4-Dimensional Objects
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tarchon
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In Einsteinian relativity, time and the three classical spatial dimensions are treated more as directionalities of a four-dimensional space-time. Time isn't exactly a spatial dimension but more of an observer-dependent preferred axis that the observer perceives as time. Different observers can however take different axes to be the time axis, which is how the Einsteinian construct fundamentally differs from Galilean relativity. Galilean relativity postulates time as being wholly independent of the three spatial dimensions and invariant.
Extra spatial dimensions come up in a number of new theories about the underlying structure of space and matter, but they're still largely speculative, and the "spatial dimensions" in those cases are usually highly constrained, so that they don't have a macroscopically apparent extent. What they really are is something like extra modes of vibration (or mathematically analogous to vibration), and one idea is that these modes occupy some sort of vestigial spatial dimensions that aren't realized in the universe as we know it. My guess is that these theories are grabbing at straws, and someone will eventually come up with a more elegant explanation for the mathematical features they explain.
Extra spatial dimensions come up in a number of new theories about the underlying structure of space and matter, but they're still largely speculative, and the "spatial dimensions" in those cases are usually highly constrained, so that they don't have a macroscopically apparent extent. What they really are is something like extra modes of vibration (or mathematically analogous to vibration), and one idea is that these modes occupy some sort of vestigial spatial dimensions that aren't realized in the universe as we know it. My guess is that these theories are grabbing at straws, and someone will eventually come up with a more elegant explanation for the mathematical features they explain.
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Brennin Magalus
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Cor Azer said:
There is nothing arbitrary about it. 0! = 1 is perfectly consistent with the Gamma function.
Chaldfont
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Yes, time can be viewed as a 4th dimension, but it is not a *spatial* 4th dimension. There is some weird mixing going on between the time and spatial dimensions, but it's not as if you could do, for example, a rotation between a time and a spatial dimension.
What's we're talking about in this thread is a 4th spatial dimension. For a long time, this was just an abstract notion. Just as we - living in 3 dimensions - can imagine objects in 2 spatial dimensions (we can draw them on a piece of paper - heck, we can ieven imagine life in 2 dimensions, as in Abbott's 1884 book "Flatland"), we can imagine objects in more than three spatial dimensions.
It's really just algebra: 2 points define a line, 3 points define a plane, 4 points define a 3D space... 5 points define a 4D space! (And if those 5 points are equally far spaced apart, they outline a pentatope, the smallest 4D "die".) Just like we can algebraically define objects in 3D, nothing stops us from defining objects in any artbitrary number of dimensions, and analyzing properties of such objects. For convex polytopes, this is fairly easy, because you only need to define a set of points in N-dimensional space. The object formed by the convex hull of these points is the polytope. In 4 dimensions, that's called a (convex) polychoron.
Now, more recently advanced theories of the natural world such as string theory have postulated that there may very well be more that 3 spatial dimensions even in our seemingly 3D world! It is thought that those extra spatial dimensions are "rolled up" very tightly, so we don't actually observe them. Just like a clothesline looks 2-dimensional to us, but it would look distinctly 3-dimensional to an ant crawling on its surface.
What's we're talking about in this thread is a 4th spatial dimension. For a long time, this was just an abstract notion. Just as we - living in 3 dimensions - can imagine objects in 2 spatial dimensions (we can draw them on a piece of paper - heck, we can ieven imagine life in 2 dimensions, as in Abbott's 1884 book "Flatland"), we can imagine objects in more than three spatial dimensions.
It's really just algebra: 2 points define a line, 3 points define a plane, 4 points define a 3D space... 5 points define a 4D space! (And if those 5 points are equally far spaced apart, they outline a pentatope, the smallest 4D "die".) Just like we can algebraically define objects in 3D, nothing stops us from defining objects in any artbitrary number of dimensions, and analyzing properties of such objects. For convex polytopes, this is fairly easy, because you only need to define a set of points in N-dimensional space. The object formed by the convex hull of these points is the polytope. In 4 dimensions, that's called a (convex) polychoron.
Now, more recently advanced theories of the natural world such as string theory have postulated that there may very well be more that 3 spatial dimensions even in our seemingly 3D world! It is thought that those extra spatial dimensions are "rolled up" very tightly, so we don't actually observe them. Just like a clothesline looks 2-dimensional to us, but it would look distinctly 3-dimensional to an ant crawling on its surface.
Mercule
Adventurer
Torm said:Right. I'm not a stupid person, or at least I don't think so, when it comes to these things, and yet I have been left by most of this thread puzzled as to why anyone would think that a "4D" object would have any more sides than a "3D" object when the 4th dimension is an expression of transition through TIME rather than spacial dimensionality......?
Okay, of the three spatial dimensions that we can perceive, which is the 1st dimension; height, width, or depth?
Time may be a dimension, but it is not necessarily the "4th" dimension. Sure, it could be one of four dimensions if you're solving an equation involving height, width, depth, and time. But, what we're talking about here is an object that involves four spacial dimensions -- or a universe in which we could control our movement across four dimensions.
In other words, we're imagining a 5D (including time) object, but the equation for which we're solving involves only four of those dimensions. And it sounds better to say "4D" than to make up a word and say, "height, width, depth, and pucth".
