For complicated reasons, I need an example of two sets (X,Y) that optimize the following criteria:
Simple to understand, including for non-mathematicians
There exists an (approximately) one-to-one mapping φ:X→Y
φ should not feel totally artificial, like don’t map R to R2. Should probably be continuous if the sets have a topology.
X and Y should be as intuitively different as possible
My current favorite is Cartesian to polar coordinates, X=R2 and Y=[0,360)×R≥0, which scores well on #1-3, but the spaces feel more similar than I’d like. Any better ideas?
Then, switch it up by pointing out this is the same as:
a, b, c, d, …, z, aa, ab, ac, …
Might not be two sets but:
The relationship between ‘long division’ and ‘synthetic division’.
Moves in two different games. This could be hard to set up, but have one of the games be a rubik’s cube. (Might be important to point out that this is a history—not the structure of the two games.)
WASD doesn’t seem to cover all the moves on a cube though.
Intuitive ‘best case scenario’ for snake makes it the same game as tron.
$ to integers. (If you can only split down to a penny, it’s an easy mapping. $ → N: drop the $, and multiply but 100. N → $: divide by 100, and divide by 100.)
Frames from one frame to frames in another movie - (more interesting for how this doesn’t work (two movies might have different frame rates)) - but if they have the exact same number of frames you can make such a mapping. Might be worth emphasizing it seems like no one would do this (if frame rates vary over the course of a movie). But if you have two movies that play at the same rate, and have the same length...you can swap the audios.
Might be better to just take two songs, and swap the music, but keep the music the same (where the meters match).
The comparison between a cylinder and a tower of coins (analogy used in calculus). There’s more surprising mappings that are related.
Intuition for the volume of a sphere, or a pyramid...
‘Double shapes/half shapes’ (like a square in a square)
Could you unpack what you mean by “intuitively different” in a bit more depth? Do I understand correctly that the way the third and fourth criteria are not in direct tension is that you’re focusing on the difference in familiarity-feel between the mapping and the sets themselves? (I think “familiarity” is probably not the right word, but I’m having trouble finding a more accurate one.)
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.
For complicated reasons, I need an example of two sets (X,Y) that optimize the following criteria:
Simple to understand, including for non-mathematicians
There exists an (approximately) one-to-one mapping φ:X→Y
φ should not feel totally artificial, like don’t map R to R2. Should probably be continuous if the sets have a topology.
X and Y should be as intuitively different as possible
My current favorite is Cartesian to polar coordinates, X=R2 and Y=[0,360)×R≥0, which scores well on #1-3, but the spaces feel more similar than I’d like. Any better ideas?
These might be too obvious or not work.
0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, …
and itself.
Then, switch it up by pointing out this is the same as:
a, b, c, d, …, z, aa, ab, ac, …
Might not be two sets but:
The relationship between ‘long division’ and ‘synthetic division’.
Moves in two different games. This could be hard to set up, but have one of the games be a rubik’s cube. (Might be important to point out that this is a history—not the structure of the two games.)
WASD doesn’t seem to cover all the moves on a cube though.
Intuitive ‘best case scenario’ for snake makes it the same game as tron.
$ to integers. (If you can only split down to a penny, it’s an easy mapping. $ → N: drop the $, and multiply but 100. N → $: divide by 100, and divide by 100.)
Frames from one frame to frames in another movie - (more interesting for how this doesn’t work (two movies might have different frame rates)) - but if they have the exact same number of frames you can make such a mapping. Might be worth emphasizing it seems like no one would do this (if frame rates vary over the course of a movie). But if you have two movies that play at the same rate, and have the same length...you can swap the audios.
Might be better to just take two songs, and swap the music, but keep the music the same (where the meters match).
The comparison between a cylinder and a tower of coins (analogy used in calculus). There’s more surprising mappings that are related.
Intuition for the volume of a sphere, or a pyramid...
‘Double shapes/half shapes’ (like a square in a square)
Noether’s theorem? Symmetries and conservation laws seem quite different.
Could you unpack what you mean by “intuitively different” in a bit more depth? Do I understand correctly that the way the third and fourth criteria are not in direct tension is that you’re focusing on the difference in familiarity-feel between the mapping and the sets themselves? (I think “familiarity” is probably not the right word, but I’m having trouble finding a more accurate one.)
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.