Sorry, I should have explained the images a little better. The x axis is the real part, and the y-axis is the imaginary part.
And yes, these are iterated natural logarithms: x, ln(x), ln(ln(x)), ln(ln(ln(x))), etc. I'm using the real interval (0, 1) as the 0th iteration, so the first iteration is the real interval \( (-\infty, 0) \), which isn't shown. The second iteration is then as described for the first image I showed, and so on.
Note that I only showed the primary branch. I'll get around to doing a version with several branches, but it's hard to decide how to do it, as the number of curves would grow exponentially if I used multiple branches for each iteration. Someday, if this still interests me, I'll write a version with GMP where I can create a large bitmap with various colors representing how many iterations are required to get back to reals. This should reveal a fractal which should cover most if not all of the complex plane, though the measure of the union of these curves should remain 0.
If my understanding is correct (and it probably isn't), for any complex number, there is a number arbitrarily close to it in the complex plane that can be exponentiated iteratively to recover a real number. This is similar to saying that for any real number, there is a number arbitrarily close to it which can be multiplied by an integer to recover an integer.
The measure of the rationals is 0, but they are dense. Similarly, the measure of the set of all complex numbers that can be exponentiated iteratively to recover a real number should be 0, but it should be dense over most if not all of the complex plane.
Not sure how this helps with tetration, but I have a rough idea (more like a vague hunch)...
And yes, these are iterated natural logarithms: x, ln(x), ln(ln(x)), ln(ln(ln(x))), etc. I'm using the real interval (0, 1) as the 0th iteration, so the first iteration is the real interval \( (-\infty, 0) \), which isn't shown. The second iteration is then as described for the first image I showed, and so on.
Note that I only showed the primary branch. I'll get around to doing a version with several branches, but it's hard to decide how to do it, as the number of curves would grow exponentially if I used multiple branches for each iteration. Someday, if this still interests me, I'll write a version with GMP where I can create a large bitmap with various colors representing how many iterations are required to get back to reals. This should reveal a fractal which should cover most if not all of the complex plane, though the measure of the union of these curves should remain 0.
If my understanding is correct (and it probably isn't), for any complex number, there is a number arbitrarily close to it in the complex plane that can be exponentiated iteratively to recover a real number. This is similar to saying that for any real number, there is a number arbitrarily close to it which can be multiplied by an integer to recover an integer.
The measure of the rationals is 0, but they are dense. Similarly, the measure of the set of all complex numbers that can be exponentiated iteratively to recover a real number should be 0, but it should be dense over most if not all of the complex plane.
Not sure how this helps with tetration, but I have a rough idea (more like a vague hunch)...
~ Jay Daniel Fox
