The language
The Turtle family is the family of formal languages over the alphabet Tt = {F, f, l, r}.
The canvas
The semantic meaning of words over Tt is defined using a turtle canvas c = (t, s), where t is the turtle, defined as t = (p, h) (with p,h∈ℝ²) and s is the set of segments drawn, defined as s⊆{{p₁,p₂}|p₁,p₂∈ℝ²}. The canvas starts in the state ((p₀,h₀), ∅).
Interpreting a word
Each letter of a word over Tt changes the canvas ((p,h),s) according to the following table:
| Letter | Transformed canvas |
|---|---|
| f | ((p+h,h), s) |
| F | ((p+h,h), s∪{p, p+h}) |
| l | ((p,rotate(h,+90°), s) |
| r | ((p,rotate(h,-90°), s) |
Thus, the interpretation of a word is simply done by transforming the canvas for each letter, and getting the end state ((p',h'),s').
Languages in Turtle
We consider actual languages L over Tt that are described in a simple recursive way, by explicitly defining a single word u₀, and giving a function f mapping from letters of Tt to words over Tt. Hence,
For example, consider the following language:
f(F) = FlFrFrFFlFlFr
This gives this graph as u₀ and the right-angle Koch islands as u₁, u₂, u₃, u₄,… (click to see them in SVG).
Python implementation
This Python program implements Turtle languages as defined by an initial word and a recursive homomorphism. The output is a set of SVG files.