Fractal Landscape Grid

Scrolling through a procedurally generated landscape.

Based on an initial grid script, modified with Diamond-Square terrain and perspective projection. (Click canvas to toggle animation)

Folded Plan Fractals and Their Dimension

• What is a Folded Plan Fractal?
  A folded plan fractal is a type of fractal that looks like a surface (a "plan") which has been folded, twisted, or crumpled in a very complex way. Imagine taking a flat sheet of paper and crumpling it lightly without tearing it — you still have a surface, but now it fills space in a much richer, more intricate way.
• Why is it Special?
  Even though the original object is two-dimensional (like a sheet), once it’s folded and twisted at every scale (even infinitely small ones), it starts behaving as if it were partly three-dimensional. It doesn't completely fill a 3D volume like a solid object would, but it "touches" much more space than a normal flat surface.
• Hausdorff Dimension Between 2 and 3
  In mathematics, the Hausdorff dimension describes how "thick" or "complicated" an object is:
  • A perfect line has dimension 1.
  • A perfect flat surface has dimension 2.
  • A solid object (like a cube) has dimension 3.
  For folded plan fractals, the Hausdorff dimension is between 2 and 3, meaning:
  • They are "thicker" and more space-filling than a flat surface (dimension 2).
  • But they are still not completely solid like a cube (dimension 3).

  Example of values: A crumpled surface might have a dimension like 2.3 or 2.7, depending on how "intensely" it is folded and packed.

• Intuitive Image
  Think of it like this:
  • A flat map laid on a table = dimension 2.
  • A crumpled map stuffed into a small ball = looks like it almost fills the space inside the ball, but it's still just made of surface material — more than 2D, less than 3D.
• Real-World Examples
  
  • Crumpled sheets of metal or paper.
  • Some types of biological tissues (like brain folds).
  • Fractal-like structures in turbulent flows or certain geological formations.

References:
- Falconer, K. J. Fractal Geometry: Mathematical Foundations and Applications
- Mandelbrot, B. B. The Fractal Geometry of Nature