Building systems for creating three-dimensional structures include a plurality of polyhedral units, each unit being hingedly coupled to at least one adjacent unit, wherein the plurality of polyhedral units is arranged to be interchangeably manipulated between various forms of three-dimensional structures. The building system includes at least two types of plate-like building elements, including a single-piece element and a double-piece element that is formed of two single-piece elements arranged adjacent each other and being hingedly attached to each other at one of the edges of each of the two adjacently arranged single-piece elements.