The Hexadragon Curve: 6-Fold Terdragon Snowflake Symmetry
The Hexadragon Curve (also known as the six-headed dragon or terdragon snowflake) is an exquisite composite fractal structure. By arranging six identical terdragon curves rotated in $60^\circ$ increments around a single starting point, they interlock beautifully to form a highly symmetric, snowflake-like tiling tile.
Construction and Geometric Assembly
The construction of the Hexadragon relies on the properties of a single terdragon curve starting at the origin $(0, 0)$. First described by fractal pioneer Benoit Mandelbrot, the terdragon curve is defined by a simple recursive L-system. When you generate the path vertices for a single terdragon of order $n$, you replicate the path six times, applying successive $60$-degree rotations around the starting point:
• Branch 1 ($0^\circ$ Rotation): $(x, y)$
• Branch 2 ($60^\circ$ Rotation): rotated by $60^\circ$
• Branch 3 ($120^\circ$ Rotation): rotated by $120^\circ$
• Branch 4 ($180^\circ$ Rotation): rotated by $180^\circ$
• Branch 5 ($240^\circ$ Rotation): rotated by $240^\circ$
• Branch 6 ($300^\circ$ Rotation): rotated by $300^\circ$
Because the terdragon boundaries have a highly irregular boundary that fits perfectly with adjacent copies, these six rotated curves interweave. As the number of folds grows, they grow outward in six directions, forming a perfectly balanced, six-fold symmetric snowflake-like structure.
Mathematical Properties and L-System
The underlying terdragon component is defined by the following Lindenmayer system rules:
• Axiom: $F$
• Angle: $120^\circ$ (or $2\pi/3$ radians)
• Rewriting Rule: $F \to F + F - F$
At each recursive depth, every segment of length $s$ is replaced by three segments of length $s / \sqrt{3}$ oriented at $120$-degree turns. The boundary of each terdragon component is a fractal curve with a Hausdorff dimension of:
Interestingly, while a single terdragon has a Hausdorff dimension of exactly $2$ (making it fill a 2D space locally), its boundary has a Hausdorff dimension of approximately:
This value is identical to the dimension of the boundary of the famous Koch snowflake!
Frequently Asked Questions
Frequently Asked Questions
What is a Hexadragon curve?
A Hexadragon is a composite fractal made by arranging six identical terdragon curves around a single starting point, each rotated by a successive $60^\circ$ angle ($0^\circ$, $60^\circ$, $120^\circ$, $180^\circ$, $240^\circ$, and $300^\circ$).
How does it differ from a standard Heighway dragon?
A Heighway dragon uses a $90^\circ$ fold angle and scales by $1/\sqrt{2}$ per step, whereas each terdragon component of the Hexadragon uses a $120^\circ$ turn angle and scales by $1/\sqrt{3}$ per step. When arranged symmetrically, the Heighway dragon forms a 4-fold Quaddragon, while the terdragon forms a 6-fold Hexadragon.
Does the Hexadragon have rotational symmetry?
Yes. The Hexadragon possesses six-fold rotational symmetry (also called $C_6$ point-group symmetry). Rotating the entire fractal by any multiple of $60^\circ$ around its central origin results in a shape that is geometrically identical to the original.
Can I download the vector layouts of the Hexadragon?
Yes! Our generator allows you to download both high-resolution PNG images and lossless vector SVG files of the generated Hexadragon curve. Vector SVGs are perfect for digital art, printing, laser cutting, and architectural design patterns.
