Consider the following equations:
$$x = \frac{2}{x}+1$$ $$ x = \frac{x}{2}+1 $$
If we plug $x$ into itself on the right side indefinitely, we get continued fractions:
$$x =\frac{2}{\frac{2}{\frac{2}{\downarrow_{\frac{2}{x}+1}}+1}+1}+1$$ $$x = \frac{\frac{\frac{\uparrow^{\frac{x}{2}+1}}{2}+1}{2}+1}{2}+1$$
And the solutions for both equations remains $x=2$ of course.
If we take limits of the right sides we see that for any $x$ on the right side, we get $2$ of course:
$$ \lim_{n\to\infty}\frac{2+2x(2^n-1)}{2^nx}=\lim_{n\to\infty}\frac{x+2(2^n-1)}{2^n}=2$$
Thus, we can write (zeroth iteration):
$$ 1 = \frac{\frac{\frac{\frac{\uparrow^{\frac{x}{2}+1}}{2}+1}{2}+1}{2}+1}{\frac{2}{\frac{2}{\frac{2}{\downarrow_{\frac{2}{x}+1}}+1}+1}+1}$$
On the right side; We can replace all the $2$'s with the ascending/descending infinite fractions and replace all $1$'s with the expression itself. Here is the first iteration:
We can repeat that replacement indefinitely to get a "fraction notation fractal".
Note that the $1$ is replaced with $x$ in the picture above. (solution remains $x=1$)
Is there a software (or relatively simple programming tutorial) that can help me explore this fractal? By that I mean that it seemingly does indefinitely many iterations and allows me to zoom in or zoom out freely. I know of online sites which provide basic fractal plotting, but what's the closest thing that can handle "notation fractals"?
If I could customize the notation itself, that would be even better.
Also,
Are there examples of nested fractions which can't be easily solved/estimated and require some tricks or other mathematic techniques? Perhaps sets of such equations can be combined into a serious fractal which wouldn't be solvable very easily as something like this?
Since this seems relatively easy to solve if you've seen at least one example, or am I wrong?
For a similar simple example, we can construct something like this:
$$ \frac{1}{2}=\frac{\frac{3}{4}}{\frac{3}{2}} = \frac{\frac{\frac{\frac{\uparrow^{\frac{x}{3}+\frac{1}{2} }}{3}+\frac{1}{2} }{3}+\frac{1}{2} }{3}+\frac{1}{2} }{\frac{3}{\frac{3}{\frac{3}{\downarrow_{\frac{3}{x}-\frac{1}{2} }}-\frac{1}{2} }-\frac{1}{2} }-\frac{1}{2}}$$
Where we can replace the $0.5$'s with the fractal itself or replace the $1$ and $2$ each with parts of the previous fractal. Then we also find a fractal structure for number $3$, and continue to build up our notation fractal.
Can we introduce a relatively symmetrical fractal (based on my latest example) where we replace the $1$'s with something that has $2$'s which is replaced with a structure that has $3$'s, and then $4$'s, and so on to cycle over all integers, such that it's total value is a finite real number?
It sure looks like you've already got the basic idea - not sure what more you're looking for. As another, perhaps simpler, example that produces a classic fractal suppose we iterate the the function $$f(x) = x_x^x$$ starting from $x$. We get
$$ \left(x_x^x\right)_{x_x^x}^{x_x^x} $$ $$ \left(\left(x_x^x\right)_{x_x^x}^{x_x^x}\right)_{\left(x_x^x\right)_{x_x^x}^{x_x^x}}^{\left(x_x^x\right)_{x_x^x}^{x_x^x}} $$ $$ \left(\left(\left(x_x^x\right)_{x_x^x}^{x_x^x}\right)_{\left(x_x^x\right)_{x_x^x}^{x_x^x}}^{\left(x_x^x\right)_{x_x^x }^{x_x^x}}\right)_{\left(\left(x_x^x\right)_{x_x^x}^{x_x^x}\right)_{\left(x_x^x\right)_{x_x^x}^{x_x^x}}^{\left(x_x ^x\right)_{x_x^x}^{x_x^x}}}^{\left(\left(x_x^x\right)_{x_x^x}^{x_x^x}\right)_{\left(x_x^x\right)_{x_x^x}^{x_x^x}}^ {\left(x_x^x\right)_{x_x^x}^{x_x^x}}} $$ $$ \left(\left(\left(\left(x_x^x\right)_{x_x^x}^{x_x^x}\right)_{\left(x_x^x\right)_{x_x^x}^{ x_x^x}}^{\left(x_x^x\right)_{x_x^x}^{x_x^x}}\right)_{\left(\left(x_x^x\right)_{x_x^x}^ {x_x^x}\right)_{\left(x_x^x\right)_{x_x^x}^{x_x^x}}^{\left(x_x^x\right)_{x_x^x}^{x_x^x }}}^{\left(\left(x_x^x\right)_{x_x^x}^{x_x^x}\right)_{\left(x_x^x\right)_{x_x^x}^{x_x^ x}}^{\left(x_x^x\right)_{x_x^x}^{x_x^x}}}\right)_{\left(\left(\left(x_x^x\right)_{x_x^ x}^{x_x^x}\right)_{\left(x_x^x\right)_{x_x^x}^{x_x^x}}^{\left(x_x^x\right)_{x_x^x}^{x_ x^x}}\right)_{\left(\left(x_x^x\right)_{x_x^x}^{x_x^x}\right)_{\left(x_x^x\right)_{x_x ^x}^{x_x^x}}^{\left(x_x^x\right)_{x_x^x}^{x_x^x}}}^{\left(\left(x_x^x\right)_{x_x^x}^{ x_x^x}\right)_{\left(x_x^x\right)_{x_x^x}^{x_x^x}}^{\left(x_x^x\right)_{x_x^x}^{x_x^x} }}}^{\left(\left(\left(x_x^x\right)_{x_x^x}^{x_x^x}\right)_{\left(x_x^x\right)_{x_x^x} ^{x_x^x}}^{\left(x_x^x\right)_{x_x^x}^{x_x^x}}\right)_{\left(\left(x_x^x\right)_{x_x^x }^{x_x^x}\right)_{\left(x_x^x\right)_{x_x^x}^{x_x^x}}^{\left(x_x^x\right)_{x_x^x}^{x_x ^x}}}^{\left(\left(x_x^x\right)_{x_x^x}^{x_x^x}\right)_{\left(x_x^x\right)_{x_x^x}^{x_ x^x}}^{\left(x_x^x\right)_{x_x^x}^{x_x^x}}}} $$
Looks like a Sierpinski triangle type limit. Your situation looks a little tougher in that you've got two fractals that are linked together.
Any software that can both express functions symbolically and has TeX export should be able to do this sort of thing. I produced the pictures above with a couple of lines of Mathematica code: