Below is an image with a variety of semi-regular tessellations that is the foundation for all the images you see in this post. You can follow the creation of the traditional dual below, but those images above (and the purplish one at the end) is the result of allowing the transition to seep further past the middle of the polygons. See the animation after the break.
Visit Geogebratube to adjust this sketch to the time of your liking. Alternatively, download the entire file below.