Order (Iteration)
1
Grid Size
2×2
Cells
4
Segments
3
Dimension
2

Locality Preservation

Colors show position along the 1D curve. Notice how nearby colors stay close in 2D space!

📏 The Paradox

Can a 1-dimensional line fill a 2-dimensional square? Yes! The Hilbert curve is a continuous path that visits every point in a square. In the limit, it has dimension 2, not 1!

🔄 The Construction

Start with a U-shape. Each iteration:

Divide into 4 quadrants
Place smaller U in each
Rotate/flip to connect

Order n has 2ⁿ×2ⁿ = 4ⁿ cells.

🎯 Locality Preservation

Points close on the curve stay close in 2D! This makes Hilbert curves ideal for database indexing, image compression, and spatial queries. IP address visualizations use this property.

💻 Modern Applications

Google uses Hilbert curves in their S2 geometry library for mapping. Graphics cards use them for texture mapping. They're also used in dithering algorithms and machine learning spatial features!