Your crumpled space-filling surface makes me think of manifolds.
Machine-learning researchers often speak loosely of e.g. the "manifold"* of possible images of naturalistic scenes embedded within the space of all possible images. The networks are presumably learning a lower-dimensional representation of the world than the dimension of all possible combinations of sense impressions. If you have a 100 pixel by 100 pixel image and each pixel can have 256 intensity levels, then that's 256 to the power of ten thousand possible distinct images. If each image is a point in an abstract space of all possible images, then that space has 256 to the ten thousand dimensions. But the vast, vast, vast majority of the volume of that space corresponds to images that just look like static to humans. So the thinking is that we internally represent images as some learned non-linear transformation to a much lower dimensional set of features that actually correspond to stuff we experience / see.
A really simple canonical example is the Swiss Roll dataset. You only need two parameters (numbers) to specify it fully, but you can embed it (nonlinearly) in a 3D space. http://people.cs.uchicago.edu/~dinoj/manifold/swissroll.html
In terms of neuroscience, there are competing ideas (as always). There's a body of work that tries to show that as we learn, the brain encodes our high-dimensional sense-impressions in the lowest possible (most efficient) internal representations. However, some recent work seems to imply that instead the brain uses as high a dimension as possible, but up to some limit that demarcates the transition between being differentiable and not differentiable. They found a beautiful power-law: https://www.biorxiv.org/content/10.1101/374090v1.full
* When ML researchers speak of such a manifold, it's kind of loosey-goosey because a set that includes isolated points that don't have a continuous region around them aren't actually manifolds. In contrast, in the computer graphics and meshing literature people speak of non-manifold geometry which is isolated points and lines that you can't triangulate with 2d or 3d elements. I.e. 1d or 0d elements.