In this section, I discuss, assess and compare, with an analytic intent, select other people’s Escher-like tessellations, namely with the works of Alain Nicolas, Alec Dixon, Andrew Crompton, Craig Kaplan, John Osborn, Makoto Nakamura, Maurits Escher, Nick Scalfittura and Jinny Beyer. This is as according to a set of my own criteria as regards an ‘ideal’ life-like tessellation (as epitomised by M. C. Escher), and as detailed below, I highlight the strengths/weaknesses of their work and examine any innovations that they show. Note that the above nine people have been chosen specially and not arbitrarily for good reason, and are biased towards quality instances. Each has a unique or interesting feature of their works. One artist, Jinny Beyer, has been selected as an exemplar in poor quality tessellations, for reasons as detailed below. Further, there are additional subtle distinctions of particular note which bear upon the issues such as the use of the computer and ‘wriggle room’, where the conditions and definitions are relaxed somewhat. Note that I purposefully exclude all other additional tessellation-like applications from this discussion. These include: ‘picture stories’ in which the tessellation plays a part, e.g. as with Escher’s Sky and Water I, application to polyhedra and accompanying verse (the last only seen occasionally). Although these additional aspects are indeed of interest in themselves (especially the ‘picture stories’), and indeed are not without value, in the context of this specific remit of Escher-like tilings only, without any adornments, these are peripheral and outside the bounds of interest, these being essentially an additional aspect to the subject.

Furthermore, no concession is made in the discussion as to name or ranking in tessellation or of art and mathematics; professors or mathematical ignoramuses, or indeed to the moving spring of Escher, who was first in the field; here, I am simply not interested in background, I assess him, and everyone else, purely by the body of their work, and nothing more. Unfortunately, there seems to be a belief that Escher did everything in tessellation, and any active followers are merely going over well-trodden ground, in effect copying him, with nothing new or innovative to contribute. However, this is clearly and demonstrably not the case, of which here I set out my case for this argument. There are many, many instances where a modern-day artist has bettered Escher, not just occasionally, but consistently. To reinforce this viewpoint, each artist is directly compared to Escher.

The featured designers (mostly artists, and not mathematicians) are all of a high-profile nature, for mostly towards the good to excellent range or rarely, poor, with either an extensive body of work on a website, or of having their work featured in books and articles, thereby establishing themselves, or purporting to be, a serious player in the field. Therefore, they are thus open to review, as against people doing tessellations for recreation as a passing interest, without proclaiming to be an expert. Note that being ‘high-profile’ does not necessarily mean of a high standard! Although it is all very well just reviewing high-standard people, it is also informative to examine the lesser lights in the field and to point out shortcomings in their work to avoid their mistakes. In particular, this refers to the work of the quilter Jinny Beyer, of a high-profile nature, with a tessellation book to her name, and so fair game for critique. As such, underpinning all this is a tutorial intention, in that upon examination of other people’s tessellations, and then understanding of the various issues, of both good and bad art arising from their work, one can then strive to create better tessellations of one’s own and avoid making mistakes by purporting bad artwork as good.

For each artist, I begin by analysing their tessellations as according to my own set of ten criteria of ability and understanding of the issues below, in broad order of importance, of which ideally the artist should include most, if not all, in their body of work. Furthermore, the list should not be interpreted as a simple, sequential progression of desire, as some of the aspects bear more weight than others. To indicate this, I put in percentage terms the approximate degree of bias.

1. The inherent quality of the motif, readily identifiable in silhouette. 50%.
As such, the motif portrayed, typically of a whole-bodied animal, although non-living motifs are also acceptable, should ideally and readily be identifiable as like a real-life motif when shown in isolation as a silhouette. I cannot overstate this aspect. This is what tessellation is all about, everything else pales in comparison. I could, and indeed perhaps should rate this aspect even higher. Indeed, one could have this as high as 90%, if not even higher, it is that important. Interior decoration, no matter how good, and indeed of which it adds to the tessellation, trails in the importance of the silhouette aspect.

2. Showing the whole motif. 15%.
The premise should, ideally, be one of whole-body motifs, in contrast to the ‘partial’ type, such as ‘heads’ which are decidedly less of a challenge, as a great proportion of the motif is omitted. This point is typically missed, with people erroneously equating the two. Ideally, one should show few, if any, of these easier types (a favourite of children, on account of their ease). However, occasionally, there are indeed exceptions to this rule, such a specific head, of a recognisable person.

3. Strictly plane tiling only, with no ‘breathing’ or ‘wriggle room’ examples. 15%.
Such ‘breathing’ or ‘wriggle’ room types (see John Osborn) is where the artist permits a degree of artistic license to occur, a typical exponent of lesser quality tessellators. Here, the artist tries to mask the deficiencies of their tessellations by the use of ‘judicious’ spaces, or of ‘rounding off’, in which the ‘tessellation’ thus lacks a shared line of double contours. Strictly, according to the definition of tessellation, this should be not even classified and dismissed summarily thereof, but as it is a common feature in many artists' works it is thus discussed below.

4. The number of tessellations in the body of one’s work. 5%.
Simply stated, as a rule, the more the better, all things being equal in quality, but unfortunately, this is not always so! Quite how high to set the benchmark for an acceptable lower number is a moot point. Quite arbitrarily, I define this as 30 or more tessellations. However, the sheer number of tessellation should not override the quality issue – a hundred (or more) poor or inferior tessellations should not be considered as equal to a single example of inherent high quality.

5. A variety of motifs. 5%.
Plenty of variety is desirable and not just relying on the simpler to achieve birds and fish. There is a whole world of animals to explore!

6. A tendency, or at least a leaning, to the more difficult to achieve motifs. 5%.
Such examples include human figures, which provide a challenge to one’s skills, rather than the simpler to achieve birds and fish. Although one could quibble with the definition here, in that aside from birds and fish, all other motifs could be regarded as challenging, and not just human figures. However, human figures are indeed a challenge and attractive for obvious reasons, and so should be shown more frequently than they are.

7. Coloured or shaded tessellations. 2%.
Simply stated, colour (or shade) the tessellations as according to map-colouring rules, in contrast to contiguous colours which do not define the outline. Many artists simply do not consider this, with the motifs otherwise invariably difficult to discern. And no examples without colour at all, namely of the ‘wireframe’ only type (another favourite of children). Again, such examples can at times be most awkward to interpret (a tangled web of lines), and so show a lack of understanding of the issues by the artist. However, occasionally, this rule can be overlooked, but only in specific instances, where ‘greater symmetry’ is involved, for example, the Penrose tilings, where for a minimal colouring contiguous colours are thus a necessity. However, generally, examples of disregarding the map-colouring rule show a lack of understanding of the issues by the artist.

8. The finished rendering. 2%.
Ideally, this should be relatively simple rather than over-detailed. As a simple statement, those in the style of M. C. Escher are favoured, of a simple colouring and basic details, such as with a bird aside from core body features such as feather and tail markings. Too detailed, as in a photorealistic approach, would render the motifs as difficult to discern, with typically a multiplicity of colours masking the motif.

9. Adding a borderline. 1%.
As a general policy, it is generally advantageous to use a unit thickness borderline, albeit this is by no means essential. Indeed, depending on the colouring of the tessellation this is not necessarily required. However, some motifs, in particular, do indeed benefit from this. For example, a human figure, with many different coloured aspects, such as flesh and different clothes, should ideally employ this feature, as otherwise the motif as an entity can be lost. Likewise for a gaily coloured bird or fish.

Note that criteria 1-6 (as indicated by the percentages), as a group, are of more importance than criteria 7-9. Though 7-9 are of concern, they are of less inherent importance than as the former, as they concern aspects of presentation rather than design. However, they should not be disregarded! Note that the above criteria should be regarded as a guideline. Different artists have their own criteria, not always made explicitly. However, that said, the above gives a yardstick of how to measure the quality of tessellations, with any differences mere nuances. However, the one aspect of the determining of true ability, of which all tessellators of worth agree, is that of criteria 1, recognising the motif readily in silhouette.

Recently it has come to light that one person, in particular, seems to have taken offence at my comments on this ‘The Tessellations of...’ section, specifically concerning the work of Jinny Beyer. This refers to the late (but not lamented) Seth Bareiss, who, in a ranting, delusional and risible piece of writing (it has to be seen to be believed), if it can be called that, took aim at me, regarding these assessments as rude where I critique other people’s work. This is not the intention at all. Indeed, I have been most careful and circumspect as to the selection of the people discussed. Most of the comments here are laudatory. ‘The Tessellations of…’ premise is predicated on people who class themselves as authorities on tessellation, self-appointed or not, with either extensive web sites and/or have written about tessellation in books or articles. Although the web abounds with people I could critique, these are excluded, as I have no issues with them, in that their intention is not to set themselves up as authorities. If children just want to do tessellations for fun, well that’s fine, I have no gripes about this whatsoever. Do I comment here upon such children’s efforts? No. If adults have the same ‘fun approach’, well again, I have no gripes about this whatsoever. Do I comment on their efforts? No. However, when people set themselves up as authorities and experts, then these are to me then fair game for review purposes. All I comment on here are authorities (or who set out to be) and not those a less serious, ‘fun’ undertaking. Of note is that on occasions where I do indeed give out stringent critique, this is only to people who show quite appalling standards, namely with Beyer, but she is not alone in this. Why then do I single out Beyer in particular for condemnation? Simple, she is brought out a book (a medium that has stood the test of time, hundreds of years, of which this should surely serve as a defining standard), and thereby setting herself up as an authority. Furthermore, she is an adult and so has the necessary age and experience behind her, and so cannot use youth as an excuse to be excepted from reproach. If you set yourself up as an authority, and especially in book and article form, then are you not fair game for fair critique? Indeed, the artist should be openly self-critical, in order to improve. Let’s look at what Escher himself had to say on the subject, in Escher: The Complete Graphic Work, p. 81:

… It is really only a question of battling on relentlessly with constant and if possible merciless self criticism.

By any standards, this is a stringent self-examination. Indeed, Escher himself even invited critique, p. 62 (A letter to Mr P. Kessler, a collector of Escher’s works, in regards to an observation of Kessler’s points concerning a print of Escher’s):

I do not entirely comprehend the error that you noted (though I very much appreciate it very much that you did note it) but I’m sure it’s there…

Well, if such critique is deemed appropriate (and indeed welcomed) by Escher when applied to his own work, then why should it not apply to his successors? What is the matter with Bareiss? Perhaps I am indeed guilty as charged? However, I am comforted in the knowledge that I am not the only one who has suffered from his poisonous pen. At least two other people, Bruce Bilney, and a Russian person whose name I forget were similarly on the receiving end of his unwarranted diatribes. I also think he may have taken umbrage at not being included in my ‘top category’ of the time, which seems to have clouded his judgement. What a loser he was. Only by subjecting one’s work to critical examination can one hope to improve. If some of the people who I critique simply do not realise/do not care and simply will not do or accept this (as with Bareiss), then why should their shortcomings not be exposed? (In any case, have you not seen theatre, book, record, and film reviews? You’ll certainly find much more stringent comments there than that here). It’s all about standards. For example, should creatures with normally four legs be allowed to be portrayed with one leg? Is it to be regarded as acceptable to portray a creature without a head? It’s unbelievable, I know, but such examples are all too frequently see by people who should know better! Why should the premise of passing these risible efforts by their designer as ‘of worth’ be deemed acceptable and allowed to pass unchallenged? Those so-called ‘authorities’ peddling such monstrosities serve no purpose whatsoever and indeed impede progress, of which impressionable, typically young people upon seeing their work will simply assume that such examples are acceptable, given the media (books) in which they are displayed, and so thus be perpetuated. And so it continues. Further, a few people also make grossly exaggerated claims as to quality, and if all the above is not enough, then make claim to other people’s work as their own! Enough is enough! Why should this not be challenged? What is wrong with straightforward, honest to goodness plain speaking? Is one not allowed any more to stick one’s head above the parapet and more and say that the Emperor is not wearing any clothes? Note that I have no axe to grind with the people I castigate, if they want to continue with such matters, then that’s entirely up to them. The intention with these essays is a service to the tessellation community, to guide, and indeed prod people if necessary in the right direction, nothing more, and certainly not personal attacks of an effrontery nature, as Bareiss seemed to think. I think I do indeed think I do so, but if you disagree, or indeed agree, do let me know! And if you like, I’ll freely post them here, for good or bad on myself. Unlike Bareiss, I welcome critique and debate.

Created: 30 April 2010. Updated, revised and extended: 14 September 2011

Update 21 July 2017. Seemingly as ever in my writings, I now see room for improvements, despite this being just six years ago. However, it is not a wholesale rewrite, but rather of ‘insertions’ and ‘adjustments’ to the existing text, although there are indeed substantial additions.

Latest incarnation 29 October 2019.

Added minor rewrites in various places throughout, but the piece remains essentially of the 2011 writing.