Space filling curves filling with water

Summary by Krater

Steve Mould explores space-filling curves like the Hilbert curve, the Gosper curve, and Celtic labyrinths, explaining how infinitely thin lines can fill two-dimensional space through fractals and why their fractal dimensions make them fill space.

From the video

Answers: What is a Hilbert curve and how does a space-filling curve work?

• Hilbert curve
• Space-filling curves
• Fractal dimension
• Gosper curve
• Celtic labyrinths
• 3D printing Hilbert curves

What it concludes

• A space-filling curve is a fractal that is a one-dimensional line with a fractal dimension of two because it can fill two-dimensional space without leaving gaps.
• The Hilbert curve has good locality, meaning nearby points on the curve remain close to each other.
• Wetting the surface of a 3D printed translucent resin Hilbert curve cube with mineral oil removes cloudiness by matching refractive indices, revealing the internal structure.
• Current mathematical tools rely on iterative definitions and cannot formally prove that curves like the mould curve fill space because they lack recursive substitution approaches.