Picture book of apportionment and voting difficulties

The geometry associated with apportionment and voting methods enables us to “see" what difficulties will arise, why certain complexities are inevitable, how to create supporting examples, and identify similarities of the problems afflicting both areas. Although several new results become apparent, such as the harsh consequences for a method satisfying IIA and a replacement for the modified divisors that are used with monotone apportionment approaches, the emphasis is to use this picture book approach to explain why certain behaviors must happen.