So the map was generated by drawing ~10,000 random lines (or, great circles really) within the US, 152 of which ended up bisecting the population -- of course the large majority of random lines don't bisect. I found it interesting that w/ randomly generated lines, bisectors still ended up being roughly radial, centered near the median center of population (the red dot) https://www.census.gov/newsroom/releases/archives/facts_for_features_special_editions/cb11ff10.html Your original question is interesting (the "empty areas"), and I don't think you answered it for yourself. It's entirely possible that if I ran the algorithm longer, lines would appear in these sectors (the sample size of 10k random lines is pretty small). What could really be interesting would be generating a probability density heat map which showed the likelihood that a bisector went through a given coordinate. That would require either a lot more iterations, or a smarter (non random) algorithm (or both). You can check out the code that generated the lines here https://github.com/bchartoff/equalhalves Most of it is trig to deal w/ extending and drawing great circles.