Everything is abstractly the same as everything else. Self-similarity has many forms and is seen in all things. This chapter is an overview of many of the expressions of self-similarity in our universe of existence.
The point of the previous chapter’s description of ‘missing the point having missed the point’ is that being is a self-referential action. Godel’s proof of unprovability is of equations or statements that are self-referential and contradictory. A self-contradictory being, as surreal and ambiguous as that seems, creates a feedback image of itself by objectively being indirectly subjective and subjectively being indirectly objective. Actions and reactions are conducted on a common field of indirect play, each axis moving rationally through the other in the motionless common here and now. This continuously reflected feedback pattern, on every level of interference, leads to similar kinds of distinctions and differentiations, on every level, through harmonic dynamics and a drive for truth and beauty. (New information that agrees with old information can be said to resonate. Truth has objective resonance and beauty has subjective resonance.) Truth and beauty are goals of objective and subjective reality. Waves that resonate are attracted to each other. A wave that resonates with truth or beauty will be drawn towards that attribute. Each uses the other for more propagation of both together.
Another way of seeing self-similar patterning is by simple reflection. The reflected image is the same as the original object image of the self, only different. Two-dimensional reflections of an image are all different from the original object in the same way. The direction of spin (a quality also known as chiraltry or handedness) in a reflection is opposite to the spin in the projected object being reflected.
Each reflected image has the same quantity of information as the object of reflection, but the quality of the information has reversed. The images are different yet the same. A reflected image of an object usually appears to be located in a virtually projected place; not where the actual self is, but on the other side of the reflecting surface, where it appears with a reversal of handedness.
The surface of a flat reflector gives an undistorted image of self-reflection. The image of the self is on the other side of the reflecting surface from the self, and at equal and opposite distance from the surface of reflection. Every point on the reflecting surface reflects directly with the image of the self. The cumulative point of direct self-reflection on the surface is perpendicular to the reflecting surface as a whole. Other than at a perpendicular, phases cancell out, and no reflection is seen. At 90 degrees, the angle of incidence coincides with the angle of reflection.
A curved reflecting surface will make the reflected image of the self seem closer and smaller or farther and larger, depending upon whether the curvature of the surface is convex or concave to the object of reflection.
If instead of central self’s eyes looking out at larger or smaller reflections, the self were the smallest eye, then the central eye that was the self before is now a concave reflection. This relationship of size, distance, and curvature exists from level to level regardless of which perspective is that of the self. The greater is the same as the lesser.
A two-dimensional reflector has a two perpendicular dimensions to curve through, X and Y. Within that x,y plane, each line can reflect the other. The two axes may be flat or may curve independently into the third dimension either toward or away from the object being reflected in the new third dimension. The two axes can each curve in separate and opposite directions. The flat surface is one example of the class of reflectors that places the virtual image on the other side of the reflector from the actual object.
There are three possible positions for the reflected image of the self, depending on the class of curvature observed in the reflector’s surface. The flat reflector is an open or hyperbolically curved (like a saddle) reflecting surface. In a reflector that is hyperbolic, the observer sees the image of the self on the other side of the reflector, in at least one dimension. The closed reflector (like a circle) is an elliptical curve, and the reflected image is focused inside the same ellipse (or circle) as the object being reflected.
Between the two broad classes of curvature, (hyperbolic and elliptical), lies the fine line of parabolic curvature, where the image focuses at infinity in both directions.
(There is another entire way of describing the properties elliptical, hyperbolic, and parabolic. Here, though, the distinction is whether the two axes both curve in the same direction, different directions, or have at least one dimension remaining uncurved.)
A reflecting surface can have any number of dimensions. When a surface curves away from the observer with one axis and toward the observer with the other axis, the curvature of the two-dimensional surface is hyperbolic. Euclidean surfaces are flat and open in at least one dimension. A surface curved the same way in both dimensions is closed elliptically.
When
the observer is very close to any reflective curved surface, the
image is so nearly flat that its reflection appears to be off of a
Euclidean surface. The apparent amount of curvature can change with
distance. Increasing curvature indicates greater distance from the
reflector. Local surfaces are flat while more distant global surfaces
are curved, either towards the hyperbolic, or the elliptical.
If the reflecting surface curves away from the self, then one’s reflection will seem, from any distance, to appear smaller and smaller as the surface’s curvature increases. Although the reflected image looks farther away (is smaller) on such a convex surface, the reflected image will actually focus closer to the reflecting surface as its curvature away from the reflected object increases. A curved surface might also curve toward the self being reflected, making the concave reflection look larger and causing the image to focus farther and farther away with increasing curvature. As long as the image of the self appears to be on the other side of the surface, this concave reflector is in the same class of reflectors (relative to an observer) as the flat and convex curved reflectors.
The flat reflection, all convex reflections, and cases of reflective curvature toward the self that still leaves the reflection on the other side of the reflecting surface are all open reflectors of the self being reflected.
Any whole dimensional surface (not a fractal dimension) is flat, smooth and open if seen from close enough. As distances from reflective surface increase, making the image reflect off a more global surface, increasing orders of curvature become apparent. Curvature is a feature of globality. Locality, or the local perspective, is flat.
As the curvature of the reflector towards the self continues to increase and the image continues to grow larger and focus farther away, eventually the receding lines of focus will not converge at any distance beyond the reflector. When the lines of reflection all become parallel, the image is resolved only infinitely far away. In this case of reflection, the image is at infinite resolution in both directions at once and is just as much on one side of the reflector as the other side.
This is a parabolic reflector and is neither an open reflector nor a closed reflector. This curve is the division between open and closed curves.
The third class of curvature is elliptical. In this class of curvature, an object’s image of reflection is formed not on the other side of the reflector, but on the same side as the object being reflected.
This reflecting surface curves back on itself into a closed loop called an ellipse. Inside the ellipse are both the object being reflected and the image of its reflection. Planets orbit the sun in an ellipse, with the sun at one focal point, and no object at the reflection point. A special case of the ellipse is the circle, which alone has the object and the focus of reflection in the same spot in space at the center of the reflecting circle.
These three types of curves; Hyperbolic, parabolic, and elliptical, are the kinds of curves that can occur when a plane is passed through a cone.
A surface that reflects an image of the self might also allow the real image of the self to be projected through to the other side of that surface.
This transmissibility of a surface allows for variations in the pattern sequence of projections and reflections of the wave that resonates and reflects into the set of surfaces that make up the past history of an entity. Moments of experience are sequenced by time and space pattern characteristics. Additional surfaces of reality are created by each new moment as it occurs.
The sequence of reflections and projections may reveal complex interference patterns to resonate possible and probable futures. A real image that is a reflection on one side of the surface is a projection on the other side of the reflector.
The surface reflecting an image might also be the surface projecting an image. A surface can do both at once. Confusions can occur. We have all looked into a mirror from an angle and thought it to be a window. In such a mixed-use medium, it can at times be difficult or even impossible to tell whether an image is a projection or a reflection.
An excellent example of this effect is at the Haunted House ride at Disneyland. There is a ballroom where the ride-goer looks down onto mechanical men waltzing around without partners. Ghostly images of dancing women then appear in the arms of the dancing men. The effect is seamless.
In actuality, the suddenly appearing ghosts are below the ride-goer, on the same side of a large glass surface as the ride-goer. The ghosts are reflected on the large glass surface that the mechanical dancing men were seen through. The men are projected to the observer and the ghosts are reflected to the observer. The reflected images of the ghosts seem co-located with the projected image of the men when a light is shined on the ghosts making them visible and reflecting their image up and back to seemingly merge with the men. Projections and reflections are both equally real. Each can influence the other indirectly. Reflections and projections generate a being and are generated by a being.
Fractals are self-similar structures of infinitely fine detail. A fractal pattern looks self-similar at different scales of magnification. Magnification of a part of a fractal does not materially change the amount of relative detail in the fractal image. An example of a regular fractal is the Koch curve.
The triangular image of the Koch curve shows finer and finer detail as one looks closer and closer. In a fractal, scale is irrelevant.
Fractals dimensions exist between whole number dimensions like a one-dimensional line or a two-dimensional plane. Instead Fractals exist in 2.45536 dimensions, or whatever portion of dimension is defined by the initial parameters and particular equation of feedback that defines that fractal structure.
A fractal image is self-referential in that it uses its own value on one scale to compute the value for its modified image on a closer scale.
Feynman diagrams show the explicit observed relationship between interacting particles. They also show the unobserved, infinitely complex self-similar potential internal matrix of relationships that are implicit in any observed relationship between particles. The infinite complex of unobserved interactions that equate to the simple observed interaction are of the same type, but at a shorter scale of time.
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When two or more waves interfere, opposing constructive and destructive conditions can form a standing wave higher order interference pattern. Place two flat combs together and the interference pattern will be straight lines because the combs’ teeth are straight. A pattern can even interfere with its own projected shadow or reflection.
An interference wave pattern is formed from two or more similar patterns on another scale and orientation., and is similar to the lesser resonant structures An interference pattern contains properties of scale larger than any of the contributors, singly.
Harmonic resonance patterns depend on similar magnitudes or scales of curvature for interfering interactions to occur.
A common example of an interference pattern in seen when traveling along freeways. Overcrossings for pedestrians are fenced on both sides. Both fences have a similar pattern. When two fence patterns are viewed, one through another, a higher order interference pattern is formed. The interference pattern is larger than the pattern of a single fence’s pattern alone, but the interference pattern is the same shape as the pattern in each of the two similar fences. If the pattern in the fence is of round holes in a hexagonal array, then, the larger interference pattern of both fences together will be of round holes in a hexagonal array.
An interference pattern is the same shape as its constituents, but on a different level. The figure and ground relationship of the interference pattern can be reversed from the figure and ground image of the separate fences.
When two fixed patterns are superimposed on the same plane, then the interference pattern is as fixed in space as the contributing patterns.
If there is a third dimension of expression in the interference pattern (if there is a perpendicular space between the patterns), then the interference pattern is not fixed In space relative to the fixed original patterns.
When an observer passes by the separated fences that make up the interference pattern, that observer will see the positions of the fences (relative to the observer) to change, just like the positions of every object the observer passes by. But as the observer moves by the fences, the interference pattern may be seen by the observer to move in a direction independent of the motionless fences. The interference pattern may move in the same direction as the observer, in the opposite direction as the observer, and can even display two different interference patterns moving in opposite directions at different speeds through each other in the unmoving fences. .
Stillness or motion for these two levels of being relative to an observer makes two frames of reference. There is stillness or motion relative to the pattern, or relative to the interference pattern. These are two opposing perspectives of stillness and motion.
Resonance of one structure with and against another structure creates a similar structure on another level. The composite structure occupies a given amount of space in a similar number of dimensions. There is a ratio between the low and the high, cause and effect, the parts and the whole, the matrix of existence. This ratio is the diagonal slope which is altered in two different ways. First, two contributors can be brought closer together to change the ratio between them and the combined whole. The other method of changing the ratio is by changing the rotational orientation between the contributors. This also changes the ratio between the greater and the lesser structures. The two ways to make change are by changing distance and by changing orientation. Change is straight or curved.
The interference pattern found in any repeated pattern is strongest when the two patterns are of the same apparent size. When the patterns become out of scale with each other, the interference pattern changed resonances drastically..
A hologram is an interference pattern. Moving the point of view across the holographic plate can be seen as viewing an image at the same time from different places, or viewing an image at different times from the same place.
In our numbering system, there are symbols that stand for amounts, (1, 2, 3, 4...) These symbols may be arranged in an incremental order according to gradations in amount, a numerical sequence. Placeholding was developed to represent new levels of magnitude of the same relative quantities. We use the number ten as the base for our numbering system, but placeholding will exhibit a self-similar pattern in any base. A coloration method of notation allows for hierarchical self-similarity to be seen on different levels of magnitude for any base’s placeholding of a numeric sequence.
Each base has a different pattern seen in its placeholding. Within each base, a new order of magnitude shows the same pattern, only on another scale. As in fractal structures, we see similar patterns at different levels of order. As the number of terms grow, so do the number of possible states. As a static image, this structure displays exclusively linear and lateral components. A dynamic system would include both components and be on the diagonal.
Life reproducing life is an example of self-similar patterning. Each new generation is the same as the old one, yet different. As a dynamic structure, life flows on the diagonal, with levels of stillness and columns of motion interacting to create the rational slopes of change. Any life flow that fails to propagate through change like its predecessors will go out of existence. New life recreates the potential for new changes that keeps life continuing to exist. Simpler life reproduces simpler relations. The more complex the relationship for compared potential states, the greater the range of options between the entities of exchange.
Even non-life objects continuously reproduce themselves at lower levels of similar kinds of action. The atomic particles themselves recreate their own opportunities and conditions for continued existence. Mass creates the space for the mass to exist in.
The abstract process of existence is one of continuing self-reproduction, or continued propagation as a wave. It is the process of allowing of a continued being by continuing to be.
Exchange and feedback are parts of awareness, being, and consciousness. Each entity operates in a relationship with all entities, including itself.
There are four types of relation between the self and the not-self. The self and the not-self can each interact with the other directly and indirectly. The four paths of exchange are; Self to self, self to not-self, not-self to self, and not-self to not-self. The objective axis and the subjective axis each have a self and a not-self as their parts
A newborn child is not aware of the self and the not-self, there is just being. Everything is a part of the experience, which is being. Soon the infant has learned many new things and compares each new thing to those things that have been previously learned. Each new thing is compared to all other things in a process of pattern recognition. Each new thing learned is similar to things learned in some way or ways, and different from things learned in some other way or ways. This every increasing ability to make distinctions gives the child an awareness of the self and the not-self. The grouping of things as ‘similar to self’ and ‘not similar to self’ is analogous to grouping them as similar to each other and different from each other, or parallel and perpendicular.
Contradictory perspectives are processed diagonally as both and neither both and neither. Objective and subjective awareness creates a potential slope of opposing tensions and flows.
Time and space flow through each other with regularity of repetition. There are two beats to keep, a time axis beat, and a space axis beat. The diagonal flow of reality goes through both axes. A change in slope changes the ratio of the two beats on the line’s perpendicular passage across the X and Y axes.
The objective feedback of the world is the physical universe, including the body. This is a description of the outside of the self. Subjective feedback of the world is the non-physical universe, the spirit. This is a description of the inside of the self. The first division of the unity is into two: the physical and the non-physical, the objective and the subjective.
The one way to see duality is to see that duality may be seen in either of two ways: either as a single thing seen from two different perspectives, or as a single perspective of two different things. The observer and the observed are not truly separate.
The objective universe and the subjective universe are interwoven. The forces of objectivity and subjectivity work together and apart to follow the dictates of persistence. Each contains the other, always.
Escher’s work on self-similar contradictory imagery is unmatched.
A diagonal can have a right hand twist from one perspective and a left hand twist from another perspective
