In the first half of the nineteenth century, designers and reformers in the United States embraced the radical notion that architecture could not only _represent_ its inhabitants but also _mold_ their bodies and minds. This architecture of reform assumed distinct geometries. Octagon houses, hexagonal cities, and oval communal mansions, it was believed, would help produce stronger, healthier citizens, as well as a more harmonious and equitable society. "The Shape of Utopia" argues that a little-known strain of radically innovative architecture emerged circa 1850—a moment when architectural geometry was imagined to possess a potency it has hardly been accorded since.