Essay 9 - Complex Tessellations

Although the composing of tessellations can be a simple process, as very basic principles of symmetry are employed (for example, using squares with translated lines), examples that are more complex in nature can indeed be composed, this depending upon one's mathematical ability. This term 'complex' here refers to tessellations that are created by more advanced mathematicians, i.e. professors and are, in general terms, beyond the understanding of most people (including myself). However, despite lacking ‘theoretical knowledge' as to their composition, it is still possible to use these for representational tessellation, and by so doing, one gains kudos essentially by default.

Complex Examples
Publications that contain such 'complex' tessellations generally appear in specialised, typically obscure mathematical papers and journals. However, for those so interested in the challenge of using examples of these types, a cornucopia of advanced examples in abundance is in the more readily accessible Tilings and Patterns. Despite being of a decidedly academic nature, much remains understandable to lesser mathematicians. Amongst the more ‘accessible' advanced tessellations in the book, include the Penrose non-periodic tiles (an excellent article of which also appeared in Scientific American of January 1974). Other examples discussed include dimorphic tessellations, in which the tile can form a tessellation in two distinct ways. Such a concept can also be extended to three, four… types (examples of my own can be seen on the ‘dimorphic' and non-representational ‘hypermorphic' pages).
    As a rule, such complex tessellations have, in everyday terminology, a unique arrangement of their lines. Consequently, these thus provide a stern challenge in which to compose representational tessellations, as, in simple terms, they do not share the same 'straightforward placement' of contour lines as with more 'orthodox' normal tessellations.
    In this field, depending upon the circumstances, the addition of a motif is, or can be, most difficult to achieve. Quite simply, one is somewhat of necessity 'restricted', and so the finding of such motifs, if of a sufficient quality, is more praiseworthy than with an arbitrary tessellation.

Escher's Usage of Complex Tessellations
Indeed, of interest is Escher's own statement denying any mathematical ability, making comments specifically of H.S.M. Coxeter's article 'A Symposium on Symmetry' that such ‘hocus-pocus' text (as he put it) was beyond him, but all the same, he still made use of the diagrams for his Circle Limit prints. Another example of Escher's usage of 'complex' tessellations was of a non-periodic tessellation devised by Roger Penrose, which led to drawing 137 (his last). As such, it must be admitted that this was arguably his poorest tessellation in terms of inherent quality, the motif being unrecognisable, described by Escher as a 'Ghost', in effect apparently testifying to inherent difficulties of composing a motif. However, this should not be regarded as a true example of his ability, as at the time he was coming to the end of his life, after much illness, and undoubtedly in his prime he would have produced a more worthy motif.

Agree/disagree? E-me.

Last updated: 14 December 2005