The Sierpinski Triangle is what interests me most in this question. Would there be equations for self-similar fractals (or any fractal types) which would work on a graphing calculator such as Desmos?
[Math] Graphing Fractals With Equations
fractalsfunctionsgraphing-functions
Related Solutions
You ask: "Is Wikipedia's definition of fractal the standard?" and right near the top of Wikipedia's page of fractals, we see the following definition:
A fractal is a mathematical set that has a fractal dimension that usually exceeds its topological dimension and may fall between the integers.
The statement that the fractal dimension may "fall between the integers" really adds nothing but, other than that, I would say that this is fairly standard; it is unquestionably the definition that was put forward by Mandelbrot around 1975 when he coined the term "fractal". He did not refer to "fractal dimension" at that time but, rather, the "Hausdorff-Besicovitch dimension" as he put it. In fairness, the usefulness of this definition has been debated with even Mandelbrot himself feeling that it might not be inclusive enough. Nonetheless, this comparison of dimension is central in fractal geometry. Gerald Edgar calls his great book, Measure, Topology, and Fractal Geometry, a meditation on the definition.
Taking this to be the definition, we can definitely say that the Cantor set satisfies it. If by "fractal dimension" you mean similarity dimension, then the Cantor set has fractal dimension $\log(2)/\log(3)$, since it's composed of two copies of itself scaled by the factor three. Also, the set is regular enough that any reasonable definition of fractal dimension agrees with that computation. (Well, any real-valued defintion.)
Topological dimension is a trickier thing, actually. It's inductive in nature. Totally disconnected sets (like single points, finite sets, or notably the Cantor set) have dimension zero. Higher dimensions are defined in terms of lower dimensions. The space we live in is three dimensions because balls in this space have a surface that is two dimensional. Because of this inductive nature, topological dimension always yields an integer.
When you write that you "do not see the irregular aspects or the complexity that is usually inherent with fractals", I think you might have a bit of a mis-understanding about fractal geometry. The Cantor set is indeed regular but, then so are all the strictly self-similar sets studied in classical fractal geometry - the Koch curve, the Sierpinski triangle, the Menger sponge, and countless others all display this regularity. Indeed, it's exactly this regularity that allows us to understand them.
To emphasize this regularity, and how it appears in not just the Cantor set, compare the following zooms of
The Cantor set
The Koch curve
Now, of course, there are "irregular" fractals - or, at least, less regular fractals. Examples include random version of self-similar sets, examples that arise from number theory, and examples arising from complex dynamics (like Julia sets). It's not their irregularity that makes these objects fractal, however. On the contrary, its the regularity that we can find that allows us to analyse these objects to the point where we can characterize them as fractal. Of course, this analysis is bit harder with these less regular examples.
My favourite book on fractals is Measure, Topology, and Fractal Geometry by Edgar. A short book and not very well known. It has a great many exercises all very suitable at undergrad. level but it requires a good mathematical background in basic analysis and topology. Apart from focusing on the geometrical aspects it also gives an excellent overview on different fractal dimensions definitions. But if I remember correctly this book does not cover fractals in the complex plane (Julia sets)
However, if you just want an overview on fractals I would start with Chaos and Fractals New Frontiers of Science by Peitgen, Jurgens & Saupe. It does not really give definitions but it guides the reader with simple examples or numerical experiments to the interesting properties of fractals.
A good understanding of Julia sets unfortunately requires a good understanding of Riemann surfaces, Hyperbolic geometry and Complex Analysis. The only book I know which also covers the necessary background is the Dynamics in one complex variable by (Fields medal winner) John Milnor. This is quite a tough book I read it when I was an undergrad. but I only fully understood it when I was a grad. student. But the material is great it covers Julia sets both from an analytical and topological view. It contains a collections of amazing proofs e.g. which Julia sets are smooth, when is a (polynomial) Julia set connected and a great many fun and insightful exercises.
I am actually not sure what level you are looking for but let me order the books based on difficulty (lowest easiest). Also perhaps you might be interested in reading Falconer's Fractal Geometry book. I only read some chapters many years ago, unfortunately the author introduces the Hausdorff dimension somewhere in the beginning which is perhaps one of the most difficult dimension definitions but after that it gets more easier.
Chaos and Fractals New Frontiers of Science by Peitgen. Jurgens & Saupe (No real proofs, but contains numerics and gives excellent intuition. Also the authors always put references in their informal statements to books/articles where you can find the details)
Fractal Geometry by Falconer (only the beginning is difficult, but contains mathematical rigour and simple proofs.)
Measure, Topology, and Fractal Geometry (nice exercises, requires good background in basic analysis and topology. Note it also contains a section on basic topology/metric spaces)
Dynamics in one complex variable by John Milnor (Only Julia sets, very difficult, but also very awesome)
Best Answer
Let me describe what the functions on this Desmos page do. Try to go through the steps with me and play around with the equations and buttons to select equations, this will make everything clear.
First, deselect everything except for the equation for $(T(t),S(t))$. Clearly, that is a parametric equation for this red triangle. Now, the equation just below gives an inverted rescaled version of that triangle. This is just an affine transformation of the first triangle. As you can see by the formula which is: $\left(2T\left(t\right)-\frac{1}{2},-2S\left(t\right)+2\frac{\sqrt{3}}{4}\right)$.
The next step is to create many rescaled versions of these triangles. This is what the functions $A_x$ and $A_y$ do. To help you visualise this, make a duplicate of the ninth equation, but drop the summation terms. you should get something like this
$$\left(A_x\left(Nt,\operatorname{ceil}\left(Nt\right)\right),A_y\left(Nt,\operatorname{ceil}\left(Nt\right)\right)\right)$$
The plot of this is a series of rescaled versions of the triangle represented by $(T(t),S(t))$. The sidelengths of successive rescaled version scale as $1/2$. But that's not all. You also see this modulo function: $\operatorname{mod}\left(3^kx,1\right)$. This makes you actually traverse the triangle several times. For every rescaled version you traverse it 3 times more. You don't see it because at this stage, the triangles overlap. But this is important, because the next step is to position each traversal of the copies in different positions. That's what the summation term we dropped is used for.
Copy the following part in a separate line
$$\sum_{n=1}^{\operatorname{ceil}\left(Nx\right)}\left(\frac{1}{2}\right)^nF\left(p\left(Nx,n\right)\right)$$
Set the slide ruler in the first equation on $N=1$. When you do that, what you will see for the interval $[0,1]$ is actually just the function $F(x)$ that is defined on the second line, which is just a step function. To be more precise, it's a discrete function, but when combined with $p(x,k)$ it becomes a step function. But if you know add this to the $A_x$ from before, this will shift the x-coordinates of all the little triangles to the positions you want to have them. The clever trick is now that again using the modulo and rescaling, you can also out of $F$ construct the positions for the triangles at smaller scales. So that's what that part does. Do the same for the y-coordinates and now putting everything together and you have constructed the Sierpinski triangle.
Hope this sets you on your way. Try playing around with it to understand how it works. Another suggestion: try first making a one dimensional fractal like a Cantor set. This makes the construction a bit simpler and let's you accomodate to the way Desmos can be used to make this kind of fractals.