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!).

0-D and 1-D objects

0-D and 1-D objects

0-D object

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
Edges : 0
Planes : 0

1-D object

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)
Edges : 1
Planes : 0

2-D object

2-D object

2-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)
Edges : 4 (2 X 1 edge from each 1-D object + 2 new edges)
Planes : 1

3-D object

3-D object

3-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)
Edges : 12 (2 X 4 edge from each 2-D object + 4 new edges)
Planes : 6 (2 X 1 plane from each 2-D object + 4 new planes)

4-D object

4-D object

4-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)
Edges : 32 (2 X 12 edge from each 3-D object + 8 new edges)
Planes : 24 (2 X 6 plane from each 3-D object + 12 new planes)

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!

6 faces folding into a cube 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!

3-D object used to fold into a terrasect 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.
Rotating terrasect (faces hidden) 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.
Rotating terrasect (translucent faces) 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!

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About Filya

'As you can see, we've had our eye on you for some time now, Mr. Jokhi. It seems that you've been living two lives. In one life, you're Firoz Jokhi, program writer for a respectable software company. You have a Social Security number, you pay your taxes, and... you help your wife carry out her garbage. The other life is lived in computers, where you go by the hacker alias Filya, and are guilty of virtually every computer crime we have a law for. One of these lives has a future, and one of them does not.' - The Matrix

Posted on August 27th, 2009, in Analysis and tagged , , , , , , , , . Bookmark the permalink. 30 Comments.

  1. I can’t get my head around it lol but when I read Ezekiel chapter one of the four headed beast and the wheels within wheel I can’t help but think Ezekiel is a three dimensional being trying to describe a 4 d one. The other thing is when I think about being a 4 dimensional being I feel you would be able to see things big and small all at the same time. So I could see an ant up close and full perspective as well as the moon without changing focal length not good at describing this and colours in four dimensions move/are alive with their own individual wave like pattern once again hard to describe but I had a vision of it once and it was mind blowing (yes was straight at the time -no substances). Finally just wanted to say that perhaps our issue with getting our head around the Trinity or triune God is the same issue a 3 D person trying to describe and vision a multidimensional God. Thanks for the information by the way I enjoyed it.

  2. This is 4 dimensional grace of Jesus christ our lord which we cannot fully comprehend its scope!


  4. Man, you blew my mind! Nice Work!

  5. really nice article. im a math graduate but i left my math far behind after university and got into computers. this article has really got me interested again. nice

    • Thanks for reading up. Glad you liked it so much. These are such fantastic theories, but most people don’t sit and think about these things. If they did, they would love to read up more and more.

  6. Wow, nicely explained but its quite hard to imagine a 4D object, and i still cant do it! Oh and that 4D animation looks quite…confusing. But it was interesting reading your article. Ah maybe I will see it in my dream while I am sleeping…oneday :p

    • Thank you for reading up and commenting. Glad you found it interesting.
      Yes, trying to imagine a 4D object is hard (as this blog already explains), and I myself find it head-beating to try and understand more than 3 dimensions :)

  7. This article looks great, but why is the text cut off by the sidebar on the right? Can you see that? Maybe it’s my own issue, browser or such.

    I’d like to direct others here….

    • Thanks so much for pointing that to me. Something in the theme must have changed, so I switched to another theme to iron that out.
      I appreciate you liked it so much!

      ps: I am a KCite too :)

  8. I really loved your narration skills.It seemed as if I was sitting in a live class and being instructed very carefully.Not even at a single point I felt that I am wasting my time on some greek scripture.After every point I felt I could understand it.But sorry to say now also I am not able to get the 4D image in my head.I will keep trying but you are superb instructor.By the way are there any 5D or 6D dimension also?

  9. Cool about the 4D Dimensional cube.

  10. I truly enjoyed this article. It attempted, and quite successfully I might add, to explain an amazingly complex concept in a way that makes sense to us mere mortals.

    I also thought it was amazing ironic, that the displays of the 4-d cube, in a 3-d representation on a 2-d display (referring to my monitor). But in reality would never see the full 4-d cube in 3-d. Think of it this way. If you were to take a sphere, and have it intersect a 2-d plane, such as a piece of paper, at any one point of that intersection you will only see a circle (2d representation of a sphere. So if you lived in a 2d world and a sphere intersected your world, you would see a circle, start at a point, get larger, then smaller, and then a point again.

    So if a 4d sphere intersected a 3d world, (such as what we live in) you would see a point appear, a sphere would grow in front of you, then smaller, until it goes back through to the point, then disappears.

    Love this stuff.

    Loved the article, thanks.

  11. maybe this explanation: just as is you put a face of a cube onto sheet of paper (2d) it extends behind the paper, if you put a one cube of a terrasect (like one face of a cube) onto our 3d world the terrasect would extend back into the 4th dimension.

    also, maybe if we could make a 3d projection of a terrsect, like a 2d drawing of a cube, we could better understand it…

    • Hello and thank you for reading up.

      Yes, that explanation works to make one understand why you would see only a small part of the whole object if viewed from a lesser-dimensional world.

      I wonder if someone has tried working on a 3-d representation of a terrasect. In my opinion, it would look like a cube I guess?

  12. How about fifth, six, …, eleven dimension, infinity dimension?

    Please explain to me.

    • Hello and thanks for reading up :)

      Very honestly, those (dimensions 5 and over) are beyond my intellect to understand and explain. And I am not even sure if there are infinite dimensions.

      There are some pretty good 2D/3D representations of 11 dimensional objects on youtube though!

  13. WOW!!!
    I see the point and understand how the 4D would be, but I still cant see a 4D (I guess I am still sane)… The images are awesome and the post quite clear.
    I know what to think of when I hear 4D – moving cubes…

  14. My head’s all messed up! I think I am going to see those 4D cubes rotating in 3D in my dreams tonight.
    A good effort at explaining stuff though…i always like explanatory images!! :D

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