Here is an overview of the sort of thing I mean when I talk about dimensions and dimensionality: Dimensionality
Dimensionality is a way of defining and manipulating reality in terms of it's dimensional complexity. Events of similar dimensional complexity will share similar mechanisms despite their differences. In other words, the mathematics used to describe similar dimensional objects or events will be similar even though the objects or events themselves are different.
A dimension refers to anything that is measurable:- length, height, time, weight, frequency, amplitude, volume, etc. It is easiest to see dimensions when they are transcribed symbolically onto an imaginary perfect graph. Although we can draw a representative graph in the real world, it will not be perfect, but we can use our imagination or mathematics to perceive it so. A line drawn to represent one dimension will have a thickness which is another dimension, which need not be interpreted for our understanding of the one dimension. In the imaginary perfect graph, there is no thickness to the line, but in the real world, such a line would not be visible.
A single point on the imaginary perfect graph has no size, no length, breadth or height, and so has no dimensions. This is called a zero-dimensional point. Although the point must exist within the context of some higher dimensional reality, the point itself is dimensionless. The weight of an elephant can be transcribed as a single point onto a line representing weight. The line representing weight is the dimensional context of the zero-dimensional point. In the real world, the weight of the elephant can be measured to any degree of accuracy required, but will always represent a spectrum of weight, for example 2 tons plus or minus one ounce and so in reality should be represented by a short line. Also it is important to note that any precise measurement of weight can only be taken in a single moment in time, otherwise ingestion, respiration, perspiration, urination, and defecation, not to mention alighting flies, ticks and settling dust, would change the measurement. In measuring only the weight we have also ignored the elephants genetic heritage, its position, sex, time of measurement, colour, sound, texture, and many other factors. Each factor represents a separate dimension and would need a separate axis on any graph we might create. From this we can see that, although we consider this to be a 4-dimensional universe, we would need more than 4 dimensions to describe any real object to any degree of precision. Similar observations will apply to any dimensional measurement and we shall see that in any real life problem, we only need to consider a small proportion of the total dimensionality of the situation.
A single zero-dimensional point shall be referred to hereafter as mono-polar and can be described by 5 mono-polar logics and their 5 inverses.
A list of objects or events also forms a dimensional context. On a list of animals, for instance, one particular point may refer to an elephant. Also, since any string of letters can be converted to a numerical binary string, and by extension to a numerical string of any base, any description of anything can relate to a particular point on the number line. For example, the phrase 'An elephant eating a banana' can be converted to binary code, as a computer does automatically, and so that phrase can be symbolised as a particular number, meaning a single zero-dimensional point on the binary number line. Hence, although it would also contain many meaningless translations, an endless binary number line will contain all conceivable descriptions of everything that exists. Although this would seem to imply that all of reality could be synthesised onto a line of one dimension, we must recognise the number of other dimensions that would be necessary for this information to have meaning, not least of which is the complexity of the years of life experience necessary to interpret and understand the meaning of the information contained.
If, however, we create a separate dimension for each measurable factor, then any description, and even the totality of reality itself, becomes a single zero-dimensional point in a multi-dimensional imaginary perfect graph. If, further, there was a set of abstract logical axioms that could synthetically, but meaningfully, be used to extrapolate an understandable sequence of infinitely varied and complex dimensions that would define this multi-dimensional graph, then we can say that the totality of reality is inherent in that set of logical axioms. 'The Truth, QED' is such a set of logical axioms. These axioms are not created but can be found by stripping away all that is not absolute from reality. When all that is not absolute is stripped away from reality, the logical axioms appear, like an adamantine perfect crystal rising from the froth of confusion, manifesting from five undeniable and eternal truths. Although these truths exist independent of the reality we know, they are comprehensible from within the reality we find ourselves in. By expansion of these logical axioms we can create a multi-dimensional description which contains, as a subset, all other logical proposals and philosophical doctrines.
Because only five truths are necessary to generate the complexity that is our reality, this set of truths are genuinely quintessential and as they are the foundation of all that is, they are existential. Although we can perceive the axioms separately, they form an indivisible whole in which each is defined by and only exist in the context of the others and so there is really only one truth. Hence the name, 'The Truth as defined by Quintessential Existential Dimensionality' or 'The Truth, QED' which also translates as 'The Truth, Quod Erat Demonstrandum' which means 'The Truth, that which was to be proved'.
To further understand the nature of dimensions, since, so far, we have only defined, to any certainty, the zero-dimensional point, there follows more detailed descriptions of the basic dimensional realities from one to four. In what follows, 'graph' refers to the imaginary perfect graph and 'point' refers to a zero-dimensional point.
One Dimension
If we have two points, we can construct a line that passes through them. This is a one-dimensional line and we must have at least 2 points on it for it to have any meaning. One point on a line tells us nothing as we have no way of telling where on the line the point is. There must always be another point to act as a reference which is commonly referred to as the origin or zero. If there is an agreed scale then we can physically measure the distance between the points to establish meaning. The norm, however, is for there to be another point to establish the scale which would in most cases be the unit one. We can then repeat the distance from zero to one to establish a scale to give meaning to any measurement we have made. Take temperature as an example. There are three major scales of temperature, Celsius or centigrade, Fahrenheit and Kelvin. One degree of centigrade is the same as one degree of Kelvin but both are different from one degree of Fahrenheit. All three place their zero point at different temperatures, but as long as you know which one you are using, there is no confusion. Each temperature scale needs two points, zero and one degree to define the scale and a third point to mark a particular temperature. This means that, although we might consider the linear dimension to be one dimensional, since we need at least 3 points for it to be useful, it is really 2-dimensional.
Although a one-dimensional line is theoretically infinite, in reality, all lines have boundaries. The two points can be used to define the line, it's scale and it's boundaries simultaneously.
'One dimension' is what we normally mean when we talk of 'a dimension', and this shall be referred to here as the bi-polar dimension as it needs 2 points to define it and, as with mono-polar logics, there are five logics and their inverses that describe the interactions within a one-dimensional reality.
Two Dimensions
3 points define a plane, a 2-dimensional reality described by tri-polar logics. Of the three points, one must be the origin and one must denote scale, but for the other point, unlike in one-dimension, there is variation which becomes more complex with each increase in dimension. The third point can denote scale in a direction at right angles to the first, which gives us the standard graph that we know and love, it can denote scale at any other angle to the first, or it can denote the measurement of an angle, in which case a point is defined by a distance from the origin and the angle from some base-line. The type of information to be encoded defines the nature of the graph to be used. A one-dimensional graph seen through time is also a 2-dimensional reality, so seeing a point moving in a straight line is a 2-dimensional event.
In the same way as 2 points define a line but an extra point, even though on the same line, takes it to a higher dimension, so it is with 2 dimensions. This is because a label would be needed to say what the point means. The same information can often be depicted graphically in different ways, for example: say we measure the height of Jack and Jill, we can have a single line of measurement on which both their heights are labelled with their names, or we can have a 2-dimensional graph with one axis of measurement and the other axis a list of people, including Jack and Jill, or we can have one axis labelled "Jack's height" and the other axis labelled "Jill's height", in which case all the information is now depicted by a single point on a 2-dimensional plane. Likewise the measurement of n number of things can be depicted as a single point in n-dimensional space.
Three Dimensions
4 points define a three-dimensional space. If those three dimensions are spacial, then that defines a spacial volume. One of the dimensions, however, could be the temporal dimension, in which case we have a planar surface, a 2-dimensional plane, which is viewed through time. Such a surface might be a movie screen where, although the screen is flat, the movement makes it a 3-dimensional reality. This is different from our rather astonishing ability to perceive a 4-dimensional reality of a drama from a flat screen. Not only do we add or imagine extra dimensions but we also remove them. Life experience is so complex that we could not possibly consider every facet of reality in one go, so we filter out information that we currently do not need; when we are measuring Jack's height, we do not care and possibly do not notice what colour his trousers were, although we might notice if he was not wearing any. Each fresh piece of information is a new dimension and it would appear that one scale of intelligence might be graded on the number of dimensions one can comprehend simultaneously.
A 3-dimensional reality is defined by a suite of 5 quadra-polar logics and their inverses. A 3-dimensional graph can be composed of 3 linear axis, 2 linear and 1 angular axis or 1 linear and 2 angular axis as well as being any variety of 2-dimensional surface seen through time.
Four Dimensions
5 points define a 4-dimensional space which is described by 5 penta-polar logics and their inverses. It is normally considered that we live in a 4-dimensional reality, three of space (length) and one of time. This may be a convenient way to describe the fabric of reality upon which existence is imposed, but, as we have seen from above, for any real event, more than four dimensions are involved. An alternative view would be to consider reality to be composed of infinite dimensions and that each part of existence is a dimensional subset of that totality and that the four dimensions of the fabric of existence are the minimum contraction possible.
We would need four dimensions to locate a particular point in space and time. If the point is moving we need another four dimensions to define it's speed and direction. If it is accelerating we need another four dimensions to define the scale and direction of the acceleration. So something as simple as an orbiting body is at least a 12-dimensional reality and if we add to that, orientation, rotational velocity and acceleration, mass, volume, shape, composition, chemical interactions, colour, sound, etc. we can begin to achieve an idea of the complexity of dimensional reality.
Our most usual view of four dimensions, with three of length and one of time is not the only one. Although it is quite reasonable to have four linear dimensions, for most people, it is difficult to visualise these four dimensions together and we have to resort to looking at only three at a time. Higher dimensions are even more complex to imagine, but fortunately mathematics and computers do not have the same problems. Although mathematical formulae can only directly solve equations with a maximum of five variables, partial solutions can usually be found for higher numbers of variables. Fortunately, although reality has a high dimensional complexity, in real interactions, real events, we usually only need to pay attention to a smaller set of variables which makes most events calculable. Even though this planet is an amazingly complex place, we only need to know it's position, speed and acceleration to be able to calculate where it will be at a certain time.
Five and more dimensions
Many are so conditioned to this being a four-dimensional reality that more than four would seem esoteric and yet we continually experience more than four dimensions without any awareness that we do so. Each sense we use to experience reality has its own dimensionality, the dimension of time is common to all the senses:
Sight- each eye sees two dimensions and, as they are distant from each other in only one dimension, they collectively constitute the three dimensional view we have. Colour is received by three different types of retinal cell and so has three dimensions. It might be argued that as any given colour can be represented on a 2-dimensional spectrum, that colour is 2-dimensional, but the wriggly line that describes the spectrum of a specific colour is more than a one-dimensional line. The line itself is one-dimensional but its wriggling across a planar surface gives it another dimension. This means that the line is 2-dimensional. This gives sight, including time, seven dimensions.
Hearing- Sound can be located in the three dimensions of space and represented as a spectrum. As we saw with colour, the spectrum denotes a three-dimensional reality. So, hearing also requires seven dimensions.
With just these two senses we are experiencing in ten dimensions! They both share the dimensions of time and space. Adding to that the dimensional complexity of touch, smell and taste and we are over twenty dimensions.
If we move anything in space, there are always six dimensional possibilities. We can move in the three dimensions of space and we can also rotate in those three dimensions. Any given starting point requires those six dimensions to describe position and orientation. When something is moved, its direction and rotation require a further six dimensions. Add to that speed, acceleration and change of acceleration for both movement and rotation. and picking up a table fork becomes a eighteen dimensional endeavour. If there is food on the fork and we are smelling and tasting, whilst also experiencing the complexity of the environment through sight, sound and touch and we could be looking at forty to fifty dimensions. In human terms of experiencing everything in one go, there seems to be a limit of dimensional experience at about fifty dimensions. It is quite amazing that we can handle this complexity of information without hardly noticing.
Many common mistakes in thinking are due to the lack of understanding of the dimensional nature of the event and of not considering all the pertinent dimensions. An example of this would be an accountant dabbling with numbers on a page while being oblivious of the ramifications to the lives of the people those numbers represent. The second mistake would be not realising that different dimensional complexities require different logical frameworks to understand them. Once the logical context has been explained, you will see that we can define any problem and any solution as a distinct sets within the logical framework and that the logical assertions that connect these two sets are the path from the problem to the solution.
AD