A joint system for joining polyhedric facet elements together. Polyhedrons have a plurality of facets. The polyhedric facet elements have facet planes that are generally congruent with the facets of the polyhedrons in which they are found. Various facet elements occupy angular relationships such that they can be joined in rigid stable joints by means of key elements where the planes of the key elements extend generally perpendicular to the planes of each of the facet elements. The joint system is suitable for inter-polyhedron joint systems where the joints include four joint elements, three of which are in the respective polyhedrons that are to be joined, and the fourth is a key element perpendicularly disposed between the other three joint elements. The joint system is likewise suitable for intra-polyhedron joint systems where there are three joint elements, two of which are in the polyhedron (in-polyhedron) and the other is a key element that extends in perpendicular relationship inter-facet between the two in-polyhedron facet elements. The intra-polyhedron embodiment of the joint system is particularly well suited to permitting polyhedron structures to be formed with alternating open and closed facets for aesthetic, structural, and other utilitarian purposes.