Hello again!
I’ve shared a special file for you to explore, in my patreon page: two versions of the Dragon Curve: one created as a point cloud (in TOP) and another 3D version (in SOP). These visualizations give a fresh spin on this fascinating fractal, and I think you'll really enjoy seeing it come to life in different dimensions!
Now, let’s dig a little deeper into the math that makes the Dragon Curve so magical. The Dragon Curve is built through a simple yet powerful process called recursion. Here’s how it works:
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Start with a Simple Step: Imagine just a straight line.
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Flip and Repeat: With each "generation" or step, you add a new line segment at a 90-degree angle to the last one, then flip a mirror image of the entire pattern onto itself.
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Rinse and Repeat: Keep doing this over and over, each time creating a new “generation” of the pattern by adding a mirror image with a twist.
This recursive process means that each “level” of the Dragon Curve builds on the last, creating an intricate path without ever overlapping itself. In technical terms, it’s known as an L-system, a mathematical model that produces complex shapes from simple rules.
And here's the kicker: as you keep repeating this process, the curve fills up more and more of the space in a unique, non-overlapping way—eventually forming the “dragon” pattern with every twist and turn.
For the 3D version, imagine the same process but in a new dimension. By lifting segments slightly or adding a vertical twist, the curve gains depth, giving it an even more dramatic, “in-flight” look.
The beauty of fractals like the Dragon Curve is that these endlessly repeating shapes are both simple and incredibly complex. You can see patterns within patterns, like a universe unfolding inside itself. There’s something almost hypnotic about it, which is why I couldn’t resist sharing this in a point cloud and 3D form!
Check out the file, and let me know what you think. Thanks for being curious explorers with me, and stay tuned for more fractal fun!