Examples of beautiful mathematics for high school level

Do you know of any good examples of beautiful mathematics that can be found (or should be found!) at the high school or middle school level?  Some answers to this question can be found here:

http://mathoverflow.net/questions/74841/an-example-of-a-beautiful-proof-that-would-be-accessible-at-the-high-school-level

But I am very happy for additional examples!  Thanks in advance!

About manyapajama

Associate Professor in Mathematics Education, and part-time poet.
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