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I'm looking for a term, which I've been referring to as the "data space", though I'm reasonably sure there's a proper term for it:

Let's say I've got two variables, and when one doubles, the other doubles. You could imagine a graph, with the line $y=x$ describing this relationship. Add in a third variable, z, which quadruples when $x$ doubles, and increases by nine-fold when $x$ increases by three-fold. That's clearly an exponential relationship, so we can extend our line into a surface which looks like $y=x$ when projected onto the $xy$ plane and like $z=x^2$ when projected onto the $xz$ plane. Add in more variables, and our surface describes the relationship between all given variables.

If you look at the space we've built up, it covers all possible values of all variables being considered, with a surface showing the relationships between them. This space, while not necessarily used by many people, exists whenever you're talking about the relationships between variables, and it's been a very useful metaphor for me in the past.

My question is, does there exist a term for:

  1. the space, and
  2. the surface?
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    $\begingroup$ Re "clearly an exponential relationship:" that's a quadratic relationship, not an exponential one. The generic term for such a space is "manifold." The kinds of surfaces you appear to describe have various properties as algebraic surfaces and differential manifolds. Spaces of such surfaces have all kinds of names: families, moduli spaces, and even "models." Even a superficial discussion of the possibilities would quickly become broadly mathematical: perhaps you should be asking these questions on Mathematics? $\endgroup$ – whuber Feb 2 at 10:31

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