Maurits Escher

The Tessellations of Maurits Escher

Maurits Escher’s tessellations are of a degree of magnitude above most other people in the field, although in terms of ability I put him below Makoto Nakamura. On eight of the nine determining aspects of ability and understanding of the issues he scores heavily, with criteria as listed in the introduction:

(1) The inherent quality of the motif (silhouette)

(2) Showing the whole motif (excluding ‘heads’)

(3) The number of tessellations in the body of his work

(4) Variety of motifs

(5) X A tendency to the more difficult to achieve motifs (human figures)

(6) Coloured or shaded tessellations (excluding wireframes)

(7) Map colouring of tessellations

(8) Finished rendering

(9) Borderline (where appropriate)

Pleasingly, Escher’s tessellations possess eight of the nine desired attributes.

In more detail:

(1)  Recognisable in Silhouette
The aspect of recognisably of the motif, as seen in silhouette is fundamental to the premise of a quality tessellation. Although not all of Escher’s tessellations are immediately recognisable in silhouette, for the most part these are indeed mostly discernible. However, this is not to say that the tessellations are overwhelmingly so, as he includes many ambiguous silhouettes, far more than I would have liked. For example, he has numerous instances of flatfishes, of which I detail below. Indeed, he is far below Nakamura in this regard, albeit still of a better standard than most people.
    Lesser artists struggle with this concept (inexplicably so, given its simple premise), and fail to recognise its importance and unfortunately delude themselves as to equating interior life–like interior detail with a exterior outline.

(2) Shows the complete motif  
Escher’s tessellations in all instances are pleasingly of a ‘complete’ motif (like Bailey, Bilney, Crompton and Nakamura). The ‘head’ only type is implicitly excluded, as this category is lacking in any challenge of worth, being all too easy. Oddly, Escher never wrote why he excluded this type
    Lesser artists frequently do not understand the difference between the two types, and undertake such ‘head’ only examples (on account of their less challenging aspect), and unfortunately delude themselves as to equating these with the more challenging whole body motif.

(3) The number of tessellations
Escher shows 137 examples, well above the arbitrary benchmark figure of 50, but considerably below Nakamura (268), who shows many more, of a like comparable standard. Given such a reasonably high number, one might expect that these would not all of the same highest quality, and to an extent this occurs. For example, Escher included many flatfish type examples that are simply unworthy of him, these being particularly easy to accomplish, as they are essentially just shapes with detail added, rather than of an identifiable motif in silhouette. However, although the quality is occasionally questionable, by far the majority show true worth, and so he is not just interested in showing high numbers with disregard for the inherent quality.
    Typically, lesser artists will show a large number of inferior examples, and consider that such large numbers outweighs quality.

(4) Variety of Motifs
Escher shows 32 different motifs, of which although relatively high is still below Bilney (39), and Nakamura (39). Although the commonly to be found bird and fish motifs occur (which accounts for approximately half his output), he does nonetheless show creatures not usually shown, such as Horseman, Unicorns, Beetles, Sea Horses, Lizards.
    Lesser artists frequently shy away from undertaking such variety, showing simpler to achieve birds and fish to the exclusion of variety, and unfortunately delude themselves as to equating these with the more praiseworthy variety of motifs.

(5) Challenging Motifs – Human Figures
The more challenging and interesting human figure is mostly ignored, with only four examples.  However, many other examples of a challenging motif can be seen, notably with Horseman and Beetles, and so he scores heavily here, despite having few human figures. Even so, I would have liked to have seen far more of the human figure motif. One would have expected Escher to have realised the fundamental importance of this motif (as exemplified by Nakamura, who shows no less than 70 examples), and addressed this. To show just four examples is not nearly good enough. Their inclusion would be of far greater importance than numerous, unrecognisable, unworthy, flatfish.
    Generally artists, for other reasons than the desire not to repeat themselves, shy away from such challenging motif examples (on account of their degree of difficulty).

(6) Coloured or shaded tessellations
All of Escher’s tessellations are pleasingly shown coloured (like Bailey, Bilney, Crompton, and Nakamura). No inferior wireframe examples are shown.
    Lesser artists frequently show wireframe examples, for no good reason (from which one can only conclude is that they do not understand the issues).

(7) Map colouring of tessellations
All of Escher’s tessellations are pleasingly shown with map colours, one of the few artists who do so (like Bailey). Of note is that other leading tessellators, such as Bilney and Nakamura do not always do so.
    Lesser artists frequently disregard this aspect for no good reason (from which one can only conclude is that they do not understand the issues).

(8) Finished rendering
Escher’s style, as regards finish, is of a consistent approach, with just the right amount of detail, with little if any variation of note. (Nakamura, for instance, shows examples varying from minimalist to highly detailed.) Generally, he uses single ‘strong’ colours without any three-dimensional shading. Some examples have a more detailed rendering, such as Butterflies, but aside from this, and negating his early work where he was still establishing his style, they are noteworthy for their sheer consistency throughout the years. For instance, there is no difference in the rendering of the early drawing 18, of 1938, and his very last, 137, of 1972. As a rule, generally a simplistic finish is to be preferred, as too much detail hinders a clear interpretation of the motif.
    Lesser artists frequently render their tessellation in too much detail, believing this ‘improves’ the tessellation, when frequently it does exactly the opposite, resulting in a most trying viewing, in which there is to much detail for the eye to take in (from which one can only conclude is that they do not understand the issues).

(9) Borderline
Escher generally uses a decided borderline, and where not, uses ‘strong’ colours that do not require such usage. Other, fewer tessellations, typically of his early studies, of 1937-38, have what I term as an ‘incidental’ borderline, of which although this is discernable, is negligible, on account of its essentially hair-like nature that is not an intrinsic feature of the tessellation.
    However, the omission of a borderline cannot be said to be a fault as such, in that the inclusion or exclusion is down to personal choice, depending on the circumstances of the tessellation. Undoubtedly, this is secondary to the tessellation itself, and so of less importance to other, more fundamental issues, as detailed above.

General Comments:
Aside from the plane tilings, Escher did not unduly concern himself with more ‘involved’ tilings devised by mathematicians, such as with the Archimedean, Voderberg spiral, and the 17 plane symmetry types as a distinct body (in contrast to Andrew Crompton, the first, I believe, to present the 17 plane symmetry types as a concerted grouping). Indeed, the only instance of such ‘involved’ types was that of a tiling of rhombs as devised by Roger Penrose, the quality of which was unbecoming of Escher, titled as a ‘Ghost’, which is nothing more than a formless shape with elementary eyes.
    Admittedly, he perhaps had less access to these papers than is possible today. On the other side though, he had many mathematical contacts, who must surely have pointed out such opportunities. Of note is that Escher did discover self similar tilings, albeit these are not discussed here, as they are outside the remit.

Picking out a single, or indeed a ‘top ten’ of Escher’s works is a somewhat invidious task, in that various aspects come into play, militating against a straightforward selection. Therefore, in no particular order of merit, particular examples of his that I like (with the drawing number in brackets): Chinaman (4), Camels (6), Squirrels (7), Dogs (16), Birds (18), Fish (20), Imp (21), Birds and Fish (22), Reptiles (25), Fish (32), Angels/Devils (45), Two Birds (47), Lizards (56), Optimist/Pessimist (63), Winged Lions (66),  Butterflies (70), 12 Birds (71), Unicorns (78), Seahorse (88), Swan (96), Bulldogs (97), Birds and Fish (120), Birds and Fish (121).  Notable, and praiseworthy here is the sheer variety.
    Pleasingly, he shows a wide variety of motifs, certainly not the highest of all, but indeed amongst the highest.

However, although I am positive on Escher’s tessellations, this is not to say that his don’t have some shortcomings, in that some examples I have reservations with. For example, I take issue with a decided preponderance of flatfishes. These are nothing more than formless shapes with eyes, and are unworthy of him. Furthermore, these are not even truly flatfishes, as true flatfish posses eyes not symmetrically placed as portrayed by Escher. This shortcoming is even more surprising, given that these date from his more mature period, when he had many years of experience, and so would be expected to be aware of their lack of worth (i.e. they lack any challenge of note).
    Of note is that some of his earliest work, of 1927, is essentially worthless, but this is only to be expected, and should not be regarded as a criticism per se. indeed, I am loathe to comment on these, these being his first efforts, more properly regarded as initial probing. I only discuss these as they are included in the numbering. Similarly worthless is the ‘Ghost’, as discussed above.
    Another shortcoming is the lack of human figures; again, one would have expected Escher to have realised the fundamental importance of this motif (as exemplified by Nakamura, who shows no less than 70 examples), and addressed this. To show just four examples is not nearly good enough. Their inclusion would be of far greater importance than numerous, unworthy, flatfish.
    However, shortcomings per se are very much in the background, with his work typically being of good quality, whatever the motif.

Escher was a very good tessellator, but by far without fault. As detailed above, he understands the various issues underpinning the composing of inherent quality of tessellations. Indeed, he explained these in writing, a feature not frequently to be found in other tessellators. That said, he never even remotely explained the genesis/creation process per se. Pleasingly, he mostly, but not invariably, composes challenging motifs (albeit generally not of human figures), or at least rarely shown, with many examples as detailed above. Against that however, there is a decided absence of the more challenging human figures, with only four examples.
    Although most of his tessellations pass the above criteria tests, these are padded to an extent by the flatfish type, of a markedly lower degree of quality (and are in themselves highly stylised, no true-to-life flatfish appear like these), few of which any other tessellation artists of worth deigns to show. However, even when not of the highest standards, then these are still generally of a higher quality than with other people.
    Although it goes without saying, the advantage Escher had was of being the first in the field (negating Koloman Moser’s handful of rudimentary efforts barely worth mentioning). Consequently, this has distorted the issue of ability per se as against simply being the first person to do this, with the first awarded undue reverence. All too often, all we ever hear about is Escher, Escher, Escher, with no one else’s work deemed worthy. When judged solely on his tessellations, then on the issue of ability there is not such a dichotomy. The facts and figures speak for themselves. Other people can indeed match and better him! 

Last Updated: 3 April 2010