Editing Dragon curve (section)

=== Properties ===
* Many '''self-similarities''' can be seen in the Heighway dragon curve. The most obvious is the repetition of the same pattern tilted by 45° and with a reduction ratio of <math>\textstyle{\sqrt{2}}</math>. Based on these self-similarities, many of its lengths are simple rational numbers.
{{multiple image|total_width=800|align=center|image1=Dimensions fractale dragon.png|caption1=Lengths|image2=Auto-similarity dragon curve.png|caption2=Self-similarities}}
[[File:Full tiling dragon.svg|thumb|upright=1.35|Tiling of the plane by dragon curves]]
* The dragon curve can [[Tessellation|tile the plane]]. One possible tiling replaces each edge of a [[square tiling]] with a dragon curve, using the recursive definition of the dragon starting from a line segment. The initial direction to expand each segment can be determined from a checkerboard coloring of a square tiling, expanding vertical segments into black tiles and out of white tiles, and expanding horizontal segments into white tiles and out of black ones.<ref>{{harvtxt|Edgar|2008}}, "Heighway’s Dragon Tiles the Plane", pp. 74–75.</ref>
* As a [[space-filling curve]], the dragon curve has [[fractal dimension]] exactly 2. For a dragon curve with initial segment length 1, its area is 1/2, as can be seen from its tilings of the plane.<ref name=tabachnikov/>
* The boundary of the set covered by the dragon curve has infinite length, with fractal dimension <math display=block>2\log_2\lambda\approx 1.523627086202492,</math> where <math display=block>\lambda=\frac{1+\sqrt[3]{28-3\sqrt{87}}+\sqrt[3]{28+3\sqrt{87}}}{3}\approx 1.69562076956</math> is the real solution of the equation <math>\lambda^3-\lambda^2-2=0.</math><ref>{{harvtxt|Edgar|2008}}, "Heighway Dragon Boundary", pp. 194–195.</ref>