Essay 19 - Pseudo Tessellations
Gaps (Breathing Room) and Overlaps
‘Pseudo tessellations’ refers to what I term as tessellations masquerading under another identity, namely with ‘gaps’1 and ‘overlaps’ types (discussed in detail below). Here, the constraints of the premise of tessellation are relaxed, in that gaps and overlaps, to greater or lesser degrees, are tacitly accepted, and so the ‘purity’ of the concept (and motif) is diluted. Note that the term ‘gaps’ should not be confused with the outlining of the tessellation, where a line of unit thickness surrounds the tiles, which is a completely different matter. The popularity of these, somewhat unfortunately, is finding favour with more and more tessellation artists. However, although frequent use of this is made by other tessellation artists, I am decidedly against examples of this type, as the instance such procedures are undertaken then one strays from the principles, and indeed challenges, of true tessellation. Further to this type, the ‘gap’ relaxation runs the full gamut, from a minimal amount, in which only minor changes are shown (typically rounding of unwanted, sharp corners), to what can only be described as ‘anything goes’, where the motif is so far from the original tessellation as to be nothing more than a ‘placement’ inside the tile. The overlap relaxation is used rather less frequently. Consequently, when one starts showing examples of this type, this can lead down a very slippery road indeed. Where does one draw the line here? Gaps/overlaps at an absolute minimum? Gaps/overlaps of a slightly more extended nature? Gaps/overlaps of vast extent?
Furthermore, disappointingly, I know of few, if any, such artists who employ this in their work who will acknowledge such inferior examples of this type. The insinuation here is that such examples are of equal worth to a ‘true’ tessellation. At the very least, one should qualify such types as inferior. Frequently, this type of ‘tessellation’ is to be seen in children’s work, where quite frankly it should remain.
Aside from me, very few people speak up against these types (probably due to ignorance on such matters). Someone of a like mind is Seth Bareiss, who posted a query on this subject on the online World of Escher ‘Our Contest’ forum, ‘Contest #19 A few of your entries are not tesselllations2, pointing out that in their tessellation contests they had posted no less than three entries that were not strictly tessellations! (The links to the drawings he pointed out are no longer operative.) Disarmingly, his objection was dismissed out of hand by the administrator, with:
the definition [tessellation] is open to interpretation
and was then followed
n 1 the careful juxtaposition of shapes in a pattern.
This ‘definition’ I have found is derived from an online dictionary3. A ridiculous definition! This is not so. A tessellation, in every day terms, is of a collection of plane figures that tiles without gaps or overlaps. A simple, straightforward, and clear premise. Any mathematics textbook will state the same thing. It is not open to interpretation! This ‘definition’ was sort of justified by stating that it is ‘…a fun contest for kids’. Nothing wrong with a fun contest for kids per se. However, this is missing the point. There must be at least a token degree of structure to any ‘fun contest’, in whatever field, or what is the point to it? Taken to its extreme, any two shapes having a single border in common is a ‘tessellation’ here.
Usage by Top Artists
Of ‘gaps’ and ‘overlaps’ usage, generally the top tessellators, such as myself, Andrew Crompton, Maurits Escher, Makoto Nakamura, Alain Nicolas and Nick Scalfittura all essentially shun any examples of this type. Indeed, Bailey, Crompton, Nakamura and Sacafittura show none at all. Escher shows just a single example of gaps (No. 81, Bat, Bird, Bee, Butterfly), whilst Nicolas shows just two, of which this is so minimal that it is inconsequential. This near complete ‘omission’ from our work as a body should speak for itself, in that we take the moral high ground. We do not demean ourselves with children’s standards. Perhaps somewhat surprisingly though, some other prominent tessellators, such as John Osborn and Bruce Bilney show more examples of this kind than I would like. Indeed, Osborn’s work in particular makes heavy usage of gaps, with approximately 40% of his tessellations being like this (Bilney’s is much less, approximately 5%). Typically, the artist who shows such examples is very much of the lower league, in that it should be obvious that these are much easier to compose than a true mathematical definition of tessellation. Rather than spending time and effort refining their tessellation as recognisable in silhouette, they take the easy way out, with their thinking running along the lines of ‘an inconvenient space, oh, I just ignore it. A protuberance, oh, I’ll disregard it’. Oh, I’ll just overlap this arm (or whatever). It’s all too easy. This procedure simply misses the point, of a skilfully crafted line, of a double contour throughout, refined to the highest standards. However, that said, if the gap is of a truly minimal nature, of a inconsequential amount, I don’t object to this type of approach too much. To be regarded as ‘acceptable’, broadly defined, then this should be at the absolute minimum. However, it is still not for me.
As alluded to above, the types of gaps and overlaps cover a wide range. To simplify the discussion, I now discuss examples of both, the former being the most common, of three different types, of a ‘minimal’, ‘middling’, and’ ‘extreme’ usage:
A typical situation of an ‘acceptable’ minimal type is that of Ozzie the Magic Kangaroo, by Bilney, where he gives this minimal gap space. Although I do not find favour with this concept, I am prepared to concede that such examples are broadly acceptable.
Two examples that hover between the borders of acceptability/unacceptability are that of Scorpion Dance by Bilney and Beetles 2000 by Osborn. Here, even more leeway is tacitly accepted, albeit admittedly the tessellation principle remains in their respective works. Strictly, I consider this inadmissible.
However, what I will not accept is vast, open spaces, essentially waste space, where no pretence is made as to tessellation. An example of a wholly unacceptable sort is Teen Centaur and Friend (No. 5) by Osborn. This has vast, open spaces, not just once but twice. Another example, possibly even worse, of an wholly unacceptable sort is that by Patrick Murphy, in Modern Mathematics Made Simple, page 204, who shows a ridiculous ‘tessellation’, which consists of a ‘demon bowler’ that is essentially nothing more than a free-form drawing inside a tile, with only the most minor literal connection to the tile (the head and part of one foot). The skill and imagination involved here is decidedly lacking. (I might just add, to further the point that advanced mathematicians very rarely understand Escher-like tessellations, is that Murphy has a MSc in mathematics.) Note that I chose this example for illustrative purposes as it was in a book, thus giving it a degree of apparent credence, rather than an essentially transient web page, or an excusable child’s effort.
Frequently, examples of this type flatter to deceive, at least the inexperienced viewer. To discuss some at first sight comparable examples of equal quality, I begin with a ‘dance’ tessellation, of the premise of two dancers facing each other, with Osborn’s Dancers (No. 11) and Nakamura’s Dance 1 (No. 230) tessellations. Osborn’s ‘tessellation’, on the face of it, is most pleasing, in that the figures actually resemble dancers, and furthermore are in proportion. However, this apparent quality is misleading, and is emphatically not so. When examined in more detail, Osborn’s has not just minimal ‘gap’ room, but noticeable, large open areas of ‘waste space’. Furthermore, this particular example does not have just the one large open area, which is bad enough, but it has no less than four! What has happened here is that the artist simply couldn’t ‘accommodate’ the motifs, in that the open spaces, of necessity, had to be. Certainly, it’s good as a ‘symmetry’ drawing, but it’s not to be compared or given the status of a true tessellation in any way. In contrast, Nakamura’s tessellation also resembles dancers, and is in proportion, but this has no gaps whatsoever. Therefore here he is on the moral high ground. There is a world of difference here despite the at first glance comparable quality. As ever, the silhouette test proves the respective qualities of these types of tessellation.
A typical situation of an minimal type is that of Aussie Rules O.K?, by Bilney, where he gives the figure for the sake of better verisimilitude an arm that extends across the body of a contiguous footballer. However, when silhouetted, a pronounced unlife-like 'gouge' is evident. Admittedly, this is only of a minor nature, but should this be thought acceptable?
Aussie Rules O.K? by Bruce Bilney
By the above it should be obvious as to the type of tessellation to aspire to. Only artists who lack skill, imagination, and innovation favour these, hence their popularity. If you just want a symmetry drawing, which strictly is what these are, well, why not do this instead? But if it’s a tessellation you want, why bother with these? That said, I am prepared to give ground on truly minimal types, as espoused by Nicolas. Do them if you must, but at least qualify them. Do you really want to show these as bona fide tessellations? Do you want to be in the first division, with Bailey, Crompton, Escher, Nakamura, Nicolas, and Scalfittura, or the second division, with schoolchildren like quality? The choice is yours…
Agree or disagree? Let me know. E-me.
1 Or ‘breathing room’ a phrase coined by John Osborn. Another, interchangeable term for this is ‘wriggle room’, by Bruce Bilney
Revised and rewritten: 28 July 2010. (Previously titled as 'Breathing Room') Bruce Bilney illustrations added: 19 October 2010