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throng
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Posted 10/25/09 - 08:22 AM: Subject: n^4	#1
What would a tetrahedron be in four dimensions? Would it be four equidstant points in two dimensions, or would a tetrahedron still be three dimensional?
I know that I don't know, so I don't know if I do.

ClaudeHooper
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Posted 10/26/09 - 10:27 AM:	#2
A triangle is 3 points in 2 dimensions. A tetrahedron is 4 points in 3 dimensions. The next step up is 5 points in 4 dimensions. --- A triangle in 3 dimensional space would still be a 2 dimensional figure. The extra spacial dimension wouldn't give the triangle any more "flexibility". Similarly, a tetrahedron in 4 dimensional space would still be an essentially 3 dimensional figure.

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Posted 10/26/09 - 07:56 PM:	#3
In four dimensions a tetrahedron is still a tetrahedron. Three dimensional shapes can be embedded in a plane of a surface, just like we can have circles, triangles and squares embedded in 2d surfaces in our three dimensional world. If you are talking about extending a tetrahedron into the 4th dimension, you end up with a "hypertetrahedron", "pentachoron", "simplex" or a "pentatope", (5 points).
Ethics is the measuring of morality. Morality is the measuring of good. Good is the measuring of benefit. Benefit is the measure of values.

throng
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Posted 10/27/09 - 06:07 AM:	#4
It's a funny thing, in three dimensional space the tet has the minimum number of points. It goes to follow that five points are the minimum quantity to create 4D space. SO... tetrahedrons don't exist as four dimensional objects. If one considers rotating a triangle around a height axis its rotation occupies a 3D space, but a tet can't occupy 4D space under any circumstances. It could occupy a 3D surface, but never be rotated, folded or otherwize manipulated to pass through 4D, or could it? Edited by throng on 10/27/09 - 06:18 AM
I know that I don't know, so I don't know if I do.

reincarnated
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Posted 11/07/09 - 05:06 AM:	#5
throng wrote: It's a funny thing, in three dimensional space the tet has the minimum number of points. It goes to follow that five points are the minimum quantity to create 4D space. SO... tetrahedrons don't exist as four dimensional objects. If one considers rotating a triangle around a height axis its rotation occupies a 3D space, but a tet can't occupy 4D space under any circumstances. It could occupy a 3D surface, but never be rotated, folded or otherwize manipulated to pass through 4D, or could it? Generating additional dimensions via rotation about an axis has some curious properties. If I have a 1 dimensional object (a line segment), I can generate a 2nd dimension by rotating that object about an axis perpendicular to the 1-D axis of the object, but NOT by rotating it about an axis which coincides with the 1-D axis of the object. In other words, I cannot generate the 2nd dimension by rotating about an axis of rotation which lies in the original 1 dimension. However if I have a 2 dimensional object (a triangle), I can generate a 3rd dimension by rotating that object about an axis parallel to (or coincident with) either of the 2-D axes of the object, but NOT by rotating it about an axis mutually perpendicular to both 2-D axes of the object. In other words, I can generate the 3rd dimension by rotating about an axis of rotation which lies in one of the original 2 dimensions. So the question is: How do I generate a 4th dimension by rotating a 3 dimensional object? Where must I choose the axis of rotation in order to generate this 4th dimension? Edited by reincarnated on 11/07/09 - 05:13 AM
crumpled bits of paper, filled with imperfect thoughts... we all talk a different language, talking in defence... and if you don't give up, and don't give in, you may just be ok... (Mike & The Mechanics, "The Living Years")

throng
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Posted 11/07/09 - 06:38 AM:	#6
Ok, how is that done? Edited by throng on 11/07/09 - 06:44 AM
I know that I don't know, so I don't know if I do.

reincarnated
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Posted 11/07/09 - 08:16 PM:	#7
I have no idea. I was hoping someone could explain.
crumpled bits of paper, filled with imperfect thoughts... we all talk a different language, talking in defence... and if you don't give up, and don't give in, you may just be ok... (Mike & The Mechanics, "The Living Years")

magpies
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Posted 11/08/09 - 03:22 PM:	#8
Ok I know nothing about this... So doesn't the 3 dimensions basicaly include all the dimensions? I don't really understand how up-down, left-right, and forwards-backwards left anything out? Isn't the 4th dimension commonly known as time? Be nice if someone could explain this to me in a way I can understand

To Mega Therion
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Posted 11/08/09 - 03:29 PM:	#9
reincarnated wrote: Generating additional dimensions via rotation about an axis has some curious properties. If I have a 1 dimensional object (a line segment), I can generate a 2nd dimension by rotating that object about an axis perpendicular to the 1-D axis of the object, but NOT by rotating it about an axis which coincides with the 1-D axis of the object. In other words, I cannot generate the 2nd dimension by rotating about an axis of rotation which lies in the original 1 dimension. However if I have a 2 dimensional object (a triangle), I can generate a 3rd dimension by rotating that object about an axis parallel to (or coincident with) either of the 2-D axes of the object, but NOT by rotating it about an axis mutually perpendicular to both 2-D axes of the object. In other words, I can generate the 3rd dimension by rotating about an axis of rotation which lies in one of the original 2 dimensions. So the question is: How do I generate a 4th dimension by rotating a 3 dimensional object? Where must I choose the axis of rotation in order to generate this 4th dimension? Uh, you can't chose an axis in 3-space that would allow you to create a 4D object by rotation. Your analogy with the triangle does't word because you can't define rotation about the axis you're talking about. In fact, the only theories I know of where the dimensionality can change are those in which spacetime is not fundamental, as in string theory for example.

reincarnated
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Posted 11/08/09 - 09:42 PM:	#10
To Mega Therion wrote: you can't chose an axis in 3-space that would allow you to create a 4D object by rotation. You also cannot choose an axis in 1-space that would allow you to create a 2D object by rotation - but this (obviously) does not prove that 2D objects do not exist. To Mega Therion wrote: Your analogy with the triangle does't word because you can't define rotation about the axis you're talking about. Why can't I "define rotation about the axis I am talking about"? Could you explain how you arrive at this conclusion?
crumpled bits of paper, filled with imperfect thoughts... we all talk a different language, talking in defence... and if you don't give up, and don't give in, you may just be ok... (Mike & The Mechanics, "The Living Years")

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