Is every cube a hyper-cube ?
Posted on: April 3, 2008 - 11:22am
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Is every cube a hyper-cube ?
I asked my self this question while learning about hyper cubes if in reality I build a cube isn’t this cube a hyper cube ? I mean time is acting like the 4 dimension for this cube and since the creation of this cube there passed some time isn’t the cube and every thing with in it memorized in time and so creating a 4 extra dimension for a 3D cube making it in fact a 4D cube ?
Warning I’m not a native English speaker.
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Yes it is by definition. The hypercube is the generalisation of the cube in n dimensions. The cube is the n=3 hypercube.
If you mean that a cube is a generalisation of a tesseract, it is, but the cube is in 3 dimensions, not 4.
If you mean, and I think you do, that a physically constructed cube IS a tesseract then you have a point, but the mathematics of a 3d cube don't reflect that point at all. In reality it is possible that you may only need to construct a physical cube in order to make observation of a possible tesseract, but in the abstract sense you must construct a mathematical tesseract to see one.
Just a suggestion, if you are interested in higher dimensionality in 3d objects you might go further by studying spheres.
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Upon completion of said cube, it would then become a hyperprism, with the base as a cube. You need to convert units of time to units of distance via c*t, where c is the speed of light. If the length of an edge is d, then c*t = d for it to be a hypercube. So, you'd have to deconstruct it rather quickly to call it a hypercube! Or at the very least, have a high-end "stopwatch" to 'slice' the cube up. Each slice of the hyperprism would then be a hypercube.
But yes, as far as I can tell you would have a four dimensional polytope. Polytopes generalize the notion of polygon and polyhedron in arbitrary dimensions. Specifically, 4-D polytopes are called polychoron (as 2-D polytopes are polygons etc...).
Although, I have always wondered: would it still be a hypercube in Minkowski space? Or the coordinates being 'flat', despite the awkward signature (+ - - -), is enough to still call it a polytope? Or are polytopes restricted to R^n? I'll have to at least google that.
I think it's rather useless to think of cubes in Minkowski space unless you're doing something with all the vectors involved (unless you're an affine person, which I think makes life more difficult). But if you must know, remember that you're supposed to be sitting in Minkowski space. That might help.
I think we've come full circle now, oddly.
Yes we've always been sitting in Minkowski space, that's not what I was questioning. What I was questioning, was whether or not a word usually reserved for Euclidean objects, namely words like polygons and polytopes etc., can be used for geometric objects in psuedo-Euclidean spaces.
In other words, call it a hypercube in R^4, but is it still a hypercube in R^(1,3)? (even if R^(1,3) is as flat as R^4)?
Or is this question what you meant already by
? If so, sorry for my confusion.
Yeah. We could steal the term polychoron while R^(n,m) for (m+n)<5, I guess.