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
(i) Self Defining, where the colouration automatically defines the edge of the
(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.
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?
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.
any standards, this is a stringent self examination. Indeed, Escher even
invited critique, page 62:
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).
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
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