[Nimmie (OP)]

How do you map from a two dimension to 3 dimension? How do you covert a value in two dimension into a value in three dimension or even four dimension?

[yor_on]

Maybe you could call it a topological logic resting behind it.  https://researchblog.duke.edu/2017/04/26/visualizing-the-fourth-dimension/

[alancalverd]

Mapping reduces the dimensionality of a variable and thus loses some information that has to be replaced with an arbitrary function to replicate the original dimensionality. If you consider even a conventional contour map, whilst the north-south and east-west distances will be mutually proportional, at say 1: 1,000,000 of the actual dimensions, the 5 meter contour lines are not spaced at one  millionth of the actual height difference. You can use the spacing to produce an arbitrary vertical profile and if you have some actual reference heights, you can make that very realistic, but the necessary information is not, mathematically speaking, part of the map.

[Eternal Student]

Hi.
   I think you've had some good answers already.

   The essence of the matter is that, in the most general case, we can't make this mapping unique.  There are usually many ways to map a 2 dimensional space into a 3 dimensional space.

    Sometimes there is some extra information we could use to help (Alancalverd gave an example of contour maps).  It could also be that we know something about the higher dimensional space we wish to map to:  For example there may be physical reasons why the co-ordinate in the 3rd dimension is a known function of the two dimensions we have information for.

   Was there a specific example you had in mind?

[evan_au]

As well as topology maps, some other ways of showing 3D data on 2D are:
- "Heat maps": The color red indicates how "hot" (high values of some variable) and blue indicates "cold" (low values of some variable). The variable doesn't need to be temperature - it could be %COVID infections, for example
- Light Cones around a black hole represent a 3D effect, on 2D paper.
- Wind tunnel graphs show the direction and speed of air flowing around the silhouette of a car or plane by using little arrows. The direction of the arrow shows the direction of airflow, and the length of the arrow shows the velocity of airflow. This maps 5 dimensions onto 2 dimensions.

But the original question was about mapping from 2D to 3D
- There is no unique solution to this, either.
- You could have a 2D drawing on a sheet of paper, and position it at some arbitrary value in the Z-Axis (3D space)
- Or you could take the original 3D data which was compressed to 2D, and actually show it in 3D
- Or you could show the evolution of the 2D data over time, to make a "movie"

Note that old-style raster TVs used a 1-dimensional video input signal
- Which was scanned back and forth to make a 2D image on the TV screen
- Which changed over time, making it 3D
- Which our brains interpreted as 3D objects moving over time, ie a 4D experience...
...But different countries couldn't even do that 2D→3D mapping in a unique way!

[wolfekeeper]

Yes, counter-intuitively, the cardinality of finite one, two and three etc. dimensional spaces are all the same. To slightly oversimplify that's pretty much why computers can produce 3D video games, even though computer memory is only one dimensional.

There's multiple ways to do the mapping. First let's map one dimensions (point p) into 2 (x and y) where p, x, y are integers

x = p mod width
y = p / width (rounded down)

where width is the width of the 2D region

To go the other way p = y*width + x

You can extend this to 3D:

x = p mod width
y = ((rounded down)p / width) mod height
z = (rounded down)(p / width / height)

And go the other way p = z * width * height + y * width + x

That's basically how Evan's TV mapping works.

You can convert between different spaces going through the one dimensional space.

There is a more complicated way to map points one to one between spaces even if they have different dimensionality and even if they are rational point coordinates but they're a rather odd mapping:

https://en.wikipedia.org/wiki/Cardinality_of_the_continuum

[Bored chemist]

Until the OP gets back and tells us what question we are answering, there's not much hope of us giving the best answer.

It could be they were looking for how you map 2D latitude and longitude onto a 3D sphere.

Or they could be talking about mathematical mapping of functions in which case you certainly can map without losing data.