02
Nov
09

Imprisoned in 3 dimensions

Almost everyone needs a part of his day just for himself, having a little personal space and a little me time in doing stuff that he assumes no one else in the world cares about. Some people during that certain part of their day sitting in their personal spaces spent their personal time thinking about the “space-time” that they were sitting in, and that’s how the world now knows many important facts about the universe. But, most of us do not spend our free time like that and we’d rather prefer watching a game than thinking about invisible and almost inconceivable stuff (although most of us would admit that it’s not that we have no interest in learning about the universe we live in). The first thing many of us with any kind of knowledge in science imagine as soon as we hear the word “space-time” is “Einstein” with his frizzy hair and crazy equations. Then for the fear of going crazy trying to figure out what the term might mean, we tend to run away from it as far as possible.  I wonder (concerning the ones who think that they might be interested in such topics but still stay away from it) if it is reasonable for one to fully blame the subject matter itself for being too difficult/inconceivable/confusing, or if it is just the physical and mental work of finding suitable sources and conceiving the subject matter that one fears. It is always possible to extract some easily understandable but interesting materials from any complicated subject matter, which is what many popular science books/documentaries/websites are trying to do these days. If anyone is interested in such topics I’m sure going through these sources would be a fun journey for them.

I thought I would start a science/philosophy series about the possible structure of the universe in terms of space/space-time, mostly from a metaphysical view point. The topic seems a little dry at first, but many who get into the details pretty much fall in love with it. Today rather than going into describing what space-time means, I’ll stick to some interesting facts only about space: what our perception of the dimensions of space is, what might be the limitations associated with that perception, and what it would be like to have a higher dimensional space in the same space we conceive to be 3-dimensional.

Is everything we see/feel/perceive all that there is in our surroundings? Is there any possibility that our perception of reality is limited and we cannot conceive beyond what our brain is tied to conceive? We think we live in a 3-dimensional space. When we ask the question how the size of the universe could be we mostly indicate the 3D spatial size of the universe in terms of light years and even then we have hard times dealing with the idea that the universe might have an infinitely large space. This is just the beginning. Scientists have started searching for quite a while if there are any extra dimensions in the same three dimensional space we live in. The search doesn’t deal with the “3D” size of the universe, but rather ponders if the universe has a bigger size in terms of higher dimensions: if the universe contains more than 3 dimensions/directions at every single point in space, or in other words, if the particles in the universe have a greater freedom of movement at any single point in space. Although our brains are not used to dealing with any extra directions other than our 3 usual dimensions and thus it is impossible for us to visualize a higher dimensional space, the fact is that there is no better scientific reasons to believe that the universe can only have 3 spatial dimensions rather than more.

Our familiar space having 3 spatial dimensions means that we can describe our positions in space using only three numbers in three mutually perpendicular directions. For example in Cartesian coordinate system, the directions are indicated by the three perpendicular, x, y and z, directions. Having an extra spatial dimension means that there would be another direction (for example, w) perpendicular to all the three x, y, and z directions in every point in space. Although it is not possible for us to visualize such a direction, we can easily understand it through reasoning. “Flatland: A Romance of Many Dimensions”, a book written in 1884 by Edwin A Abbott about a 2D land, might help understand this concept better. This book depicts a 2-dimensional ‘flat-land’ (flat in Euclidean sense, like any completely flat sheet of paper with no dip in it) where everything is 2D including the houses, people and even the perception of people about the universe. These 2D people cannot see, feel, or imagine the 3rd dimension in any way because all their body features are 2D and thus their eye sight and consciousness are restricted in 2D. Now let’s consider the 2D beings’ experiences in the universe and how we can describe those experiences from our 3D point of view. (Not all of the following points are taken from Flatland, but from other sources as well).

1. The 2D beings’ senses can only experience a 2D space making them think that the universe is a 2D universe.

2. The 2D beings are not able to see the full shapes of any other 2D beings or objects the way we, the 3D beings, are able to see from above. We are able to see the full shape of a 2D square drawn on a sheet of paper from above. But a 2D being residing on the same plane as the square will rather see only the sides of the square with his 1D eyesight than seeing the full square shape. Everything a 2D being sees is just a 1D line, and he tries to understand the shape of that line from comparing the brightness of the different parts of the line (in a similar way we, the 3D people, understand 3D shapes and distances with our 2D binocular eyesight).

3. For the same reason stated above, a 2D being only sees the boundary of a closed 2D square (or any other closed shape) and is never able to see the inside part of the closed sided square that we can easily see from above. It’s like saying that we (the 3D beings) can see inside a 2D room which, from a 2D being’s perspective, is closed from all directions. We not only can see things inside a closed boundary 2D room but also can pick something up from inside the room and put it outside the room using the 3rd dimension without making it cross through the boundary walls of the room.

4. If a 2D solid square (or being) is somehow rotated through 90 degrees angle about its 2D plane (suppose, the xy plane) so that now it enters the 3rd dimension in the z-direction, and rests on the yz plane in such a way that half of the square’s body is above the xy plane (in +z direction) and half is below the xy plane (in –z direction),  the other 2D beings of the xy plane will no longer be able to see the sides/boundaries of the square and will only see a 1D line that contains a 1D segment of the middle part of the square’s body containing the square’s center. The 1D segment is one of the parts of the square that the flatlanders normally weren’t able to see when the closed sided solid square resided on the xy plane.

5. In a 2D land a ‘p’ and a ‘q’ are two similar looking shapes facing opposite directions (mirror images of each other), and however hard a 2D being tries to move/rotate a ‘p’ on the 2D plane, they can never turn it into a ‘q’. But a 3D being can pick up a ‘p’ from the 2D plane and can easily flip it in the 3rd dimension to make it face the same direction as ‘q’ and put it back on the plane so that now the ‘p’ looks like a ‘q’ to the 2D beings. It means that a 3D being can turn a 2D being/object into its mirror image by flipping it using the 3rd dimension.

Now let’s use these ideas to see how our 3-dimensional existence might look from a 4th dimensional point of view so that we can get an idea about the characteristics of a 4th dimension. (Here I am considering the 4th dimension as being a 4th Euclidean spatial dimension).

1. Our senses experience a 3D world and we perceive the universe to be 3D.

2. We are not able to see the full shapes of any 3D beings (e.g. people, animals) or objects the way hypothetical 4D beings are able to see them from the 4th dimension. We see only one side of the body (parts that are on the same side) of any 3D object at a time with our 2D eyesight. But the 4D beings would be able to see the full shape of each of us – front, back, top, bottom, all at the same time from a completely different angle. A 4D being seeing the shape of the top surface of a 3D table is at the same time able to see the shape of the bottom surface, the sides, the legs etc.

3. A 3D being only sees the surface area of a closed 3D solid shape (a room, a person’s body, or any other closed shape) and is never able to see the inside part of the closed surface unless there is an opening on the surface. But a 4D being can easily see the inside of a closed solid even if the surface doesn’t have an opening. For example, a 4D being can see the surface area of our body as well as the parts and processes inside our body – our kidneys, lungs, heart pumping blood etc at the same time without cutting through any part of our body; he can read a book that’s sitting on a bookshelf directly without even opening it. Not only can a 4D being see things inside a closed surface but also can pick someone up from inside a closed room (or any type of closed body) with all the walls, doors, windows closed, and put him outside the room without making him cross through the walls or doors of the room, using the 4th dimension. Magic!

4. If a 3D being’s body is rotated 90 degrees about our 3D world in such a way that now half of his body (head to waist) resides on the ‘+’ direction of the 4th dimension and the other half (waist to feet) resides on the ‘– ‘direction of the 4th dimension, only the cross section of the middle part of his body (the waist) will reside on our 3D world and we will ONLY see a 2D cross section of his waist (a cross section that opens up the internal part of his waist to our vision) and nothing else!

5. Our left and right hands are two similar looking shapes facing opposite directions (mirror images of each other), and however hard  we try to move/rotate our left hand in our 3D world, it’ll never turn into a right hand. But a 4D being can pick up a left hand, can easily flip it in the 4th dimension, put it back and make it a right hand in the 3D world. It means that a 4D being can transform me (or anything else) into my mirror image only by flipping me using the 4th dimension, and then my heart and all parts in the left part of my body will be in the right side of my body and vise versa.

There are many more unusual consequences like these when we consider a higher dimensional universe. I am not claiming that if there are higher dimensions in the universe they have to be Euclidean ones. Physicists are considering as many as 11 dimensions with many logical shapes (i.e. curled up microscopic dimensions, 4D sinkholes in a higher dimensional space etc) for their work in string theory. But I thought it would be a good practice to begin building our intuition for a 4th Euclidean spatial dimension before getting into more complicated structures of dimensions.

The following youtube link is a short, nice and simple visual explanation of the discussion above about flatland and its features:

Some books on this topic:

Flatland – Edwin A Abbott

Geometry, Relativity and the Forth Dimension – Rudolf v. B. Rucker

Hyperspace – Michio Kaku

Warped Passages – Lisa Randall


1 Response to “Imprisoned in 3 dimensions”


  1. January 30, 2012 at 10:16 pm

    This is my “me time” unfortunately i have only 1 minute left so on my next me time i’ll put my comment🙂


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