Josef Kurschak constructed a beautiful tile that constructs a dodecagon that makes several interesting observations possible. (See this sketch for that construction: [url]http://www.geogebratube.org/material/show/id/49694[/url]) 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: [url]http://www.geogebratube.org/material/show/id/49720[/url] More GeoGebra at [url]mathhombre.blogspot.com[/url].