- #1
Nothing000
- 403
- 0
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?
- Context: Undergrad
- Thread starter Nothing000
- Start date
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- Tags
- Space
Click For Summary
Discussion Overview
The discussion centers around the geometric representation of four-dimensional space (R4) and how it can be conceptualized or visualized, particularly in relation to functions of multiple variables.
Discussion Character
- Exploratory, Conceptual clarification, Debate/contested
Main Points Raised
- One participant expresses difficulty in visualizing four-dimensional space and questions its geometric representation.
- Another participant suggests using level surfaces from functions of three variables as a way to understand R4, proposing that these surfaces can represent points where a function has the same value.
- A third participant challenges the explanation, indicating confusion about how this relates to four-dimensional space.
- A subsequent reply provides a specific example using the function f(x,y,z) = x^2+y^2-z^2, explaining how different values of the function correspond to different level surfaces in three-dimensional space, suggesting this as a method to conceptualize R4.
- One participant humorously acknowledges prior knowledge of the concept presented.
Areas of Agreement / Disagreement
Participants exhibit disagreement regarding the clarity and effectiveness of the proposed methods for visualizing four-dimensional space. No consensus is reached on how best to represent R4 geometrically.
Contextual Notes
The discussion includes assumptions about the understanding of functions and their level surfaces, but does not resolve the complexities involved in visualizing higher-dimensional spaces.
- #2
radou
Homework Helper
- 3,149
- 8
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.
- #3
Nothing000
- 403
- 0
What? I don't get how that would describe four dimensional space.
- #4
radou
Homework Helper
- 3,149
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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.
- #5
Nothing000
- 403
- 0
I knew that. 
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