The Tessellations of...

The Tessellations of…


In this section I discuss and assess other people’s tessellations as according to a set of my own criteria as regards an ideal life-like tessellation, as epitomised by M. C. Escher, as detailed below, and highlight the strengths/weaknesses of their work, and examine/highlight any innovations that they show. 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 (however, for certain artists on occasions I have to relax this rule)

Application of tessellations to polyhedra, as with Doris Schattschneider in M.C. Escher: Kaleidocycles

Accompanying verse, as with Bruce Bilney

Not truly plane tessellations with instead ‘mathematical shapes’ e.g. such as an ellipse (however, for certain artists on occasions I have to relax this rule)

Application to the hyperbolic plane, e.g. as with Douglas Dunham

Although these additional aspects are indeed of interest in themselves, in the context of this specific remit of plane tilings as epitomised by M. C. Escher, there are outside the bounds of interest, these being essentially an additional aspect to the subject. Furthermore, no concession is made as to the moving spring of Escher, who was first in the field; I assess him (and everyone else) purely by the body of their work, nothing more.

Unfortunately, there seems to be a belief that Escher did everything in tessellation, and any followers are merely going over well-trodden ground, with nothing new or innovative to contribute. However, this is clearly not the case, of which here I set out my case for this argument. There are many instances where a modern day artist has bettered Escher. To reinforce this viewpoint, each artist is directly compared to Escher.

The featured artists are mostly of a high profile nature, with either/both extensive websites, or of having their work featured in books and articles, thereby establishing themselves as a serious player in the field, and so open to review, as against people doing tessellations for recreation. Note that being ‘high profile’ does not necessarily mean of high standard! Although it is all very well just reviewing such people, it is also informative to examine lesser lights, and so by means of contrast, I include a lesser tessellator (but of a high profile nature, and so fair game), Jinny Beyer. As such, there is a tutorial intention behind all this, in that upon examination of other people’s tessellations, and then understanding of the various issues arising from their work, one can then strive to create better tessellations of one’s own.

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 simple, sequential progression of desire, as some of the aspects bear more weight than others. To give an indication of this, I put in percentage terms the approximate degree of bias.

1. The inherent quality of the motif. 50%. As such it should ideally and readily be identifiable as like a real-life motif when shown in isolation as a silhouette. I could, and indeed perhaps should, rate this aspect even higher. Indeed, one could have this as high as 90%, if not even higher. This is what tessellation is all about, everything else pales.

2. Showing the whole motif. 15%. The premise is one of whole body motifs, in contrast to the ‘partial’ type, such as ‘heads’ which are decidedly less of a challenge. Ideally, one should show few, if any, of the easier types (a favourite of children).

3. No ‘breathing’ or ‘wriggle’ room examples. 15%. Such types are where the artist permits 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

4. The number of tessellations in the body of one’s work. 5%. Simply stated, the more the better. Quite how high to set the benchmark is a moot point. Quite arbitrarily, I define this as 50 or more tessellations. However, the sheer number of motifs 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. Variety of motifs. 5%. Plenty of variety is desirable and not just relying on the simpler to achieve birds and fish.

6. A tendency 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.

7. Coloured or shaded tessellations. 1%. Simply stated, colour (or shade) the tessellations. No examples of the ‘Wireframe’ only type i.e. without colouration (a favourite of children). 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.

8. Map colouring of tessellations. 1%. When the tessellation is map-coloured (contrasting colours) this renders the motifs as distinct, in contrast to contiguous colours which do not define the outline. Occasionally this rule can be overlooked, but only in specific instances, where ‘greater symmetry’ is involved, for example the Penrose tilings. However, generally, examples of the latter show a lack of understanding of the issues by the artist.

9. Finished rendering. 1%. Ideally, this should be relatively simple. Too detailed, as in a photorealistic approach, would render the tessellations as difficult to discern; whilst cartoon-like would be too simplified. As a simple statement, those in the style of M.C. Escher are favoured.

10. 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 do indeed benefit from this. For example a human figure, with many different coloured aspects such as flesh and clothes, should ideally employ this feature, as otherwise the motif as an entity can be lost. Likewise for a gaily coloured bird or fish. For this, I make three distinctions:

(i) Self Defining, where the colouration automatically defines the edge of the tiles

(ii) Incidentally, where a hair-like thickness of line is evident, essentially left over from the design process

(iii) Borderline, where a decided border line is evident

Note that criteria 1-6, (as indicated by the percentages), as a group are of more importance than criteria 7-10. Though 7-10 are of concern, they are of less inherent importance than as the former, as they concern aspects of presentation rather than design.

Note that the above criteria should be regarded as a guideline. For example, Bruce Bilney, a tessellator of true worth, purposefully disregards criteria 4, and so he would have a different set of criteria. However, that said, the above gives a yardstick of how to measure the quality of tessellations, with any differences mere nuances. One could also quibble with the definition of criteria 6 ‘challenging motif’ categories, in that aside from birds and fish, all other motifs could be regarded as challenging, and not just human figures. However, human figures are attractive for obvious reasons, and so should be shown more frequently. 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 some people seem to have taken offence at my comments on this ‘other people’ section, regarding these 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. ‘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 OK, I have no gripes about this whatsoever. Do I comment here upon children’s efforts? No. If adults have the same ‘fun approach’, well again, I have no gripes about this whatsoever. Do I comment upon 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. Of note is that on occasions I do indeed give out stringent critique, namely with Natalie Sirett and Jinny Beyer, who show quite appalling standards, but they are not alone. Why then do I single out Beyer  in particular for condemnation? Simple, she is bringing out books (a medium that has stood the test of time, thousands 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 specially 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, page 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 even invited critique, page 62:

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… (a letter to Mr P. Kessler, a collector of Escher’s works, in regards to a observation of Kessler’s points concerning a print of Escher’s).

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? 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 this, 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 here). It’s all about standards. For example, should creatures with normally four legs be allowed to be portrayed with one leg? Is it acceptable to portray a creature without a head? Why should the premise of passing these risible efforts as ‘of worth’ be deemed acceptable and allowed to pass unchallenged? Those authorities peddling such monstrosities serve no purpose whatsoever, and indeed impede progress, of which impressionable people upon seeing their work will simply assume that such examples are acceptable, given the media in which they are displayed, and so thus be perpetuated. And so it continues. Some people here also make grossly exaggerated claims, and if all the above is not enough, then make claim to other people’s work as there 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 say that the emperor’s 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 as 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.


Created: 30 April 2010
Last updated: 14 September 2011 Revised and extended