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November 19th, 2017, 08:01 PM   #1
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question about representation of spatial dimensions

Hi friends, I'm not on math field, but I have a question:

it's possible to represent 3 dimensions in space using only 2 dimensions, but

1- is it possible to represent 2 spacial dimensions using only 1 dimension?

2- if it's not possible, why? Because people can represent a tesseract on 3d and a cube on 2 D, why not?

Thanks for answering.

Last edited by skipjack; November 20th, 2017 at 08:16 AM.
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November 20th, 2017, 02:10 AM   #2
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It is possible but you still need to convert the dimension
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November 20th, 2017, 08:12 AM   #3
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could you give me some visual representation, on how would that be, on 1 dimension only , representing 2 dimensions?

thanks
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November 21st, 2017, 04:42 PM   #4
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When you say "it is possible to represent 3 dimensions using only 2 dimensions" but that is not really true. When you project from 3 dimensions to 2 you lose information. There are several different ways to do such projection. One is an "orthogonal projection". Draw the line, from each point on the three dimension figure, perpendicular to the plane on which you are projecting the figure. The foot of that orthogonal represents the corresponding point on the three dimensional figure. Or you could use "perspective projection. Choose a single point on the opposite side of the plane from the three dimensional figure and draw the line from that point to a point on the figure. The place where the line crosses the plane is the point representing the original point on the three dimensional figure.

For example suppose the three dimensional figure is a cube with two faces parallel to the plane you are projecting to. An orthogonal projection would project that to a single square identical to the two faces of the cube parallel to the plane. The other parts of the cube are "hidden" behind that. A perspective projection would project that to two squares, one smaller and inside the first with lines connecting the vertices of the two squares. The larger square is the face closer to the plane, the smaller square is the face farther from the plane. Whether or not you can see the smaller square depends upon whether you are thinking of the sides of the cube as transparent or not.

In the "two dimensions projecting to one', You can the analogue of each of those. For the "orthogonal projection", think of a line drawn beneath the figure and draw line from each point on it perpendicular to the line. For example, a square with two sides parallel to the line would project to a single line segment on the given line with all other points "hidden" by that first line. For the "perspective projection", imagine a point on the opposite side of the line from the two dimensional figure. Draw lines from that point to points on the two dimensional figure. Where those lines cross the given line is the point tnat "represents" the point on the two dimensional figure. For the square, the front side would be a single line segment, slightly smaller that a side of the square, with the back side an even smaller line segment inside the first, perhaps "hidden" by it.
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November 22nd, 2017, 12:02 PM   #5
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could you provide some images, this way you explain is very hard to grasp, thanks
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