Minor Books and Articles

‘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 (Article)

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 tessellation…

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
Unacceptable. 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…


4. 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.


5. 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.


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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 addressed.

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 the drawings.

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.


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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:
A short, clear discussion showing how you or your students can create tessellation art
The 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 constructed this!

A distinct dichotomy exists here, in that the instructions are obscure, of no practical use. However, in contrast, the three tessellations are all of relative quality!

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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.


1. ‘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…


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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.



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Created: 27 September 2009. Minor revision 17 July 2012