Terdragon

The terdragon is a self-similar fractal curve on the triangular grid introduced by Davis and Knuth (1970). It is a ternary relative of the dragon curve: its recursive construction uses three copies, shown in different colors above, where the usual dragon curve uses two (Davis and Knuth 1970). It can be generated as a Lindenmayer system with initial string "F", replacement rule "F" -> "F+F-F", and angle .

Using coordinates on the triangular grid with basis vectors meeting at , the -coordinates of successive points begin 0, 1, 0, 1, 0, 0, , 0, , ... (OEIS A349040), and the -coordinates begin 0, 0, 1, 1, 2, 1, 2, 2, 3, ... (OEIS A349041).

After replacements, the approximant consists of equal line segments. With the initial segment from 0 to 1 in the complex plane, scaling the th approximant by and rotating it by fixes its endpoints and gives a limiting curve of Hausdorff dimension 2. Complete triangular folding curves extend to coverings of the triangular grid in which each edge is traversed exactly once. Rounding the angles at the vertices makes the curves disjoint (Oger 2023).