The dragon curve
The dragon curve can be generated from repeated folds, turn sequences or affine transformations. Its sweeping coils look organic even though the rule is discrete and exact.
From a strip of paper
Fold a strip repeatedly in the same direction, unfold each crease to a right angle and read the resulting turn sequence. Successive approximations develop the characteristic paired spirals of the Heighway dragon.
Fold a paper strip repeatedly in the same direction, then unfold every crease to a right angle: the result is a sequence of left and right turns. The nth fold produces 2ⁿ segments, and their orientation can be generated by a simple recursive word rule. The paper model makes the local instruction tangible, while the drawing reveals how it accumulates into a compact and increasingly nested whole.
Two transformations build the same set
An iterated function system can construct the curve from two rotated and scaled copies. This exposes its exact self-similarity and connects symbolic turn rules with plane transformations.
The same limiting object arises from two contracting affine maps with scale factor 1/√2 and appropriate rotations. This IFS view exposes self-similarity more directly than folding: two reduced copies join to form the next stage. A renderer can therefore generate line segments recursively or sample points with a chaos game. At sufficient depth, both methods should recover the characteristic dragon shape.
A boundary that tiles
The limiting dragon has positive area and can tile the plane in related constructions, while the curve itself does not cross. Finite raster versions can appear to overlap when the pixel scale is too coarse; that is a display artifact rather than the defining geometry.
The Heighway dragon has an interior of positive area even though it is constructed as the limit of a line. Suitable copies can tile the plane without gaps while retaining a fractal boundary. The distinction between space-filling interior and intricate edge matters visually: an outline and a filled tile reveal different properties of the same construction and require different antialiasing strategies.
Sources and further reading
This article summarizes the following specialist sources in original wording. Accessed and editorially reviewed 12 August 2026.
- Dragon CurveWolfram MathWorld
- Fractal Geometry: Mathematical Foundations and ApplicationsWiley
Continue the idea
Space-filling curves
How a one-dimensional interval can map continuously onto a square and why finite Hilbert curves help order multidimensional data.
Iteration and recursion
How repeating a rule produces fractal detail, why iteration is not the same as recursion and how limits enter the definition.
Fractal art: mathematics as material
How artists turn formulas, parameter search, color, texture, lighting and export decisions into authored images.