Polyhedra

An additional aspect of tessellations is the possibility of their application to their three-dimensional cousins, namely polyhedra. Now, it may be thought that the "transfer" of both "non-representational" and "representational" examples would be a straightforward task, but in general terms this is not actually so. As such, for a given life-like tessellation, the underlying polygon may indeed be "utilised" to tessellate the polyhedra in the form of nets, from which when duly assembled one finds that the life-like tessellation does not in fact tessellate in three-dimensions. Also, even when a net is suitable the coloration may also need to be "adapted." Therefore, in essence each tessellation has to be investigated afresh for this purpose.

Concerning Escher�s efforts in this field, he somewhat neglected the possibilities, with only very few completed examples, of which a Rhombic Dodecahedron (of "bat, fish, lizard") and Dodecahedron (of "turtles") can be seen on page 246 of Visions of Symmetry. As such, I find this most surprising, as such "decoration" of polyhedra undoubtedly can be regarded as a natural progression of ones studies. Some further examples concerning the "adaptation" of Escher�s tessellations to polyhedra (by Doris Schattschneider) can be found in the book M. C. Escher Kaleidocycles pages 22-28, where the five platonic solids as well as a cuboctahedron are thus duly shown. However, the above is only really a snapshot of the possibilities, and indeed a book on the subject per se is decidedly overdue. As such, it is most hard to find studies in this field, and so I thus consider that this has been somewhat neglected in the mathematical literature.

Below I show some arbitrary examples of my own, of the "representational type."

Kaleidocycle

Tetrahedron

Octahedron

Rhombic Dodecahedron

Square based Right Pyramid