Infinity is a mathematical construct, yes. But it is a very powerful one which does have merits in "the real world".
All that infinity means is that something does not end, that to any set of examples you have, you can add another example.
In case of the natural numbers, that means, that if you have the set {0,1,2}, you can add 3 as another example of a natural number, to make that set {0,1,2,3}.
Now you could add 5 and your set becomes {0,1,2,3,5} - note that the order here doesn't play a role. You might as well add 100 or 456451692060887 as two more examples.
The natural numbers have an additional property of being enumerable, listable or countable (all synonyms).
You can order them 0,1,2,3,4,5,...
and there is a simple algorithm to get from one natural number to the next (namely "add 1").
Since this algorithm exists, and no matter what natural number you have, you can add 1 to it, you get a countable infinity.
If you try that with real numbers, you will fail. You cannot list all possible real numbers.
E.g. there is no way for any algorithm to create an exhaustive list of all reals.
The real numbers are uncountably infinite.
This notion of infinity is trickier.
However, your given example of a sierpinsky triangle is very rigid. It repeats in a simple, obvious manner. (It is strictly self-similar) In that example it is very clear that:
- you will NOT, in fact, get all shapes imaginable
- the repetition can be executed indefinitely and thus no matter how deep you zoom, you can see more of the same.
The Mandelbrot set, however, is only "pseudo-self-similar" - it "almost" repeats itself over and over again.
There are an infinite number of shapes in it.
This can also be easily proven: There are sequences where, if you zoom in, you get patterns that double their symmetry each order of magnitude. You have an (almost) bilaterally symmetric pattern which, if you zoom in, becomes an (almost) 4-way-symmetric pattern, which, as you zoom in, becomes an (almost) 8-way symmetric pattern, etc. You can see such paths in lots and lots of deep zooms out there.
What is not guaranteed is, that the M-Set actually contains all possible shapes or, more precisely, that it may approximate all possible connected shapes arbitrarily well.
It certainly does approximate all possible Julia-sets. But you can classify Julia-Sets by overall appearance and there are only a few types that "feel different" from each other with smooth transitions in between.
Non-Julia-Set-like shapes are a bit harder to find but if you look for it, you can often find it anyway, given some experience. There are pictures of the Mandelbrot set approximating all letters in the alphabet for instance. And a bunch of faces too.
Come to think of it, I'd love to see the corresponding Julia sets. Technically, they *should* show those same patterns, right? The deeper you zoom into the M-Set, the more it should look like the corresponding Julia set.
But Julia sets are way less variable in their appearance, than the M-Set. Could you get Julia sets that look like letters?
That being said, there are so-called Compositional Pattern Producing Networks (CPPNs) which are like Iterated Function Fractals where you use a different function (from a predefined set of functions) each time rather than using the same function over and over.
There are some fractals that come very close to this, like randomly parametrized möbius transforms (a x + b)/(c x+ d) with a, b, c, d random complex numbers each iteration. Instead of only allowing that family of functions, you could allow any other set of functions to be used.
What's possible with this is demonstrated in projects like
http://picbreeder.org/or
http://endlessforms.com/It's very clear that you can, at least in principle, construct any shape you can possibly think of in this way.
It's not entirely clear, though, whether it is sufficient to only iterate z²+c to accomplish this amount of complexity.
Though if that is not the case, it certainly comes close.
There also is the Zeta-Function (and a whole family of functions to which it belongs) which, in its major strip, approximates a small circular cutout of any other function arbitrarily well.
It is an open problem if it also approximates itself arbitrarily well in that strip, though if it does, that the Riemann Hypothesis is true.
This means that the Zeta-function *should* also approximate (sections of) the M-Set arbitrarily well, and the Zeta-Function isn't even defined recursively.