"Natural" objects such as clouds, mountains, trees, snowflakes, coastlines, galaxies, plants, vascular systems, river deltas, smoke, turbulence, and percolation are not easily described with "traditional" geometry.
The geometry of the smooth and regular is called Euclidean Geometry.
The geometry of nature is called Fractal Geometry.
There are quite a few definitions out there:
Self-similarity means "looks roughly the same" at different scales. We see this in nature; here zooming in on a cauliflower (from H.-O. Peitgen, J. Jürgens, and D. Saupe, Chaos and Fractals: New Frontiers of Science):
In nature, the self-similarity ends after a few orders of magnitude of zooming in. But we can give mathematical definitions with extremely simple rules that generate objects that are infinitely self-similar. Here's an introductory example:
There are many ways to define dimension. Topologically, the dimension of a set of points is the number real numbers required to unambiguously denote a point in the set. But there's something called the fractal dimension, which this video will tell you about:
Fractal dimensions need not be integers. Many shapes with a fractal dimension strictly larger than their topological dimension are called fractals.
Do a web search for fractals in nature and you'll find a zillion images. Or better, watch this short narrated slide show from Nicole Kershaw:
There are quite a few ways to describe (or draw) fractals.
These are described here.