## Designing and Drawing Tessellations, by Robert Fathauer

Although on the face of it the title would suggest a book on tessellations per se, the premise is of overwhelmingly towards designing Escher-like tessellations. A better title would have been to include a reference to ‘life-like motifs’ or ‘Escher-like’, although admittedly the cover picture is of a life-like motif.

As a generalised statement, the advice given by Fathauer is very good indeed, with many useful hints and tips as to the ways and means of creating Escher-like motifs. Although primarily aimed at a school-age level (12-16), anyone with an interest in creating Escher-like tessellations will find it advantageous, as it broadly addresses the all-important understanding of the underlying issues concerning the creation of life-like motifs. Very few books concern themselves with this matter, even in passing, and of which lies at the crux of the problem of designing high-quality motifs, and so this aspect in particular is warmly welcomed.

Broadly, upon the reader having absorbed each chapter, a series of activities are then given, suitable for the above age range.

A brief overview of the chapters:

Chapter 1, Introduction to Tessellations, contains illustrations of tessellations in the real world as a historical context, leading onto forerunners of Escher.

Chapter 2, Tessellations in our World, shows tessellations from the natural world.

Chapter 3, Geometric Tessellations, details the underlying mathematical aspects behind tessellations.

Chapter 4, Symmetry and Transformations in Tessellations, discusses mirror symmetry and translation, glide reflection, and rotation.

Chapter 5, Tips on Designing and Drawing Escheresque Tessellations, gives a series of tips on designing Escher-like tessellations

Chapter 6, Special Techniques for Solving Design Problems, gives a series of techniques for improving on a initial tessellation

Chapter 7, Tessellations Based on Square Tiles, a step by step guide to creating tessellations

Chapter 8, Other Tiles Based on a Square Grid, uses right triangles and kites with a series of templates

Chapter 9, Tessellations Based on Equilateral Triangles, uses equilateral triangles with a series of templates

Chapter 10, Tessellations Based on 60°-120° Rhombus Tiles, uses rhombuses with a series of templates

Chapter 11, Tessellations Based on Hexagonal Tiles, uses hexagons with a series of templates

As such, one could argue that Chapters 1-4 could very well be excluded on the grounds of familiarity, as previous books have covered the same ground. Still, for an easy, and convenient, ‘basics to hand’ covering of such matters, there is nothing to fault here. However, for those familiar with such matters, the book only really begins much later, with Chapter 5, this concerning the Escher-like aspect, with a series of tips on drawing and designing. Chapter 6 then gives a series of techniques, in effect attempts at improving upon the initial tessellation. Chapters 7-11 are concerned by more specific matters, with tessellations based on specific tiles that frequently occur.

A feature of the tessellations is that many are shown as wireframes, which as I discuss elsewhere, in Essay ** renders the motifs as not as readily identifiable as could otherwise be. This could, and should, have been shown in different shades, especially as the tessellations were drawn on a computer, from which such matters are quite easy to accomplish. Cost seems to have been a consideration; there are no colour pictures in the book.

As stated above, the essence of the book is Chapters 5 and 6. Chapter 5 consists of a series of tips, or statements, with illustrations, giving advice on the creation process. Chapter 6 then essentially refines the tessellation with a variety of special techniques. As such, as they are the core of the book, I now examine these tips and techniques in relative detail, firstly with a quote from the introductory text of Chapter 5, and then the following tips, and then Chapter 6, of techniques.

Chapter 5 Specifics:

Quote 1

Designing an Escheresque tile is easier if you choose motifs to be flexible. Some animals can be twisted around and distorted quite a bit, while others can’t be. Lizards, birds and fish seem to work particularly well, and M.C. Escher used these motifs for many of his tessellations. As a motif people are especially challenging. Not only are they not very flexible, but the human eye is more sensitive to whether people look right than to whether or not other animals that we don’t see as often look right.

I take issue here with the term ‘flexible’. I am not certain as to what the author intends here – does he mean inanimate (non-flexible) and animate (flexible) motifs? Inanimate motifs, although indeed generally inflexible, are not necessarily non-flexible. For instance, a book can be shown opened or closed, a car can have the doors, bonnet, or boot opened, and so have a degree of flexibility. Admittedly, most inanimate motifs lack this, for example a soccer ball, or a house. Just by changing the qualifier to animate or non-animate clears the issue, and removes any uncertainty.

Another aspect I take issue with here is the specific motifs used to illustrate this point. Broadly, with few exceptions, any creature can be twisted around, and adopt different poses. For example, in contrast to Fathauer’s assertion that a human figure is, in so many words, ‘not flexible’, it can adopt a whole myriad of poses, such as standing, lying down, jumping, stretching, crouching or bending in half. Other creatures have different degrees of flexibility. Birds can have their wings raised, lowered. Fish can have their body twisted, and fins of just about any shape. However, all this does not explain why birds and fish are easy motifs to tessellate with, in contrast to human figures which are relatively difficult.

An exception to this ‘flexible’ rule would be creatures with a shell, such as turtles, of which by their very nature such poses above are impractical. However, it is still possible to design plausible motifs with shells. Therefore, as argument per se, this ‘flexible’ description does not convince me.

Also, I take issue with lizards being the first motif mentioned; as this gives the impression that this is one of the easiest motifs to accomplish. This motif is found less frequently in tessellation, with birds and fish dominating, and so I would emphasise these motifs, and would have lizards third in this list.

Tip 1, Page 49
To the extent possible, the outline of the tile should suggest the motif even without interior details
Exactly! I cannot over emphasise the importance of this point. This is the essence of life-like tessellation, a true test of understanding of the issues and of one’s ability to compose motifs that are recognisable by their outline or silhouette alone. Furthermore, this point is clearly illustrated with two examples, of which I describe as inferior and superior, thus clearly showing the differences in quality. In silhouette form, the inferior T. Rex (left) is not particularly suggestive of a recognisable motif. In contrast, the superior winged dragon (right) is instantly recognisable as of a motif in profile, albeit of an imaginary nature. Note that although the T. Rex is not readily identifiable, with the addition of interior detail it can be made into a ‘respectable’ motif of relative quality.

Tip 2, Page 50
The tiles should make orientational sense
As such, I take issue with this statement, as it is open to interpretation, albeit it does contain merit, but in a different context. The author is apparently referring to what I prefer to term as views, in that the motifs should be consistent, such as all seen from above, or from the side, but never combined. If indeed shown combined, then such a presentation is obviously inconsistent.
This also implies that the motifs are ideally portrayed in an ‘upright’ orientation. However, in the course of designing tessellations, one can use various symmetries, some of which naturally ‘force’ the tessellation motif to appear in many different orientations. For example, Escher has tessellations that have as any as six orientations. Does this make the ‘six orientation’ type inferior to a ‘single’ orientation per se? I think not. Certainly, this issue has implications when a story is being told, as with Escher's prints, in which he used elements of his tessellations. For example, Day and Night, where it is of crucial importance for the motifs to be upright, as otherwise the composition is absurd. However, this is a different matter from a tessellation per se, where such concerns do not come into play.

Tip 3, page 51
When using more than one motif, choose motifs that go together
Excellent! This shows true understanding of the issues. Although it is possible to have creatures in a ‘multi-motif’ tessellation that have no connection, such as a dog and a book, such tessellation are lacking in an aesthetic sense. Better is to have those with connections to each other, such as a dog and bone or with natural contrasts, such as angels and devils. However, although desirable, this should not be undertaken at the expense of quality of motif. A lesser quality example is not made ‘better’ by just having connections to each other.

Tip 4, Page 52
The different motifs should be commensurably scaled, unless a special effect is desired
Excellent! This shows true understanding of the issues. Although it is possible to have creatures in a tessellation that are not commensurably scaled, such as a dog larger than a human, such tessellations are lacking in an aesthetic sense. Better is to have those that are more or less true to scale. Again, this should not override matters of quality though.

Tip 5, page 53
For real world motifs, it is important to use source material to get the details right
This really concerns the drawing aspect, and is sound advice. As one’s tessellation will mostly concern the animal world, of which there is a great many different motifs, with literally thousands of different species, who can claim to be an expert in drawing all possible creatures? For example, although a tessellating motif may look like a certain creature, say, a Gila monster, how many people could draw this accurately enough for it to be regarded as true without reference to a picture? Not many. Certainly, not myself. Even Escher himself used reference material.
Also, as the motif can adopt different poses, say with a bird in flight with different wing positions, or a fish twisting, of which again appropriate reference material is desirable.
If one is serious about creating life-like motifs, one should bias this learning towards birds and fish, as tessellations mostly consist of these particular motifs.

Tip 6, Page 54
Stylize the design. Don’t try to make it too realistic
I take issue with this. Again, it’s not clear as to what the author has in mind. The outline? The interior design? On the contrary to the given statement, the desire of utmost realism, as regards the outline, should be the main aim. However, although a laudable aim, this ideal is seldom accomplishable. The author gives a frog example to illustrate his point. As can be seen, the frogs leg/foot consists of a series of very fine detail, with many convoluted curves that are simply not ‘appropriate’ for tessellation purposes. Of necessity, these feet/legs have to be simplified if it is to tessellate. As can be seen, upon so composing a tessellation, these are now heavily stylized, but to an acceptable degree, as the motif is still broadly recognisable as a typical frog. Incidentally, Escher’s own frogs are shown in the same stylized manner as here. In short, make the outline as realistic as possible; and only stylized if not.

Tip 7, Page 55
Choose a style that fits your taste and abilities
This refers to the style, as regards the degree of realism and stylization of the finished artwork, either by hand or computer, and which can range from a wireframe, cartoon, to a photorealist finish, with nuances in between. To illustrate the point, six different examples are given with different drawing styles. However, the issue of ‘recognition’, as in a ‘complete’ tessellation, with many motifs is not addressed. For example, a tessellation with a photorealist finish, which may be thought to be the epitome as regards realism, when shown as a actual tessellation will be found to be most unclear, as the various colourations tend to mask the individual motifs. Therefore, a generally stylized finish is to be preferred, of generally one coloured motifs.

Tip 8, Page 56
Choose colours that suit your taste and that bring out the tiles
Excellent! However, the author somewhat dilutes this by stating in the text that it is usually desirable, but not essential to use different colours for adjacent tiles…For me (and Escher), it is essential; as such uncoloured (wireframe) types lack what I term as recognition. For example, to compare colour and non-colour types, I use the seahorse tessellation. This is shown uncoloured page 63, and coloured on the back cover. Is the motif readily identifiable with no colour? I think not. When coloured, immediately so.
As a rule, colour your tessellation. By so doing, this will aid recognition of the motifs, as the colour defines the motif, in contrast to a wireframe example which is implied here.
Another area of ambiguity is in colouring adjacent motifs the same colour. Generally, I frown upon this, as it does not aid the recognition. However, on occasions, it is possible to override this rule. An example of my own is of the Penrose tiling, of birds and fish, where I have two motifs. These are coloured in two colours. However, it can be seen that this violates the map-colouring rule of non-contiguous colouring, where three colours are necessary in this instance. However, as three colours are incompatible with two motifs, and so there is a trade of between minimum numbers of colours and map colouring. However, as a rule, recognition of the motif should come first.

Chapter 6 Specifics:

Technique 1, Page 64
Distorting the entire tessellation
Excellent! This technique distorts the ‘finished’ tessellation slightly, either by stretching or shearing, with the aim in mind of improving upon the original. However, the examples here, of rabbits and a chameleon barely require this feature. Indeed, at first glance the chameleon is no different.
To give perhaps a better illustration, beginning with a square, let us say that this may be suitable, but not ideal, for a proto human figure. By its very nature, the motif will be somewhat dumpy. If this is then stretched to a rectangle, then a more realistic human figure will appear.
However, although the premise is good, this technique is very infrequently required. Only on a handful of occasions will it be found necessary.

Technique 2, Page 65
Breaking symmetries
This could perhaps more better to have been included in Technique 4, Splitting and Moving. As such, this is of limited use, and will only be used very infrequently.

Technique 3, Page 66
Splitting a tile into two or more smaller tiles
Excellent! This is a device that is of some importance as regards design, and is in common usage (in contrast to other techniques mentioned above). I frequently use this, of which Escher did so likewise. Therefore, this is of more interest than otherwise.
Although the principle is correct, this is unfortunately illustrated with a relatively poor quality tessellation that does not serve the argument well, of frogs, toads, and snakes, but it will suffice for discussion purposes. The premise here is that the frogs and toads are of an acceptable standard, whereas the snake is not. Accepting this premise as so, can this be made better? Instead of adding snake-like detail to essentially unsuitable raw material, a different approach is possible, with the tile spitting procedure. This is now divided into two identical tiles from which a ‘better’ creature emerges, a tadpole. In reality, this is not really any better than the snake, but the principle is germane.
Frequently, a situation will arise where in a tessellation of ‘multiple tiles’ where for one tile the motif will be of good quality, but for the other tile nothing suitable can be found. By so ‘tile splitting’ this thus gives a possibility of a better motif, and so a higher quality tessellation than otherwise, rather than using an essentially unsuitable one.

Technique 4, Page 67
Splitting and moving vertices
Excellent! This is a device of some importance, albeit of a secondary aspect to the initial design, in that it can be use frequently to improve upon the initial tessellation. Escher frequently made use of this somewhat more advanced technique, especially at the beginning of his studies, perhaps somewhat surprisingly, given the relative degree of complexity involved. Broadly, the procedure involves ‘releasing’ the motif from its original vertices, and by so doing thus, in theory at least having the potential to improve the original motif.

Critiques of the Individual Tessellations:
A pleasing variety of motifs are shown, with no less than 43 different motifs, this in contrast to the more commonly found birds and fish, to the general exclusion of others. However, this as an ideal per se should not be at the expense of quality – a quality motif, say a bird, is worth more than a hundred (or more) poor quality, of rarely shown motifs. Quality is everything…
Many of the tessellation shown here are in outline (wireframe) mode, a type of presentation that I dislike, preferring coloured, or shaded examples. Although this is fine for studies, I do not consider it so for finished work. However, this aspect does not affect the quality of the motif per se, as it could be coloured. For example, some of the best examples namely Men in Coats, Seahorses are presented as such. Where examples of this type occur, I simply state wireframe, with the aim in mind of signifying my disapproval of presentation.

1. Dinosaur (T. rex), Page 37
Reasonable. A lower tariff viewpoint tessellation, in that its outline is not immediately recognisable as a dinosaur. Better would have been from the side. That being so, the artist nonetheless manages to create a respectable tessellation. The shading is poor, not following the map-colouring rule.

2. In the Garden, Page 37
Poor. I don’t like this. A themed tessellation, of three garden-based insects, with bugs, ladybirds and leaves. Although as a concept aesthetically satisfying, as might be imagined, this is of a lower quality, due to the self imposed themed restrictions. Indeed, one of the motifs is unidentifiable.

3. Horned Lizard and Gila Monster, Page 38
Poor. I don’t like this. As can be seen, both of the creatures have legs that are merged with the body, thus rendering their outline indistinct. A decidedly lower quality example. The shading is also poor, not following the map-colouring rule.

4. Ankylosaurus, Page 39
Reasonable. A passable Ankylosaurus, although the view shown is not the best, as this would be better portrayed from the side. Pleasing elements include the plate at the rear of the tail, echoing the real motif. However, this would be lost on most people, as whom, save for a dinosaur expert, would know this? (I myself only knew this for a picture search for these comments.)

5. Pteranodons, Page 39
Poor. I don’t like this. Strictly speaking, this is not a Pteranodon, in that creature has amongst its elements a longish neck and legs, where this has these elements more restricted. Also, it’s shown from the above rather than ideally from the side, where its ‘Pteranodon elements’ would be more readily visible. Another example where the elements merge with the body. Of lower quality. Also note the shading, where contiguous regions share the same shade. However, in this instance, this was probably undertaken to emphasise the glide reflection symmetry, and so has a purpose.

6. Squids, Rays, and Sea Turtles, Page 40
Reasonable. A themed tessellation, with a sea theme, with squids, rays, and sea turtles. A better tessellation, in that the outlines are more recognisable.

7. Firemen, Page 42
Reasonable. A lower tariff of difficulty type of motif, of heads. Of its type, reasonable.

8. Winged Dragon, Page 49. Also see page 56 (Wireframe)
Good. A good tessellation, the motif being readily recognised in silhouette, which as stated above, is the true test of quality, and is shown in its most typical representation, from the side. However, it is of a lower tariff of degree difficulty, being an ‘imaginary’ creature. Would benefit by shading/colouring.

9. Rays, Page 50, 115 (Wireframe)
Poor. I am not convinced by the portrayal of the rays here. As such, these seem highly contrived, of which after a picture search was unable to find any that adopt this position (although this is not to say that they might). Would benefit by shading/colouring.

10. Beetles, Moth, and Bumblebees, Page 51
Reasonable. A themed tessellation, with three insect motifs. As such, although the motifs are identifiable with the detail, in silhouette this is less so, as the outlines lack the necessary indentation.

11. Cats and Mice, Page 52
Poor. A themed tessellation, with two related motifs, cats and mice. Although at first glance a respectable tessellation, with both cats and mice readily seen, I have reservations here. The legs of both motifs merge with the body (as does the mouse’s tail), and so is not ideal. As such, I am dubious of this type – although of a theme, and so aesthetically pleasing as a concept, the trade-off is one of quality – are the motifs good enough here? A further drawback is the incongruous scales used, although I have less of an issue with this. Also, the colouring is a concern, which has adjacent areas the same colour, albeit again, of lesser concern than the quality per se.

12. Monkeys (and Bananas), Page 52
Poor. A themed tessellation, with two related motifs, of monkeys and bananas. Although at first glance a respectable tessellation, with both monkeys and bananas readily seen, I have reservations here. Three of the four elements of the monkey merge with the body. A further drawback is the incommensurate scale of the respective motifs, although I have less of an issue with this.

13. Frogs, Page 54
Reasonable. A better tessellation, in that the silhouette is instantly recognisable, with the elements (limbs) made distinct. Although the legs/feet are shown somewhat stylized, this is a necessity, and should not be thought of as a weakness in this case. Also note the shading, where contiguous regions share the same shade.

14. Cats, Page 54
Reasonable-Poor. A somewhat unusual portrayal, with the cat shown on its back. Of its type, quite good, although again with considerable surface decoration. Also note the shading, where contiguous regions share the same shade.

15. Rabbits, Page 64 (Wireframe)
Reasonable. Quite pleasing, although the motif is seen from the front, rather than the ideal side. A drawback is the ears, which merge. Would benefit by shading/colouring.

16. Chameleon, Page 64 (Wireframe)
Good. Pleasing, in that a recognisable motif is discernable, of the ideal representation, from the side, and furthermore with all its elements readily identifiable. Would benefit by shading/colouring.

17. Pteranodons, Page 65
Reasonable. Strictly speaking, this is not a Pteranodon, but rather a generic dinosaur, in that former creature has a longish neck and legs, where this has these elements considerably more restricted. Perhaps I’m being over fussy here as to title. Still, if a specific creature is given, one should expect a degree of resemblance. Also, it’s shown from the front, rather than ideally from the side, where its Pteranodon elements would be more readily visible. Certain aspects of this merge. Average. Also note the shading, where contiguous regions share the same shade.

18. Frogs (and Toads, Snakes), Page 66 (Wireframe)
Unacceptable. I don’t like this, albeit themed. The frog’s front and back legs merge with the body, whilst the snake is most contrived, with a pseudo coiled effect. The tadpoles are vague, essentially unrecognisable. Would benefit by shading/colouring.

19. Seahorse, Page 79 (also see page 63 and coloured version, back cover) (Wireframe)
Good. A very pleasing tessellation, of undoubted high quality. Attention has been paid to fine detail, and so thereby this aids recognition. Indeed, all the elements are readily visible, such as the elongated nose, fins, curly tail eye region, all of which echoes the subtleties of the real motif. Would benefit by shading/colouring.
Also see a six-fold version, page 106.

20. Seahorses and Eels, Page 80 (Wireframe)
Reasonable. A themed tessellation, with two related motifs. However, I am not convinced by the eel drawing. Is this truly eel-like? Of note is the quality of the seahorse, which is of a lesser quality than with single example tessellations of this motif on pages 79 and 106, this caused by the demands of the themed tessellation. Would benefit by shading/colouring.

21. Flowers 1, Page 88 (also see coloured version, back cover) (Wireframe)
Reasonable. Of its type, quite pleasing, however, the category itself, of an ‘amputation’ type, is of a decidedly lower tariff. Would benefit by shading/colouring.

22. Seahorses (Six-fold Seahorse), Page 106 (Wireframe)
Good. Very pleasing. As can be seen, this uses many of the same lines as of the four-fold example (page 79), and is also instructive in showing how an initial suitable line can be used again with different symmetries. Would benefit by shading/colouring.

23. Manta Rays, Page 115 (Wireframe)
Reasonable. As such, an exotic creature, of which most people will be unfamiliar with its exact shape. However, upon picture research, this looks broadly correct. Would benefit by shading/colouring.

24. Geese, Page 116 (Wireframe). Also see frontispiece
Good. Quite pleasing. Of note here is premise of ‘same orientation’ that in this instance I consider to the detriment of the tessellation. Note that this tessellation consists of a single tile, it appearing in two orientations. Consequently, if the same interior was employed, one motif would be upright, the other upside down. As has been previously stated, the author does not favour such ‘orientation discrepancies’, of which to overcome this uses a different interior for the ‘upside’ down tile. However, the query is to whether this justifiable, as I consider that bird to be noticeably weaker. Would benefit by shading/colouring.

25. Men in Coats, Page 118 (Wireframe)
Good. I like this. From the outline alone it is recognisable broadly as a human figure, despite the arms merging with the body. Every element is in proportion. This is arguably the best tessellation here. Would benefit by shading/colouring.

26. Largemouth Bass, Page 120 (Wireframe)
Reasonable. Quite pleasing, as by the outline alone, if not readily apparent, it can be said to be undoubtedly fish-like. Would benefit by shading/colouring.

27. Dragonflies (Bug Reflections), Page 124 (Wireframe)
Reasonable. However, examples of this type, with ‘imaginary’ bugs, are quite easy to do, and are of a lower tariff of difficulty. This is despite them being of three motifs, of which as detailed elsewhere in my essays, is generally difficult to accomplish, but in instances of this type, is relatively easy. Would benefit by shading/colouring.

Summary

As such, I am broadly positive about this book, albeit with some reservations. Pleasingly, the author has addressed some of the key issues underlying the creation of Escher-like motifs, such as the outline or silhouette test; this being a key aspect. This is something which is conspicuous by its absence in other people’s writings (I even include Escher in this), and so on that account the book is warmly welcomed. However, even here, other key issues have been omitted, such as why birds and fish are so suitable, as well as an assessment as to inherent worth of different motifs. (Some motifs are more ‘worthy’, or of higher tariffs than others.) Furthermore, there are other aspects I am less keen on. Prominent throughout is the insistence on ‘themed tessellations’, of which I consider detracts from issues of quality of motif. Numerous instances are shown where such themed examples lack inherent quality. Although as an ideal the aim is laudable, the tessellations of this type are much lower in quality, as might readily be imagined by such restrictive choices. Here, I believe that this factor has overridden matters of quality assessment, with many examples of at best questionable quality, and at worst unworthy of the art.
Although many different motifs are shown, of which again is a laudable aim, in the interests of variety, many of these fail the outline or silhouette test, the true test of quality. Again, I believe this desire to show many different motifs have overridden matters of quality assessment, with many examples unworthy of the art.
Nonetheless, there are indeed some of true worth here, such as the Men in Coats, Chameleons, and Seahorses.

Last Updated: 30 September 2009. Revised 10 July 2012, 13 September 2019

Minor revision of 13 September 2019

Life is a lifelong education! Upon recent correspondence with Alex Dixon, of the UK, a dinosaur tessellation expert, I had occasion to revisit this page, aware of Fathauer’s dinosaur interest. With my new interest, and at pains to be correct in nomenclature matters, I now see I committed the cardinal sin of incorrectly detailing the binomial nomenclature. Not only did I capitalise the species, but I also added a hyphen, with T-Rex! Likely, back in 2009, when I wrote this, I must have presumed this layman's use was correct. Possibly, this was misremembered from my youth, of the pop group ‘T. Rex’, albeit spelt without the hyphen (although still incorrect), popular in the early 1970s. This I now (2019) correct. And then further to this, by a remarkable coincidence, I found, in the same passage, a reference to a ‘Largemouth Bass’, also in Fathauer’s book! Previously, I thought the ‘largemouth’ part was his appendage to the title, to account for the large mouth, as a generic Bass with a large mouth, but there really is a ‘Largemouth Bass’! https://www.quora.com/Why-do-people-hyphenate-T-rex-even-though-T-rex-is-the-species-name-and-hyphenation-is-not-a-taxonomic-convention-for-species