I’m not sure if you mean a true mathematical mapping, or a conceptual mapping with the math as an analogy only. If the former, this should perhaps be a sphere (or hypersphere) inversion, rather than a box. If the latter, are there aspects of the circle (or sphere or box) mathematical definition that you want to preserve, in order to clarify other aspects?
For instance, does the un-mappable nature of the origin have meaning in these mappings? Does the fact that outside distances are non-linearly related to inside distances (inside things near the center are close together, but map to things far apart outside, and inside things near the edge stay roughly the same distance from each other in the outside mapping) mean something in this model?
I’m not sure if you mean a true mathematical mapping, or a conceptual mapping with the math as an analogy only. If the former, this should perhaps be a sphere (or hypersphere) inversion, rather than a box. If the latter, are there aspects of the circle (or sphere or box) mathematical definition that you want to preserve, in order to clarify other aspects?
For instance, does the un-mappable nature of the origin have meaning in these mappings? Does the fact that outside distances are non-linearly related to inside distances (inside things near the center are close together, but map to things far apart outside, and inside things near the edge stay roughly the same distance from each other in the outside mapping) mean something in this model?