« The Fourth Dimension

Analogy is often drawn between one, two , three and higher dimensional beings, and their behaviour, in one two or three dimensions , not respectively.

I will try to express my views on one such analogy which everyone is so grossly mistaken about.

The dimensions in increasing order are : zero, one, two, three, four….etc.

The examples of figures in these dimensions are point, line, square, cube, tesseract… etc respectively.

SECTION 1 : DIMENSIONAL BEINGS

ZERO DIMENSIONAL BEING

Let us imagine a zero dimensional being. He ( no gender implied ) would be a point, and there would be nothing towards either direction of him. A very uninteresting universe. But consider, could there be more than a single, zero dimensional beings? If so they would be able to see each other but not move in any direction. They would look like tiny specks scattered everywhere. Did I say everywhere? Yes, even randomly ( or for that matter, evenly ) distributed in space, in three dimensions. But neither of them would know that they are in three dimensional space. So an infinite number of zero dimensional beings could exist in either a line ( 1 dim. ), a plane ( 2dim.), or space(3 dim), without them having the concept of any movement or imagining to join themselves together with lines etc.

Now , when we say that a being is N dimensional, we automatically imply that the being has conception of dimensions 0 to N. We are 3 D beings so we can deal with and “see”, 0, 1, 2 & 3 dimensions. ( You will find out later that an N-D being can “see” only dimensions ranging from 0 to N-1. )

Let us assume that our 0 dimensional being has got eyes to see. Therefore the only dimension which a zero D being will be able to see will be the zero dimension. ( we’ll deal with negative dimensions some other day ). Since he is able to see only the zeroth dimension, he will be able to see only in zero directions. What? This means that he will be able to see in NO direction at all. Which means that he will not be able to see at all. Implying that our zero dimensional being would be only a thinking being without eyes to see. And even though he is a thinking being he would have no concept whatever of more dimensions.This means that there could be many such beings without them seeing each other. That gives us a striking corollary – if we can imagine N dimensions, we are N dimensional beings. Note that we cant draw such a being in our world, either on a paper or elsewhere, because theoretically a zero dimension has NO dominion at all. Even a small point has dimensions. So our zero dimensional being has no chance of existence. Case closed.

ONE DIMENSIONAL BEING.

Things begin to get interesting. The one dimensional measure we would talk of here is a length. ( Breadth and Height are also, after all lengths in other directions ).

So we have some one dimensional beings which have defined lengths. These beings would always continue to move in the same direction. To them it would appear that they are always in a straight line but that might not be always so.

Let us say these beings are moving in a straight line which extends from -∞ to +∞. The scenario would look like this. ( Note that you being a 3D being can see these beings but they cannot see you ).

a,b,c and d are four one-dimensional beings.

Consider four one-D, red colored, (a,b,c and d,) beings traveling in a straight line, towards right. They all can see only in one direction, the one they are moving in. The being a can see b, b can see c, c can see d. But they cannot see the others length. They see the being in front of them only as a single point. If any of them is conscious of its own length, it would consider itself gifted, coz it would see the others only as points. Let us take an example out of this. B sees C as a point, and feels proud that he is gifted with a length. And he knows how long he is. Now if suddenly C were to die and vanish, B would see D as a point and will be unable to differentiate between the “point” C and “point” D. So even if C magically vanishes, B would see the same thing in front of him, namely the point end of D.

a,b are one dimensional beings.

Now consider two, one-D red colored beings, a and b, moving along a curve, towards right. At every point of their journey it would seem to both of them that they are moving along a straight line. This is coz, they can see only in one direction, which would be the tangent to their position on the curve at any given instant. Note that we are 3D beings so we can see these two beings, a and b. Also if b could turn his head he would see himself. Since b cannot do this he is only probably conscious of his own length, and has never seen himself .

What happens when b moves in a closed curve, which has a perimeter comparable to its length?

Length of being is comparable to the length of its path

The red colored being moves along a rectangle(black in color), clockwise , all the time thinking that it is moving in a straight line. Also the being is conscious of his own length, but has never seen himself( as said above ). X is the head of the being, and Y its tail. (Remember the game Snake?). Now when it turns around the top left corner of the rectangle, it would see its own tail, which will appear to be a point. (Note that it cannot see itself from opposite edges, since it cannot turn its head. It will be able to see itself only when it comes face to tail with itself, in a single line). Since the being has never seen itself, it would never know what it looked like. As it continues to trace the upper edge of the rectangle towards right, its tail would move away from its head ( the being is of constant length ). And so our being would assume that its tail is just another “point” being, moving ahead of him in the same direction as he is. Of course , the “point” being would stop whenever he stops and start to move whenever he does. And also the closed path that our being traverses cannot be less than his own length. If not the being would have to adjust his own length ABOVE or BELOW himself in the closed figure, which is not allowed.

Wait a minute. We said that our one dimensional being has length and is conscious of it. It means he can measure. Let us make our being more intelligent. He can now measure length. If he starts measuring, the distance he has traveled along a closed rectangle, then when he comes face to face with his tail, the being would reason that the distance between his face and tail is his own length. Which would contradict the fact that his length which he is conscious of, is really the perimeter of the rectangle minus the distance between his head and tail ( when he sees it ). Now our being would be totally confused. This gives us a deduction that such a being would not be able to recognise his tail, and even if he is conscious of his own length, there would be no way for him to verify it. Note that when a one dimensional being comes face to tail with itself, it does not leave room for any other being in the closed curve.

What if there are a number of different closed curves in space and they can only be traversed along their perimeter and cannot be crossed? This would make way for infinite one dimensional beings.

Movement constraints in 1-dimension

Now let us assume that there are infinite number of closed curves in space, oriented along different planes but all of these inhabit one dimensional beings, moving along their perimeters.

infinite number of one-dimensional beings, all in different planes

Let us say that these three closed curves are along the planes xy, yz and xz, in three dimensions. And each inhabit a one dimensional creature. ( I am fed up of using the term being ).

If a creature reaches the top of circle A and sees straight, it would see one edge of circle B. If another creature is traversing circle B on this edge, creature A would only see one point in front of him ( he can only look in a straight line). The same goes for creature B seeing creature C on his circular trip.

These circles now, cannot intersect. Because if they do, a creature moving on a path would be faced with two choices of movement , two degrees of freedom, at the intersection, which, in our definition of one dimensional world is not allowed.

My point here is that, a one dimensional creature will see another one dimensional creature as a point ( zero dimension ), and would not even know that others like him have a defined length. Remember this for further discussion.

TWO DIMENSIONAL BEING.

Now let us consider that we have a two dimensional being in a plane. Meet TODIM. That is our name for our two dimensional creature. Note that Todim may be of any shape and size, a square, rectangle, circle , oval etc and may even be a mixture of different shapes, for that matter.

Introducing - Todims

Now let us say, that out of these multifarious beings, the one resembling PACMAN, is Todim. We will now see what Todim’s world looks like.

Note that if we draw a smiley , then our smiley looks at us, hence it cannot be considered to be a 2D creature, since it is able to look into the 3^rd dimension, which is perpendicular to the screen you are looking at. If it cannot look into its own plane and cannot turn around it would be only a one dimensional creature ( since it keeps looking front ), but then, since it has a radius and a dimension in a plane it would be a contradiction. Therefore a smiley is a two dimensional being which can look in to the 3^rd dimension. ENLIGHTENED. I will not deal with enlightened beings here.

Here is the point I wish to make. Todim would always see everything as a straight line.

What a two dimensional being would see

...a straight line.

This gives us a very very important deduction. It implies that in a two dimensional world, no creature would know its shape or size. ( Does Pacman know that its round ?). Therefore if I were to draw any 2D figure in front of Todim, he would not know what it was, since he would always see a straight line. Todim is capable only of seeing a line and anything which is of a lesser dimension than a straight line, which of course is a point. Todim, however, would have the conception that if he sees a number of points in front of him, he could join them to make a line. He would also know that if you join to lines perpendicularly it would form a plane, but he would not be able to see that plane. All he can ever see is a straight line.

Todim can construct 2-dim planes...

...but always see a straight line, no more.

A curve becomes a straight line...

Hence a two dimensional creature would have concept of two dimensions, but it would not be possible for him to see planar figures.

Technically speaking, even if a three dimensional object casts a two dimensional projection on Todim’s native plane, Todim would never be able to see it. Remember this for future reference.

SUMMARY OF DIMENSIONAL CREATURES.

A zero dimensional creature would never exist.
A one dimensional creature, conscious or not, of its own length, would never be able to see a length projection. It would see everything as a point. Likewise it may be conscious of some other one dimensional creature’s length, but would only see it as a point.
A two dimensional creature, conscious or not, of its own size or shape, would never be able to see a plane projection. It would perceive everything as a straight line. Likewise it may be conscious of some other 2D creature’s size or shape, but would only see it as a line.

Let us write the above summary as an analogy for a creature of the third dimension. A three dimensional creature, conscious or not, of its own size, shape or dimension, would never be able to see a 3D projection. It would perceive everything as a plane. Likewise it may be conscious of some other 2D creature’s size or shape, but would only see it as a plane. But wait a minute, we are 3D beings. It means that we cant perceive a 3D projection? Yes. And that is what I wish to prove now.

SECTION 2 : INTER-DIMENSIONAL INTERACTIONS.

Leaving the Zeroth dimension alone I begin with my explanation of the interaction between higher dimensions.

For reference sake, lets name our one, two and three dimensional creatures as ODIM, TODIM AND THREDIM. And an additional creature of the fourth dimension called, FODIM.

ODIM

Let us say that Odim_1 and dim_2, are two odims which are moving in a straight line. Suddenly Odim_2 becomes enlightened and hence conscious of the second dimension. It starts moving in a different trajectory.

The strange vanishing of a one dimensional colleague

Odim_1 does not realize that his friend has started moving in the second plane, until the entire length of Odim_2 vanishes ( moves entirely in the other direction ), and so does the point that Odim_1 has been seeing in front of himself ( and may know of the point to be Odim_2).

Odim_1 will not be able to explain the sudden vanishing of the “point” Odim_2. Also if the line Odim_1 is moving on, is part of a closed curve then Odim_2 can turn back and start following Odim_1…and so on.

If a two dimensional object trespasses the one dimensional world all creatures in the one dimension would see their projections as a single point.

Does a 1-dimensional creature see a plane?

Does a 1-dimensional creature see a cylinder?

This reasoning is perfectly consistent with our reasoning in Section1, which was that a one dimensional being is capable of seeing ONLY A POINT PROJECTION. ( ZERO DIMENSION ).

TODIM

Let us now discuss interaction between the two dimensional world and other dimensions. For reference we would use our pacman-like friend, Todim.

We have already seen that Todim will see anything in his plane as a straight line. But what if something in another plane intersects his plane? What does Todim see? I shall prove that in any case he will always see a line, no matter what.

And here is the proof.

Does a 2 dimensional creature see a plane?

Also let a 3D object intersect Todim’s plane…

Does a 2-dimensional creature see a cylinder?

In this case of a cylinder intersecting Todim’s plane, Todim would see a line suddenly appearing out of nowhere in his plane and then suddenly vanishing.

What if the object intersecting Todim’s plane is a sphere? Todim would see a small point appearing out of nowhere, in front of him, the point would gradually grow in to a line which would grow in size, and then finally reduce to a point and disappear.

What if the object intersecting Todim’s plane is a Cone, which enters his plane apex first? Todim would see a point appearing in his plane, which would grow in size into a line and then the line ( equal to the diameter of the base of the cone ) would suddenly vanish, when the cone has completely passed through his plane. You get the idea.

This is perfectly consistent with our observation in Section 1, that a 2D being would be capable only of perceiving a line or a point and NOT any kind of shape, whether it is a projection of a “higher” object or any shape in its own plane.

One very common thought experiment in analysis of dimensions is this. Suppose Todim is surrounded by a rectangle in his plane, which is his house. There is no way for him to come out of it without either, opening or closing a door, or damaging the boundary of the rectangle.

Now suppose that you are witnessing this entire scenario and you “pluck” Todim out of his plane and put him outside the square. Todim would be totally confused. For him he would have accomplished an impossible task. It would be magic.

You can find innumerable examples on the internet about this experiment. But there is a basic issue that this experiment fails to address. What would Todim feel like while he is in your hand, being “plucked” out of his dimension, and waiting to be placed back. What would the world around him seem like to him?

Although we will come to THREDIM shortly, but it seems apt to refer to an analogy to the above “plucking” effect. Suppose you are in a steel box, with no windows or doors. Ok, the box is lighted. Now in the world of three dimensions that we know of, it would be impossible for you to get out without breaking some part in the six walls that surround you. If a four dimensional being is watching you, he could “pluck” you out of your dimension and put you outside that box. What would the world seem to be, when you are in four dimensions.

THREDIM ( There is a reason why, I am not yet introducing THREDIM as a human. )

From our previous discussion, we would remember that

A one dimensional being, Odim, is capable of perceiving the ZEROTH DIMENSIONAL PROJECTION only.
A two dimensional being , Todim, is capable of perceiving the ZEROTH AND THE FIRST DIMENSIONAL PROJECTIONS only.

Now I will prove that a three dimensional being is capable of perceiving ZEROTH, FIRST AND SECOND DIMENSIONAL PROJECTIONS only.

Let us see what happens when we see a 3D object. Light falls on the object and gets reflected onto our retina. There it forms a TWO DIMENSIONAL image of the object. This image is sent to the brain via the optic nerve, and the brain translates it into a three dimensional image of the object.

Remember that I earlier said that a one dimensional being can be conscious of its own length but would never be able to see it. It would only see everything as a point. Similarly a two dimensional being could be conscious of its own size and shape but never would be able to verify it. It would see everything as a line or a point.

Now, when I say “see” I mean, the projection that the reflected light casts on your retina. So when “seeing” three dimensional, I would mean that the light casts a THREE DIMENSIONAL PROJECTION on your retina. Which is never so. Hence we can perceive a three dimensional object, but not “see” it.

“Projection” can also be understood of as a “shadow”.

If you hold a long stick perpendicular to this screen, and shine a torch on it from your end, then you would see a point shadow of the stick on the screen. So the one dimensional stick causes a zero dimensional shadow. Or, its projection on a plane perpendicular to it, is a point.

Now hold a paper in front of the screen with the writing side facing upwards ( or downwards ) and either of the edges towards you. Shine a light on the paper from your end, and you will see a one dimensional “line” shadow on the screen. Thus its projection on a plane perpendicular to it, is a line.

Now hold a box in front of the screen in whatever way you like, and shine a torch at it from your end, and you see a “plane” shadow on the screen, a two dimensional shadow. So its projection in this case is a plane. This is a two dimensional projection, or shadow.

So we, being three dimensional beings can “see” zero, one, or two dimensional projections.

But what is a THREE DIMENSIONAL PROJECTION?

As far as the term projection is concerned, you can practically and safely replace it with the term shadow. Now, when was the last time you saw a THREE DIMENSIONAL SHADOW ? Huh? Four dimensional beings see it all the time!!!

Precisely my point. We can say that we are 3D beings if we are able to “see” 3D projections. As I said above, we can only perceive, the third dimension, but actually our eyes ( the retina ) receives only a two dimensional input from the external world.

But we can also work with three dimensional objects right? We can hold a book in our hand or throw the television set out of the window. Of course, the situation is similar to a very early example in this text that I demonstrated. That of Todim, pushing a line to join it with another line, in order to form a corner, but when he “sees” it he finds it that it is again a straight line. He knows that he just invented a corner, but cant “see” it. He can push it further, pull it, move around it, maybe even break a line in it, but cant “see” it. Which of course is because, his retina ( if he has one ) registers only one dimensional images. Let us extend his capabilities. Our Todim has eyes and a brain. The eyes could register one dimensional images on the retina and transfer it to the brain, which would now convert it into a two dimensional a” picture”. Finally Todim can distinguish between a circle and a square. So our eyes register a two dimensional input and the it is the brain that converts it into a three dimensional perception.

THE QUESTION

We saw earlier about objects of any of the dimensions crossing the world of another dimension. Now that we have given all dimensional beings retinas and brains, our concepts of inter-dimensional interactions will have to change.

I pose a question. A sphere crosses a two dimensional world. What would be the experience of Todim?

Todim’s retina registers :

a point appears out of nowhere.
the point becomes a line
the line grows in length
it reaches a maximum and then starts diminishing
the line reduces to a point
the point vanishes.

Now it is upto Todim’s brain to receive the information and process it. If it can make out that the “line” he saw was actually a circle, which started as a point, grew in diameter and finally reduced to a point and vanished, well and good, our Todim is a thinking creature. But he will never be able to figure out that even those circles were the projection of a SPHERE in his plane. He just cant imagine 3D. What a pity.

Does Todim see the "sphere" ?

A clearer picture

Again, if a sphere passes through one dimensional world then what would Odim “see”?

A sphere through a 1-dimensional world

The one dimensional red colored being, Odim, would now see a series of points, which conceal different lengths behind them. Now if Odim has a brain which can distinguish these points and dictate the lengths of the lines of which these are a projection on his retina, well and good. Now Odim would infer that a series of lines crossed his world, but could never imagine, that all these lines were the chords of a circle.

More interaction...

Can we draw any conclusion from the above ?

CONCLUSION ONE

A THREE DIMENSIONAL object , sphere, passes through a ONE DIMENSIONAL world. The ONE DIMENSIONAL creature “sees” a series of points ( ZERO DIMENSIONAL ). His brain might infer that these ZERO DIMENSIONAL points are in fact projections of ONE DIMENSIONAL lines of varying lengths, passing through his world. But he would never reason that these are chords of a TWO DIMENSIONAL circle. He would never be able to imagine a TWO DIMENSIONAL object.

CONCLUSION TWO

A THREE DIMENSIONAL object, sphere, passes through a TWO DIMENSIONAL world. The TWO DIMENSIONAL creature “sees” a series of lines ( ONE DIMENSIONAL ). His brain might infer that these ONE DIMENSIONAL lines are in fact projections of the diameters of TWO DIMENSIONAL circles of various sizes, passing through his world. But he would never reason that these are in fact projections of a THREE DIMENSIONAL object. He would never be able to imagine a THREE DIMENSIONAL object.

Since most of the examples in study of more than 3 dimensions are based on analogies, let us now write a conclusion three, using the two conclusions drawn above.

CONCLUSION THREE

A FOUR DIMENSIONAL object,4D sphere, passes through a THREE DIMENSIONAL world. The THREE DIMENSIONAL creature “sees” a series of CIRCLES ( TWO DIMENSIONAL ). His brain might infer that these TWO DIMENSIONAL circles are in fact projections of THREE DIMENSIONAL spheres of various sizes, passing through his world. But he would never reason that these are the projections of a FOUR DIMENSIONAL object. He would never be able to imagine a FOUR DIMENSIONAL object.

Means that if a four dimensional object were to visit us from the fourth dimension, it would generate two dimensional projections but we would “perceive” the projections as three dimensional.

Perfect conclusion! With a very important and interesting result. Just like the projection of a THREE DIMENSIONAL object in a TWO DIMENSIONAL world is ONE DIMENSIONAL (conclusion two).

HENCE THE PROJECTION OF A FOUR DIMENSIONAL OBJECT IN A THREE DIMENSIONAL WORLD IS TWO DIMENSIONAL.

So the next time you “see” a FLAT SHADOW ( and perceive a correlated three dimensional object ) moving around anywhere, you know that it is from a four dimensional world. Note that the everyday shadows that we see, are always on a two dimensional plane, a wall, the floor etc, and not on a three dimensional space. Suppose if you are casting a shadow on a wall, and you start rotating, the shadow does not start rotating with you, it just remains on the wall and keeps on changing form. However, a two dimensional shadow ( or projection ) in a three dimensional world would be free to change its plane of movement ( or rotation if you please ), so you would see it “hanging” in space.

IF WE GENERALIZE THIS RESULT : THE PROJECTION OF An N-DIMENSIONAL OBJECT IN A (N-1)-DIMENSIONAL WORLD IS (N-2)-DIMENSIONAL.

Which was what I wanted to prove.

So, people people people! When you say that “If we see a sphere that starts at a small size, grows and then reduces back to small size, we have seen the passing of a 4-D sphere into our 3-D space…” you are WRONG WRONG WRONG. The correct version is that “If we see a circle (not a sphere) that starts small, increases in size and then goes back to small size, THEN we have been visited by a FOUR DIMENSIONAL SPHERE in our 3-D space.”

Don’t scare us plebians by using spheres and crap. Circles are scary enough.