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Thursday, August 4, 2011

How to make your own, 'Nth' dimension?

So we start off by making a 2-d figure which we are very familiar with.. Now take two separate lines on a sheet of paper not intersecting.
Now join the start points and the end points(vertices) of the two, you have a '2-d' figure.
 Make the lines straight and you have a square.
Next make two squares, close to each other as shown
 and join the respective vertices, ones lying on one side join with one lying on the same side.

Now you've a 3-d figure (a cube) on paper, or a projection of a 3-d object on 2-d paper.
Now comes the interesting part, to make your fourth dimension and view it, find out the point of origin of the cube, or the hidden edge so as to say the one that you cannot see if you made a solid cube. This point is actually from where all three of the mutually perpendicular axes originate on paper. Then take your paper and pierce a pencil (or a thin pen refill) right through the sheet perpendicular to it.
 So now you can view the fourth dimension in the 3-d world.
How about taking it further ? Make a 4-d cube, which is quite simple again. Like we did above, following steps joining 2-2 vertices to make a 2-d figure on paper, then joining 4-4 vertices to make a 3d figure on paper, now take two cubes 
and join the 8-8 vertices (respective) of the two cubes and in effect you have a 4-d cube also called a hypercube. 
This is a projection of the 4-d cube on 2-d paper. This hypercube also has a point of origin from where you can draw all the axes. Now take your 'refill' again and pass it through this point on the paper.

In effect you have the projection of 5-d object in the 3-d space. Now take a picture of the whole process and you have the projection of the 5-d object on 2-d space. Take a print of this....

Pass your pencil through the point of origin on the print and you have the projection of a 6-d object in 3d space.
You can keep on doing this for 'N' dimensions.

We realise that the origin is a point which we consider to have no dimension, but in fact lies on all possible axes that can exist in an N dimensional space. It's almost as if the N dimensional space has stretched out or pulled apart the origin along all possible axes making it itself 0-d but nonetheless lying on all possible axes. Making the origin  or the 'Point' the most effective dimensional tool.   
Here's my 5 dimensional object projected on 2-d page (sample not to scale).

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