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Tesseract


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#1

agarrett
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Posted 17 October 2010 - 06:57 AM

http://en.wikipedia.org/wiki/Tesseract

Oh man, I am consuming myself trying to comprehend a fourth dimension that is not time. It is hard, very hard. I just can't imagine and put in my minds eye a three dimensional net of several cubes forming together to get a shape.

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In that image, all angles are perfect right angles, although it does not appear to be so.

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This image shows the net of a tesseract, when morphed together, would form a four dimensional object. I've seen drawings of 2 cubes, connected by an invisible fourth dimension. That fourth dimension can only be represented by a 3 dimensional line.

The way I try to understand it is how when we draw a cube on a sheet of paper, we aren't seeing 6 faces, we are seeing 7 shapes.

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Count the shapes, there are 7. Unfortunately, what makes understanding a tesseract even that much harder, is the fact that you are ALSO drawing it on a 2 dimensional plane, that is, the computer screen or a piece of paper. What really would help in understanding this illusive dimension is having it drawn in 3rd dimensional space.

Of course, we wouldn't be seeing the individual cubes forming a shape (the equivalent of faces for a tesseract.) instead all we would see is separate cubes, not really bound together. It's hard to explain, please refer to the above analogy.

But think about it, that's all the tesseract is, an analogy based off patterns discovered in the three dimensions we can be sure of.

A bit off track with the analogies, I was looking at a book, I didn't read it, but the concept in the book basically describes something like a square, looking at a three dimensional object, pulling a circle out of box, without fracturing the box. This is impossible for the square to comprehend, for it does not see, or understand the third dimension.

Converting that to the third dimension, imagine one of us, looking at someone pulling out a ball of a cube, without fracturing the cube. Now, this brings me to another concept, are video games gateways to the fourth dimension?

We have all seen glitches, part of your arm is in a door, a ball flies through a wall that appears to be solid. physically this should be impossible, but with virtual imagery, it can happen, as physics do not apply. But instead, imagine your character is reaching into the fourth dimension, the fourth axis that cannot be seen, and there is a gap that he sees in that dimension, that we cannot reach into or see, but that in which he can put his hand through.

:D  :Red:  :Red:
:Red:  []  :Red:
:Red:  :Red:  :Red:

Imagine these are cubes, and this is what he sees, and he can put his hand through it, but from our perspective, all we see is a wall.

I don't know how to describe it any better than the post I just made. I think just by writing this I have gained an understanding of this fourth dimension. Anyways, I know most of you won't read all of this lengthy and garbled post of my brain trying to decipher what I have read, so:

TL;DR Discuss the fourth dimension and the tesseract. Perhaps this will help:

http://teamikaria.com/hddb/classic/
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#2

Guke
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Posted 17 October 2010 - 07:23 AM

Right..
Gary Fucking Oak

#3

SchrodingersHat

Posted 17 October 2010 - 08:12 AM

No, video games as you describe them are not gateways to the fourth dimension. Just because there is graphics clipping does not mean part of the videogame is four-dimensional.

#4

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Posted 17 October 2010 - 11:23 AM

A 4D object has a 3D shadow. That 3D shadow can have a 2D shadow, wich is basically a 3 dimension object drawn on paper.
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#5

Yourself
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Posted 17 October 2010 - 06:21 PM

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In that image, all angles are perfect right angles, although it does not appear to be so.

No they aren't, for exactly the reason that they don't look it.  You're looking at a projection of a tesseract.  Indeed, the angles in the actual object itself are 90 degrees, but the projection clearly doesn't have that property.

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This image shows the net of a tesseract, when morphed together, would form a four dimensional object.

Morphed is misleading, there's nothing really tricky about it.  The net is the 'unfolded' version of a tesseract.  Just as you can unfold a cube into 6 squares connected together, so can you also unfold a tesseract (or 4-hypercube) into its 8 cubic faces.  To turn it back, you fold each of those faces up through the 4th dimension and glue them together.  

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That fourth dimension can only be represented by a 3 dimensional line.

Lines are 1D objects.  You can embed them trivially in higher dimensions.  You can only represent 4D in 3D in the same ways that we represent 3D in 2D (for computer graphics): cross-sections and projections.  One is a 3D slice of a 4D space and the other is the 4D space mapped onto a 3D one.  In either case there's a loss of information (like above, the angles between edges wasn't preserved).

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Count the shapes, there are 7.

There are 8.  One of the cubes is in the middle.  It's actually rather easy to understand the properties of the tesseract by analogy to the jump from a square to a cube.  Each time we want to generate a higher dimensional hypercube, we do it by sweeping a lower dimensional one through the new dimension.  For example, to create a cube (3-hypercube) from a square (2-hypercube), we take a square and sweep it through the dimension perpendicular to it.  So this means the number of vertices doubles.  Since a square has 4, a cube has 8.  Likewise, since a cube has 8, a tesseract has 16.  A 5-hypercube 32, and so on.

Another thing that happens is each vertex creates a new edge (connecting the old square to the new one).  So the number of edges in a cube is the number of edges in both squares (4+4 = 8) plus the number of edges created by the vertices (4), so there are 12 edges in the cube.  Likewise, we find that there are 24 (double existing edges) + 8 (add one edge for each vertex) = 32 edges in a tesseract.

We can keep going, there's one face in a square.  When we do a sweep, each edge in the square becomes a new face (notice how a sweep promotes each entity of a particular dimension to one of a higher dimension).  So the 4 edges produce 4 new faces.  Plus the 2 from the original and new squares, gives us the 6 faces of a cube.  Since a cube has 12 edges, the sweep through a 4th dimension produces 12 new faces, plus the 6+6 new and original faces, we have 24 faces on a tesseract.

Now, keeping with the theme of promoting entities by 1 dimension, the square (which has one face; itself) will promote into a cube when we do the sweep.  So the tesseract has a set of cubic facets (which can be unfolded into the net).  How many does it have?  Well, the cube has 6 faces which will sweep out 6 cubic facets plus the original cube and the new cube after sweeping leaves us with 8 cubic facets.

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What really would help in understanding this illusive dimension is having it drawn in 3rd dimensional space.

This is often the least helpful way to understand it.  You won't get anywhere trying to visualize the actual shape, your brain simply does not have the capacity to do it.  The best anyone can do is understand the properties and relationships to other shapes.
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#6

agarrett
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Posted 17 October 2010 - 07:04 PM

Thanks, your descriptions are much better than mine. I wrote that at midnight and I was exhausted, thanks for clearing some stuff up.
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#7

agarrett
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Posted 17 October 2010 - 07:57 PM

Sorry for the double post, but a thought just occurred to me. We all should know that a square has no height, it is infinitely small in that sense, and since it is infinitely small, an infinite amount of square planes can be stacked upon each other in the third dimension.

As this is true, so should the fact, that since a cube is infinitely small in the fourth dimensions sense, we can stack an infinite amount of cubes upon each other in the fourth dimension. But this raises other questions. Would we exist in a universe with a fourth dimension? Just as a two dimensional object cannot exist in our universe, as its volume is always x*y*0, resulting in its volume always being nonexistent.

In a tetra-universe, perhaps volume would be determined by x*y*z*h, and h would always be 0 for us, therefor our volume would be nonexistent in a four dimensional universe.

The whole concept is really interesting to think about.
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#8

killer2329

Posted 17 October 2010 - 10:04 PM

if i remember right the current theory is 11 dimensions in our universe

#9

WrenAkula

Posted 19 October 2010 - 12:58 AM

Yourself said:

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Count the shapes, there are 7.

There are 8.  One of the cubes is in the middle.

Actually, I think he counted that one. The one that's tough to spot (even with an animated rotation) is the 'exterior' cube, since it occupies all other space in the 2D and 3D representations.

#10

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Posted 19 October 2010 - 04:31 AM

WrenAkula said:

Yourself said:

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Count the shapes, there are 7.

There are 8.  One of the cubes is in the middle.

Actually, I think he counted that one. The one that's tough to spot (even with an animated rotation) is the 'exterior' cube, since it occupies all other space in the 2D and 3D representations.

I assumed that he was referring to the net rather than something else.
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#11

WrenAkula

Posted 19 October 2010 - 08:45 PM

Oh right, in that case yeah the inner cube was probably overlooked.