Dual Dilations
This sketch has a section of an Archimedean tiling (rhombitrihexagonal) and its dual.
The dual is found by taking the center of each polygon, then connecting those as vertices if the polygons they're in are adjacent (share an edge).
When you move the slider or hit the play button, the sketch will shift between the original and the dual.
It's called the dual, because if you do that again, you get back to the original (or a variation of the original.) You can use this technique to find the structure of tessellations.
If you download it, this sketch has tools to make your own. One tool finds the center of a polygon (barycenter), and the other family of tools is for making the animated dilations.
Inspired by bmk sketches like at http://geogebrart.weebly.com/blog/duality-2
More GeoGebra at bit.ly/mh-ggb