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I have a very sophisticated mental picture of higher dimensions and I really need some guidance in correcting my wild imagination.

Is it ok to visualize $ \mathbb{R}^4 $ like a regular 3D space filled with infinitesimal tiny points each of different color? The color being the 4th dimension? Isn't such an approach more accurate than the standard old school of schematically squashing and heavily distorting $ \mathbb{R}^3 $ into a plane and extending the forth dimension as height?

Imagining for example a filled sphere made from(or with) infinitesimal colored points that change over time($ \mathbb{R}^5 $) and we can freely move with our camera(or our mind) in-between those infinitesimal points. Of course to be able to move our point of view "in-between" those infinitesimal color changing points, then obviously those points will have to live in $ \mathbb{Q}^5 $ let's say and our point of view (camera) will live in $ \mathbb{R}^3 $ superimposed or living inside somehow in between this $ \mathbb{Q}^5 $ space (all mixed in just like $ \mathbb{Q} \in \mathbb{R} $).

Bare with me here. Then to go even higher with my wild imagination to visualize $ \mathbb{R}^6 $ I see myself zooming in to a "infinitesimal" point in 3D($ \mathbb{R}^3 $) then somehow I'm able to cross its "event horizon" of that pseudo-"infinitesimal" point and find myself inside another completely new UNIQUE 3D space where my position in there or "address" is given by 6 unique coordinates (in reference to my old home 3D universe). I seriously need some guidance.

Combining and mixing all these pictures one can embed in their mind (by a fractal self-nesting of some sorts) infinite dimensional space $ \mathbb{R}^{\infty} $.

Is my wild method of picturing in my mind higher dimensions flawed? If so, why? If one pushes the tiny little argument that each higher dimensions axes are 90 degrees(orthogonal) in respect to all the other, then...(crying)... all my Sci-Fi movies in my mind crumble to pieces...and are completely trash. Please help me!

I'm not a mathematician, so type slowly. I'm beginning a course in math and I'm struggling to understand and visualize.

I guess one can see a changing isothermic weather map as seen on TV as an actual $ \mathbb{R}^4 $ plane! A graph of x,y,temperature,time on the same flat plane. Cool.

But what if we could take a 6 component (or coordinates) vector (a,b,c,d,e,f) (a vector living in $ \mathbb{R}^6 $), take its first 3 components, plot them in a regular cartesian 3D world($ \mathbb{R}^3 $) and then, at that point [a,b,c] we imagine an invisible "intangible" tiny imperceptible SUBSPACE where the rest of the 3 components are plotted on/in another 3D cartesian coordinate system. Obviously this sort of subspace nesting can go to infinity, but is it correct, or even useful somehow? I'm just asking.

Because even if it's sort of cool one can think of higher dimensions nesting like this, I'm not entirely sure it's correct. That's why I'm asking the question and asking for guidance. Because even though I haven't studied this part yet, I wonder, don't these coordinates of vectors have to obey some rules? Doesn't this example violate some rules? Don't the coordinates have to live somehow in the same space, or have each an axis, or each I don't know obey some rules I yet don't know? That's why I'm wondering if this kind of visualization is correct or even practical.

One example for 7 dimensional space I could think of is what if we could encode 7 components (or 7 types of information into a single point) in a 3D map for let's say statistical purposes. Imagine a collection of dots in 3D space ($ \mathbb{Z}^3 $ built on or with integers). Each dot having its own size ($ s \in \mathbb{R} \mid 0<s<1 $) size being a function of time, its shape factor ($ k \in \mathbb{R} \mid 0<k<1 $ cube=1, sphere=0), its own color as a function of time and the rate time passes controllable with an imaginary virtual slider under each dot.

This could be a very practical application visualization technique useful for statistics and data analysis. But I don't know it this is a correct visualization. My mind is very wild and untamed. Is it even valid to insert 4 additional $ \mathbb{R}^1 $ subspaces for each [x,y,z] point in $ \mathbb{Z}^3 $ and call that a $ \mathbb{Z}^7 $ domain or is it $ \mathbb{R}^7 $ ? I'm pretty sure I'm breaking some rules here. I don't know. My mind is swarming with unanswerable questions. I need guidance. Am I making confusions here? between dimensions and parameters or something else? Can someone enlighten me where my mind is in all this? I'm kind of lost...

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closed as unclear what you're asking by Morgan Rodgers, Thomas Shelby, Cesareo, ancientmathematician, ncmathsadist Mar 8 at 15:50

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    $\begingroup$ Colouring is quite commonly used for an extra dimension in complex analysis. Now, you can visualise in your mind whatever you like for artistic purposes or your own pleasure, but how would this particular proposal help to visualise actual mathematical statements in $\Bbb R^n$? Can you give a few examples? Also, what happens if you visualise three-dimensional space as a plane filled with coloured dots, does this give some new insight? $\endgroup$ – Torsten Schoeneberg Mar 7 at 18:31