Introduction to Tessellations

Introduction to Tessellations by Dale Seymour and Jill Britton. Dale Seymour Publications, 1989

This book is aimed primarily at a ‘layman’ in tessellations (as given in their preface), although perhaps more correct is that of school age level, of the 12– 14+ age range. As the title suggests, this primarily concerns the basic mathematical aspects of tessellations per se, such as the symmetry operations involved, and simple tessellation aspects, such as why all triangles and quadrilaterals tessellate. As such, for this age range, this is fine, and as an introduction to the subject serves admirably.

Another aspect of the book of more concern to readers of this site is creating Escher-like art, Chapter 7, ‘Creating Escher-like tessellations’, of which, in contrast to the above, I discuss in detail and depth. Pleasingly, this is at least of a substantial nature, of no less than 53 pages, and this in itself is to be welcomed (with other publications treating the matter decidedly more cursorily, often in a handful of pages, which is wholly inadequate). However, this is somewhat disappointing in its inherent quality, with some poor illustrative material, almost all lacking in quality. Broadly, the chapter discusses two aspects of Escher-like tessellation, somewhat confusingly intermingling, rather than as distinct aspects. Firstly, this discusses (or purports to) the 'creation' of 13 of Escher's periodic drawings. Secondly, the creation of original tessellations, with numerous examples from a variety of nine different students with an artistic/mathematical bent. However, although in theory this should make for an interesting discussion as to the procedures, in practice this is lacking in many ways, and is strictly incorrect. Below, I discuss these two aspects:

1. Escher's Periodic Drawings
The all-important attempts at unravelling Escher's methods fall woefully short. Seymour and Britton purport to illustrate this with 13 examples of Escher's periodic tessellations (see table below for details). For each periodic drawing, this is then analysed, with a line-for-line recreation. The impression is that Escher spontaneously and simply chose an appropriate ‘squiggly line’ that was immediately and without further refinement proved suitable for an animal-like motif. (Indeed, such a method is the premise of the whole chapter, with other artists' examples shown following the same procedure.) Quite simply, this is not how Escher went about such matters of designing his motifs, the method given being far too advanced, even for Escher's powers of imagination. The imagination required to draw at least two distinct lines that are 'perfect' i.e. of life-like form, when assembled of a given geometric tessellation is simply not possible. Granted, on page 188, Britton states
…modify the sides of the polygon until its contour resembles that object…
but not a single diagram shows this! All the diagrams are of a finished line. Curiously, it is little-known how Escher went about this. More correctly is Escher started with a geometric shape that tessellates, and then by trial and error developed and refined this, until finally arriving at a finished motif. Visions of Symmetry, page 111, shows how the tessellation of ‘Horseman’ developed. The procedure of Britton’s interpretation at prima facie is simply incorrect.

2. Students’ Tessellations
A curious occurrence with these students’ efforts is the sheer number of the lower category of difficulty of ‘head’ type (see my Essay 6, Categories of Difficulty). Indeed, no less than 22 out of the 32 tessellations are of this type, a massive 68% of the output, with no concession made to this lower quality category in the writing. A moment’s consideration will show that these are much simpler to achieve than a whole-body motif. Compare the complexity of a whole body motif, with say, a quadruped, with head, neck, body, legs, and tail, with just a single head. Largely here, this difficulty is swept away, with just the head shown. Furthermore, even within this basic category, there are many undeserving examples here, these mostly being embellishments. No other tessellation artists, even those who lean towards this category, come anywhere near this total in percentage terms. Compare this to the top tessellators, Bailey, Bilney, Nakamura and Scalfittura, who do not show any. Other top tessellators, such as Crompton, Escher, and Nicolas, show either one or two, and for specific, recognisable motifs, such as Sherlock Holmes, rather than these ‘arbitraries’ here. As a genre, we ignore these, with the implication that these are unworthy of the art. Perhaps one could argue that in a book of this nature, aimed at the layman, one is cavilling here. However, it would surely not be asking too much to qualify these with just a few lines that, as a category, they are decidedly lesser in comparison to a whole-bodied motif, but no such qualification occurs.

Another concern is with the inherent quality of the tessellations. Most of the examples can be described derogatorily as embellishments, or ‘shapes with eyes’, with all that the artist has done is to add animal-like elements to an outline that does not bear any resemblance to the creature it is portraying. Indeed, I classify a substantial number of these, no less than 20 out of 32, 62%. To give an actual example, Dog Head with Bow Tie, page 196, by Lyda Kobylansky. At first glance, does the tile outline give the impression of a dog’s head? No. It is just an arbitrary shape, with the addition of dog-like detail, i.e. embellishment. (Many others are of the same ilk, see table.) There is a world of difference in quality and difficulty between a tile outline that resembles a dog and a tile with dog embellishment! Ironically, on page 234, Britton gives an illuminating quote from personal correspondence by George Escher (Escher’s son) on this matter:
Do not confuse the creation of a meaningful contour with highlighting of the interior tile. These are fundamentally different things. Almost anyone can take a random shape and draw something lifelike inside its outline. But it is entirely a different story to push a recalcitrant outline into a pattern that suggest, without highlighting, some living thing. Highlighting may be necessary to clarify a decision, is it a bird or fish? But it is often not even necessary, if the contour is characteristic enough. This discussion is not new…
Excellent advice! Why didn’t these artists listen? Again, perhaps I am being unfair here, in that it is difficult to emulate Escher in quality. However, that should not mean that one should not at least try! Even if one can only do inferior categories and examples, these should not be passed of as ‘equal’ to Escher’s in worth, which is the general impression given here.

Another feature of the tessellations is the lack of colour or shading to aid in distinguishing the motifs. Indeed, not a single tessellation is of this type; all these are the inferior wireframe type. Again, the reason for this is not given. The impression thus given is as a type i.e. wireframe, these are acceptable. Unfortunately, this is not so, as by their very nature, the drawing is rendered inconvenient to view, in that the motifs are not readily discerned. This aspect is all the more important when lower-quality tessellations are shown, as here. A typical example is ‘Cartoon Head’ by Steve Dawson. Can you determine what this is at first, or indeed second, glance? No. The eye has to ‘struggle’ as it were. As a courtesy to the viewer, it should be discernible immediately. Simple contrasting colours or shading would greatly assist here, even for lower-quality examples. In connection with this, better would have been better to emphasise the outside line of the motif with a thicker line than the interior. All the diagrams are of lines of unit thickness, which, again, with issues of discernment. As guidance for best policy, compare these tessellations with Escher’s own. Did he do any wireframe types? No. There is a reason. However, in the artist’s defence, this may have been occasioned by the editors of the book requiring a simple format suitable for printing in a largely uniform manner, these being taken from a previous competition, of a standardised format, of the above wireframe type. Even so, this should not negate the inherent quality aspect afforded by such types. These should at least be qualified, but again, not so.

A slightly lesser concern is that none of these are titled, or catalogued, which makes referring to them inconvenient. Consequently, for descriptive purposes, I give these my own titles. Although it may be thought unnecessary, when lower quality examples are shown, as here, titling is even more necessary. For example, how best to describe the lower example on page 188 by Steve Dawson? Pig snout fish with wings? Seahorse with wings? It might even be something else.

The main shortcoming of the tessellations shown is their inherent lack of quality, with the various artists lacking any insight in Escher-like art. Furthermore, the presentation is lacking, with the diagrams mostly of lines of unit thickness, rendering the motifs difficult to view with ease, forcing the viewer to scrutinise these as to ‘what’s what’. These would have been improved by the simple process of emphasis in their outline. However, this may have been occasioned by the editors of the book, a detailed above. Even so, this should not negate the inherent quality aspect.

Critiques of Tessellations
Having assessed the chapter in a broad-brush manner, below I give a general discussion of each individual tessellation. For the sake of succinctness, I summarise in a few stock phrases:

§ A shape with …-like elements added.
This is intended to be of a derogatory nature, applied to a tessellation of the lowest possible quality. All the artist has done is to add life-like interior detail to a shape bearing not the slightest resemblance to the motif it is portraying.

§ …but only due to the talents of the artist in extracting something recognisable from an arbitrary shape
This, in essence, follows on from the ‘a shape with ...-like elements added’ in that the artist has been able to extract something recognisable from poor raw material

§ Somewhat far-fetched…
Applied derogatorily to a tessellation where a ridiculous, contrived creature is formed of different contrary elements, such as a woman’s head with a pig’s snout.

§ Whimsy
Applied to a tessellation of ‘disjoint’ aspects, with elements taken from different creatures, which although possessing shortcomings, has some limited degree of merit, in that it is at least broadly ‘believable’.

§ What can I say…
Applied to tessellations that are so bad that words should not be necessary, in that the shortcomings are so obvious as to not need commenting upon.

1. Horse Head. Anonymous. Page 137
Poor. A shape with horse-like elements added. Cartoon-like, with conflicting perspectives, with the head seen in profile, with the eye straight on. The underside of the head has a strange gouge. Furthermore, a horse’s head in profile is not like this. Note the difficulty in identifying the motif here.

2. Witches Head, by Steve Dawson. Page 188
Poor. A shape with witch-like elements added, but only due to the talents of the artist in extracting something recognisable from an arbitrary shape. Loosely, yes, as titled, but of the easiest tessellation type (head).

3. Winged Seahorse by Steve Dawson. Page 188
Poor. A seahorse with wings? Possibly, but only as a fantasy. Somewhat far-fetched…

4. Dog Head, by Tracy Steszyn. Page 189
Poor. A shape with cartoon-like, dog-like elements added.

5. Woman’s Head, by Christina Jams. Page 189
Unacceptable. A shape with woman-like head elements added. The body is formless…

6. ‘Mutant Beast with Open Mouth’ Head, by Steve Dawson. Page 190
Unacceptable. A shape with ‘mutant beast-like’ elements added. Slightly like a ‘mutant beast with open mouth’, but still broadly a shape with beast-like elements added.

7. Mutant Elephant Head With Human-like Features, by Steve Dawson. Page 190
Unacceptable. A shape with mutant elephant-like elements added. Various uncertainties arise - does this possess horns, or is this horn-like appearance actually part of the neck? An emphasised outline would negate this ambiguity. Has a human-like mouth. Somewhat far-fetched…

8. ‘Hockey Players of Unidentifiable Creature’, by Steve Dawson. Page 191
Poor. The creature is unrecognisable, loosely described as of a ‘mouse-like head, wearing a floppy hat and scarf, with a human body holding a hockey stick’. Assuming that this is as I presume, a most unlikely combination, with no unifying features – just why should a mouse-like head, complete with a floppy hat and scarf, with a human body holding a hockey stick, be… However, as a ‘whimsy’, this is quite acceptable and is one of the ‘better’ tessellations here, as anatomically, despite the unlikely constituents, there are no obvious inconsistencies – everything is in proportion. Somewhat far-fetched…

9. Dog Head. Anonymous. Page 194
Poor. A shape with dog-like elements added. A lack of dog anatomy is evident here – the mouth is portrayed incorrectly, and the left ear is somewhat human-like. However, this matter aside, the detail does indeed resemble a dog’s head. However, this is only due to the talents of the artist in extracting something recognisable from an arbitrary shape.

10. Eagle Head, by Lyda Kobylansky. Page 195
Poor. A shape with eagle-like elements added.

11. Horse Head with Antlers, by Henry Furmanowicz. Page 195
Poor. A reasonable attempt at a horse head, as it does indeed bear a likeness, albeit let down by strange antler-like protrusions above the head.

12. Dog Head with Bow Tie, by Lyda Kobylansky. Page 196
Poor. A shape with a cartoon dog-head-like elements added. Why should a dog be wearing a bow tie…? Loosely identifiable as a dog head. However, this is only due to the talents of the artist in extracting something recognisable from an arbitrary shape.

13. Pirate Head, by Nick Zannella. Page 196
Poor. A shape with pirate-like elements added. This example is worth more praise than others are, as by the skill of the artist, a reasonable pirate emerges from unpromising raw material.

14. Cartoon Head, by Steve Dawson. Page 197
Poor. A shape with human-like cartoon elements added. However, this is only due to the talents of the artist in extracting something recognisable from an arbitrary shape.

15. Bird, by Henry Furmanowicz. Page 197
Reasonable. In contrast to most of the examples, one of the better ones here, with an outline that does indeed truly resemble a recognisable creature, a bird. A pleasing aspect to detail is that the wings are, or at least implied, anatomically correct, with serrations to the rear, as is the tail. This, though, may just be accidental… However, it's let down somewhat as being over-detailed, as a bird is not evident at first glance, being lost in a myriad of detail.

16. ‘Rabbit Head’ and ‘Creature with Bowtie Head’, by Steve Dawson. Page 200
Reasonable. A shape with head-like elements added. A relatively pleasing innovation is of two different interiors, but this is militated against in that the tile is really just an arbitrary shape. Both shapes with head-like elements of a cartoon-like nature added. Although of a weak category, as a ‘whimsy' the rabbit-like creature is pleasing.

17. Bird, by Jill Britton. Page 204
Reasonable. Loosely bird-like, especially so concerning the head.

18. Moose Head, by Steve Dawson. Page 205
Poor. A shape with cartoon moose-like elements added. However, this is only due to the talents of the artist in extracting something recognisable from an arbitrary shape. As a ‘whimsy’, reasonable.

19. Rogue in a Plumed Hat Head, by Steve Dawson. Page 205
Unacceptable. A shape with a few rogue elements added. However, this is only due to the talents of the artist in extracting something recognisable from an arbitrary shape. As a whimsy, reasonable.

20. ‘Pirate’, by Jill Britton. Page 208
Poor. A shape with pirate-like elements added. A lack of a defining outline renders this most inconvenient.

21. Bird, by Steve Dawson. Page 209
Reasonable. Loosely bird-like. The detail on the wings displays a lack of anatomical knowledge.

22. Dog Head –‘Tisha’, by Stephen Makris. Page 212
Poor. A shape with cartoon-like, dog-like elements added. However, this is only due to the talents of the artist in extracting something recognisable from an arbitrary shape. As a whimsy, reasonable.

23. Duck, Bird, Mutant Horse, Goblin, Dragon, and Cuckoo Heads, by Steve Dawson. Page 213
Unclassified. Firstly, as this is in two parts, with single motifs and a tessellation, I discuss them separately.

Single motifs ‘Possibilities’:
A shape with head-like elements of different creatures added, of five heads, and one whole bird creature. Mostly dreadful. The one example of relative worth is ‘duck’, which as a whimsy, is reasonable. All Dawson has done here is to add multiple head details to an arbitrary shape. The impression given here is one of inventiveness, as a single tile has multiple motifs. However, many other heads can be added in a like manner, rendering the number meaningless.

Tessellation:
A shape with dragon-like elements added. The skill of the artist results in a reasonable dragon head.

24. Hummingbird, by Jill Britton. Page 216
Poor. A shape with a hummingbird-like beak element added. At least the beak is portrayed accurately. This has similarities with one of Peter Stevens’s birds, in a different book, in which he is listed in the references.

25. Clown Head, by Jill Britton. Page 219
Reasonable. A shape with cartoon-like, clown-like elements added. However, this is only due to the talents of the artist in extracting something recognisable from an arbitrary shape. Of its type, of a lower tariff of difficulty, quite pleasing, with no obvious shortcoming per se.

26. Devil Head, by Steve Dawson. Page 222
Reasonable. A shape with cartoon-like devil-like elements added. Again, of its type, of a lower tariff of difficulty, quite pleasing, with no obvious shortcoming per se, albeit it is not clear as to outline, as the beard and hair are vague.

27. Fish, by Jill Britton. Page 225
Reasonable. Fish-like in outline, of a reasonable quality fish, albeit the upper fins are not streamlined.

28. Dogs, by Jill Britton. Page 229
Reasonable. A dog lying down. This has a likeness to the St Bernard tessellation of Joseph Teeters. A reasonable tessellation if indeed of originality, less so if ‘after Teeters’. One of the better ones, if indeed it is original.

29. Owl, by Steve Dawson. Page 230
Poor. A shape with owl-like elements added. However, this is only due to the talents of the artist in extracting something recognisable from an arbitrary shape The one redeeming feature of this is the head, with tufted ears, as in a real-life owl.

30. Sea Lion, by Steve Dawson. Page 230
Poor. A shape with sea lion-like elements added. However, this is only due to the talents of the artist in extracting something recognisable from an arbitrary shape

31. Birds, by Jill Britton. Page 233
Reasonable. At first glance, reminiscent of Escher’s No 44, but is indeed distinct, with different symmetries.

32. Woman’s Head with Pig-like Nose, by Sheila Le Blanc. Page 233
Unacceptable. A shape with a cartoon-like pig-like snout and human elements added. Somewhat far-fetched. What can I say… 

Summary
The main shortcoming of the Escher-like tessellations is their inherent lack of quality, with the various artists lacking any insight in Escher-like art. The majority are essentially worthless. Very few are of any note, and even when recognisable, they still lack inherent quality. Predominant are the inferior category ‘head’ types, with no concession made to this easier category. Worse, the various artists still struggle with achieving believable ‘heads’. Furthermore, the presentation is lacking, without colour or shading, with the diagrams, wireframes, of lines of unit thickness, rendering the motifs difficult to view with ease, forcing the viewer to scrutinise these as to ‘what’s what’. These would have been improved by the simple process of emphasis in their outline. However, as discussed above this may not have been dictated by the artists themselves.

A lack of critiquing is evident throughout (though that said, nearly all artists do not critique their tessellations. Why not?). This is regrettable, in that the impression given is that these are the ideal to aim for, and so inferior ones are perpetuated by the reader with the belief that these are ‘good’.

Perhaps one could argue that due to the youth and inexperience of the artists one could, and indeed should expect inferior standards. Well yes, but why show these? Al this is all the more surprising given the quote near the end of the book, page 234:

Do not confuse the creation of a meaningful contour with highlighting of the interior tile.

Where all the evidence is that the artists have!

12 June 2025. Updated text from Classic Sites. Much to my embarrassment, the text was liberally sprinkled throughout with poor grammar, poor English, spelling errors and typos. All this must have been per Grammarly. Quite frankly, I am shocked!

Created: 26 September 2009. Updated, revised and enlarged: 30 August 2010, 25, 29 June 2012