Thanks for the post. What are the special icosahedron related functions one commenter said could be used to solve quintics?
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!).