Concepts of higher dimensions – Part 3 : Higher dimensional objects
Welcome back! This article shall give you your first glimpse at a 4-dimensional object.
Wait! Is that even possible? How does one look at a 4-D object in this world? Didn’t the previous article explain how one cannot see the higher-dimensions in a lower-dimensional world?
Correct, you won’t really see the 4th dimension; but I can still show what it would look like in this world. Just like you can see a 2-D photograph of a 3-D object you can see a 3-D visualization of a 4-D object.
First things first, let me explain how we arrived at the conclusion of how a 4-D object would look like. Obviously no one has ever seen a 4-D object, so this is pretty much theoretical (but interesting!).
Start off with a 0-dimensional object – a point (shown in red). It has (obviously) 1 point only. No edges and no surfaces (or planes).
Points : 1
To create a 1-dimensional object out of this, we take two of the 0-dimensional objects and join them with an edge (shown in green). This 1-dimensional object now has 2 points, 1 edge and 0 planes.
Points : 2 (2 X 1 points from each 0-D object)
Now to form a 2-dimensional object, we take two of the 1-dimensional objects we created earlier (shown in red), and place them side by side. We use 2 new edges (shown in green) to join the end points of the objects.
This 2-D object now encloses a new plane within it (shown in blue).
Points : 4 (2 X 2 points from each 1-D object)
Continuing the same trend, we now take 2 of the 2-D objects created earlier (shown in red) and place them one over the other (at a little distance from each other).
This time, we add 4 new edges (shown in green) to join the points of the 2-D objects. In addition to the 2 planes we initially had from the 2-D objects, the new edges now enclose 4 new planes (shown in blue). If the initial planes formed the floor and ceiling of the box, these 4 new planes form the wall.
Points : 8 (2 X 4 points from each 2-D object)
This is how we arrive at the formation of a 4D cube (it’s not called a cube though).
We take two of the 3-D objects (shown in red & blue) and place them one inside the other. We use 8 new edges (shown in green) to connect the points of the inner object to the outer object. What do we get?
Finally, a 4-D object! Note that the 8 new edges we added now form an additional 12 new planes (click here to count the 12 new planes formed)
This 4-D object is very commonly known as a ‘hypercube’ or ‘tesseract’. It is simply a 4-dimensional cube.
Points : 16 (2 X 8 points from each 3-D object)
|Tesseract (or Hypercube)|
|If you were alert enough (or from a science background), you should have guessed by now – Obviously the construction of the tesseract I explained is flawed! Placing a 3-D object inside another 3-D object doesn’t make it 4-dimensional! It is still only a 3-dimensional object.
True, very true. But what if (and this is going to bore into your brains like a horrific worm in a bad sci-fi flick!) both the cubes shown were of the exact same size? Well then, you would say – both the cubes would fit together pretty snug, leaving no room for the 8 new green edges.
Now what if I told you (this is the part where the worm starts chewing your brains!), the 8 new edges are not only present, but also of the same length as the existing red edges! Obviously this would also imply that the 12 new faces created (the green ones) are perfect squares, and of the same size as the existing faces (ones in blue/red).
This is where our imagination fails us, and our mental image of the tesseract implodes. Sadly, we have the same chances of understanding the geometry of a tesseract as a 2-D ZED had of understanding the 3-D cube – both of them pretty close to zero!
Let me try and explain the geometry of the tesseract in another way. This may or may not help you imagine it better, but hey, at least I tried!
|Here is an animation showing how a 6-face surface folds to form a 3-dimensional cube. Conversely, a 3-dimensional cube can unfold to form a 6-face cross-like surface.
This is self-explanatory. We have all studied how plane surfaces can fold in various ways to form 3-dimensional objects (at the lower end of the skill/imagination/art spectrum, we can crumple a piece of paper to form a 3-dimensional sphere … and then toss it at the person farthest from you!)
We started off with the dark blue face touching 4 other faces. We folded the object in such a way that each and every face now touches 4 other faces (and shares a common edge with it). This is how we form a 3-D cube from a 2-D object.
But what happens if you already had a 3-dimensional object with you and you fold it?
Answer : You either lose your mind trying to imagine what happens, or you lose your mind celebrating your success at imagining it!
|The 3-D object we start off with is a bit complicated, so let me explain that first. Visualize it first as a 3-D cross (without the pink and green cubes). So this time, instead of 6 faces, we have 6 cubes forming the cross. So the dark blue cube now touches 4 other cubes (and shares a common face with each of them).
Add the pink and green cubes to each free side of the dark blue cube. Now, the dark blue cube touches 6 other cubes, sharing each of its 6 faces with other cubes.
Now for the fun folding part!
You have to somehow fold this object in such a way that each of the 8 cubes now touches 6 other cubes. If you do manage to do this, sadly, you are messed up pretty bad! This is not a possible fold in our 3-D world at least. You could bend and mold the cubes to achieve something similar, but the cubes won’t be cube shaped any longer.
But if you were in another world with 4-dimensions, you would have been able to fold it such that each cube of the given 8, touches 6 other cubes and shares a common face with each of them. You just produced your first tesseract!
|Here are some cool animations of a rotating tesseract I googled up. Again, a tesseract can only exist in a 4-dimensional world, so its rotation too can only be explained in the same world. We can only be content with watching a 3-D representation of it.|
|It is obvious from watching the animations that the cube really retains its shape only when it is either in the exact center of the structure, or when it’s forming the outermost cube. But this is because you are watching it in a 3-dimensional projection. It is the same way perspective works in distorting a 3-d object’s dimensions & shapes in a 2-d photograph.|
|So in a 4 (and more)-dimensional world, each cube would retain it’s exact size and shape irrespective of where it is during the rotation of the tesseract. Simply amazing!|
|This was my attempt at making you understand what a 4-D cube would look like, how it could be created and its behavior. If you understood even some part of all this gibberish, it would make me happy!|
|In my next (and I think final) article, I shall explain how extra dimensions account for some weird but amazing concepts like time-travel and teleportation Stay hooked!|
Posted on August 27th, 2009, in Analysis and tagged 4th dimension, dimensions, extra dimensions, filya, firoz jokhi, teleportation, terrasect, tesseract, time travel. Bookmark the permalink. 30 Comments.