1. Creating Escher-Type Drawings by E. R. Ranucci and J. L. Teeters

This book is aimed apparently primarily at a school-age level, of approximately the 12-14+ year age range. As such, it has a pleasing structure, as it progresses from first principles of basic tessellations to Escher-like tessellations in a sequential manner, albeit whether it is quite necessary for the early chapters to be so relatively lengthy, containing elementary symmetry matters that have been discussed elsewhere in a multitude of books is doubtful, the material covering nothing new. However, a case can be made for a self-contained book containing this material, gathered in one place, of which I have no qualms about.
Annoyingly, who actually created the many Escher-like tessellations designs is not always explicitly stated – two main authors appear on the cover, Ranucci and Teeters, whilst on the title page two further people are credited, Dale Seymour and Bob Fuller. Joseph Teeters is credited on some, but not all of the drawings. I now address each chapter in turn, relatively briefly for the non Escher-like aspects, and in more depth where these Escher-like art is shown.

Chapter 1 Plane Tessellations, Pages 1-18
Starts with the basics, of regular and semi-regular tessellations, and generalises quadrilaterals, pentagons and hexagons, and some simple geometry aspects as to why certain polygons tile.

Chapter 2 Transformations and Symmetry, Pages 21-36
This discusses the three basic transformations, of translation, rotations and reflection.

Chapter 3 Altering by Translation, Pages 39-59
This addresses the beginnings of Escher-like creation, by the alteration of the sides of various tessellating polygons by translations, with in some instances this illustrated by life-like motifs, unfortunately of a very poor standard, and a recreation of some of Escher’s own tessellations. Here, the method given for the like-like tessellations is somewhat simplistic - a ‘squiggly’ line is given, from which, when duly applied to the prepared tessellation, is then added suggestive life-like markings. However, what is lacking here is any suggestion of refinement of the original line - that given remains implicitly inviolate. Such matters effectively serve to restrict the quality of the subsequent motif. However, as a general premise, this is indeed a viable way in which to begin. Such examples are additionally accompanied by showing examples of (non life-like) tessellation and symmetry, and as such are not per se concerned with the creation process.

Chapter 4 Altering by Rotation, Pages 61-101
This continues on the altering theme, this time by rotation with a few examples of Escher’s own tessellations and the author's own.

Chapter 5 Altering by Reflection and Rotation, Pages 103-117
This continues on the altering theme, this time by reflection and rotation with a few tessellations of the authors own devising, and this time (inconsistently) excluding Escher’s own examples.

Chapter 6 Analysing Escher's Tessellations, Pages 123-134
This analyses four of Escher’s tessellations in terms of how he likely created them, in order of sequence Nos. 20 (Birds), 1 (Creature), 96 (Swans), and 21 (Imp). The selection appears to have been arbitrary, and includes three examples (Nos. 20, 1, 21) that are relatively complex which are beyond the understanding of the intended audience. Indeed, some of these even confuse Ranucci, as he gives spurious constructions.

Regrettably, Ranucci errs here in the analysis of No. 20. Although I will not go into detail here, he draws the wrong conclusions from his analysis. Paradoxically, he apparently inadvertently gives the correct fundamental construction in Fig.6-4 and then promptly asserts that this is not the true construction! What he thinks is the correct fundamental region of Fig.6-6 is incorrect… The analysis of No. 20 is somewhat involved, of which I will discuss separately elsewhere. To choose this particular tessellation for analysis was unwise, as it is beyond the understanding of the intended audience.

No.1 is poorly explained, and again requires a more lengthy description. Ranucci states, page 128 … a quadrilateral ABCD. Paradoxically, some of Escher’s earlier examples are relatively complex in their construction (as here), involving what I term as ‘overlapping of lines’, and the explanation given is far too simple, and again is inappropriate for the intended audience.
No. 21 is again lacking. Ranucci states:
The basic cells for the drawing are distorted hexagons
Whilst Escher can be seen to have overlaid his motifs with a rhombus grid, rather than a distorted hexagon.

The construction given for No. 96 is true as to how Escher did this, albeit the impression given is one of ‘inviolate’ lines, achieved at the ‘first instance’ of drawing, which is not how Escher did this – he refined the initial line.

Chapter 7 Using Imagination and Special Techniques, Pages 137-149
This chapter is the main aspect for this review, and so is thus discussed in greater depth than with other chapters, discussing as it does the creation of original Escher-like tessellations. In contrast to other books, this is at least a concerted effort, of both depth and variety of method, and is by far the most informative as regards the design process, albeit the examples arising from this are not up to Escher’s standard. Two different methods are given:
(i) of deforming a given polygon, (Ranucci calls this ‘using imagination’) and
(ii) of ‘adapting’ a non-tessellation figure into a tessellation. (Ranucci describes this as a ‘special technique’).

Deforming
The deforming method, although somewhat brief, meets with my full approval, and is undoubtedly how Escher went about his. Firstly, a given tessellation, Fig. 7-1 is altered, with opposite sides of a hexagon translated. This is then examined, with the question posed by Ranucci, page 138 concerning Fig. 7-2:
Now, the imagination is required. What does this outline remind you of?
Six possible designs are given, five of which are of the type I describe as 'cartoon-like heads' (a lower category of difficulty), and one of an American football player, albeit the latter in a somewhat ‘rough and ready’ state at this initial stage. For example, the head is unnaturally pointed (with what Teeters then describes as a 'triangular hat’), the shoulder protrudes at an unnatural angle, and the legs and feet are poorly defined. Although this does indeed broadly resemble a (American) football player, a more correct anatomical appearance would ideally have been accomplished, along with all the accoutrements of a football player i.e. appropriate clothing, such as a helmet. The football player is then separated as Fig. 7-4, with the intention of modifying, i.e. improving the existing design. Teeters states that
The player is running. The triangular hat needs to be changed to a helmet. The legs need to look more like legs. The arm on the left needs to be more like the one on the right... and so on.
This is good advice, as it shows that Teeters understands the issues, by assessing the weaknesses and addressing these concerns, and not simply accepting the first design, a typical beginner’s shortcoming. Such changes are thus duly made, albeit unfortunately, for the sake of the design process, the ‘experimental phases’ are not shown. Still, it should be obvious as to how he proceeded. The refined tessellation, Fig. 7-5, can thus be seen to be a distinct improvement, as when compared to the original of Fig. 7-2, the figure is now more anatomically correct and appropriate - for example, the protuberance of the shoulder has been removed, and the 'triangular hat' is now a helmet, thereby reinforcing the impression of a football player. However, weaknesses remain, in that the figures’ right leg is lacking in definition. This definition aspect will typically be found in tessellation matters, in that no matter how hard one tries, some elements will remain weak, of which the question the tessellator has to bear in mind is to the degree of acceptability – does the tessellation overall broadly retain artistic integrity? In this instance, yes.

Adapting
A different method given by Teeters, with Fig. 7-6 is to begin with a given, non-tessellating figure (coincidentally of another American footballer), it being 'adapted' to a tessellating tile, Fig. 7-7.
....A completely different approach is to start with a design in mind and to use your knowledge of mathematics of tessellation to create a figure that tessellates.
Let’s consider the football player again. You may have seen a photograph or an artist’s sketch that could be a beginning. The general outline of what you have in mind might be in a more triangular shape like the one in Figure 7-6.

This method is less frequently encountered, and indeed is the only example of its type in the book. Whether this is a ‘better’ method than the other is a moot point. The decisive test is in the final tessellation – namely in the quality of motif.

The adaptation method has a weakness in that one has to first simplify it into a tessellating polygon, and then perform a symmetry operation on it to accommodate the new lines. By contrast, using the deform method; this aspect is alleviated, as the tessellation operation (when undertaken appropriately) performed is certain to tessellate.

The premise of this method is that is different from that previously given. Stated simply, this (adaptation) can be described as ‘top down’, whilst the former is ‘bottom up’. The adaptation will result in a motif that of necessity will be weaker than the initial one (of a real motif), whilst the ‘bottom up’ will always be striving for improvements. As such, I am reluctant to too dogmatic as to which is the best, as I have shied away from using the ‘top down’ method. Preferences aside, the tessellation Ranucci shows as a result of this method is indeed another fine example, albeit again I have minor quibbles, with the right leg lacking definition, but again, this broadly retains artistic integrity, and so shows that this method is viable.

Teeters briefly introduces yet another technique during its creation (that can be applied to both methods), namely that of 'overlapping', illustrated by Fig. 7-9 and Fig. 7-10. (This is a device Escher frequently used in his earliest tessellations, of 1938, but noticeably less so from 1940 onwards.) Disappointingly, these aspects are discussed in a most brief manner, it being summed up in a handful of lines and diagrams, whereas a more in-depth treatment is required.

Teeters concluded the chapter with some further analyses of Escher’s tessellation No.113 (Fish and Boat), and unfortunately gives a spurious interpretation of the construction, that is not only long-winded, but wrong. Various unit cells are given as potential underlying polygons, before he eventually settles on a parallelogram as definitive. No. In actuality, this was based upon a hexagon, and can clearly be recreated – just locate the vertices of both motifs, and join accordingly. The hexagon can be either convex or concave.

Chapter 8 Worksheets for Developing Techniques, Pages 151-170
Broadly, ‘things to try’, based on chapters 2-5.

Chapter 9 Selected Grids, Pages 171-187
A variety of grids to use for experimentation purposes.


Critiques of the Original Tessellations:
Unacceptable, Poor, Reasonable, Good,
1. Man with ‘Banner Board’. Page 1
Reasonable. Although not shown entirely as a complete tessellation, the implication is indeed of that premise. This was probably designed with a specific purpose, signifying the chapter number (and is used for succeeding ones), of a figure holding a ‘banner board’, and as such, is a relatively pleasing example. This is let down somewhat by the drawing, as it is clear that it is intended for the figure to be mirror symmetrical, but some careless drawing (of shoulders and head) departs from this. However, this is a minor drawback, relating to accuracy in drawing, and is not to the detriment of the intrinsic quality of the concept, and could easily be corrected.

2. ‘Woman’? Page 44
Unacceptable. A shape with a few woman-like head elements added. This is truly dreadful! The only remotely woman-like aspect of this is the head. The mouth or nose (I can’t tell what this is representing) looks like a bird’s beak…. And as for the body… Anybody who does examples like this and perhaps thinks they are good does not understand tessellation.

3. ‘Clown’? Page 44
Unacceptable. A shape with clown-like elements added. Again, this is dreadful. The only remotely clown-like aspect of this the head. What is the region below the head…?

4. Birds. Page 47
Poor. A shape with one bird-like element (eye) added. Birds? No. This is just a shape with an eye. No other interior details are included…

5. Unidentifiable? Page 48
Unacceptable. A shape with a dot for an eye and mouth-like elements added. What sort of a creature is this? Dreadful.

6. Head (with feather in cap). (Teeters) Page 51
Reasonable. This head is reasonable within the context of a lower degree of tariff. A shape with head-like elements added. A reasonable head, in that the interior details are at least clear. However, it is of a lower quality category, lacks a defining outline necessary for such wireframe types, and it fails miserably the silhouette test, the test of quality.

7. Wrestlers (Teeters) Page 51
Reasonable. This does indeed represent a pair of figures in a typical wrestling pose. However, the figures have shortcomings, with general awkwardness and anatomic inconsistencies, with the legs of the left hand side figure being particularly poor.

8. Flying Owl (Teeters). Page 52
Reasonable. Broadly owl-like, albeit with reservation. Pleasingly, rather than a generic bird, this is designed specifically of an owl, in that the ears are showing, a typical identifying feature, as in a real-life example. A drawback is that the wings are out of alignment. Being a wireframe type, would benefit from outlining.

9. Dog (Teeters). Page 52 (and repeated page 144)
Reasonable. Broadly dog-like, with an outline that clearly resembles a dog, albeit with reservations, as it is still anatomically deficient, especially of the front legs which are somewhat awkward, and mouth region which is awkward. Even so, although not without faults, it is one of the more worthy examples, and has some degree of artistic integrity.

10. Spaceship. Page 53
Unacceptable. A shape with supposedly ‘space-like’ elements added. Really? This looks like a shape to me with obscure markings… Dreadful.

11. Dog Head, of Doberman Pinscher and An Old-Fashioned Shoe. Page 97
Reasonable. Both motifs are shown in combination, the head being of a lower tariff of difficulty. Unfortunately, the presentation of the symmetry is poor, with no apparent ordering - the motifs appear to have been placed randomly… The Doberman pinscher, albeit of only a ‘cartoon head’, is reasonably good, in that it is identifiable as a dogs head. The shoe is somewhat less so in quality, as there is no space between the heel and the sole as in a real-world example.
Furthermore, to a less important degree (as sometime incongruous motifs are unavoidable), the above motifs in combination are somewhat incongruous (albeit Escher included some like incongruent examples in his own work) – there is no connection between these. Better would be to have the respective motifs as ‘single’ tessellations.

12. Mr. Frog, Mr. Golliwog, Knight, The Executioner, Ms. Fish, H. Dawg (and an unstated ‘long nose’). This also appears on the front cover, in colour. Pages 98 and 99
Unclassified. A somewhat confusing sequence of two related pages, with single tiles with motifs on page 98, and a combination of tessellations in a triangular composition on page 99. Unfortunately, these do not marry, as Ranucci and Teeters give 6 different motifs, page 98, of which they use three for the triangular composition on page 99, whilst introducing an additional motif not on page 98. What can one say… To ease the analysis, I discuss each page separately:
Page 98
Various cartoon-like motifs of six different motifs, of four ‘heads’ of various motifs, and two whole animals, a frog and fish. Two of these are very pleasing examples of their type, namely Mr. Golliwog (which bears no resemblance to a golliwog…) and H. Dawg, whilst the others are less so – the Mr. Frog especially is very poor, a shape with frog-like elements added…
Page 99
A composition of a 4-unit sided triangle of 16 equilateral triangles, based on three of the preceding motifs, with, oddly, bizarrely even, of one new addition, of ‘long nose’, which incidentally is quite good quality. Disappointingly, the composition shown is very poor, as although four different motifs (‘Long Nose’, Ms. Fish, Mr. Frog, and H. Dawg) are shown repeated four times, there is no apparent orderly placement of these, resulting in an uneven arbitrary distribution of motifs. Also, the artist selects inferior examples in the composition, including the very poor Mr. Frog, whilst excluding the excellent Mr. Golliwog…

13. Heads - Two Different Heads and ‘Unidentifiable Creature’. Page 100
Poor. A tessellation of two different tiles with three different motifs…. A shape with head-like elements added. Truly dreadful. Shapes with eyes, nothing more. The third motif is probably intended as a ‘complete’ animal, albeit unidentifiable, and is unlike the other two in its nature. This being so, why include in an obvious incongruous composition? Furthermore, again, the composition is lacking…

14. Bird. Page 113
Reasonable. A better tessellation. The bird is broadly recognisable in silhouette.

15. St. Bernard Dog (Teeters). Page 114
Reasonable. A better tessellation, albeit with reservation. 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 lower part of the dog, with legs, belly is entirely mere surface decoration. Furthermore, 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.

16. Heads - 6 Different Heads - Freckle Face, Football Player, Animal, Young Girl, Weird Sailor, Big Mouth. Page 138
Unclassified. Although not shown as a tessellation per se (aside from the football player on the next page), the various possibilities are clear in their tessellation premise. Five heads (of a cartoon-like appearance) and one whole football figure are shown, of various qualities. ‘Weird Sailor’ (albeit nothing like a sailor...) is probably the best.

17. Football Player (Also see my comments of Chapter 7). Page 139
Reasonable-Good. A better tessellation. Here the authors refine the football player of Figure 7-4, and it must be admitted, successfully. This actually resembles a football player. Although the resulting figure is not outstanding, it is considerably better than the initial drawing of Figure 7-4, and is one of the better ones, and is outlined.

18. Football Player (Also see my comments of Chapter 7). Page 142
Reasonable. A better tessellation, albeit with reservation. Again, another success in relative terms. The figure is in proportion, and has no stand out weaknesses. However, both arms are mere surface decoration, and a leg is decidedly awkward.
Ideally, this would have been coloured in either three or four colours, as two colours as here are inadmissible for map colouring, but this is a relatively minor matter in the context of the book.

Summary
Overall, the book has many shortcomings. Quite simply, the authors plainly do not truly understand the issues of composing Escher-like tessellations, as they include many examples that are unworthy. The number of quality tessellations amount to a mere handful, with dogs on pages 52 and 114, and football players on pages 139 and 142 being the best. Conspicuous throughout are those of the ‘head’ type, the easiest type to compose, requiring little or no ability. However, some of these, within a whimsy, cartoon-like category, are quite pleasing, such as Mr. Golliwog or H. Dawg, page 98. The authors do not question why Escher only showed a single instance of this type (and myself none. There is a reason…). Noteworthy is the exclusion as to what it is that makes birds and fish so suitable for motifs, to the exclusion of other motifs. This would entail understanding the issues, which the authors plainly lack.

A lesser criticism is that occasionally Teeters shows compositions which are particularly poor, in both concept (with unrelated motifs, albeit this is a lesser criticism), and symmetry arrangements, with the motifs added at whim, in an arbitrary manner, with no consideration for symmetry. This aspect, in contrast, is a major criticism. This is very surprising, given Ranucci’s mathematical background.

Not only is there general poor practise, but there is mathematical inexactitude, and furthermore in more than one place, such as with the bird and fish analysis on pages 146-149 and 'Day and Night birds, pages 123-126. Again, this is inexcusable.

Although the authors purport to show how Escher did his tessellations, this is only very loosely so. When analysing/recreating Escher’s tessellations, the impression given is that of a single instance of a line drawn in a ‘fully refined’, state, whereas Escher reworked the initial line. Oddly, the authors state this was what they did for their own tessellation of the football player, but do not do this for their other tessellations.
However, aside from the above, overall the book is an ideal introduction for the intended age range if due allowance is made for the above aspects. In all likelihood, one would not expect the intended audience to be overly concerned with the production to an Escher-like standard, and although written for a younger age group, this should not necessarily mean that a less than exact approach should be adopted than would otherwise occur.

Agree or disagree? Email me