbsoluteValue[Y/X] and
    Theta = negative ArcTangent[Distance].

Quadrant II: Points (X, Y) such that X is negative and Y is positive
    maps to CircularPoint'
    R = negative Y/X and
    Theta = Pi minus ArcTangent[Distance].
    
Quadrant III: Points (X, Y) such that X is negative and Y is negative
    maps to CircularPoint'
    R = Y/X which is positive because both X and Y are negative and
    Theta = Pi plus ArcTangent[Distance].
    
the Mathematica code defining this transformation follows:

take an XValue and YValue in the linear mapping of the Cartesian plane.
CalculateTransformation[XValue_,YValue_]
:=(

the new point in polar coordinates has a r, radius, equal to the magnitude of the slope of the original point to the origin, for any point {X,Y} with X not equal to zero.
    r = M = Abs[YValue/XValue];

the radius of the linear Cartesian plane point itself is its Distance from the origin.
    Distance = (XValue^2 + YValue^2)^(1/2);

If function is undefined, return Indeterminate.    
    {XPrime, YPrime, Theta} = {Indeterminate, Indeterminate, Indeterminate};

Theta in the case of Quad I:
    If[XValue>0,
      If[YValue >0,
        Theta =  ArcTan[Distance]]
      ];
    
    Theta in the case of Quad IV:
    If[XValue>0,
      If[YValue < 0,
        Theta = -ArcTan[Distance] ]
       ];
       
    Theta in the case of Quad II:   
    If[XValue <0,
      If[YValue>0,
        Theta = Pi  - ArcTan[Distance] ]
      ];
      
    Theta in the case of Quad III:  
    If[XValue<0,
      If[YValue <0,
        Theta = Pi + ArcTan[Distance]]
      ];
      
    convert r and Theta back into Cartesian coordinates.  
    XPrime = r * Cos[Theta];
    YPrime = r * Sin[Theta];
    
    return the transformed point.
    {XPrime, YPrime})

Figure 3: The Linear model on the left and the Polar model on the right. Points are colored to match on each side. On the linear side are the equations y = x and y = (-1) * x. The Polar model shapes the Unit circle.
Bleeding I: the further from the origin in the linear mapping, the closer to the Y-axis is the cooresponding point in the circular model. So, the circle is never closed on the Y axis, as the quadrants are unbounded on two sides.
Bleeding II: the closer to the horizon, the x-axis in the linear map, the smaller the radius of the circle in the circular model. Any point on the x-axis itself has a slope of zero to the origin, and slope becomes radius in the circular model, yielding a circle with radius zero, a single point. The point center origin in the circular model expresses the entire line horizon x-axis of the linear model, allowing for a point as a circle with zero radius. If this circle cannot exist, than the entire x-axis, the real number line in the complex plane, maps to nowhere.

Figure 4: In grey, in the O mapping, the unit circle and its X mapping on the left. In color on the X side lines with slopes of +/- 2 and +/- 1/2.
Relation of Two: As the slope of one bisects the horizon and the infinite vertical slope, the slope of two bisects the without, slope of one to infinite vertical slope. The reciprical of two, in the model the reflection within of this second, 1/2 similarily bisects the within of horizon and the slope one boundary. If 2 divides half a quadrant and 1/2 divides half a quadrant, this second's presence within and without contains the area of half a quadrant, just the same as each of the within and without of the one, shown in different colors, contain the area of half a quadrant.

This makes five equal marks of
    

the unapproachable infinity;

 two;

 one;

one-half;

zero line -> point.

The following functions are defined and usable after the initialization cells of this notebook are evaluated.

1. WithinUnitCircle[{X_,Y_}] - takes a point {X,Y} on the Cartesian plane and returns True if it lies within the Unit Circle and returns False if the point lies on the boundary or without the Unit Circle.

2. WithoutUnitCircle[{X_,Y_}] - takes a point {X,Y} on the Cartesian plane and returns True if it lies without the Unit Circle and returns False if the point lies on the boundary or without the Unit Circle.

3. CircleBoundary[{X_,Y_}] - takes a point {X,Y} on the Cartesian plane and returns True if it lies on the boundary of the Unit Circle and returns False otherwise.

4. CircleRecipricalPoint[{X_,Y_}] - takes a point {X,Y} without the Unit Circle and returns the point {XPrime, YPrime} which is the reciprical point within the Unit Circle, or takes a point within the Unit Circle and returns the reciprical point without. Points on the boundary of the Unit Circle are mapped to themselves.

5. CalculateTransformation[XValue_,YValue_] or    
    CalculateTransformation[{XValue_, YValue_}] - takes a point {X, Y} in Cartesian coordinates on either the linear or circular mapping and returns the point {XPrime, YPrime} in the opposite mapping.

Converted by Mathematica      October 9, 2005