Abstract: A rotary piston engine has a plurality of explosion chambers situated around the periphery of the housing and includes within the housing a piston. According to the preferred embodiment, a housing having H recesses would contain a piston having P = H-1 lobes. According to the preferred embodiment, the structure of the housing relative to the piston, and vice-versa, has been improved to give optimum results. The relationship between the housing and the piston is such that it may be characterized by an optimum mathematical relationship which can make the construction of such machinery simpler than has been heretofore known. Additionally, the relationship between the improved piston and housing configuration is related to other engine parameters such as the compression ratio and efficiency.