Author here, in case anyone has any questions. :)
> Now this formula looks right, since x_1x and x_2x are at the same coordinates as r_1r and r_2r.
I assume you mean "since before they have been moved...", but by the time the reader gets to that point you've already told them to move r1 and r2 and x1 and x2 obviously aren't at the same coordinates.
> But that means that our candidate solution cannot be the quadratic formula! If it were, then x_1 and x_2 would have ended up swapped, too.
Why?
It would be more satisfying (and more general) to exhibit a specific unsolvable quintic.
In practice, there is a general quintic formula. It just needs one extra operation.
If you are free to add any additional operation, this whole thing becomes meaningless. You can simply define your operation "NewOp(a0,a1,..,a4)" to something like "the smallest root of the quintic a0+a1x+...+a4x^4+x^5".
(Here, "smallest" can be anything as long as it is a completely defined tie-breaker, such as: the value with the smallest real part, and among those the one with the smallest imaginary part.)
But it's a _long_ time since I've done Galois theory and I can't find a decent math exchange answer for it right now, so don't treat this as gospel.
Edit: Oh, but that only lets you solve some quintics. https://news.ycombinator.com/item?id=14686886 describes the functions you need to solve all quintics.
On the other hand, the topological method can be easily extended to prove stuff about formulas including other continuous, single-valued operations, like exponential and trigonometric functions, and Galois theory has a harder time with this.
Can these topological methods be generalized to arbitrary fields?
The special icosahedron functions can be expressed in many ways but perhaps the most elementary _explicit_ representation is as particular Gaussian hypergeometric functions. See this repo for example: https://github.com/ocfnash/icosahedral_quintic The special functions are used here: https://github.com/ocfnash/icosahedral_quintic/blob/master/q...
Geometrically these functions locally invert the (branched) covering represented by the diagram at the top of this page: http://olivernash.org/2012/02/05/on-kleins-icosahedral-solut...
Maths is just the extreme example of science, where to make communication possible between its practitioners, new words encoding known facts are constantly created. Then another layer of new words with definitions based on the first layers are defined... and so on. Rapidly, we end-up with total gibberish for the non-initiated.
For example on that page, we start with quintic and radicals (I can grok that). Then icosahedral functions, then hypergeometric functions, finite monodromy, 60-fold branched covering of the complex projective line (lost!).
I'm reading through it now, and I'm confused by this statement: "Second, and more surprisingly, if you swap r_1 and r_2r, x_1 and x_2 also exchange places, seemingly contradicting Theorem 1!"
Why is there any connection to theorem 1 here? We're talking about a radical, and theorem 1 is concerned with rational expressions.
I'm not entirely happy with the framerate I get with the button-triggered animations (especially on mobile) but lack the front-end expertise to track down the problem.
If there truly are roots of multiplicity >= 2, then preserving the order of the multiple roots doesn't matter. Another issue is that Newton-Raphson might fail if the roots are too close.