Memes

at the higher levels you start to see all kinds of crazy stuff, here are some examples:

• mathematicians abstracted the idea of measure and then found out not everything can be measured
• we know there are different sizes of infinity, and we know what the "smallest" infinity is, but it's impossible to "know" (ie prove in ZFC) what the "second smallest size of" infinity is
• we took the regular number line and made it longer just to see what would happen
• The Hairy Ball Theorem, which says "you can't comb a hairy ball flat without creating a cowlick" (quote from source)

but as with any discipline, a big part of how much fun it is to learn has to do with how it's taught. i think it's possible to teach middle school/high school geometry in a way that makes it fun and engaging, but it's often not taught in this way. there's a great article/paper that talks about this. it's written to be very readable and accessible, although it is a bit long (but you can get the basic idea in the first 5-7 pages). he talks about how terribly math is taught in school and how it's no wonder so many people hate it as a result.

he also talks about how learning math could be much more fun if it was taught differently. he gives a really great example of this when he discusses something as simple as the formula for the area of a triangle (on the bottom of page 3 to the end of page 4). i tried to summarize it for this post, but i don't think a summary would do it justice, so i strongly encourage you to read it if you're interested.