Zing Origami - 2002
I became interested in enumerating all the unique polyhedra in terms of their maps, starting with the low ones and seeing how far it could go. There is only one tetrahedron, composed of four triangles. There are two pentahedra, a prism and pyramid. The triangular prism is presented here is a "regular" triangular prism, having all edges of unit lengths and faces of regular polygons, and only a single type of vertex. I' not sure why is it not considered an Archimedean solid; I guess because it bears no particular relationship to any Pythagorean solid. The helf-trahedron faces include a square, two equilateral triangles, and two trapezoids whose long edge is twice as long as the other three edges. It has the interesting property of being able to form a tetrahedron when joined to another of the same shape by the square face. Lastly the "Nugget" has as faces two equilateral triangles, two rhombi, and two trapezoids, and it has the interesting property that it can be used as a module to tile space. I do not know if it has a formal name among mathematicians.
Crease pattern for Triangular Prism
Crease pattern for Half Tetrahedron
Crease pattern for "Nugget" Hexahedron