Is it possible to convert a 4-dimension space to a 3-dimension space in the same way we can convert our 3-dimension world into a 2-dimension map?

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TopNewestWhy of course. Just read

Flatland– Bob Krueger · 3 years, 12 months agoLog 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. – Mark Hennings · 3 years, 12 months agoLog 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. – David Mattingly Staff · 3 years, 12 months ago

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

(mathematicsand_physics) – Hafizh Ahsan Permana · 3 years, 12 months agoLog in to reply

– Okay Nho · 3 years, 12 months ago

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

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. – David Mattingly Staff · 3 years, 12 months ago

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– Tan Li Xuan · 3 years, 12 months ago

Then could we make a 3-d map of the surface of a 4-d object?Log in to reply

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. – David Mattingly Staff · 3 years, 12 months ago

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http://en.wikipedia.org/wiki/Projection_(mathematics) – Okay Nho · 3 years, 12 months ago

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– Wayne Zhao · 3 years, 12 months ago

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?Log in to reply

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). – Jon Schneider · 3 years, 12 months ago

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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! – Arkady Arkhangorodsky · 3 years, 12 months ago

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It's like converting .doc file to .txt. Some data (one dimension) would be incomprehensible and hence lost. – Lokesh Sharma · 3 years, 12 months ago

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Of course! – Jeremy Shuler · 3 years, 12 months ago

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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. – Cole Wyeth · 3 years, 12 months ago

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One fun way to do this is by Schlegel diagram. – Eric Edwards · 3 years, 12 months ago

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Sure, project it! Projection on the Euclidean space is a continuous function. – Okay Nho · 3 years, 12 months ago

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just make 4 th coordinate constant like we make 3rd coordinate constant to make it 2d – Amankumar Jha · 3 years, 12 months ago

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What is the fourth dimension that you have assumed? – Tamoghna Banerjee · 3 years, 12 months ago

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may please know my dear able friend where does this 4D exist in this world – Arpan Vishnoi · 3 years, 12 months ago

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– Hafizh Ahsan Permana · 3 years, 12 months ago

i think you have to read some theories firstLog in to reply