Hi Jacob! Thanks for the question! A fractal is an object that looks the same at different length scales. This means that when you zoom in on something with a fractal form, you see the same shape that you did when you were zoomed out. For example, you might look at a tree and see a trunk with branches stemming off. Then, if you look at a single branch, it has the same pattern (a main stem with branches coming off). If you zoom in on a single leaf, you'll see the same pattern again, a large vein down the middle with smaller branches stemming off, and even tinier branches stemming off of those small branches. I hope that makes sense! Here are some examples of fractals in nature:

fractal image1
fractal image2
fractal image3

Fractals form in nature because it is easier for living structures to function if they are made up of many tiny parts, rather than one single large mass. A mass of clustered cells that are individually regulated is more easily maintained than one giant cell would be. For example, imagine if a tree was made up of one giant cell. It would be much more difficult to transport nutrients from one end of that cell to the other than it is to transport nutrients from tiny cell to tiny cell all the way up the tree. So, natural structures tend to be made up of fractals rather than large pieces.

To put it very, very simply, a fractal is something that looks the same no matter how much you magnify it or de-magnify it. The formal definition of fractals is hard to understand, but if you look at the gif on the Wikipedia page, you can see an example of a fractal that might be easier to grasp. For an example of a natural fractal, see the first picture on this page.

The reason we call this branch-and-leaf a fractal is that the overall shape of the branch is reflected in the shape of each leaf, which is in turn reflected in the shape of each distinguishable part of each leaf. Trees can be another example, where each large branch has many small branches arranged in a certain pattern, and each small branch has smaller branches arranged in a similar pattern. As to how fractals form in nature, we don't really know. Our understanding of how organisms build their parts does not adequately explain how this building process leads to apparent fractal structures in some organisms (in other words, it is not adequate to say that the biochemical/biological processes within our understanding naturally leads to the appearance of fractal structures because there is no innate, logical connection between our understanding of the details of these processes and the final appearance).

Why and how fractal structures form in nature may fall beyond the realm of science, and the willingness to entertain the possibility that science is not absolutely all-encompassing is part of the attitude of being a scientist.

Fractals are self-similar patterns, meaning that the pattern at small scales is repeated in the shape at larger scales, and this is true at all length scales. One can "make" a fractal by adding a piece to an existing object by repeatedly applying the same rule. I was not able to find an explanation of how fractals form in nature, but perhaps the idea is the same - performing the same sort of action at multiple length scales (e.g., to build the famous Mandelbrot set ). There also does not appear to be a good reason as for why nature forms fractals, but they may be more efficient than other patterns.

For example, when shown a new image a human's eye will jump across it in a series of short and long jumps which are more efficient than random jumps or a series of side-to-side movements. As an aside, fractals in nature are not "true" fractals because fractals are mathematically infinite (there is no limit to how small or large the scale can be, the pattern will always be the same).

Fractals are patterns that look mostly the same as you zoom in on them at smaller and smaller scales. Trees are fractals. I like this picture of a tree-like fractal with 1 stem that branches into 2 stems and each stem branches into 2 more stems, and the stems and branches get smaller and smaller.

Trees are fractals, too, with a trunk that branches into branches that branch into smaller and smaller branches and finally twigs. Rivers are fractals - as you go upstream, the rivers branch, and then the branches of the river branch into smaller rivers and then creeks and streams. Coastlines are fractals. You can look at a map of the world and see that the coastlines turn this way and that way. You can look from an airplane and see one coastline that turns this way and that way. You can walk along the beach and see how the coastline turns this way and that way.

Mathematically, fractals are anything that forms a repeating pattern at different scales. There are a lot of things in nature with fractal-like qualities, from the arrangements of flowers in broccoli to the uneven nature of coastlines to the shapes of successively larger clusters of galaxies, all of which appear similar over a range of scales. However, none of these things are truly fractal, because above or below a certain scale they stop behaving in a fractal-like pattern.