When you say we make 2D maps of our 3D world, you are saying that the surface of the sphere is what is called a 2D-manifold, in that at every point on the surface there is a neighbourhood which can be bijectively and smoothly mapped to a region in \(R^2\). Each such bijection is called a "chart", and the image of each chart, being a subset of \(R^2\), could be printed on paper. You can have maps of KL, maps of London, etc. No single chart can exist for the whole surface of the sphere. "Maps of the world" are generally maps of the world minus a line running from pole to pole through the Pacific Ocean --- in other words, a chart.

There are many different ways to construct these charts and therefore many different ways to construct maps of the world.

If you want to generalise our 2D-manifold world up a dimension, then the 4D hypersphere \(x^2+y^2+z^2+u^2=1\) is a 3D-manifold --- at any point on the hypersphere there is a region which can be mapped smoothly and bijectively to a region in \(R^3\). You would need a 3D printer to handle these charts!

An interesting chart is so-called stereographic projection from \((0,0,0,1)\), which is the map
\[
f\;:\;(x,y,z,u) \; \mapsto \; (\tfrac{x}{1-u},\tfrac{y}{1-u},\tfrac{z}{1-u})
\]
If \(\mathbf{x} \neq (0,0,0,1)\) lies on the hypersphere, then \((0,0,0,1)\), \(\mathbf{x}\) and \((f(\mathbf{x}),0)\) are collinear. This is a smooth bijection between the hypersphere minus one point (remove the north hyperpole \((0,0,0,1)\)) and the whole of \(R^3\). This chart would give a fairly good picture of what things were like near the south hyperpole, but would give a very distorted picture near the north hyperpole. If you wanted a good chart of life near the north hyperpole, take a stereographic projection from the south hyperpole.

Yes, in many ways. The fundamental question about any such reduction is whether or not you want to try to preserve all/some/none of the information in the original dimensions and what that information is.

This is correct as far as I know, but someone should check. And this is one of the essential distinctions you need to make when you ask "can I convert?" If you don't mind losing some information/distinction between points (because you need to have projection operators in there somewhere) then you can take a 4 d space and squish it down to 3 d. If you want to preserve all the information then you're stuck I think.

Note: Check the comment below by Wayne Z, I always think in terms of continuous maps first cause I'm a physicist. Anyone want to teach me something?

Two additional comments. One, in map making we really project our 2-d world (the surface of a sphere) onto a 2-d map. A 3-d to 2-d map would take the entire earth (including the inside) and put it on a 2-d surface. Again, you see the difficulty - each ray from the center of the earth gets projected to a point (for example).

Second, we often define theories in physics that naively look 4-d or n-d or whatever. However, many of those theories actually have symmetries, and if you choose your projections right you can reduce those theories to lower dimension theories. You just have to align the projections with the symmetries so you don't, in the end, lose any information.

@Tan Li Xuan
–
yep. And there's a real map that we see all the time. In general relativity we work with spacetime, a 4-dimensional space that consists of time + the ordinary three spatial dimensions. Objects and 3-d regions of space trace out "world-volumes" through the spacetime, which are four dimensional regions (just as if you plot x vs. t for a 0-dimensional point mass you get a 1-d line, if you trace out a 3-d object you get a 4-d region).

If you consider the 4-d description fundamental then, in a perverse way of thinking about it, every 3-d region you look at at an instant in time is the 3-d map of part of the surface of the 4-d world volume. Voila! 3-d maps of surfaces of 4-d regions are all around you, every instant of the day.

Isn't that possible? I don't know if a continuous bijection is possible, but if you can biject \(\mathbb{R}\) and \(\mathbb{R}^2\), can't you biject similarity with different dimensional spaces?

@Wayne Zhao
–
A continuous bijection from \(\mathbb{R}^n\) to \(\mathbb{R}^m\) for \(m \neq n\) is impossible; this follows from the fact that topologically, \(\mathbb{R}^n\) and \(\mathbb{R}^m\) are not homeomorphic, which to the best of my knowledge requires some pretty advanced algebraic topology to prove formally.

But yeah, as you pointed out, if you remove the constraint that the bijection is continuous, then it is possible to biject them. One (fairly) explicit way to do this is to note that \(\mathbb{R}\) has the same cardinality as the set of infinite binary sequences, and you can biject \(\mathbb{R}^n\) to \(\mathbb{R}\) by "interleaving" n binary sequences; that is, bijecting the \(n\) sequences

Then, once you've bijected \(\mathbb{R}\) and \(\mathbb{R}^n\), by composing these bijections its fairly easy to biject \(\mathbb{R}^m\) and \(\mathbb{R}^n\). (Of course, this non-continuous map is pretty ugly/useless for visualizing properties of the original space, but it exists).

This is an excellent question. What may interest you is a subject known as 'Dimensionality Reduction'. It's a bit more general than what you're talking about, because it talks about mapping from some arbitrary high dimension to any dimension less than that, so you can even make 2-D plots of 10000 dimensional data! There are ways to do this without completely throwing out all of the information from some of the dimensions, as people have suggested here. A lot of different techniques exist for this, and since it's impossible, unless you have really nice data, to do it 'perfectly' i.e. losing absolutely no information, it's still an open problem with new techniques being proposed. The wiki page for it is a good place to get started, and here are some links to useful papers if you're interested:

I always think of four dimensional space as a series of models of three dimensional space- Take time, for instance. We track it by looking at how things were, in three dimensions (i.e, Bob is standing) and then how things are in the next second (e.i. Bob is raising his hand) and that's how we can tell that time is actually passing. So every dimension above the last is just a series of the one before lined up... See David M's comments, he explains it better and I didn't get the last part until I read a little of what he posted.

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TopNewestWhy of course. Just read

FlatlandLog in to reply

When you say we make 2D maps of our 3D world, you are saying that the surface of the sphere is what is called a 2D-manifold, in that at every point on the surface there is a neighbourhood which can be bijectively and smoothly mapped to a region in \(R^2\). Each such bijection is called a "chart", and the image of each chart, being a subset of \(R^2\), could be printed on paper. You can have maps of KL, maps of London, etc. No single chart can exist for the whole surface of the sphere. "Maps of the world" are generally maps of the world minus a line running from pole to pole through the Pacific Ocean --- in other words, a chart.

There are many different ways to construct these charts and therefore many different ways to construct maps of the world.

If you want to generalise our 2D-manifold world up a dimension, then the 4D hypersphere \(x^2+y^2+z^2+u^2=1\) is a 3D-manifold --- at any point on the hypersphere there is a region which can be mapped smoothly and bijectively to a region in \(R^3\). You would need a 3D printer to handle these charts!

An interesting chart is so-called

stereographic projectionfrom \((0,0,0,1)\), which is the map \[ f\;:\;(x,y,z,u) \; \mapsto \; (\tfrac{x}{1-u},\tfrac{y}{1-u},\tfrac{z}{1-u}) \] If \(\mathbf{x} \neq (0,0,0,1)\) lies on the hypersphere, then \((0,0,0,1)\), \(\mathbf{x}\) and \((f(\mathbf{x}),0)\) are collinear. This is a smooth bijection between the hypersphere minus one point (remove the north hyperpole \((0,0,0,1)\)) and the whole of \(R^3\). This chart would give a fairly good picture of what things were like near the south hyperpole, but would give a very distorted picture near the north hyperpole. If you wanted a good chart of life near the north hyperpole, take a stereographic projection from the south hyperpole.Log in to reply

Yes, in many ways. The fundamental question about any such reduction is whether or not you want to try to preserve all/some/none of the information in the original dimensions and what that information is.

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If you mean time is the 4th dimension space than can we put an example as 3D move object?

in this link i thought not explained 4th dimension as time http://en.wikipedia.org/wiki/Dimension

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Isn't it impossible to preserve all the information? Because that would require bijection from high dimension to any lower dimension.

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This is correct as far as I know, but someone should check. And this is one of the essential distinctions you need to make when you ask "can I convert?" If you don't mind losing some information/distinction between points (because you need to have projection operators in there somewhere) then you can take a 4 d space and squish it down to 3 d. If you want to preserve all the information then you're stuck I think.

Note: Check the comment below by Wayne Z, I always think in terms of continuous maps first cause I'm a physicist. Anyone want to teach me something?

Two additional comments. One, in map making we really project our 2-d world (the surface of a sphere) onto a 2-d map. A 3-d to 2-d map would take the entire earth (including the inside) and put it on a 2-d surface. Again, you see the difficulty - each ray from the center of the earth gets projected to a point (for example).

Second, we often define theories in physics that naively look 4-d or n-d or whatever. However, many of those theories actually have symmetries, and if you choose your projections right you can reduce those theories to lower dimension theories. You just have to align the projections with the symmetries so you don't, in the end, lose any information.

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If you consider the 4-d description fundamental then, in a perverse way of thinking about it, every 3-d region you look at at an instant in time is the 3-d map of part of the surface of the 4-d world volume. Voila! 3-d maps of surfaces of 4-d regions are all around you, every instant of the day.

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http://en.wikipedia.org/wiki/Projection_(mathematics)

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Isn't that possible? I don't know if a continuous bijection is possible, but if you can biject \(\mathbb{R}\) and \(\mathbb{R}^2\), can't you biject similarity with different dimensional spaces?

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But yeah, as you pointed out, if you remove the constraint that the bijection is continuous, then it is possible to biject them. One (fairly) explicit way to do this is to note that \(\mathbb{R}\) has the same cardinality as the set of infinite binary sequences, and you can biject \(\mathbb{R}^n\) to \(\mathbb{R}\) by "interleaving" n binary sequences; that is, bijecting the \(n\) sequences

\[(a_{1,1}, a_{1, 2}, \dots), (a_{2,1}, a_{2, 2}, \dots), \dots, (a_{n,1}, a_{n,2}, \dots)\]

to the single binary sequence

\[(a_{1,1}, a_{2, 1}, \dots, a_{n, 1}, a_{1, 2}, a_{2, 2}, \dots, a_{n, 2}, a_{1, 3}, \dots)\]

Then, once you've bijected \(\mathbb{R}\) and \(\mathbb{R}^n\), by composing these bijections its fairly easy to biject \(\mathbb{R}^m\) and \(\mathbb{R}^n\). (Of course, this non-continuous map is pretty ugly/useless for visualizing properties of the original space, but it exists).

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This is an excellent question. What may interest you is a subject known as 'Dimensionality Reduction'. It's a bit more general than what you're talking about, because it talks about mapping from some arbitrary high dimension to any dimension less than that, so you can even make 2-D plots of 10000 dimensional data! There are ways to do this without completely throwing out all of the information from some of the dimensions, as people have suggested here. A lot of different techniques exist for this, and since it's impossible, unless you have really nice data, to do it 'perfectly' i.e. losing absolutely no information, it's still an open problem with new techniques being proposed. The wiki page for it is a good place to get started, and here are some links to useful papers if you're interested:

http://www.math.uwaterloo.ca/~aghodsib/courses/f06stat890/readings/tutorial

stat890.pdf http://homepage.tudelft.nl/19j49/MatlabToolboxforDimensionalityReductionfiles/TR_Dimensiereductie.pdf http://www.purdue.edu/discoverypark/vaccine/assets/pdfs/publications/pdf/Understanding%20Principal%20Component.pdfEnjoy!

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It's like converting .doc file to .txt. Some data (one dimension) would be incomprehensible and hence lost.

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Of course!

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I always think of four dimensional space as a series of models of three dimensional space- Take time, for instance. We track it by looking at how things were, in three dimensions (i.e, Bob is standing) and then how things are in the next second (e.i. Bob is raising his hand) and that's how we can tell that time is actually passing. So every dimension above the last is just a series of the one before lined up... See David M's comments, he explains it better and I didn't get the last part until I read a little of what he posted.

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One fun way to do this is by Schlegel diagram.

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Sure, project it! Projection on the Euclidean space is a continuous function.

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just make 4 th coordinate constant like we make 3rd coordinate constant to make it 2d

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What is the fourth dimension that you have assumed?

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may please know my dear able friend where does this 4D exist in this world

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i think you have to read some theories first

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