The AMS have a post explaining a pretty sweet result about the best octagon packing. But there was no way to play with it. [url]http://blogs.ams.org/visualinsight/2014/10/15/packing-regular-octagons/[/url] I added the slider so you could make the approximation more accurate.
More GeoGebra at [url]bit.ly/mh-ggb[/url]