Practical application for n-dimensional transformations.

Higher-dimensional geometry (previously "Polyshapes").

Practical application for n-dimensional transformations.

Postby Roberto » Wed May 02, 2007 11:58 pm

Hi, i have red recently about n-dimensional tranformations, like scaling, translation, rotation.


And i'm thinking, how this mathematical formulations can be applied in a practical or real application. the unique example that i have found is to graph a complex function in 4D, but what about in higher dimensions ?


I want to make my final project of my bachelor's degree about this. Do you have ideas.


Thanks.
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Postby Roberto » Fri May 04, 2007 1:24 am

no one ? :\
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Postby Nick » Fri May 04, 2007 10:11 am

I'm no good at geometry :(
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Postby wendy » Sat May 05, 2007 1:07 am

Much of the practical applications of N dimensional geometry rely on the relation of Rn <=> En, that is to say, a coordinate of n terms is a point in N dimensions.

For example space-time is (x,y,z,t). The geometry here is not euclidean but Minokoskian.

There is physics 'phase-space', which maps all possible coordinates into a sequence and maps, eg the state of a gas of N particles onto a single point in 3N dimensions.

There is also "group theory" which is usually implemented as points and lines of objects in multiple dimensions. There are some rather interesting relations between eg the "double-sixes" of intersecting cubics, and the 27 vertices of a 6-d polytope.

The desarge configuration corresponds to a kind of projection of the N+1 simplex.

Many relations are just plain easier to understand if one takes the full expansion.

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Postby PWrong » Sat May 05, 2007 3:08 am

the unique example that i have found is to graph a complex function in 4D, but what about in higher dimensions ?

Well, 4D can also be used to represent quaternions. There are also octonians, which you can graph in 8D. These things are probably less useful than extra dimensions themselves.

You could use 5D to graph a function from the complex numbers to 3D, i.e. f : C -> R<sup>3</sup>. There's other functions you could try, which would give you manifolds of different dimensions.

There are plenty of applications in physics, like many particle systems, relativity, and string theory. However on the whole, 5D space is probably less useful than infinite dimensional space.

There is physics 'phase-space', which maps all possible coordinates into a sequence and maps, eg the state of a gas of N particles onto a single point in 3N dimensions.

I think that should be 6N. In phase space, you count velocity as a dimension.
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