Space-filling surfaces

[fractower]

Quote
Is it possible to construct a surface such that the shortest path on the surface between any two points is always infinite
Yes.. for instance my 'volcanic surface' fractal: http://www.fractalforums.com/new-theories-and-research/new-fractal-needs-a-name/msg54859/#msg54859

This is an interesting fractal, but the operation appears to preserve edges. If this is a general property then points on an edge have a finite connection.

Since the edges added at each generation are exponentially shrinking, it seems that the path between any two arbitrary points will also have a finite connection.

[Tglad]

Ah true, certain paths are finite in this case. Well, other fractals have the property that any two points have an infinite surface path between them... like cumulus clouds, broccoli, fractal mountains (e.g. diamond square algo), etc.