Could you unpack what you mean by “intuitively different” in a bit more depth?
I mean it pretty literally, i.e., the first reaction when someone looks at them (especially if that someone isn’t a mathematician) should be “okay, these are definitely totally different things”.
Maybe #3 and #4 are in conflict, mostly #4 is just more important. Like, any continuous function is probably good enough, and there have to be some more “different” looking continuous deformations than what I can think of. (Doesn’t have to be topological spaces, but that would be one approach.)
What about something like the faces of a polyhedron to the vertices of its dual? Additionally, would you count those as highly different, perhaps because a face and a corner feel and look very different when observed physically? Or would they count as similar, perhaps because they’re both geometric ideas used to describe parts of polyhedra and are concepts that are frequently used together rather than being totally unrelated?
That’s really good! I think they count as quite different.
The one thing I don’t like about it is that the dual of the entire geometric object is another, similar geometric object, so on level it only shuffles around vertices and faces. But the vertices themselves become radically transformed, which is great. It’s definitely a better solution than polar/Cartesian coordinates.
I mean it pretty literally, i.e., the first reaction when someone looks at them (especially if that someone isn’t a mathematician) should be “okay, these are definitely totally different things”.
Maybe #3 and #4 are in conflict, mostly #4 is just more important. Like, any continuous function is probably good enough, and there have to be some more “different” looking continuous deformations than what I can think of. (Doesn’t have to be topological spaces, but that would be one approach.)
What about something like the faces of a polyhedron to the vertices of its dual? Additionally, would you count those as highly different, perhaps because a face and a corner feel and look very different when observed physically? Or would they count as similar, perhaps because they’re both geometric ideas used to describe parts of polyhedra and are concepts that are frequently used together rather than being totally unrelated?
That’s really good! I think they count as quite different.
The one thing I don’t like about it is that the dual of the entire geometric object is another, similar geometric object, so on level it only shuffles around vertices and faces. But the vertices themselves become radically transformed, which is great. It’s definitely a better solution than polar/Cartesian coordinates.