No matter how closely you zoom into the image, it always shows up more detail. So what do these mathematical curiosities have to do with the real world. Well, everything as it turns out. Such objects turn up all the time in the natural world. Animals, plants, rocks, crystals and liquids all exhibit fractal properties and self similarity.
Lets take a look at a common plant, the fern.
The fern is typical of many plants in that it exhibits self similarity. A fern consists of a leaf, which is made up from many similar, but smaller leaves, each of which, in turn, is made from even smaller leaves. The closer you look, the more detail you see. This is a standard fern, which you may well find while being dragged on long walks in the country by your parents long before you're able to fully appreciate the beauty of nature. You will see the overall theme of repeating leaves.