Abstract We present a new algorithm for modeling a self-consistent set of global plate polygons. Each plate polygon is composed of a finite list of plate margins, all with different Euler poles. We introduce a "continuously closed plate" (CCP), such that, as each margin moves independently, the plate polygon remains closed geometrically as a function of time. This method solves emerging needs in computational geodynamics to combine kinematic with dynamic models. Because they have polygons that are too widely spaced in time and have inconsistent motions between margins and plates, traditional global plate tectonic reconstructions have become inadequate for geodynamics. The CCP algorithm has been incorporated into the GPlates open-source paleogeographic system. The algorithm is a set of procedures and data structures that operate on collections of reconstructed geometric data to form closed plate polygons; the main data structures used for each plate polygon are based on a nested hierarchy of topological elements. Reconstructions with CCPs can be created, edited, visualized, and exported with GPlates. The native storage of the dynamic reconstructions is the GPlates Markup Language, GPML, which uses an XML-based file format called GML. We demonstrate the utility of the CCP method by creating a global reconstruction with continuously closing plates from 140 Ma to the present using data from existing, traditional reconstructions.