‘Minor’ Books and Articles
This page consists of what I
term as ‘minor’ books and articles, as according to their depth. As the minor
articles do not really warrant a page for each author (largely for the sake of
convenience and space saving, in contrast to the ‘major’ articles), these are
combined as a single page. Note that ‘minor’ does not necessarily mean an
article of lesser worth; this just refers to the treatment given by the author,
i.e. of the utmost brevity.
6. Modern Mathematics Made Simple, by Patrick Murphy (Book)
7. Regelmatige vlakverdeling, by M. C. Escher (Book)
8. How to Draw Tessellations of the Escher Type, by Joseph L. Teeters
9. Rolling a Tetrahedron, by Kodi Husimi (Article)
10. Tessellations File, by Chris De Cordova (Book)
6. Modern Mathematics Made Simple
by Patrick Murphy, pages 194-205
As such, I am somewhat reluctant to critique this book on the grounds that Patrick
Murphy, a mathematician of some renown, is writing almost entirely on
mathematics, and so is not specifically discussing or ‘how to do’ Escher art
per se, as only Chapter 10, pages 194-205, is concerned with tessellations
per se, and even this is mostly of the mathematical aspect rather than of the
Escher aspect. However, as it does include a few instances of life-like
motifs, it is thus reviewed.
In the relevant chapter, Murphy purports to give a general study of
tessellation, of a relatively more in-depth mathematical treatment than with,
say, Ranucci and Teeters. Unfortunately, much of the material is overblown,
in which the tessellation aspects are unnecessarily made more complex than
necessary. To apparently lighten the study, Murphy introduces Escher-like art
to some of the tessellations, of which these fail miserably. Indeed, read
these as how not to do
1. Bird and Frog, page 197, fig. 10.5
Unacceptable. A frog, where?
There is nothing frog-like about the outline or the interior detail. Indeed,
the shape is unrecognisable as anything! The bird is faintly recognisable,
but even so, leaves a lot to be desired, and is very far from a true outline.
2. Dog, page 197, fig. 10.6
This is potentially deceiving as to quality. Now, upon examining the example
by Murphy, it may at first glance appear to be a perfectly acceptable, if not
good dog to the novice, as it possesses, or at least has interior markings,
of a head, body, legs and tail. However, such apparent 'quality' is
misleading, as upon comparing the line drawing with its silhouette, it can be
seen that the latter is now barely recognisable as a dog. Indeed, to my eye
the silhouette is more reminiscent of another creature altogether, a frog,
with reservations. So, what is it that explains the difference? Essentially,
Murphy's dog is lacking in definition, as the elements are very poor in this
matter. Essentially, he has added detail (embellishments) to poor raw
material. Shortcomings include:
• Poor definition of the underside of body and legs. Here the underside of
the body and legs combine as a single mass, thus rendering no distinction
between the elements. As such, this is the major shortcoming.
• Poor definition of neck. Essentially, the neck is ignored, with the head
joined immediately at the body.
• Poor definition of chest. Essentially, the chest is ignored, as the head
immediately joins the legs.
• Poor definition of the head. The head possesses no muzzle or stop or chin
(although some breeds do not have a stop, and so it could be argued that this
example is thus not a shortcoming. However, as most breeds do, this should
ideally be included).
• Poor definition of body. Essentially, elements are drawn onto a body that
lacks clarity. Furthermore, towards the rear of the dog is a pronounced
'gouge' in the back before the tail.
Aside from the inherent shortcomings is the lack of anatomical correctness
concerning the drawing of the details:
• The hind legs are shown incorrectly, being far
too much towards the top of the dogs' back
• The legs are shown as mere appendages
• Conflicting perspectives, with the eye shown as seen from the front, whilst
the rest of the detail is as portrayed from the side
In short, this is as poor an example of a ‘dog’ as there could be.
3. Cross-legged Chicken, page 197, fig. 10.6
Unacceptable. Well, at least the head is reasonable,
but as for the rest…
Bird, page 203, fig.10.15 (ii)
Unacceptable. Well, the head
is reasonable, but as for the rest… From the back of the wings to the tail
has no definition whatsoever, and is just surface embellishment. Admittedly,
Murphy states that the bird ‘… is merely decoration’ to the tile, but even
so, why add such a poor decoration? Incidentally, a better motif for the tile
would have been a fish.
Demon Bowler, page 204, fig.10.17
Unacceptable. Oh dear. This
example is of a type that is occasionally seen that is perhaps misunderstood,
in that at face value this might be taken as ‘good’, or indeed desirable by
the novice, in that a complete ‘demon bowler’ is portrayed, albeit of a disproportionate
scale, with the head too large for the body. Also, the legs and feet are
somewhat odd, albeit the latter two points are of secondary concern here.
However, examples of this type are amongst the very worse. Upon inspecting the tile more closely, it can be seen
that Murphy has taken liberties with the interior design, this not matching
the tiles outline, resulting in ‘negative space’ between tile and motif (with
the exception of the top of the head). Therefore, no skill whatsoever is
shown here. As an analogy, take a square, and draw a lifelike motif in its
interior. Easy, beyond drawing ability, no skill whatsoever is required, the
tessellation principle is neglected. In principle, that is all Murphy has
done here. When the tile itself is examined in details, as a silhouette, this
does not resemble a ‘demon bowler’ in any way, and so is yet another failure.
6. Fish (Plaice)
Poor. Well, very loosely a flatfish can indeed be
seen, but as discussed elsewhere, such a type is of a decided lower tariff of
difficulty, so much so that it is essentially worthless as a motif.
Well, a very sorry state of affairs indeed. Not a single tessellation can be
said to be in any worthy. Indeed, five of the six are assessed as
unacceptable, with only one ‘attaining’ the heights of ‘poor’. By showing
these execrable examples without qualification Murphy show that he doesn’t
understand issues of life–like tessellation.
Agree or disagree? Email me.
Last updated: 27 September 2009. Revised 17 July 2012, with simpler assessments,
and occasional minor revision of text
7. Regelmatige vlakverderling (The Regular Division of the Plane, Escher's Own Writings on Tessellation Translated
into English in Escher: The Complete Graphic Work)
Of note is Escher's writings (and more rarely,
film) on tessellation, of which, perhaps surprisingly, do not in any way
concern themselves with the creation process per se. For example, his major
work on the matter, in Regelmatige vlakverderling (The Regular
Division of the Plane) is essentially a comment on the examples given (with
frequent digressions of unrelated matters), albeit these were specifically
designed for the purposes of the book. How the designs came into being is not
Other writings by Escher include comments he composed for his book The
Graphic Work of M.C. Escher, of which this includes 14 of his numbered
tessellations. However, these appear very much merely as background sources
for the tessellating-based prints, albeit admittedly, being not graphic art in
themselves, than their omission of discussion can be argued as natural, being
intrinsically subsidiary to the book. Whatever, these are not discussed in
the book. Another of Escher's major writings concerned a series of lecture
notes for an American tour reproduced in Escher
on Escher, in which he discussed both prints and his tessellations.
However, once again these do not address the 'how' aspect, bur rather are
comments, of a decidedly brief nature. Aside from these three books, Escher's
other writings on the 'How' aspect is conspicuously absent. Why should this
be so? As such, perhaps unfairly, I am somewhat suspicious as to Escher's
motives in all this. Being generous, it is indeed possible that he was more
interested in producing the tessellations, rather than in trying to explain
how they came about. Indeed, his active period in tessellation was relatively
short, curtailed in later life by ill health. Consequently, it may well be
that he simply considered detailing such matters as a matter of being sidetracked,
the drawings themselves being of more inherent importance. However, a counter
argument can be put forward, in that as Escher frequently wrote about other
graphic issues, well why not detail by far the most important aspect of his
life's work, namely the tessellations themselves? Rightly or wrongly, I
believe that Escher wanted to keep the 'secret' to himself, thereby not
diluting his star. Furthermore, the preparatory drawings/sketches that
underlie the genesis of the periodic drawings, that again would reveal the
'secret' rarely, if at all, are shown. Again, I am most suspicious as to why.
In just about all publications on Escher, these are notably conspicuous by
their absence. Indeed, most books completely disregard them. As such, the
only source for such material that I am aware of is Visions of Symmetry
by Doris Schattschneider, who at least does show a few examples, pages
106-108 and 111, albeit even so, only page 111, showing the Horseman drawing,
directly relates to the genesis of that particular periodic drawing. During
the course of designing the periodic drawings, Escher must have composed many
hundreds, perhaps even thousands of such sketches. Therefore, what has become
of this material? Did Escher mostly dispose of such studies upon having
completed the definitive periodic drawing? This seems highly unlikely given
his general approach, with thoroughness in attention to detail and record
keeping. Oddly, in contrast to the paucity of the above material, the
preparatory drawings for Escher's non-tessellating 'spatial structures' (such
as Depth and Spirals) are shown in various books in abundance - why should
there be such a discrepancy? Again, rightly or wrongly, I believe that the
purpose behind all this is to keep Escher's star solely in the ascendancy, to
the exclusion of other people. However, with the publication of all the
periodic drawings, the 'secret' can be, if not gleaned, than glimpsed from
So, How Did He Do It?
So, using Schattschneider's opening line in Visions of Symmetry, how
did he do it? Without having seen the preparatory drawings underpinning the
tessellations, it is not possible to state categorically that he used a
certain method/s. However, the morsels given in Visions of Symmetry
are most illuminating, of which beyond all reasonable doubt show that he
started with no predetermined motif in mind, in which a framework of a
tessellating tile had added to it a gently indented line, either of straight
lines (as in the case of Horseman), or curved squiggly line. From this
initial effort, if he could vaguely discern that the shape had some sort of
likeness to a representational motif, he then continued to seek improvements
in increasing its inherent quality by successively refining it. Quite how
long this took, as a rule, is unknown. Possibly, some drawings were devised
more quickly than others. Furthermore of interest are those examples that he
considered as failures, which did not lead to a definitive periodic drawing.
Interestingly, these are not shown. Quite simply, it is unbelievable that
Escher attempted 137 efforts and had 137 successes, with no failures.
Interestingly, the 'random line' method appears to be the only method Escher used,
as the periodic drawings are quite similar in this regard, in that a simple
underlying tessellating unit underpins the drawings. So, are there other
methods? Well, yes.
Agree or disagree? Email me
Last updated: 29 March 2010
8. How to Draw Tessellations of the
Escher Type, by Joseph
L. Teeters. Mathematics Teacher April 1974. 307-310
A brief (four page) article on creating Escher-like tessellations. Teeters
states in the subtitle and introduction:
short, clear discussion showing how you or your students can create
writer presents here a technique that has proved useful in creating
fundamental regions of the Escher type...
If only this was so! Unfortunately, I found that the article does the exact
opposite! (Albeit the tessellations are of a higher standard than normally
seen, and indeed, are relatively pleasing.) It’s indeed short, but not at all
clear, being confused, and decidedly hard reading. The pictures lack
captions/titles and the article has tessellations appearing without
explanation or reference in the text, resulting in a veritable mess. The
‘procedure’ explanation given is unnecessarily convoluted and unclear, and of
what it is referring to as well – the unicyclist accompanying it? Indeed,
even after looking at this several times since I began doing tessellation, it
remains obscure. To try and unravel this is a veritable nightmare. Has anyone
actually tried these techniques (and found them useful), and indeed at least
understandable? Let me know if so.
1. Unicyclist, page 307
Good. A better tessellation. Of its type quite pleasing, in that it does
indeed bear a resemblance to a unicyclist, and as it is of a less frequently
occurring type of motif (indeed, it is unique), this is worthy of more praise
than of more common motifs. However, there appear to be instructions accompanying
this as to how Teeters did this!
2. St. Bernard Dog, page 309
Reasonable. A better
tessellation. A reasonable portrayal of a dog is shown, and more specifically
of a specific breed as well, with the pleasing factor of a rum barrel beneath
the St. Bernard's collar adding to the interpretation. However, the
position adopted is not a true representation of a sitting dog (if this
interpretation is correct), albeit this is masked to some extent by the
relative quality of the interior elements, whilst the lower half of the dog
is entirely surface decoration, and lacks articulation.
3. Red Indian with Tomahawk, page 310
Good. A better tessellation. A reasonable portrayal of a Red Indian is shown,
and more specifically accompanied by appropriate dress, thereby reinforcing
the tessellation. However, there are no instructions as to how Teeters
A distinct dichotomy exists here, in that the instructions are obscure, of no
practical use. However, in contrast, the three tessellations are all of
Agree or disagree? Email me.
Last updated: 29 March 2010 revised 17 July 2012
9. Rolling a Tetrahedron...
In M.C. Escher
Art and Science: 'Rolling a Tetrahedron on the Plane to Produce Periodic
Patterns of Symmetry P2 and Drawing Dragon Curves as Backbones of Escher
Figures', by Kodi Husimi
Although the article is mostly concerned with producing patterns by rolling
around a tetrahedron on the plane, the article ends with a brief guide to producing
Escher-like art, of which the motif, probably of a dog, is most poor indeed. Husimi
'proposes', page 183 of:
'... an easy way of inventing Escher figures,
without spending much time and energy'.
The motif is unrecognisable, and so Husimi has failed in his assertion above.
A very sorry state of affairs indeed. A single tessellation of an execrable
standard. By showing this execrable example without qualification Husimi show
that he doesn’t understand issues of life–like tessellation.
‘Dog’, page 185
Unacceptable. A shape with cartoon-like dog-like elements. Husimi purports to
show an Escher-like tessellation, arising from his ‘rolling’ process,
described above, of a creature probably intended as a dog. A dog? Where? Admittedly,
it is indeed vaguely reminiscent of a dog, with dog-like elements, a head,
ears, four legs, and tail, but this really is stretching the definition
somewhat, to say the least. The perspective conflicts are numerous here, so
much so one could almost say that it is an (unintended) cubist type picture.
Just about every element is in conflict, of a variety of viewpoints, so
jarring on the senses. Also, the ears are anatomically incorrect, the front
and back legs combine. The head has no stop, and if all that is not enough,
it is sloppily drawn. It’s all so horrid. Is this really what tessellation art
is about? Can we not aspire to higher standards? What can one say…
Agree or disagree? Email me.
Created: 27 September 2009.
Revised 17 July 2012, with minor addition to text
10. The Tessellations File
by Chris De Cordova. Tarquin Publications 1983, page 6
A beginner’s guide to
tessellations, from the basics, ‘what are’ tessellations, regular and semi
regular tessellations. Even as a guide to tessellations per se, decidedly
lightweight, of just six pages, with the rest of the book consisting of
tessellating grids. A very minor guide to producing Escher-like art, with
just one instance, not really worth the dignity of qualifying as a guide, nor
indeed reviewing. Again, it’s perhaps slightly unfair of me to review the
book, given that it’s not leaning towards the Escher aspect, but as it does
purport as to how to do Escher art, I do so nonetheless.
1. Human Figure, page 6
Unacceptable. A shape with child-like human-like elements added. The body
is a rectangle, the shoulders are square, the ‘arms’ are ludicrously long,
and the legs are too short for the ‘body’. Almost entirely, save for the
head, any detail is mere surface embellishment. What can one say…?
A very sorry state of affairs indeed. A single tessellation of an execrable
standard. By showing this execrable example without qualification De Cordova
shows that he doesn’t understand issues of life–like tessellation.
Agree or disagree? Email me.
September 2009. Minor revision 17 July 2012