You'll never be able to zoom in on a little piece and just see a line or a curve. As fascinating as I find all of this, this is not why fractals became well known. While fractals were the first inroad that chaos theory had into the popular consciousness, it wasn't the biggest. If that's true, then the wind created by the flapping of a butterfly's wings could ultimately result in a hurricane ravaging the South Pacific and bringing great surfing conditions to Hawaii. Life itself is a chaotic system.
Small events can easily lead to major life changes.
Take Amanda for example. Having nothing better to do, she went with us. Amanda is a big Talking Heads fan and that sucked her in. She ended up really enjoying the show and started seeing a lot more concerts. She now spends her time on tour with a band that will remain nameless to protect her taste.