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Incongruities In Spatial Geometry

by Roy D. Follendore III

Copyright (c) by RDFollendoreIII

I am an adjunct Professor at George Mason University. Among other disciplines, I have been educated and worked in the fields of cryptography, metrology, cryptography and fine art. Within my varied studies I have observed that spatial geometry is critical to understanding and the accumulation of knowledge. It plays a vital role in the logical core of technical security and cryptography, it is at the heart of physics of measurement and it has everything to do with our perception of the visual arts. This essay is a tour of my current way of thinking about the incongruities of spatial geometry and the future that I see which which can and will eventually change the way that we perceive this complex universe in which we are immersed.

July 17, 2006

Algebraic notions that we derive from geometry both originate and result from the analytical portrayal of visual space. We have little choice in this. We are primarily biological visual creatures, not creatures born of cold analytical logic. On the other hand, Biology is the foundation of logic. The intuitive understanding of geometry is built within biological systems. The physiology that provides us the means by which we are able to understand the relationships of geometry is also the basis of logic. But in order to extend logic further we must learn algebraic logic from basic basic visual representations of geometry. We would be considered abnormal if as infants we instantly understood notions of algebra before we came to recognize the qualities of a toy block or ball. Children must first be able to play with shapes in their minds to be able to understand them. How well we are able to play with the visualizations determine how we tend to perceive the logic that may be derived from geometric perspectives.

The origins of human understanding of algebra arises from our natural senses and the way in which our minds are able to become wired in context with the structure of the visual cortex. For instance, one way to define dimensionality is to say that it is essentially the result of our biological ability to sense potential temporal differences. For each of us it is the perspective of the mind's 'I' which is the reference point for 'there'.  We could say that for 'I', object 'a' is closer than 'b' because 'b' appears to take more time to reach 'there'. What is important to recognize is that in this simple concept of geometry as well as in every possible geometrical measurement, no matter how much we might choose to distance ourselves, there always exists the implication of the biological perspective of human perception.

The biological basis for logic is the ability of the human brain to manage uncertainty as an object with physical existence. The philosophical belief that the universe is composed of independent objects which have geometric attributes is essentially derived from our biological ability intellectually disassociate our thinking from our senses. The suspension of disbelief is something that we choose do this all of the time for personal entertainment. For instance, as we watch television, we choose to ignore the physical discontinuities that are observed and which appear to violate the cannons of physics. The willingness of the brain to be able to do this represents a learning process involving the comparison of the benefits and rewards of suspending disbelief to the the potential risks and liabilities of ignoring physical reality. The ability to trick the biological mind into ignoring the 'reality' of the senses is inherently reflected in the fact that biological senses are limited.

For example, it is well know that the human eye is constructed with a blind spot, the reality of which is concealed except through specific circumstances of perspective. To compensate for this loss of data, the eye produces essentially vibrates around objects that are being visualized so that cognitively we are not allowed to consciously 'witness' this physiological error.  The vibrating biological design of the eye not only covers the blind spot, but also has the benefit of refreshing the stimulus so that we are far more sensitive to physical changes within our physical environment. The comparison of physically tangible things to intangible ideas such as numbers becomes a tricky phenomena to consider because the eye and the 'I' are different version of the reality of things. Our cognitive brains normally define ourselves as biological entities that do not have blind spots in our vision, even though that is exactly what we are. It is ultimately who we cognitively are that defines the 'I' that we are. On the other hand, the physical manifestation of 'I' is the cognitive basis for persistence. 

There is an indivisibility in the physical manifestation of 'I' even though there we may perceive a divisibility within logical computations of dimensional physics. As we define relationships that exist within a physical dimension, the mind is choosing to ignore the true physical process of reality, in favor of the minds visualization of possible alternate realities. Look in the mirror at you reflection and simultaneously attempt to maintain the conscience thought of what you perceive and how you perceive yourself. Doing so produces the same vibratory feeling that results from simultaneously perceiving two forms of a reversible illusion.

The conscience mind does not easily track many alternate dimensions. Instead it searches for opportunities to  favor of the specific one it perceives as most beneficial at the particular moment  Geometric relevance can be represented in terms of the comparison of the qualities of visual space. With regard to perspective there is a constant relevance of the relationship of the eye to the mind. First of all this means that with respect to the notion of physical measurement we can not get away from the notion of biological perspective, and with respect to the notion of perspective we can not get away from the notion of the all seeing 'I'. Second, this also means that whether we are aware of it or not there exists a bias of perception. Visual preferences, based on context, physiology and psychology are constantly affecting our perceptions of dimensionality and space. Since the ordered measurement of dimensionality is the basis of geometry, and geometry is the foundation of mathematical logic, it stands to reason that the historical choices that have been made in representing logic is also affected. 

Image:Kurt Gödel.jpg

Kurt Godel, one of the most significant logicians of all time and a colleague of Albert Einstein proved that when describing our universe, any given set of axioms mathematics will always be incomplete. This can be interpreted as meaning that the particular way in which human beings have constructed logical relationships is incomplete with regard to modeling our observations. It also means that there are alternative ways to define logic, some of which may be more useful in context with the context and perspectives of particular circumstances but none of which are complete for all perspectives and circumstances. The way in which we are constructed as well as the cognitive interpretation of simultaneous perspectives define our sense of logic. This also means that intelligent creatures who potentially exist in other Galaxies may will probably not have evolved with the same set of environmental and evolutionary constraints, so that their sense of logic may have been constructed differently. Their sense of 'I' could also be unique so that their interpretation of dimensionality, even if their senses are similar could be different.   

It therefore seems natural for us to assume that the primary way in which human beings are able to innately visualize dimensional perspective of things that exist around us, also affects the way that we are able to logically deduce the potential existence of dimensionality. If we can not measure, we can not perceive. Many different aspects of dimensions may well exist but remain transparent to our 'natural' infantile perceptions. It is also quite plausible and probable that significant incongruities may exist between the boundaries of algebraic and geometry theories, which though also unnoticed may describe incongruities within the domain of spatial geometry. These most fundamental concepts deserve rethinking. Our ability to bear witness to this spatial geometry is more of a matter of creating a language for visualizing. Two 'languages' may be able to communicate similar ideas, though one may do so better.  Perspective changes the language which we use to communicate and communication changes our visual sense of understanding. Beneath the process of visual thinking may exist expressions of a new form of logic and the potential of creating a visual language which then perhaps may even supersede traditional geometric spatial logic.

In order to produce a new spatial logic it becomes necessary to first create a theoretical 'laboratory' within which we may place a primitive rigid model of 'a' universe to interpret and explain. Instead of the four dimensions that exist within our universe, we may choose to reduce our model to two dimensions and instead of distances there there may be only probabilities of geometric lines and points that exist on the plane of our universe. If this is the way in which we would define our model then we must also include the definition of our presence within the model. If the context of geometry is to become the basis of our two dimensional model then dimension is bounded by the fact that probability is defined by the improbable. An infinite number of possibilities exist for drawing a line on a two dimensional surface but there are infinite more ways which can not be drawn because of the constraints of that surface. 

Because the classical utopian sense of the universe is 'theoretically' defined to be perfect and without external bias of 'I', the existence of a line existing in a way that is congruent with all dimensions that we can observe within our actual physical universe becomes an improbable artifact. From our perspective, the existence of a line is a process, not an instantaneous event and yet from the perspective of our universe it must be so. The theoretical denial that bias does not exist becomes impossible without acceptance of an active role in ignoring the fact that as the creator of this universe, we are the bias. We profess to have logical objectives which involve antithetical relationships of completeness, scalability and flexibility while knowing with certainty that this analogy would be tantamount to having the free lunch of having our philosophical cake and eating it too. Somehow we want to expect that the universe to be less complicated than we are and upon acting on this philosophy we attempt to reduce complexity regardless of the fact that in doing so we are permuting our relative perceptions about everything we logically operate on. 

We may attempt to intentionally choose to create this model universe with constraints, but then it seems that at least in a conventional sense we must also choose to deny that those constraints which we have chosen so carefully; also of course precluding the our future choices. For instance, the choice of two dimensionality affects what we should be able to observe within our universe model. By denying time as a integral part of our model, we force ourselves to either deny the potential of probability as a 'resident' aspect of our model or bias our perspective. We must be either willing to accept the notion that there exists a probability that points and lines 'magically' appear within our two dimensional universe, or that one universe suddenly ceases to exist and another perfectly integrated universe replaces it. The model of this two dimensional universe that we have chosen to consider is one which can not be observed from the perspective of its theoretical residents, not only because of the nonexistence of intelligent life to be present within such constraints, but also because without time, every change would represent an 'instantaneous' new universe.

Within classical geometry and mathematical logic we are representing the perspective of that omnipotent creator who is allowed to observe our domain. The properties of every exercise of geometry and therefore mathematical logic gives rise to this notion but offers no scientific credence to it. (In the field of psychology we might say that the 'id' is being ignored as a factor in the operation.) The potential inference of this way of thinking about mathematics is that Algebra based on Geometry is incomplete because it chooses to ignore the fact that the creation of a new universe requires the simultaneous recognition and resolution of the differences between the notions of least three separate perspectives. It is as though there is the perspective of the creator, the theoretical perspective from which the logical universe being modeled and there is also absence of the perspective of that creator as an integral part of that model.                     

Within classical Cartesian logic we have a grid within which linear horizontal and vertical lines always intersect at right angles like railroad tracks on ties. Obviously the more and the closer the tracks and ties exist, the less empty space there will be. Lines drawn in this manner create the surface of a two dimensional plane. Cartesian logic assumed that if enough infinitely thin lines are drawn infinitely close together, they will eventually cross ever potential point thereby producing an infinite plane. The square nature of Cartesian geometry requires logical evidence that the God of logic must remain in his heaven and all will be well. Another way one might put this is that Cartesian logic is based on the instantaneous expectation of infinite lines producing infinite planes. But what if this were not true? What if planes were to be defined as an incremental inversion of dimensional space? How might the incremental dualism of dimensionality fundamentally change classical algebraic expressions? Where would such reasoning come from? It would obviously come from the recognition of the importance of perspective to the foundations of logic.   

Let us assume that we choose to model a 'universe' in terms of the existence of an infinite circular plane at the center of which exists as an infinitely small point. From the practical perspective of a 'native citizen' on the surface of our model it is impossible to draw a line in a random direction through that point or thereafter to draw another line which is intentionally either tangent or perpendicular to the first line through that center point. At least in our natural universe the existence of an infinite point in space and time becomes moot by a lack of a physical means of observation but this is not necessarily true for the hypothetical two dimensional universe that we have chosen to create. The incongruity that we either must face or choose to ignore is that from our natural and tangible perspective, if we are to assume that an infinitely small spot can be accurately and precisely identified as a point we might as well choose to believe in 'tiny fairies'.

Spatially there exists this dichotomy of simultaneously looking at the infinitely macro and micro perspective of this experimental 'telescope' that we have chosen to construct. We may choose to ignore the incongruities but this does not mean that they do not exist or that we can not better understand the nature of what we are attempting to do with geometric logic by incorporating it into our conception of what we attempting to functionally achieve.     

From the perspective of our universe, when we break the process down too far, we as their creators face a similar epistemological dilemma as those tiny fictional fairies. When exactly is a geometric line to be defined as having existence? Does its existence occur after it is drawn, while it is being drawn, or the instant that consideration was given that it is to be drawn? Assumptions of probability become limited by these decisions and for the sake of rhetorical argument it becomes easy to take upon ourselves the burden of inconsistent perspectives.

Obviously tangibility has a lot to do with the nature of geometric probability because otherwise we would constantly be dealing with the potential of potentials without limit. The question of quantum physics is really all about the notion of dimensional tangibility. Perhaps the reason why we do not perceive dimensions is because we have deliberately chosen a visual language which limits our perception. The points and lines become the active performers, not human beings. Procedurally, our conventional notion of geometric randomness depends on the the notion of procedurally modeled predetermination. Thinking about two lines and points in a two dimensional universe as having the independent capability of intelligently inducing themselves is an attempt to change the way in which we are able to work within our modeled universe and ignore our bias as deity, even though it also inevitably requires the introduction of an additional dimension of time. 

Let us now move on and assume that all points on a circular plane have an equal probability of spontaneously coming into existence. Considering the infinite notions of precision and accuracy, the infinite potential number of points on a plane without prespecification of reason, and given that a single random event might somehow naturally exist at some explicit location is a extremely small probability which is more than zero.  The probability that a line made up of an infinite number of these points might exist through the center point of a circular plane would be less than that. Because of the infinite number of potential directions though which a line could find itself on the center point, the probability that the second line might not only also cross the first line at the center point of the circular plane on the first attempt, as well as be perfectly perpendicular to the first random line would obviously be infinitesimally smaller than the probability of the first line going through the center point. The probability of the second line is limited by the probability of the existence of the first. However fine grained, once again this probability must be considered to be infinitesimally small though more than zero.   

Next let us specify a particular situation where this infinite circular plane exists with two perpendicular lines that coincide at its center. By this prespecification of our plane it is also possible to randomly designate a quadrant as well as a random angle along with a random distance from the center of the plane. After selecting our random point, suppose we then begin adding more and more randomly directed lines through the center point of our plane. Let us then assume that this process was repeated an infinite number of times. In doing this is it then possible to create a two dimensional plane where lines cover every infinite point?

If you sit down with a pencil and piece of paper and begin doing this on a quadrant you will begin to see the implications of this question. If this answer is yes then we must account for our observations with our concept of geometric convergence. If the answer is no then the part of geometric logic that we have based logic upon has not properly defined negative space. The correct answer is both yes and no. We must redefine our expectations of the dimensionality of space. We obviously should be finding ways of observing missing dimensions by challenging the way that we choose to manage our perceptions of the incongruities within spatial geometry.

The complementary question to be asked is this; What is the probability that one of these random lines might coincide to cross over the random point? Coinciding with the answer to that question is this. Assuming we were to randomly repeat the process of establishing yet a third point at half the interval from the center of the circle to the second point. We can now ask the question in a slightly different way; "Can a one of an infinite number of random lines drawn from a particular point on a two dimensional surface cross any given particular point on the same plane?" The obvious answer is yes. But this does not answer the question of relative probability with regard to spatial geometry. The question should be: "What is the probability that one of the infinite number of random lines drawn through the center of the circular plane would cross through a randomly selected point at a location that is different from the center point on a two dimensional surface?"

At first it might seem reasonable to assume that an infinite number of lines through a single point on an infinite circular plane could completely cover the surface of plane. The problem with thinking of the model in this way is that we can demonstrate through observation it also doesn't. Lines which pass through any specified point on a plane continue to diverge in an orderly, continuous and progressive manner regardless of the lines width. This divergence establishes the calculus of a fundamental geometric requirement that in order to maintain an equal probability between the closer and further away point being intersected by any given random line, the number of infinite lines must be larger relative with respect to the area covered by each points distance from center point of the infinite circular plane. Infinities need not be equal and proportionality between infinities need not exist but we may be able obtain a far better understanding of the relative nature of infinite relationships through notions of probability.  Indeed, it should be obvious that by assuming the probability of individual abstract infinities are equal, the probability of two randomly selected infinities being exactly equal would be infinitesimal. This does not mean that geometric spatial infinities can not be indirectly compared through probability.

In other words, our little thought experiment appears to have have shown that proportional expressions of randomness change with respect to the relativity of dimensional space from each point. The interesting notions of incongruities that I we have discussed with respect to the title of this essay perhaps would have better began with the idea that negative space exists within the nature of its process. Changing our perspective of spatial geometry changes the way in which we may come to expect to see and broadens the the geometric and logically algebraic nature of our universe. Notions of negative space is in between the relative degrees of perspectives that each line is conventionally allowed when specified by a common point and this is something that is essentially profound which conventional geometry overlooked. Regardless of the angles of the lines, at some measure of resolution negative space always appears to be approaching its center point and it has a proportional expression regardless of the number of intersecting lines. We we should observe from the starting point of our theoretical model that this inverted (negative) perspective can also be interpreted as the aggregation of probabilities. Perhaps along with with the physical notions of dark matter and dark energy, there are also darker dimensions which penetrate to the geometric core of our notions of spatial logic. To understand things that we do not directly perceive we need better models to understand ourselves. We need a comparative models which will invert our way of thinking.

There are also any number of other potentially interesting questions that arise from this particular model. By convention we have traditionally chosen to define geometric compositions in the form of their binary placement, rather than the probability of their existence. By this I mean that a point or a line is predetermined by the use of our logic to either exist or not exist. The existence of points in or plane are traditionally defined in terms of either zero or one. In the physical universe there are alternatives of may, may have or may shall and this of course returns us to proposition of the importance of including time as a third dimension. Our perspective affects our ability to understand the geometry and of course time affects our perception. 

The changes that we must be willing to consider when fine grain probability is involved changes our rational expectations of our geometric model. We need to further consider if there fundamental theoretical conflicts existing within the concept that two dimensional plains can be comprised of an infinite number of parallel lines or are irregularities brought about by the probability of the position of lines as being fundamental to the conceptual nature of a geometric plane.   

In conclusion I wish to allude to the fact that we have been essentially taught that geometric lines are defined as being imaginary devices that describe the perspectives of shapes. As my daughter interjected when I described the model previously described within this essay, the traditional definition of geometric lines means that they can have no mass. This is true when the objective of the the two dimensional universe is isolated without the benefit of the objectives of a three dimensional observer. To turn this two dimensional a model of spatial probability into three dimensions involves a matter of rotation. One must assume that the perspective from our three dimensional observation changes the model that we might construct. For this reason it may very well be impossible for a three dimensional being to construct a model of a two dimensional universe but it does not mean that attempting to do so does not produce knowledge that is intellectually productive. This is an interesting potential area of theoretical research.  

I feel that a future redefinition of the relationship between the principles of geometry and algebra we will find far better alternative ways to discuss dimensional qualities that we can only indirectly observe. Our physical observations with respect to the actual application of physical geometry is quite different from the abstract theoretical definitions. Within our universe we can never witness a truly one dimensional line and there are no one dimensional lines that pass through a common point can never overlap to form a plane, but perhaps that does not mean that we can not understand an essential aspect of it.

I therefore speculate that if it is to maintain progress, the fundamental logic of Mathematics must evolve to find ways to better track and keep relative the fact that points and therefore lines, planes, geometric shapes and even numbers are relational probabilities rather than fixed associations.   




Copyright (c) 2001-2007 RDFollendoreIII All Rights Reserved