Today: Hyperbolic Geometry, and why it is awesome.
So a quick recap on the different kinds of geometry we generally deal with. Euclidean geometry is the kind we all know and love, where, if you have a line and a point outside a line, exactly one line can be drawn through that point that is parallel to that line. Hyperbolic geometry falls into Non-Euclidean geometry, where an infinite number of lines can be drawn through that point such that it is parallel to that line.
Now despite the fact that hyperbolic geometry has existed as mathematical concept for an extremely long time, it's only until recently that mathematicians were able to come up with a tangible model for it. The coral reefs, of all places, are one of several examples of hyperbolic geometry in nature (crochet is another method of recreating hyperbolic geometry).
If the mathematical concept is a little hard to visualize, watch this TED talk on the subject, which has a wealth of information on the subject: Margaret Wertheim on the beautiful math of coral.