Output a geometry rather than a number

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So, in my studies of math so far, we can take a certain geometry (or just any function) and analyze it, and ask questions about it. The answer that's given in all the work I've done so far is a number (or a function that will reduce to a number when solved). Now, let's say I want to do the opposite (or the inverse operation): start with a number and have a function reduce to a geometry rather than another number. Here's an example (actually why I'm motivated to ask this question):

Niels Bohr and those before him predicted a circular (spherical) orbit of electrons in the hydrogen atom. Starting from the idea that the orbit would be circular, we can use this geometry to give an expected value (number) of the energy that would be emitted (or have to be absorbed) in a change of state. The measured values correspond with the expected values which correspond to the geometry of a sphere, which supports the hypothesis.

So, after we've done all this and we have a basic template for finding the numbers from the geometry, what kind of math does it take to work backwards, starting from the numbers, to output the geometry? You can certainly use the same example as I did, but it's not necessary, any example would be great.

 

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To go from a list of numbers or list of lists (position vectors or other types of vectors) to geometry requires some guesswork or extra empirical measurements. A list of numbers is missing a topology that relays how the numbers are connected to each other and a metric that relate distances between the numbers to create a geometry.