Nothing000
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I just can't picture it in my mind. Is it possible to have an idea what a 4 dimensional space actually looks like?
How is R4 space represented geometrically?
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For a function of two variables, you can consider all the points in the domain at which the fuction has the same value to retrieve a family of curves.
Analogically, you can actually consider all the points in space at which a function f(x, y, z) of three variables has the same value. These points form a surface, in general. So, that could be a way.
Analogically, you can actually consider all the points in space at which a function f(x, y, z) of three variables has the same value. These points form a surface, in general. So, that could be a way.
Nothing000
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What? I don't get how that would describe four dimensional space.
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Okay, take, for example f(x,y,z) = x^2+y^2-z^2.
f(x,y,z) = -1 => -x^2-y^2+z^2 = 1 ;
f(x,y,z) = 0 => x^2+y^2=z^2 ;
f(x,y,z) = 1 => x^2+y^2-z^2=1 ;
...
Every of these three surfaces represents the set of points in space in which the function f has the same value, as stated before. These surfaces are called level surfaces. Theoretically, plot all the level surfaces, i.e. all surfaces f(x,y,z) = c (where c is in the codomain of f), and you have a mapping between every value c and it's coresponding surface z = z(x,y). So, you visualized* R^4 in R^3.
* Conclusion: of course you didn't, and of course you can't, but creating level surfaces is the only way to 'deal' with R^4.
f(x,y,z) = -1 => -x^2-y^2+z^2 = 1 ;
f(x,y,z) = 0 => x^2+y^2=z^2 ;
f(x,y,z) = 1 => x^2+y^2-z^2=1 ;
...
Every of these three surfaces represents the set of points in space in which the function f has the same value, as stated before. These surfaces are called level surfaces. Theoretically, plot all the level surfaces, i.e. all surfaces f(x,y,z) = c (where c is in the codomain of f), and you have a mapping between every value c and it's coresponding surface z = z(x,y). So, you visualized* R^4 in R^3.
* Conclusion: of course you didn't, and of course you can't, but creating level surfaces is the only way to 'deal' with R^4.
Nothing000
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I knew that. 