Twindragon

The Twindragon, also called the Davis–Knuth dragon, is what you get when you take two copies of the Dragon Curve and place them back to back. The second dragon is the first one turned by 180° around the middle of its chord, so it runs from the end of the first dragon back to its start. The two halves fit together without crossing or sharing a single edge.

Each half is the same folding rule as the Heighway dragon:

const twindragonHalf = {
  axiom: "F",
  replace: {
    F: "F+G",
    G: "F-G",
  },
  angle: 90,
};

Here the two halves are drawn in different colors, so you can see where one dragon ends and the other begins.

The filled shape is a self-similar tile. It is made of two smaller copies of itself, each scaled by $\frac{1}{\sqrt{2}}$ and turned by 45°. In the complex plane it is the set of all numbers

T = \left\{ \sum_{k=1}^{\infty} d_k \, (i-1)^{-k} \;:\; d_k \in \{0, 1\} \right\},

which means it plays the role of the "unit digit square" for numbers written in base $i - 1$ . Turn on showTiling to see one of its best properties: copies of the Twindragon, shifted along the chord $c$ and along $i \cdot c$ , cover the whole plane with no gaps and no overlaps.

Because the tile has a real area, its Hausdorff dimension is exactly $2$ . Its boundary is a fractal curve with dimension

D = 2 \log_2 \lambda \approx 1.5236, \quad \lambda^3 - \lambda^2 - 2 = 0,

which is the same as the boundary dimension of the Heighway dragon. Other path-based relatives in the garden are the Terdragon and the Lévy Curve.