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Infinite Dimensions


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Can there be an infinite amount of dimensions? Example: The 1D is an infinitely long, infinitely thin line. 2D is an infinite amount of those lines next to each other. This is infinitely flat. 3D is an infinite amount of those 2D flat sheets on top of each other. This is an infinitely 'big' space. 4D is an infinite amount of 3D stacked ana and kata direction 'next' to each other. Would 5D be infinite 4D also 'next to each other. Then 6D to 5D etc.

 

For a video on 4D click this

. Notice how 4D is 'cubes' that are infinitely big in a line. Would 5D be that line stacked infinitely 'next' to each other?
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Each time you get a new dimension, you draw a new axis perpendicular to each other axis (well it just needs to have a component perpendicular to them). You can't draw a 4D object in 3D because you're missing a dimension. There's no limit to how many dimensions you can have, but they won't necessarily correspond to reality just because you can describe them mathematically.

 

To imagine multidimensional things, I suggest you start with points equidistant from each other. In 0D you have a point, in1 D you get a line segment with two points, in 2D an equilateral triangle with 3 points, in 3D you get an equilateral tetrahedron with 4 points, in 4D you get an equilateral shape with 5 points equidistant from each other, etc.

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In physics, we have to contend with infinite dimensional spaces. For example, the configuration spaces of fields.

 

You can deal with these spaces without much worry using local coordinates and then taking any sums to be integrals. You can be a bit more rigorous by considering finite dimensional subspaces.

 

The trouble with infinite dimensional spaces is that many of the theorems for finite dimensional spaces do not carry over. You will have to be a bit more careful in general.

 

Modelling infinite dimensional manifolds via Hilbert spaces is hard. I am not very familiar with this, other than you won't get too far in general.

 

As I am usually interested in the mathematical constructions arising in physics, I would usually consider the spaces to be finite dimensional in order to avoid any technical problems. That said, much of my work uses local coordinates so I expect most of it could simply be amended accordingly to take into account infinite dimensions. Good notation for this (once you understand it) is de Witt notation. This allows you to write things down "as if they were finite".

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As I am usually interested in the mathematical constructions arising in physics, I would usually consider the spaces to be finite dimensional in order to avoid any technical problems. That said, much of my work uses local coordinates so I expect most of it could simply be amended accordingly to take into account infinite dimensions. Good notation for this (once you understand it) is de Witt notation. This allows you to write things down "as if they were finite".

 

I agree the 'infinite' part is a twist and is probably not possible to show on paper. I mean infinity times zero is zero which is the logic flaw in my thinking. I can't seem to find a way around that.

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