Are there any unique places in the Mandelbrot Set that have not yet been seen graphically?

Question

I'm aware that the Mandelbrot Set is an infinite set of numbers, and that there are many beautiful patterns that can be found within it.

So my question is, are there any more beautiful patterns that are yet to be seen in the set; but we just can't due to how much computational power it would take to calculate/go that deep within the set?

For example, here we see the elephant valley:

Now, is it possible, that somewhere hidden in the Mandelbrot Set, there is a Man Riding Horse Valley with impeccable detail that we just haven't seen yet, because it is hidden so deep?

Answer 1

Maybe dense parts of the parameter plane

In generally one can zoom in infinitely many places which takes time ( limited !) and precision, so there are infinitely many such places. See also perturbation method for some improvement.

Similar interesting problem is on the dynamic plane : there are some Julia sets ( Non-computable Julia sets ) which were note yet been seen graphically ( even without any zoom) : Cremer Julia sets

Answer 2

Yes and there always will be.

Any graphical image is a raster of coordinates sampled in x and y. As such it is only ever a finite countable subset of what it represents. The real image has a cardinality greater (reals vs rationals) and no matter how many (countable) times you may zoom and pan, you will never be able to reach the uncountable level of detail of what you are sampling.

As such this is true of any digitised image, whether computer generated or captured from the real world. For a photo for instance you will be limited in resolution by convolution, the beam width at the focus and aberations, all combining to hamper the ability to gain more detail beyond a certain point.