Essay 6 - Categories of Difficulty

Without doubt, the ‘acid test' of a tessellation is to be judged on its resemblance to the creature(s) it is supposedly portraying, of which the likeness should at least be vaguely recognisable. Furthermore, ideally this would be indistinguishable from the real-life creature, in effect the tessellating motif being ‘assumed' as a picture in its own right, without any ‘shortcomings' that usually occur, due to the demands imposed by the symmetry arrangements of the lines. However, in practise such a desire is rarely accomplished, with the tessellation being a compromise, to greater or lesser degrees in this matter. Regrettably, all too often, I see motifs that are lacking in this most basic aspect, bearing only a passing resemblance, unworthy of the description of a bird, fish or of whatever creature are supposedly being portrayed. Consequently, ‘false credit' often arises, with no distinction being made between motifs that are obviously inferior and superior when duly compared with each other. Therefore, ideally some sort of ‘quality test' should thus be applied to a tessellating motif in order to differentiate unequivocally between inferior and superior examples. Such a test should ideally be relatively simple to undertake, without the necessity of a long and convoluted process. Consequently, I have below placed tessellations into six types or categories, as according to descending degrees of worth. Each of the categories motifs has been discussed in more detail in Essay 3, Motif Choice.


Category 1 - Animal World
Examples of creatures derived from the animal world, say, birds, fishes or human figures, to name but few, showing the complete figure, with the elements, such as legs, arm, wings in proportion. Such examples are the most difficult to achieve due to the demands of the different elements, all of which have to be taken into consideration. Needless to say, no grotesque ‘protuberances' that bear no relation to the motif can be admitted.
    However, not all of these are to be regarded as of the same intrinsic difficulty. For example, birds and fish are amongst the easiest types of motifs to compose, whilst in contrast, say, a human figure poses more of a challenge. Therefore, the human figure is thus more ‘worthy' because of its inherent difficulty in composing.

Category 2 - Mythology 
Examples of creatures derived from the world of mythology, such as with centaurs, or simply of an ‘imaginary nature', typically with elements from two or more distinct creatures. This category is typically relatively easy to compose, as previously discussed, primarily due to their outline possessing potential for variety than with a strict portrayal as with the birds, fish… above, which of necessity consist of exact outlines. Although these motifs can indeed be shown in a highly detailed manner, equal in respect to the above, the variety of such possibilities renders these as of a distinctly lower intrinsic ranking.

Category 3 - Cartoon-like
Examples where the motifs are of a cartoon-like nature. As such, although clearly of a lower inherent quality than with categories of 1 and 2 (as considerable variation of the motif is possible), as these nonetheless consist of a whole figure they thus have some inherent worth. Interestingly, Escher composed no examples of this type, and, in contrast to what I believe was a likely ‘planned omission' of inferior types (of heads, below), I consider that he simply did not recognise the possibilities here. Although examples of this type are relatively few, those that do exist are pleasing.

Category 4 - Heads
A decided lower category of motifs are ‘heads’, where these are shown in isolation aside from their parent body, mostly of humans, but also of a wide variety of other creatures. This is undoubtedly the simplest type of ‘motif’ to compose, as essentially only a single aspect is under consideration i.e. a head, in contrast to the multi-elements of a whole body  (head, body, appendages) of a whole creature. Somewhat annoyingly, examples of this type abound in other people's work, with the impression being given that these are in some way ‘equal' in worth to the more exacting ideals as above. Unfortunately, they are deluding themselves, as can readily be imagined with a moment’s thought.
    However, that said, I find this type acceptable if the head is of an actual portrait of somebody, as this is obviously more of a challenge, rather than just a generic head. For example, Andrew Crompton has done one of Sherlock Holmes, with all the accoutrements, such as a Deerstalker hat and pipe. Alain Nicholas has done one of Escher, with his typical beard. In these instances, this is acceptable, in that the artist has striven for a particular portrayal, rather than just a generic head.
    Interestingly, examples of this type are conspicuous by their absence, or near absence in the top tessellators; Bailey, Bilney, Crompton, Nicolas, Escher, and Nakamura. Indeed, Bailey, Bilney, Nakamura show none. The other artists are essentially of a like mind. Escher composed only one example, namely the first, rudimentary effort of Eight Heads of 1922. Crompton (Sherlock Holmes) and Nicolas (Escher) also show just the one, as detailed above. Why should we only compose either none or this solitary example? Are we lacking in ability? No. The implication is that these are beneath our dignity, lacking any challenge of worth. Putting it simply, such types are really only for children, who lack the ability of composing whole body motifs. Undoubtedly, all the people here could have produced hundreds of such examples, and yet we show none. The conclusion is, or should be, obvious.
    All too frequently, examples of this type appear on the web (to be found on a picture search for tessellation), typically in school websites such as ‘Mr Smith's 7th grade class'. Unfortunately, the quality is generally most poor, even when exceptions are made for their age. Astonishingly, examples of this type have even won competitions on the World of Escher website! This simply shows a lack of understanding by the judges of their inherent merits. That said, I do not object to this type out of hand, even if the head is of a generic type. Certainly, one or two are acceptable, but when such examples consist of a sizeable percentage of one’s work, say five per cent or more, then this simply signifies a lack of ability, in effect giving up on the harder to achieve whole body motifs in favour of easier, simpler types.

Category 5 - Unrecognisable
Examples where the motif is unrecognisable as of a creature, consisting essentially of a ‘shape' with an eye added to supposedly ‘animate' the outline, along with further decoration of vague lines denoting ‘body markings'. Such examples are essentially to be regarded as unacceptable, as they require no skill to compose. All too often, examples of this type abound, arising because of essential ignorance in composing representational tessellations. Frequently examples of this type are to be seen on websites of schoolchildren, as above. Although the interest is commendable, such types should not be encouraged.

Category 6 - Copying Escher
Examples where the motif is a blatant copy of an example of Escher's, with only the most minor changes being made to the interior design. Somewhat disturbingly, examples of this type are appearing increasingly. What is the point? Although I am critical of the types of categories 4 and 5, at least those are of an original nature, whilst in contrast these are plainly not. By all means, replicate Escher's drawings for the purpose of study and analysis, but for then to try and claim (false) credit for an inconsequential change of interior design is to be regarded as quite frankly unacceptable.

Unclassified Category - 'Tessellationesque'
Another type of tessellation occasionally encountered is what I term (of my own devising) tessellationesque, a term that requires definition. This refers to examples where the motifs are not strictly interlocking, and so consequently cannot be considered as a tessellation in the normally accepted sense. An example of this type to show as a model is by Escher, with drawing 81, of bats, birds, bee and butterfly. Although this is undoubtedly based upon tessellation principles, Escher here has widened the term slightly, as certain of the motifs; specifically of the butterflies tail, the bats head and bee's head and tail do not intrinsically interlock. For these motifs, in the interests of a better resemblance, he has departed from the true outline of the tile (discussed below). To some extent, this aspect is 'disguised' by Escher, as he employs a decidedly thick line surrounding the extremities of all the other motifs here. In contrast, all other drawings of his have a delineating outline (if at all) that is of a less substantial nature. Therefore, strictly this should not be regarded as a tessellation.
    Examples of this type can be seen in other people's work, of which due to the 'widening' of the term better resemblance to the motif portrayed can be accomplished. However, what is thus shown, no matter how good in quality, is now a motif placement type of pattern, and is not therefore a tessellation. A typical occurrence is when the head of a motif, of whatever creature, meets at a vertex. Invariably, this will result in a somewhat unnatural 'pointed' appearance. Consequently, practitioners of the 'tessellationesque' type will then simply 'round the head off', ignoring the vertex (as Escher has done with the bees head in the above example). Although it may be tempting to 'improve' on a given tessellation in this manner, the concept of representational tessellation, namely the duality of the line, representing different aspects, is thus lost. To emphasise, it is not a tessellation. Recently, on the World of Escher website, this was a topic of discussion on one of the forums. A correspondent had written in pointing out that an example of this type had won a tessellation competition organised by themselves that was not actually a tessellation! Somewhat alarmingly, this objection was dismissed out of hand. By all means, have a separate section for examples of this type, but to regard these fundamentally different types as being of no real difference is to miss the point.
    As regards placing this in an category, due to its nature, the above ratings are simply inapplicable in this situation, as one is not comparing like for like. Therefore, it is thus placed aside from the above. However, if examples of this type are indeed contemplated as a separate entity, then the above categories ratings can indeed be transposed.

To summarise, quality, as regards resemblance is everything, of which even a single example of true worth is to be preferred to, say, a hundred (or more) poor quality examples of the inferior types.

Agree/disagree? E-me.

Last updated: 28 February 2006 (minor addition)