Demonstrating Klein's icosahedral solution of the quintic equation
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The icosahedral solution of the quintic in Python

Although the general quintic equation cannot be solved 'in radicals', it can be solved using some additional special functions associated to the icosahedron.

The scripts here implement the icosahedral solution for quintics that take the form:

  • y^5 + y + c = 0 in
  • y^5 + 5ay^2 + 5by + c = 0 in

Note that it is possible to reduce a general quintic equation to either of these forms and it requires solving only one auxiliary quadratic equation to reduce it to the latter form.

I waffled on about this in a bit more detail in the notes published here available for free at this link.