I'm trying to construct examples of combinatorial group theory flavor to introduce the ideas to young students. More specifically, I'm trying to introduce the idea that you can have strings of letters that have some replacement rules (such as "you can delete $\texttt{XX}$" or "you can replace $\texttt{XY}$ with $\texttt{YX}$"), and that you can just play with these things as formal symbols.
But it would be nice to also have real examples that the students can act/draw out to verify their computations with the abstract pencil-and-paper manipulations. It struck me that turtle graphics could be one such example. Here's the idea:
Let's say that the turtle graphics group $\mathcal{T}$ is generated by elements $L,R,F,B$. In the typical setting of a turtle with a pen moving in the standard integer lattice in the plane, $L$ means to make a quarter turn left, $R$ a quarter turn right, $F$ move forward one step drawing e.g. a black line, $B$ move backward one step drawing e.g. a blue line. We'll keep track of the multiplicity of lines between the lattice paints, and let's say that black lines and blue lines along the same edge cancel. Then $L=R^{-1}$ and $B=F^{-1}$, and I think it follows that this is a group.
Can anyone give a presentation for this group? I.e. what are the relators? It's clear that $R$ has order $4$ and $F$ has infinite order, so a presentation might start at $\mathcal{T}=\langle R,F \mid R^4,\dotsc \rangle$. But it's unclear to me whether $R$ and $F$ have any relation. Is the group in fact just the free product $\mathbb{Z}/4 * \mathbb Z$?
Edit: to make the question more rigorous along the lines of the comment from @Kyle Miller, there are a few things at play here. There is a monoid generated by the symbols $R$ and $F$. There is a set $D$ (for drawings?) that indexes possible states of the turtle graphics system. This set can be described as follows:
Let $V=\mathbb{Z^2}$ denote the set of integer points in the plane ($V$ for vertices). This is the set of possible positions of the turtle.
Let $E$ denote the set of edges in the standard square grid. If you like, the elements of this set are all intervals of the forms $\{x\} \times [y,y+1]$ and $[x,x+1] \times \{y\}$, where $x,y \in \mathbb Z$.
Then $D$ is the set $V \times \{0,1,2,3\} \times \mathbb{Z}^E$ where the first coordinate is the position of the turtle, the second is the turtle's orientation, and the third tells you how many times each edge has been drawn with multiplicity (positive for a forward step, negative for a backwards one).
Then there is an action $\langle R,F \rangle \to \operatorname{Sym}(D)$, and I suppose $\mathcal{T}$ is the group that is the image of this representation.
This should be a comment and not an answer, but it is with photos for demonstration, and I want to point out a fact that the turtle graphics group $\mathcal{T}$ is not a pure algebraic object, it is related with the underlying geometry space the turtle is in, for most of cases, it should be a homogeneous space. With different underlying space the group is different.
case 1: Euclidean plane
It is easy to see, rectangles(as a special parallelogram) can be constructed by turtle.
case 2: Hyperbolic plane
No way to construct a parallelogram. and in fact we can construct a cayley 4 tree, or an order 4 apeirogonal tiling.