Purely out of curiosity, this is a summary of all the space filling polyhedra; forms that tesselate in 3D. It should cover the basic variety of forms but there are several additional variations from these which can be achieved by bisecting, elongating, twisting or distorting the basic units. Some but definitely not all are noted here.
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| Space Filling Polyhedra |
The cube is the only platonic solid to tessellate in 3D. The triakis truncated tetrahedron is the closest form to the tetrahedron. Shapes which tessellate in 2D work as space filling polyhedra when extruded into prisms. Slicing, bisecting and skewing these prisms create further space filling forms such as scutoids. There are also other variations of gyrobifastigium, hendecahedra and hexahedra which form their own families of polyhedra. The square based pyramid is a very versatile form for mediating between the cube, rhombic dodecahedron, dodecahemioctahedron, elongated square pyramid and other forms. More complicated geometries such as the plesiohedron and stephenoid hendecahedra accumulate into dynamic collections of forms. All of this is worth an investigation in relation to architecture for its innovative value to create economical and dynamic spacial arrangements of buildings.
