Space, Dimensions, and Manifolds

February 27, 2012

By: Paul R. Martin

This essay will examine the concept of space. Space seems very familiar to all of us as being the place where things exist and where events happen. Immanuel Kant claimed that space was one of only two "transcendental aesthetics", the other one being time, which are what he called a priori notions that must exist in our minds before we can imagine, or conceptualize, anything happening at all. It's not hard to agree with him that space is a notion we pretty much have to have in order to comprehend our world at all. And we do have such a notion.

What we will attempt to do in this essay is to pin down some of the features of the concept of space, not only as it appears to us as part of our real world, but also as it appears in the purely conceptual world of mathematics. The primary features we want to look at are dimensions and manifolds.

We are going to examine these two features in both of the two different contexts: both as mathematical concepts and also as real-world concepts.

We have to be careful with the "real-world" concepts, however, because we really don't have access to the "real world". We only have access to phenomena and not the real world. Phenomena are sense impressions that we are aware of and which seem to originate in some "real world" that we believe exists "out there". That might seem rather picky, but at the depth we want to cover these concepts, the role of our perception and consciousness comes into play in important ways.

So before we proceed, let's set the stage a little more carefully. What we are ultimately investigating is reality. That is, we want to know as much as we can about whatever it is that exists. Recently Sir Roger Penrose has divided all of reality into three separate worlds. His distinctions don't seem to have caught on among mainstream commentators on Theories Of Everything, but in my opinion, his distinctions are logically unassailable. We are going to observe them in this essay. Before we describe them, though, let's review the history of how reality has been viewed around the world over the past couple of millennia.

It has been common for people ever since ancient times to believe that in addition to the physical world they lived in, some nether world, or spiritual world, also exists. This belief persists to this day. It is believed in various and disparate ways among the many different religions and superstitions that are believed by a great many people around the world. On the other hand, this "spiritual world" is dismissed or ignored by the current official scientific community.

To briefly summarize the Western Tradition of thought that has lead up to modern science, let's start with Plato and Aristotle. As Rafael clearly depicted in his famous painting, Aristotle was of the opinion that we could learn most about reality by examining the world we see and live in. Plato, by contrast, is shown pointing up to some domain not of this world, but a world of ideals containing the fundamental forms which are ultimately responsible for how things appear to us in the world. He introduced Platonic Dualism.

Centuries later, Rene Descartes introduced a different form of dualism by dividing reality into two worlds: the physical and the mental. Even though Descartes also believed in a spiritual world, philosophers seem to have ignored this and labeled his viewpoint as "Cartesian Dualism", thus acknowledging only two of the three worlds Descartes actually believed in.

In Cartesian Dualism the Physical World was similar to Aristotle's Physical World, but his Mental World was different from Plato's Ideal World by being more personal and dynamic. Plato's Ideal world consists of timeless and unchanging forms while Descartes' Mental World consists of the personal mental experiences of a single individual person.

A couple centuries later, Emanuel Kant argued that we really can't access or know anything about the "real" physical world; we can only know and observe what we experience, first as sense impressions, and second as thoughts that are not dependent on sense impressions at all. He called these phenomena and noumena respectively. He claimed that the "real" Physical World is completely inaccessible and unknowable in principle.

Now several centuries later yet, Roger Penrose argues that at least three worlds clearly exist. He calls them the Physical World, the Mental World, and the Ideal World. (I would prefer to call them the Aristotelian, the Cartesian, and the Platonic Worlds, respectively--and also respectfully to give their originators due credit—but we'll stick with Penrose's terms in this essay).

Kant's phenomena clearly fall within the Mental World, being the mental apprehension of sense impressions. But the noumena might exist in either or both of the Mental World and the Ideal World. For example an individual thinker might apprehend the concept of a perfect circle which would place the concept in the Mental World of that individual. Yet the concept of a perfect circle seems to exist in a much broader Ideal World that goes far beyond that individual in both time and space.

Penrose argues that you can't logically deny the existence of any one of these three "Worlds". First, there obviously is some kind of Physical World out there which is somehow responsible for generating the sense impressions that we interpret as physical phenomena. Second, each of us obviously, intimately, and inescapably experiences conscious awareness of our thoughts and feelings, which constitute an undeniable Mental World, albeit it is unique to each individual. And third, there are concepts, primarily mathematical concepts, which "exist" in some sense prior to any person actually experiencing them in their Mental World. These are the mathematical concepts that are "discovered". The fact that they are discovered implies that they existed prior to the "discovery". Plato's Ideal World is where those concepts reside, which obviously cannot be either the Physical World or the Mental World.

If you buy into Penrose's argument, then reality consists of at least these three "Worlds". In addition, reality may also contain some sort of Spiritual World which may or may not be reasonably described by some religion or superstition.

So, in order to keep our minds as open as possible as we probe into the true nature of reality, we should keep open the possibility of the existence of all four of these "Worlds" as possible components of reality. We needn't posit anything specific about any of them, but we shouldn't preclude any of them out of prejudice either.

Now let's begin our discussion by examining the first feature of space, the mathematical concept of "Dimension". Dimension is a fairly simple concept so it's a good place to start. Being a mathematical concept, it resides in the Ideal World, but with this essay, we hope it will also become part of your Mental World. The concept is rather straightforward: dimension is simply an independent variable. Or in other words, dimension is a degree of freedom. That's it.

This definition is deliberately left general, or wide open. We don't specify what it is that varies, or exactly what it is that enjoys the freedom. The variable may be almost anything you can imagine. The variable might be quantitative, like a number which counts things, or a number which measures the amount of something, or it could even be qualitative like color or softness.

Mathematically it doesn't matter. A dimension is just an independent variable in the system we are considering. Nothing more; nothing less.

If the system we are considering contains more than one independent variable, then we say that it has more than one dimension. If it contains four independent variables, then it has four dimensions.

In mathematics, an independent variable is usually designated by a symbol such as 'x'. Mathematicians typically enclose the symbols for the variables in a list separated by commas inside of parentheses. This is probably most familiar to math students in the two-dimensional case. With two independent variables, e.g. x and y, we designate points of the form (x,y) which we plot on what is called the Cartesian plane on which we have defined a Cartesian coordinate system. (Yes, the same Descartes who came up with his brand of dualism also came up with the familiar x,y coordinate system.) The x and the y are variables which specify how far along the respective x and y axes the orthogonal projection of each point falls. Familiar stuff if you have studied much algebra at all.

Slightly less familiar, unless you have delved a little deeper into mathematics, is the expression of a function of a single variable as f(x). Here f denotes the function, and x enclosed in parentheses denotes a single independent variable. This is the one-dimensional case.

Similarly, points in a three dimensional system would be designated as the triple, (x,y,z). Lists of variables like this, separated by commas and enclosed in parentheses are commonly called vectors, except in the one-dimensional case. In the one-dimensional case, the one variable is called a scalar instead of a vector.

Vectors behave much like numbers in mathematics in that they can be added, subtracted, and multiplied, and they form an algebra by themselves. We won't go into vector algebra here. Just know that mathematically, the number of dimensions is the same as the number of variables listed in a vector expression: e.g. (x) is one-dimensional; (x,y,z) is three-dimensional.

Now, departing from the mathematical world, let's look at the "real world". Dimension in the "real world" is a special case of the mathematical notion. In this case, the variables are more limited. Instead of the variables representing any old quantity, they refer only to position in space. (For the present we will ignore the extension Einstein made to space in his Theory of General Relativity by adding time as a fourth dimension. We will consider only the position of rigid bodies in space, not the position of events in space-time.)

Position is a variable which specifies where something is, or where one may find it. The concept of "position", or "where" is familiar to each of us in our phenomenal worlds. It is not so clear what the concept might mean in the inaccessible underlying real world. We'll confine our discussion to the phenomenal world.

In the one dimensional case, one specific value of the variable means the location of a particular point on a line which has been "marked off" by the real numbers. In the two-dimensional case, there are two of these number lines, or axes (I mean the plural of 'axis', not big hatchets), which are perpendicular to each other and intersect at their respective zero points. An ordered pair of variables, or a two-dimensional vector, say (a,b), then specifies a particular point in the plane containing the two axes in the familiar setting of a Cartesian coordinate system.

Although harder to draw on paper, this concept can be extended to three dimensions. The set of numbers over which the variables can range is called a space. More accurately, the range is a subset of the space, meaning that the space is formed from the entire set of numbers while the range is some limited set of those numbers which we are considering in our mathematical system.

In the "real world", we have examples, or approximations, of spaces that are of 0, 1, 2, and 3 dimensions, so we have quite a bit of information to use in trying to visualize space. I say that these examples are approximations because first of all, as I said before, we don't have access to the "real world" but only to our sense impressions in a phenomenal world, and secondly, all "real" examples are imperfect if you look closely enough. Here are some examples of these "real" approximations.

A single fixed point, such as a sharp corner on a stone monument, is an example of a 0-dimensional point. Unless you move the monument, there are no degrees of freedom for this point. It stays right where it is; it is not variable. But under a microscope, the "point" is seen to be a bumpy surface of some size no matter how highly polished the stone corner is.

A line, such as the sharp edge of a counter-top, is an example of a 1-dimensional space. Here there is a range of points which exist on the line which we can locate by specifying values for the one variable, the one variable being, for example, the number of inches along the edge from the left-hand corner of the counter-top. Here again, a microscope would show that the line isn't really straight and uniform like a number line should be.

Moving up a dimension, a sheet of paper is an example of a 2-dimensional space, called a surface. Here again we have to disregard the thickness of the paper and its rough texture. The familiar Cartesian coordinate system shows how two variables are used to locate distinct points on the sheet of paper.

Up one more dimension yet, solids, like bricks, or even liquids like a still bucket-full of water are examples of 3-dimensional structures existing in 3-dimensional space.

And, up one more dimension yet, four-dimensional spaces are harder to come up with in "reality". In fact, it seems that there aren't any. Henri Poincaré gave a method of determining how many dimensions there "really" are in our world and he concluded that there are only three. Kant agreed with Poincaré on this question, as do most scientists today. I don't.

Poincaré observed that 0-dimensional points can bound and separate a 1-dimensional space. That is, if you have a line, or a 1-dimensional space, you can form "boundaries" by placing two distinct points on the line, and by so doing, you have bounded the space between them. You have separated the space into at least two separate subsets. The two points are the boundary, and the two subsets are 1.) the interval between the two points, and 2.) all the points on the line that are not on the interval. His reasoning was that if you placed a pen tip on the line anywhere in the interval and tried to move the tip, keeping it in contact with the line, to any point outside of the interval, you couldn't do it without having your pen tip coincide with one of the "boundary" points. This is true whether the line is "straight" and goes off to infinity in both directions, or whether the line curves back and joins itself to form a circle or other closed loop.

But if we have a 2-dimensional space, like a sheet of paper, then two points cannot bound a subset of that space. Following the same rule of keeping our pen tip in contact with the space, i.e. the paper surface, we are able to move the pen tip from a point in the interval, off the line, go around the erstwhile boundary point, and back onto the line on the other side without the pen encountering the "boundary" point.

However, a closed 1-dimensional curve, like a circle, on that same sheet of paper can bound and separate the 2-dimensional surface. If we start with our pen tip inside the closed curve, we cannot move it to any point outside the circle without either lifting our pen from the paper or having the pen contact some point on the boundary circle. This demonstrates that the paper is 2-dimensional.

Moving up one more dimension, closed 2-dimensional surfaces, like the surface of a balloon, can bound and separate 3-dimensional space. The space inside the balloon is bounded by the balloon and is separated from the space outside the balloon. This, according to Poincaré, shows that we live in at least a 3-dimensional space.

Poincaré went on to conclude that since we can't bound a portion of our "world" with a closed 3-dimensional structure, our "world" must consist of only and exactly three dimensions. But, since we can only deal with appearances, or phenomena, in our "world", we really don't know whether or not 3-dimensional structures might exist in higher dimensional spaces in which our apparently 3-dimensional world is embedded. All we might need is a 4-D pen and we could show how to trace a line from inside the balloon to the outside without popping the balloon. Unfortunately, all of our pens are 3-dimensional structures.

Most scientists and mathematicians would consider the idea of extra dimensions, or "hyperspace", to be outlandish fantasy not worthy of our consideration, but I suggest seriously that we cannot rule it out so easily.

The second, and last, feature of space we want to look at is the concept of manifolds. Even though this concept may be less familiar to most people, it is not all that hard to comprehend, unless you want to be mathematically rigorous. It just takes a little thinking about some familiar objects.

A manifold is roughly defined to be a subspace of a "bigger" space. "Subspace" means simply that it is part of a space. "Bigger" usually means a space of higher dimension, but "bigger space" can also mean a space of the same dimension but greater in extent. In other words, a manifold is a subspace with dimension less than or equal to the dimension of the space in which it exists. That's the definition you should burn into your brain at this point. (It will be on the test!)

In the case where the dimensions are different, the manifold is said to be "embedded" in the higher-dimensional space. (It is embedded manifolds which will be of most interest to us in this essay.)

In order to be somewhat systematic, let's begin a list of features of manifolds that will be important in the remainder of this essay:

I. Manifolds have dimension less than or equal to the embedding space. This feature comes straight from the definition.

II. Since manifolds are subspaces, they have all the same features and characteristics as spaces do. What that means is that any mathematical structures defined in a space will behave exactly the same as if they were defined in a manifold of the same dimensionality. I fear that this feature is commonly ignored or overlooked by most scientists causing significant errors in their thinking.

III. Manifolds need to be "smooth". Mathematically that means that they need to be everywhere differentiable. But for our purposes, you can just think of that requirement as as meaning that the manifold can't be kinked or creased. What is important is that if you zoom in on any part of the manifold far enough, it will eventually appear to be smooth and flat.

The concept of flatness is very important to our discussion, so let's elaborate on the idea now. One of Euclid's axioms, that lasted without question for a couple of millennia, was that any two points determine a straight line. The concept of a straight line was accepted without questioning what it really meant. In Euclid's time, a sufficiently taut string was about as close as you could get to a straight line. Later, the path of a ray of light was taken to be ultimately straight.

Nowadays, we know that the influence of gravity will deflect both the string and the path of a photon by some small amount. So what is a straight line in our phenomenal world anyway?

Here again Euclid offers an answer. A line is said to be straight if and only if it is the shortest path between two points. That's a good enough definition in the realm of mathematics, but it leaves open some questions when we try to apply it to the "real world". We can see the problem when we consider navigation on the surface of the earth. Ships on the high seas follow great circle routes in order to minimize the travel distance between ports. So are those great circles straight lines? The answer is yes and no: "Yes" in spherical 2D geometry; "No" in Euclidean 3D geometry.

If you consider the space available to the ship to navigate in to be the 2D surface of the ocean, which is a perfectly reasonable assumption, then the great circles are straight lines. But if you consider "reality" to be the 3D space in which a spherical earth with its surface oceans exists, then, as the name implies, the great circles are circles and not straight lines. A straight line would be defined as a line that goes down at some angle into the ocean at each port, heading straight toward the second port. If the ports are far enough apart, the line may even go through the mantle, or the core of the earth. That is hardly a path the ship could take even though geometrically it is shorter than the arc of the great circle connecting the two ports.

This example is a perfect illustration of a manifold. The surface of the earth is a 2D manifold embedded in 3D space. It is 2D because there are only two degrees of freedom available to a ship's navigator as he/she plots the ship's position, namely latitude and longitude. And it meets the two conditions we have mentioned so far. First, it is smooth (you have to ignore the waves or think of a dead calm day), and second if you look at a small enough portion of the ocean surface, the earth's curvature cannot be detected so the portion seems to be flat. There are no kinks or creases in the surface of the ocean (on a dead calm day). On the large scale, the surface is curved, but on the small scale it is not. This is an important feature of (some) manifolds which we should emphasize here. A manifold may be "bent" or "curved" even though the space in which it is embedded is not bent or curved. That is worth repeating (in fact, it will probably be on the test.): A manifold may be "bent" or "curved" even though the space in which it is embedded is not bent or curved.

When I was in graduate school, sitting in a class on Differential Geometry, I vividly remember the professor demonstrating the proof of a theorem which said that A space cannot be bent unless it is a manifold which is embedded in a space of at least one higher dimension. I have not been able to find a reference to that theorem since then, but I still believe it is true. It seems obvious when you think about it, but of course, that is not a proof. The theorem says, for example, that you can't bend a sheet of paper that is lying on a flat tabletop unless you lift some portion of the paper up off the surface of the table and into the 3D space above the tabletop. I believe that this theorem holds the key to our ability to understand how space might exist in greater reality, so I'll repeat it: A space cannot be bent unless it is a manifold which is embedded in a space of at least one higher dimension.

IV. Manifolds may or may not have boundaries. There are three cases involving boundaries:

a) The manifold has a boundary. N.B. (note bene – that means, this is important so make a note of it and remember it! It will be on the test!) The boundary of a manifold is necessarily an embedded manifold itself, and it must be exactly one dimension less than the manifold of which it is the boundary.

b) The manifold does not have a boundary, and the manifold "closes in on itself". I suspect that this is the most common type of manifold in our phenomenal world, and maybe even in the "real" world.

c) The manifold does not have a boundary, and is "infinite" in extent, i.e. it does not close in on itself. A common example is the 1D straight line in 2D Euclidean geometry. The line is considered to go off to infinity in both directions.

It's easiest to grasp these ideas by considering some specific examples of manifolds in various dimensions. Here are fourteen more examples with the concept and a real-world approximation for each: (As an exercise for the reader, it might be a good idea to consider each of the fourteen examples below in the context of each of the four features I through IV above and satisfy yourself that each of the four features appears in each manifold example.)

Embedded in 1-D space:

A. 0-D manifold – a point on a line – a dot on a piece of thread

B. 1-D manifold – a line segment on a line – a contiguous colored section of a piece of string

Embedded in 2-D space:

C. 0-D manifold – a point on a plane – a dot printed on a sheet of paper

D. 1-D manifold – an unconnected curve on a surface – a spiral drawn on paper

E. 1-D manifold – a connected curve on a surface – a circle drawn on paper

F. 2-D manifold – a finite subset of a surface – the disk inside a circle drawn on paper

G. 2-D manifold – an infinite subset of a surface – the part of a sheet of paper above a horizontal line drawn on the paper

Embedded in 3-D space:

H. 0-D manifold – a point in space – the point of a needle in a room

I. 1-D manifold – an open curve in space - a strand of curly hair

J. 1-D manifold – a connected curve in space - a loop of string

K. 2-D manifold – a closed surface in space – the inside surface of an inflated balloon

L. 2-D manifold – an "incomplete" surface in space – the inside surface of an intake manifold with one valve open

M. 3-D manifold – an infinite subset of space bounded by a surface in space – the space below a horizontal plane

N. 3-D manifold – a finite subset of space bounded by a surface in space – the air inside a basketball

With this many examples available to us, it shouldn't be too hard to grasp the general concept of manifolds. Now let's revisit some of the specific features of manifolds in the context of some of these examples. If you did the suggested exercise, you may have some questions we'd like to clear up now. If you didn't do the exercise, this might make up for it.

To start with, let me admit that I had a hard time coming up with the word choices I used in the examples. In mathematics, there are some terms which have very rigorous definitions and have very specific connotations. They are not used casually in mathematics, but I used them rather casually in the examples above. The terms are,



compact/ non-compact


I will not use, nor try to explain, the mathematical definitions of these words here; that would take us far beyond the scope of this essay. Instead I'll use some of them in commonsense vernacular. The problem is that the meanings of some of the terms changes as you consider different dimensions. We'll try to keep things straight.

So now let's revisit our list of features of manifolds with examples A through N in mind.

Feature I. Examples B, F, G, M, and N are all examples of manifolds with the same dimension as the embedding space. As I said, these are not of too much interest to us. The rest of the examples are of manifolds embedded in spaces of higher dimension than the manifold. Those are the juicy ones.

Feature II. Manifolds act just like spaces of the same dimension. In other words, a 2D manifold acts just like a space of 2-dimensions. (Examples F, G, K, and L) In any of those examples, if you had a suitable pen, you could draw pictures and write words on the manifold just the same as you could on a tabletop, or other 2D space. We have to be careful, however, to note that the manifold may be curved, or bent, so that it is different in that respect from the tabletop. But Feature III comes to the rescue here and allows us to zoom in far enough so that the curvature becomes negligible. And, anyway, we don't really know that any true space is flat and straight. Einstein was the first to show that our 3D space is undoubtedly curved. So spaces and manifolds of a specific dimensionality act alike. It is very important to understand this.

Feature III. Manifolds are smooth. Except for the 0D manifolds, for which smoothness seems meaningless, all the examples are smooth if you focus on a small enough part of it.

Feature IVa. Manifolds with boundary. In B, D, and I, the boundaries are the end points of the string, hair, or spiral. In F the boundary is the circle. In L the boundary is the valve seat, and in N the boundary is the sphere forming the inside surface of the basketball. The important thing to note in these examples is that the boundary is exactly one dimension less than that of the manifold.

Feature IVb. Closed-in Manifolds without boundary. These manifolds may enclose or isolate a portion of the embedding space as long as (or as the mathematicians say "if and only if") the dimension of the embedding space is exactly one dimension higher than the manifold. Thus even though these manifolds don't have boundaries, they themselves are, or form, boundaries for subsets of space one dimension higher. (This feature provides the basis for Poincaré's method of counting dimensions.) In E, even though the circle has no boundary, it forms the boundary for the disc inside which is a dimension higher (the circle is 1D and the disc is 2D). In K, the balloon has no boundary but it forms the boundary for the space inside the balloon which it encloses. (the balloon is 2D and the space inside is 3D). J does not enclose a portion of space because the embedding space is two dimensions higher than the manifold, not one. That is why you can inflate a balloon but not a loop of string.

Since, as I mentioned earlier, this sub-feature is so important in our phenomenal reality, I will digress and mention some phenomenal facts. If you let matter just "do its thing" in our 3D phenomenal world, in many cases it will form spheres that are 2D manifolds without boundary. Examples are the surfaces of tiny water droplets that make up fog, bubbles that get loose and float around in the air, the surfaces of planetary and stellar objects that are massive enough that gravity levels the really big bumps. Planets, stars, and many moons form spherical surfaces which are 2D manifolds without boundary (e.g. planet earth has no "edge" that ancient mariners were afraid of falling off of). Smaller, lumpy bodies, like asteroids and smaller moons still form 2D surface manifolds without boundary, although you have to zoom in at some places in order to meet the smoothness requirement. In addition to these examples, almost everything we see and interact with in our earthly phenomenal world is a 2D surface manifold without boundary. The basketball player touches and interacts only with the outside surface of the basketball. The outside surface is a 2D manifold without boundary. The painter will only apply paint to the 2D surface manifold that forms the boundary of your house. (Yes, you have to think a little about this one, but it's good exercise.) When we take pictures, we get the image only of the boundary surfaces of the objects we think are "out there". And so on....

Feature IVc. Open-ended Manifolds without boundary. G is probably ambiguous and I may be wrong in my characterization of it. But you could consider the "half-plane" above the horizontal line to be "half bounded" by the line. The line is one dimension lower than the half-plane so it qualifies as a boundary in that way, so maybe it belongs in IVa. I'll leave that determination as an exercise for the reader. M is the same ambiguous case as G except it is one dimension higher. I'll leave that to you as well. This category also includes A, C, and H as special or degenerate cases simply because 'boundary' is undefined for 0D manifolds. That is because boundaries must be one dimension lower than the manifold, and negative dimensions are not defined.

My profound apologies are extended to all readers who slugged their way through all of that and found it tedious or unrewarding. If it seemed too overwhelming you can try going over it one more time, or you can just focus in on one or two examples and realize that the concept extends into more combinations of dimensions. To simplify things, just think of 2D manifolds embedded in 3D space. They are very familiar to us. They are the surfaces in our world that you can paint or write on. They can be bent, like a sheet of paper, or they can be stretched like a sheet of plastic wrap. That will allow you to contemplate manifolds adequately, but you need to keep in mind that manifolds exist in higher and lower dimensional combinations as well.

Now, to review the important features of manifolds, they are always smooth down at the small scales, they can be gently curved and they may exist inside spaces of higher dimensions. They may or may not have boundaries, and they may themselves be boundaries of higher dimensional manifolds.

Our purpose in this essay was to examine the concept of space. We see that space comes in various dimensions, our familiar space seeming to be 3D. We also realize that manifolds act just like spaces, except that they can be gently curved, so we really can't be sure whether our 3D space might be a manifold embedded in a higher dimensional space or not. The fact that we observe a gentle curvature of space at the astronomical scale seems to suggest that it just might be an embedded manifold. This opens up some powerful possibilities for extraordinarily complex structures that might exist in the higher dimensional space in which ours is embedded. It also means that since all of our instruments and all other means of observing are parts of our 3D manifold, that our access to the higher dimensions is severely limited.

Having reached this level of understanding of dimensions and manifolds, let's close the loop and consider some observations and implications for the four possible components of reality we discussed at the beginning: The Physical World, the Mental World, the Ideal World, and the Spiritual World. As we noted earlier, all these components are beyond our grasp. Besides our own individual access to our own thoughts in our private Mental World, the only access to information about the components of reality are in the phenomena we experience. So let's start with this "Phenomenal World".

Everything we observe, or measure, or describe with any clarity and specificity at all is part of our phenomenal world. And it is only via information about the phenomena in this world that we can communicate anything about reality to one another so that we can share these concepts in our respective Mental Worlds. So what have we observed and documented about this phenomenal world that we can share and ponder? Well, that is exactly what the body of science is.

So let's start there. What does science have to say about the space we seem to live in? Geographers and Astronomers have given us a good picture of our 3D phenomenal world and how it moves and changes over the course of time. As we noted, Poincaré has demonstrated that the phenomenal world is 3D. Einstein has demonstrated that this 3D space is curved. The curvature, however, is on a fairly large, even astronomical scale, so that at small human-size scales, it appears to be flat.

With just that amount of information, and from the theorem mentioned under Feature III above, we can conclude immediately that since our 3D space is curved, it must be a 3D manifold embedded in a higher dimensional space. So the embedding space must be at least 4D.

From Feature II, we know that our 3D phenomenal world, since it is a manifold, must behave exactly like a space of the same dimensionality, namely 3D. What that means is that it is not possible for 4D or higher structures to exist in our 3D manifold. And that means that we can't access anything outside of our manifold even though there is a lot of extra 4D and maybe higher space existing "out there". In our phenomenal world, all of our access to information about the physical world comes to us via sense impressions. These are all features and manifestations of 3D structures and effects. This includes the objects we have access to as well as the means by which the information about these objects gets generated and transmitted to our sensory organs and even the pathways and mechanisms from these organs to our brains and from there to our minds, where it becomes accessible to our Mental World. The manifold itself limits what information we can have access to via this set of mechanisms.

The important conclusion to draw from this is that we should not expect to be able to "see" or otherwise detect higher spatial dimensions if they exist. Therefore the commonly accepted notion that any extra dimensions must be "compactified" or "rolled up" so small that they are undetectable is a completely unnecessary approach. The reason we can't "see" higher dimensions is that all of our "seeing" capability depends on 3D structures and effects, and these are confined to our manifold. Otherwise, it would be like expecting Edwin Abbott's flatlanders to be able to somehow "see" objects in the 3D room in which their 2D world is an embedded manifold.

This realization opens up a host of interesting possibilities. First and foremost, it provides a "place" for some, or all, of those mysterious components of reality to exist. If there is a Heaven, for example, as many religions maintain, then it certainly could exist in higher spatial dimensions which are inaccessible to us. Similarly, if there is more to human beings than simply our 3D phenomenal bodies, such as a soul or a mind, then the extra parts could exist in the higher dimensions and still be inaccessible to us. This extra space could also house Plato's Ideal World somehow, and it could house the ontologically ultimate physical world which Kant as well as modern scientists find unreachable and inaccessible.

From a physics standpoint, higher dimensions of space open up new possibilities for explaining phenomena in our 3D world. String theorists are busily exploring some of these possibilities now. Early in the 20th century, Theodor Kaluza showed that 5 dimensions unifies gravity and electromagnetism. Extra spatial dimensions provide extra directions in which vibrations may occur. Boundaries of higher dimensional manifolds might provide the "aether" or medium in which waves might "wave". For example, the surface of the ocean is a 2D manifold in our 3D world, and that surface is host to familiar waves and swells on the ocean. A similar thing could be happening in higher dimensions.

Roger Penrose has ruled out higher dimensions because he believes there to be too much freedom out there and he finds the possibilities too overwhelming. I, on the other hand, see these enormous possibilities as exciting ones to explore. And I think we can enlist mathematics to reduce the complexity and number of possibilities down to something comprehensible.

It could be, for example, exactly as Plato described with his cave analogy, in which the inhabitants only knew of the shadow world (2D) on the wall of the cave when in reality, the shadows were projections on the wall of real 3D objects in an invisible and inaccessible more real world behind them. Similarly it could be that our 3D phenomenal world is a projection of a more real world of 4D or higher.

Here we might pause and think about how higher dimensional objects might be projected into lower dimensions. There are several ways. In thinking about reducing 3D objects to 2D, which is common in our world, we might section the object, project the object in any of several ways, or we might take only the boundary of a manifold in our space.

Sectioning means slicing through an object using a "knife" that is one dimension lower than the object. We commonly reduce 3D objects to 2D manifolds by sectioning them. The sections carry some of the information from the 3D object.

Sections are common in preparing specimens for microscopic study by slicing them in thin sections. Sections are also common in drawings of man-made structures like houses or engines. Sections of geological strata appear at outcroppings or canyon walls which reveal something about the 3D structure of the ground behind them. It takes a little imagination to visualize it, but our 3D world might be a section of a 4D structure.

Projections also reduce objects to manifolds of lower dimensions. Of the many types of projections, the most familiar one to us is the perspective projection of the boundaries of 2D manifolds embedded in our 3D space. Let me explain. When we look at a scene, say of a basketball game, we see the floor of the arena, the bodies of the players, and the basketball. But do we?

We see only the surface of the floor which has been varnished or painted. This is a 2D manifold which forms part of the boundary of the 3D floorboards. Similarly we only see the skin of the players, and not even that in areas covered by the clothing and shoes. And of that, we see only the half that extends toward us. We can't see the back side unless the player turns around. We are seeing the 2D manifold of the boundary of the player and only half of that at one time.

Similarly the ball appears to us as a 2D disk. But with our binocular vision and some brain tricks, we know that the surface of the basketball disk bulges out toward us. And here again, we see only half of the surface of the ball. The surface of the ball is a curved 2D manifold which is the boundary of the 3D object called a basketball which is composed of rubber and air. I mentioned that all these projections are perspective projections, which is due to the geometry of the light rays from the objects to our eyes. This type of projection makes parallel lines seem to converge at a distant point. I think you are familiar with this effect so I won't elaborate.

Other similar projections that are not perspective, such as orthogonal and isometric are used in 2D drawings of 3D objects. Mercator projections, among others, are used to project our 3D earth onto 2D maps.

Yet another type of projection is shadows or silhouettes. This is the type of projection Plato used in his cave story. Shadows and silhouettes can carry some information about the shape, size, and motion of higher dimensional objects and deliver it to a lower dimensional manifold.

Here again, it is a stretch of our imaginations to try to visualize how any such projections might explain how our phenomenal 3D world might be produced by more complex objects in a 4D world. But I think it might be worth the exercise anyway.

In addition to sections and projections, a third possibility for our 3D manifold being a reduction of a 4D object, is that it might be the boundary of a 4D manifold. We don't have any good ways of visualizing how a 3D structure could bound a 4D structure, but there are some simple ones. I won't go into them here, but they do exist. It just takes a lot of imagination and work to visualize it. This, by no means however, is a reason to rule it out as a real possibility.

In summary, we have learned so much about our phenomenal world through the methods of science, and we have learned so much about the implications of myriad mathematical concepts, that we now seem to be in a position to explore possibilities for a better explanation of reality than ever before. In my opinion, the greatest impediment to making progress is the set of limitations imposed on us by institutions that have grown to believe that they alone can distinguish between crackpots and serious thinkers, or between heretical thinkers and orthodox thinkers. Fortunately we live in a time when it is fairly safe, at least for our lives if not for our grant money, to venture beyond these constraints and let our minds run free.

To paraphrase Sir James Jeans (I think it was he), The universe is not only more wonderful than we imagine, it is more wonderful than we can imagine. I say let your imagination run wild and consider all possibilities you can think of and try to see if they can't make sense.

Please send me an email with your comments.

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