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The answer is “it depends on how you defined the operation that acts on the circle”
As long as, C*Cinv = I, where C is the circle, Cinv is the inverse of circle, and I the identity. You're right. C, I, and * are entirely up in the air.
Basically: Given a function `f`, then the function `g` is the inverse of `f` if `g(f(x)) = x`.
This article is interesting but not rigorous I think. * The inverse of a geometric shape makes no sense. We only inverse operations. * aa^-1 = 1 only if you consider the multiplication over reals. * 1/0 is not equal to infinity. Because the article is interesting but some people might be put off by the first few sentences, I suggest to had a disclaimer that this article lean on edutainment to the detriment of rigorou…
For example you can find Riemann Sphere in wikipedia - https://en.wikipedia.org/wiki/Riemann_sphere
on 1.2) is 1 divided by zero not undefined instead of infinite?
This article is interesting but not rigorous I think. * The inverse of a geometric shape makes no sense. We only inverse operations. * aa^-1 = 1 only if you consider the multiplication over reals. * 1/0 is not equal to infinity. Because the article is interesting but some people might be put off by the first few sentences, I suggest to had a disclaimer that this article lean on edutainment to the detriment of rigorou…
Earlier quoted context omitted.
As long as, C*Cinv = I, where C is the circle, Cinv is the inverse of circle, and I the identity. You're right. C, I, and * are entirely up in the air.
An inverse is pretty easy to generalize, and it need not rely on multiplication or the identity. Basically: Given a function `f`, then the function `g` is the inverse of `f` if `g(f(x)) = x`.
I did some wacky inside-out pics of planets years ago, inspired by the inside-out Mandelbrot set looking like a leaf/teardrop. https://www.adamponting.com/inside-out/ Related: Not Knot , 1991 Thurston-ish short film about knots and knot complements - "the space where the knot isn't". https://www.youtube.com/watch?v=zd_HGjH7QZo https://en.wikipedia.org/wiki/Not_Knot https://en.wikipedia.org/wiki/Knot_complement
In polar coordinates a circle can be defined by all points where r = a. The inverse would be all points where r doesn't equal a.
I interpreted the question myself from another angle: a circle is a function where every f(x) is an equal linear distance to an arbitrary fixed point z. So, the "inverse" to this function could be a function where every f(x) must have a different linear distance to z.
However, as someone said above, f() is the inverse of g() if g(f(x)) = x. When put into practice, this means that the inverse is the reflection of the original function over y = x.
However, there's one problem with looking at the problem this way: A circle is NOT a function. Therefore, it does not have an inverse as we are thinking of it. A circle can be described by two functions, and both of these inverses combine to form the same circle. So, the inverse of a circle is (sort of) itself.
I did some wacky inside-out pics of planets years ago, inspired by the inside-out Mandelbrot set looking like a leaf/teardrop. https://www.adamponting.com/inside-out/ Related: Not Knot , 1991 Thurston-ish short film about knots and knot complements - "the space where the knot isn't". https://www.youtube.com/watch?v=zd_HGjH7QZo https://en.wikipedia.org/wiki/Not_Knot https://en.wikipedia.org/wiki/Knot_complement
Is it just me, or does it look like black holes hovering over the planets? I guess that is to be expected, if a planet with black background is used. Pseudo-scientific question: Could black holes have anything to do with something being inverted?
Earlier quoted context omitted.
As long as, C*Cinv = I, where C is the circle, Cinv is the inverse of circle, and I the identity. You're right. C, I, and * are entirely up in the air.
In general for inversion, we have object A (argument) and object I (identity) and a function F of two arguments, so we have equations: `F(A, X) == I, F(A, I) == I, F(X, I) == I, A != I, A != X`, where A, I, and X are objects in the same category, i.e. they must be circles `(x² + y² == r²)`. If F is defined as `ra•rx`, then `ri == 1`, and inverse will be `rx = 1/ra`. If F is defined as `ra + rx`, then `ri == 0`, and i…