The observation that spatial dimensionality is limited to three dimensions has been for long a puzzle to scientists. Our mathematics do not limit us to three dimensions. Why are there only three dimensions? We are 3D observers, and this makes it easy for us to conceive the observed reality as 3D. We can also quite easily conceive a 2D universe as a subset of our 3D universe, and we see how complex the explanations can get in a 2D universe, for events we find simple in our 3-D universe.
The relativity theory made spatial dimensionality elastic. The space-time continuum was conceived. Four dimensional space-time was proposed and attempts to visualize a 4-D space, as an extension of our 3-D world, became popular. We talk about 3-D space being curved around some 4-D sphere like the atmosphere around the earth.
In science fiction, discussion of alternate planes, or dimensions of existence, have become ingrained. Religious "Heaven" has been moved from the stars and galaxies to these alternate dimensions. In this section I will show you how to scientifically understand higher dimensions, which will hopefully lead you to better understand the higher dimensional universe which we all form part of.
Many modern physicists, in their attempts to unify theory, have proposed the existence of many space dimensions beyond three. The multi-dimensional efforts at grand unification have indeed mathematically helped describe theory and predict experimentally observed facts, but attempts at 4D visualization seem hard indeed. We talk of extra dimensions being curled into minute 3D spaces.
One should keep in mind what we are with respect to the space around us. The answer is that each one of us is a 3 dimensional spatial observation point in space, and that dimensionality is not a property of 'reality', but of the being, the observer. Instead, our spatial dimensionality is a characteristic of our conceptions, our mind. This means it is a characteristic, or property, of knowledge rather than of reality. Spatial dimensionality is a property of the observer rather than of the observed.
So, is everything observed around us just an illusion? Not at all, the things around us will still exist even if no one looked at them. To say spatial dimensionality is a very powerful tool may be one of the all-time greatest understatements. However, if spatial dimensionality is a property of our knowledge, then it is not a complete universal truth, but just a shadow of the truth. The answer, of course, is that our spatial dimensionality is based upon what we see. Of all the senses which a typical person possesss, sight is the one which plays the greatest role in the perception and conception of reality. The perception of spatial dimensions does not have to be based upon sight, hearing or any of the other senses. Our eyes are essentially 2D arrays which sense light reflected from viewed objects. Therefore, we never actually 'see' three spatial dimensions. We see (perceive) stereographic 2D pictures. In our mind, we conceive the existence of a third dimension using two stereographic pictures. As you see, our mind is already 'too busy' converting 2D sensed data to reconstruct a 3D observation picture of reality. For humans to visualize a world in more dimensions than 3D is no trivial task. It may even be impossible, without physically modifying ourselves. If dimensionality is not a property of the universe, but of ourselves, then our attempts to 'visualize' 2D and 4D universes in terms of our 3D abilities is not only futile, it is nonsense. The reality perceived by a 2D being is the same reality as perceived by a 3D being and a 4D being. Their methods of description will vary greatly, but they are each attempting to describe the same thing.
This alternative perspective on spatial dimensionality has offered a rational answer to the question of why do we conceive the universe to be limited to three spatial dimensions. The answer is the universe is not limited to 3D, and most scientific evidence points to higher dimensional universe, but it is we who are limited, due to our senses. Another dimensionality issue that is answered is that of the co-existence of multiple dimensions beyond three. This issue becomes nonsense. An object cannot pass to another plane or dimension of existence, because these planes or dimensions do not exist. No dimensions exist except in our minds.
Dimensions are powerful tools which we use to organise, live and understand the universe. It seems reasonable to believe that a being who can conceive an "n"*D universe can develop more elegant knowledge that a being who can only conceive an "n-1"*D universe. In essence, the more dimensions we can conceive, the more about the universe we can understand. TIME is only a way to organise information about the n*D universe, for all those mysteries which we have not been able to fit entirely into our (n-1)D spatial dimensionality framework. Remember the initial hypothesis was that a being who perceives "x" dimensions, will conceive the universe in "x+1" dimensions. We are now expanding the hypothesis to say that a being who perceives "x" dimensions, will conceive the universe in "x+1" dimensions where the "+1" is "time." Therefore, as beings may increase the total number of dimensions in which they perceive and conceive the universe, there will always be a temporal dimension to the universe for the beings. In the case of a limited dimensional universe of n*D dimensions, then the universe (reality) will be the being (the n*D observator) itself and that is the only possible non-temporal dimension.
If we could increase our perception to 3-D so we could then conceive a 4-D universe, many phenomena which we now describe as occurring at different times would then be described as occurring at different spatial locations. The progressive increase in spatial dimensionality moves explanations from the infinite reservoir of "time" to spatial locations. However, even though the number of spatial dimensions may increase without bound, the conception of "time" remains constant for all beings, from 0D to 3D to "n"D.
From these ideas one can deduct, that we are 3D spatial observation points observing a multidimensional universe around us. For us 3D observers, the "+1" dimension cannot be spatially observed, so our mind perceives different 3D pictures changing through 'time'. Time being the "+1" dimension is so embedded in our minds, that subconscious brain functions may be "hardwired" to better enable its "conception".
Current scientific knowledge is based on a 3D based reality which seems to get in trouble when small dimensions of length or time are involved. Science is now talking of energetic particles that randomly pop in and out of existence, which doesn't make sense if we do not try to understand how higher dimensional universe may work. It is a fact that at the time of writing, the best candidate unified theory is fully compatible with this higher dimensional space theory, namely the supersymmetry.
Supersymmetry is an idea that has been around for decades. It states that every boson has an associated fermion and vice-versa. So a quark, which is a fermion, has a supersymmetric imaginary partner called a squark, which is a boson. Likewise a photon, which is a boson, is teamed up with the photino, a fermion. None of the proposed supersymmetric particles have ever been detected. Scientists say this is because current particle accelerators are just not powerful enough. Science knows that these imaginary components MUST exist, but will never be able to detect/isolate them with the current methods, for the simple reason that they are imaginary. Note that the term 'imaginary' is a mathematical term and does NOT mean 'non-existent'. Any form of matter interpreted in our space-time dimension can be mathematically expressed as a complex (Complex = Re+Im) function of space and time. Lately, some evidence that supersymmetry is real may have emerged from a study of gold and platinum atoms. Teams from the Ludwig-Maximilians University in Munich and the University of Kentucky in the United States have used the Tandem accelerator in Munich to bombard gold atoms with sub-atomic particles. The results of the interactions between the targets and the projectiles, they say, can only be explained by supersymmetry. This is the way to go, since we can only observe these imaginary particles through the motion of the real part.
Understanding 1 dimensional space
Supersymmetry involves the concept of multidimensional space. In order to understand dimensional spaces higher than three, let's start with the simplest 1D case, that of a 1D observer - a line. You might think, well that's quite easy. In fact it is quite easy, but if you really understand it, you might use your knowledge to understand higher dimensions. The amination below shows the observer as a grey line, who is trying to percieve a reality (a 2D circle in this case) in his 1D limited mind. The animated blue line is what he perceives. Note that the reality, the circle, is not changing in time, its radius, colour and all other properties are a part of the reality. The observed thing is very different from this, it is a blue line varying in length WITH TIME. For the observer, it remains a mystery as to what happened to the original full length of line, why and how it changes length and 'pops in and out' of his 'observed reality'.
Understanding 2 dimensional space
Let's now start analysing a 2D case, that of the classic Flatland example, in which a person lives in a 2D universe and is only aware of two dimensions (shown as the blue grid), or plane, say in the x and y direction. Such a person can never conceive the meaning of height in the z direction, he cannot look up or down, and can see other 2D persons as shapes on the flat surface he lives in.
Now we know that 3D space exists, and can conceive that, because we see each other in 3D space. So, what does a 3D reality sphere look like into a 2D plane? The answer is again graphically shown in the animation, which shows a circle expanding and contracting depending on which slice of the sphere intersects the 2D observation plane. In the 2D plane, the thickness of the plane tends to zero, but is not absolute zero. There must be enough thickness for the circle to form and be observed. Thus, the 3D sphere is being differenciated with respect to one of its spatial dimensions (z in our case) across its diameter. Actually, in the special case of a sphere, it could be intersecting the plane at any angle to the z axis, and still be perceived as a perfect circle in 2D. For the person that lives in 2D, the only way to recognise such a 3D structure is through integrating all the circles he sees, on top of each other. But here is the problem, he cannot imagine anything 'on top of each other'. A clever 2D guy has just one simple way to refer to this z-axis, which is constantly differenciating the 3D object, and that is TIME.
I admit this concept is quite hard to grasp, especially when one moves on to describe a 4D universe differenciated by a 3D space, with both real and imaginary axis. The imaginary space dimensions can be pictured as follows. Just try to imagine a person in front of a 2D plane surface, but this time a mirror surface. The person is equivalent to the real part and his image in the mirror is equivalent to the imaginary part. Imagine also that such a mirror is present everywhere he can possibly move. So, the person becomes DEPENDENT on the existence of his imaginary component. That is, if the image is no longer present in the mirror, then one can deduct that the person can no longer exist in reality! Now this was an example of a 3D image reflected on a 2D plane (the mirror).
Understanding 4 dimensional space
Recall ages ago, when most people believed that the earth was flat. Some thought that they would "fall off the edge" of the earth if they went out too far. Little did they know, that if they kept on going, they could possibly end up where they started, having experienced the entire trip as being in a straight line! No matter how far the subject travels (by boat, train, or plane), he will never come to a boundary: there is no "edge" to fall off from!! It is because the earth exists on the surface of a sphere that these properties hold true. Let us now take this a step further.
Launched from the earth is a rocket ship that is travelling out into space. Its mission is to continue outward in a straight line in its current direction until it reaches the "outer edge" of the universe. When will the rocket ship reach the outer edge of space? In the previous example we find a similar situation: the concern of "falling off the edge" of a flat earth - an earth that in reality has no "edge" to fall off from. Now, if our universe reality is not 3D we will find out that the ship will never encounter an outer edge. Not only that, but it could also possibly end up where it started, having experienced the entire trip as being in a straight line! No matter how far the rocket ship travels through space, it will come across no boundary of any kind. These properties would hold true if the universe existed on the surface of a hypersphere in the same way that the earth exists on the surface of a sphere.
The hypershpere is the 4D analogue to a circle in 2D or of a sphere in 3D. How would we picture a hypersphere? The key to approaching something of the fourth dimension is by means of the tool of analogy: we rely upon corresponding lower-dimensional structures we have studied as the means by which their 4-dimensional analogue is constructed. A solid circle is a 2-dimensional object. When cut into 1 dimensional slices, you will see a line, that varies in length between the size of a single dot to its full length. A solid sphere, as shown above in the flatland animation, is a 3-dimensional object. When cut into slices, we find that a solid sphere is in essence an array of solid circles that increase and then decrease in diameter. Having obtained the knowledge we have so far, we now possess the ability to bring these lower-dimensional structures "up a notch" through analogy to envision a 4D hypersphere.
|We cannot directly visualize a hypersphere for the very reason that it is a 4-dimensional object and goes beyond our senses. What we can visualize, however, is a hypersphere in the form of 3-dimensional slices (as is displayed to the left). A hypersphere is in essence an array of 3 dimensional solid spheres that increase and then decrease in size. This would represent our basic conception of the hypersphere, and is shown in the animated picture here.|
As I have said, in 4D space, our 'time' is integrated in a space dimension, and then action at a distance (gravity being the purest example), becomes much clearer to us. Just imagine, in the classic 2D example shown above, that the 2D person is somehow able to impart a force on the circle he sees on the plane. What would the consequences be? He would eventually move the whole sphere and would also change the position of the future circles in the plane. He would also move all points on the circle, as if all points are 'entangled', and the transmission of this force from the point of action to any other point on the circle does not depend on the time it takes for the sphere slice to pass through. So, to the question, is gravity a push, a pull or both, or does gravity act on a body, or is gravity generated by the mass of a body, there is no answer if the problem is analysed only in 3D space, as the interaction between two bodies is just an effect we see due to the interaction on a single body existing in a higher dimension. The interaction between the two different dimensions takes place in the 'mirror plane' where the time dimension does not exist, but is rather a perception of the observer. That also means that issues like 'the finite speed of gravity' clearly make no sense.
If you try to extend this to our existence and to the existence of all matter, you will find that all actions (including gravity) are at work at a higher dimension and we are here in 3D space observing the effects that are being played at this higher dimension. The 4D I am referring to, is quite different from Einstein's 4D Space time, in that it is a 4D space and no time. The time coordinate comes in as a false perception of the 4th space dimension, which we are unable to imagine, analogous to the flatland man who cannot understand height and depth. In this figure, you can see what a 4D sphere looks like when differenciated in 3D space. When one differentiates this 4D dimension with respect to an infinitely small mirror thickness (Plancks length being the best candidate), then you get the universe we observe, with Plancks time being the time taken for each 3D slice to pass through the 'thickness' of the mirror, and such universe is equivalent to Einstein's Space-time.
So, what is the speed of light? The speed of light can now make more sense, it is the thickness of the mirror divided by the time it takes for the next slice. It is the maximum speed of differenciating the 4D reality from a 3D spatial point of view. In our context, the value would be equal to Planck's length/Planck's time which is equal to c, the speed of light. That's why Einstein's theory of relativity although correct, CAN NEVER give us all the answers to our questions, because it is NOT COMPLETE. As Rudolf Steiner stated: "Anything dead tends to remain within the three ordinary dimensions, while anything living constantly transcends them". Applying the same rule to everything, we may modify this statement as "Anything stationary exists in the ordinary 3D, whilst anything moving is being constantly differentiated in each 3D plan,e and hence exists in the fourth dimension". This statement is thus defining the 4D space as space in motion with respect to itself. Click here for an excellent site discussing Space in motion.
What's the evidence for the existence of higher dimensions?
In physics, the inverse square law relation is quite common. This relation is valid for the gravitational attraction between matter, for the electrical forces between charges and for magnetic forces between moving charges. A force that varies with the square of the distance means that the force will increase with the square of the distance if we reduce the distance, and it will decrease with the square of the distance if we increase the distance.
Electromagnetic energy decreases as if it were dispersed over the area of an expanding sphere, 4piR2where radius R is the distance the energy has travelled. The amount of energy received at a point on that 3D sphere diminishes as 1/R2. This clearly shows the origin of the inverse-square law.
Here is a table showing the volume and surface area of hyperspheres of different dimensions:
|Dimension (n)||Shape||Volume||Surface Area|
|2||circle||π r2||2π r|
|3||sphere||(4/3)π r3||4π r2|
As a result, a force that varies with the square of the distance can be considered as a conventional 1-dimensional force vector (x-axis) that is scattered into 2 additional dimensions (y, z) due to the 3-dimensional nature of space. The square power of the distance indicates the projection of such a force over a 3D spherical surface area. But what happens if the force is also acting in higher order dimensions? What if the force is originating force is being projected on a higher dimensional surface area? Are there forces which vary to other powers than the inverse square law?
The Casimir force related by the above equation is known to vary as the inverse d4, which is two orders of dimensions higher than the more common forces, and coincides with a force projected over the surface area of a 5D hypersphere (see table above). Such force that varies with the fourth power of the distance can be thus considered as a force vector that is scattered in a 5-dimensional space. Therefore, it is evident that the field that originates the Casimir force is a 5-dimensional field, that it is in fact a hyperspace field that produces the corresponding effects in our restricted 3D vision of our universe.
Can dimensions be limited, or is the universe really infinite
From our point of view, the universe seems to be infinite, and it seems that it's not only infinite but even ever expanding. Now that you should be able to understand how our seemingly 3D space time universe can all fit in a 4D hypersphere, which in turn can fit on a surface of a 5D hypershere and so on, where a difference in time is equivalent to a different point within its volume, you can understand why the universe as seen by a 3D observing creature/mind has no limits.
Just imagine one of those 2D creatures who cannot understand what is height in the z direction and put him on the surface of a sphere. He would walk round and round searching for an edge for ever, and finally he may conclude and even prove that the path is infinite. Same applies to a 1D creature going round a simple circle, and therefore same applies to us 3D creatures living and travelling around in a 4D universe! In general we can say that a creature with n*D observation capability, will observe an (n+1)D dimension universe as infinite. We also learn that for an n*D observer, the only way to observe a universe of a higher dimension than himself is to 'walk around it' and memorise. A 1D creature cannot understand what is a circle other than observing all the points making it up, one by one. Similarly a 2D creature cannot understand what is a sphere other by observing the flow of circles making it up. We see that in all cases, walking around, or observing the flow through time, is necessary to observe a higher dimensional space.
The question is, how can we know how many dimensions is the universe made up from. All the arguments mentioned above can be applied to any dimension and would imply the possibility of an infinite dimension space. But mathematics shows us that there are yet unknown reasons for which an ultimate dimension may be reached. One very interesting curve is the plot of surface area of hyperspheres of different dimensions, shown below. One would easily think that as we go higher in dimensions, the surface area of the n-sphere would increase at each stage, and yet, something very strange occurs, as a maxima in its surface area is reached at the 7th dimension. Could this indicate the real ultimate dimension of the universe?.
What would an n*D observer see if the universe in which he lives in is his own n*D dimensions ? - the answer is 'a still, or static (frozen in time) spatial shape of n*D dimensions'. A 2D creature does not need to move around the circle to recognise it or know anything else about it, and a 3D creature does not have to flow through circular slices of a sphere to recognise a sphere. Note that the actions move and flow both require the time dimension to make sense, but recognise is an act that reacts to the shape of a static structure and needs no time. For an n*D observer, the n-dimensional universe is static, lifeless, and does not change through time, but has all the knowledge of what's within all lower dimensions. Let's name this ultimate n*D observer as the universal observer. For the universal observer, time does not exist, since both himself and the universe are the same thing and neither himself nor the universe is effected by time in lower dimensions, and from a lower dimensional observer point of view he can be said to be existing from eternity to eternity.
For those mathematically minded, let's take a car accelerating in a road. If we integrate the observed acceleration m/s2 with respect to time we get a car driving at a velocity measured in m/s. We have thus moved the motion of the car one dimension up with repect to time. If we integrate further the velocity with respect to time, we get the total distance covered in metres, no time. So did the road distance exist before or after the car started acceleration? As you see the 'road', the time independent dimension is NECESSARY for all other actions (differentiations with respect to time) to take place, and hence the universe should be limited in its number of dimensions, with the highest dimension being time independent, and being the universal observer itself.
Hypersphere - Wiki
Hypersphere - Wolfram
Hypersphere volume derivation (pdf)- A.E. Lawrence
Global spec CR4 - Hyperspheres general equations