Could there be an "arctic geometry" by raising the rank of all operations?

[Syzithryx]

Tropical geometry studies algebraic varieties with exponentiation replaced with multiplication, multiplication lowered to addition, and addition to max (or, zeration minus one?). So "arctic geometry" might result if one were to raise the rank of operations, so that exponentiation becomes tetration, multiplication becomes exponentiation, and addition becomes multiplication.

I don't know anywhere near enough to guess how such a "geometry" would behave (or, tbh, why even tropical geometry exists or what it's used for), but I did notice that the "circle" produced by doing this to the equation x²+y²=n has some interesting properties as n is raised or lowered. To make it symmetrical I used absolute values: |x|^|x| * |y|^|y|=n. It looks vaguely like a four-petaled flower at n=1, but near 0.5 it resembles four separate blobs, one in each quadrant. At around n=0.4791417087 the blobs disappear (at least according to Desmos - the possibility of float rounding errors is ever on my mind when I use that app), but I don't know of any explanation or closed form for that number.

I guess that "n" is really radius squared for a circle - so, symmetrically, for the "tetration circle" I would assume n to be the radius to the radius power - but there is no real number r such that r^r = 0.4791417087. So basically, I'm just going on analogies with completely no idea what I'm doing, as I know basically no analysis...

[11Keith22]

Tropical geometry studies algebraic varieties with exponentiation replaced with multiplication, multiplication lowered to addition, and addition to max (or, zeration minus one?). So "arctic geometry" might result if one were to raise the rank of operations, so that exponentiation becomes tetration, multiplication becomes exponentiation, and addition becomes multiplication.

I don't know anywhere near enough to guess how such a "geometry" would behave (or, tbh, why even tropical geometry exists or what it's used for), but I did notice that the "circle" produced by doing this to the equation x²+y²=n has some interesting properties as n is raised or lowered. To make it symmetrical I used absolute values: |x|^|x| * |y|^|y|=n. It looks vaguely like a four-petaled flower at n=1, but near 0.5 it resembles four separate blobs, one in each quadrant. At around n=0.4791417087 the blobs disappear (at least according to Desmos - the possibility of float rounding errors is ever on my mind when I use that app), but I don't know of any explanation or closed form for that number.

I guess that "n" is really radius squared for a circle - so, symmetrically, for the "tetration circle" I would assume n to be the radius to the radius power - but there is no real number r such that r^r = 0.4791417087. So basically, I'm just going on analogies with completely no idea what I'm doing, as I know basically no analysis...

Although I don't have enough knowledge, as of right now, to know if arctic geometry could be a thing, I do know the closed form of the number you found. Firstly, the minimum value of |x|^|x| * |y|^|y| clearly must be the minimum value of |x|^|x|, squared. That value can be found by taking the derivative and solving for 0, which results in a minimum of (1/e)^(1/e) (source: wolfram alpha). Therefore, the minimum you found must be ((1/e)^(1/e))^2 = (1/e)^(2/e) = 0.47914170878... (source: google calculator) I hope that helps a bit, at least for now. I hope others can comment as well, as this isn't a real answer.

[Syzithryx]

Thanks Keith! I guessed it would have something to do with e, but I couldn't figure out exactly what. And yeah, I don't expect there to be much which can be said about this topic - I was just putting the idea out there.