A little Math problem...

        I have been studying the Mandelbrot Set for a number of years, and during that time I've learned many wonderful things.  The Set features a filligree of interwoven spiral patterns in seemingly endless variations.  But it turns out that the branch points and centers of spirals display a mathematical regularity that makes them Misiurewicz points - places where the generating function does not immediately repeat, but eventually falls into a repeating pattern.  Such points are called pre-periodic, and they are classified by how often they repeat and how long they take to begin repeating.  So how do we find them?

       When we look at the generating equation for the Mandelbrot Set and Julia Sets, it reads `z->z^2+c` and this becomes `z->z^2+z`, or more specifically `z_n=z_(n-1)^2+z_0` in the case of `cc(M)`.  So we expect that the Misiurewicz points will have terms of this same form, repeated a specific number of times on either side of the equation.  One such point of interest satisfies the equation `((c^2+c)^2+c)^2+c=(c^2+c)^2+c` where `c in bbb(C)` - the complex numbers*.  We see that this is the Mandelbrot formula iterated three times on the left hand side and twice on the right hand side of the equation.

       We could expand this out to get `c^8+4c^7+6c^6+6c^5+5c^4+2c^3+c^2+c=c^4+2c^3+c^2+c` by squaring the terms inside the parentheses and then combining like factors, but we are looking for roots of the equation.

       When we solve for `c` we find a trivial solution (which is actually a periodic or repeating point).

       We get two non-trivial results with imaginary components.

`c=(sqrt(11)/3^(3/2)-17/27)^(1/3)*((sqrt(3)i)/2-1/2)-(2*(-(sqrt(3)i)/2-1/2))/(9*(sqrt(11)/3^(3/2)-17/27)^(1/3))-2/3`

and

`c=(sqrt(11)/3^(3/2)-17/27)^(1/3)*(-(sqrt(3)i)/2-1/2)-(2*((sqrt(3)i)/2-1/2))/(9*(sqrt(11)/3^(3/2)-17/27)^(1/3))-2/3`

       But the solution we are interested is a non-trivial result where `c in bbb(R)`  i.e. - a Real numbered value.

       We note that the expression `sqrt(11)/3^(3/2)-17/27` can be simplified.  Since `3^3=27`, the first segment becomes `sqrt(11)/sqrt(27)` and using the fractions of square roots rule, we can write this as `sqrt(11/27)`.  Then our solution becomes

       This has a numerical value around

       But can we get closer to an exact value?

       This extended precision result may suffice: