Kurschak's General Tile 2
Josef Kurschak constructed a beautiful tile that constructs a dodecagon that makes several interesting observations possible. (See this sketch for that construction: http://www.geogebratube.org/material/show/id/49694) In that sketch he inscribes four equilateral triangles in a tile. Well one of the things mathematicians love to ask is "What If?" What if it they weren't inscribed in a square? This sketch lets you change the number of sides.
See also the sketch that goes deeper into the possibilities with a square: http://www.geogebratube.org/material/show/id/49720 More GeoGebra at mathhombre.blogspot.com.