Mathematics of The Multi Dimensional Universe

  • #1
What are the mathematics behind multiple dimensions? Does mathematics allow for the existence of more than four dimensions? What allows the ability to possess more that four dimensions, if there is already proof? I looked around, but I didn't find much by way of the answer I was looking for.
 

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  • #2
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Yes, mathematics allows as many dimensions as you desire. Even an infinite amount.
 
  • #3
FactChecker
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Suppose you agree that you can add time to the known physical (x,y,z) dimensions to get (x,y,z,t). What stops you from adding more: (x,y,z,t,a,b,c,d,...)? You might object to the idea that they can all be at right angles to each other, but that is just because you are trying to imagine it within a 3 (or 4) dimensional space. That leads to the question of whether so many dimensions can all be at right angles to each other. The answer is yes. In theoretical mathematics, angles can be defined using "inner products" and then there are examples where there can be any finite or even infinite number of dimensions, all at right angles to each other (see https://en.wikipedia.org/wiki/Orthonormal_basis )
 
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  • #4
mathman
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The question of multiple dimensions that you are posing is a physics question. As others have noted there is nothing in mathematics that restricts you to any number of dimensions.
 
  • #5
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The question of multiple dimensions that you are posing is a physics question. As others have noted there is nothing in mathematics that restricts you to any number of dimensions.
But even within physics, the decomposition of a signal into its frequencies is a good example of using an infinite dimensional space.
 
  • #6
Cool, thanks FactChecker! That's the simplistic answer I was looking for :D
 
  • #7
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Multidimensional space are very common in mathematics. Part of models describe also physical theories, as example the Hilbert spaces that are infinite dimensional for the quantum mechanics or Calabi Yau manifolds in string theory ...
 
  • #8
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In fact, any time a new bit of information is added to a tuple and that information was not already determined by the prior information in the tuple, the dimension has been increased. If I was keeping track of an object, I might include its position at a time, it's weight, its dimensions, etc. (x_position, y_position, z_position, time, mass, height, width, length). This is a 8-dimensional state-space. So you can certainly make high dimensional state-spaces with ease. Things get more complicated if you want a metric for that space. There might not be a meaningful metric.
 

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