Introduction to Tessellations by Dale Seymour and Jill Britton
This book is aimed primarily at a school age level, of the 12-14+ age range. As the title suggests, this primarily concerns the mathematical aspect of tessellations per se, rather than that of dedicated Escher-like tessellations, albeit Chapter 7 is indeed devoted to 'Creating Escher-like Tessellations'. However, this is somewhat disappointing in its inherent quality. As such, this chapter purportedly discusses both the 'creation' of some of Escher's periodic drawings, along with examples from other people, these being somewhat confusingly interspersed, rather than as distinct aspects. The all-important attempts at unravelling Escher's method are incorrect. Given are 13 examples of Escher's periodic tessellations (with numbered drawings in brackets), pages 187 (25, Reptiles), 193 (90, Fish), 199 (75, Lizard), 202 (104, Lizard), 207 (25, Reptile), 211 (44, Birds), 215 (35, Lizards), 218 (4, Chinaman), 221 (117, Crabs), 224 (97, Dogs), 226 (63, Pessimist/Optimist), 228 (96, Swans) and 231 (67, Horseman), purporting to 'show' how Escher did this. However, as discussed in the Essay *, it the more correct would be to say that these diagrams show the 'modified line' in its perfected state; with the impression being given that Escher simply chose an squiggly line which immediately and without further refinement proved suitable. Indeed, such a method is the premise of the whole chapter, with other artist's examples shown following the same technique. Quite simply, this is not how Escher went about such matters, 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. (Instead, Visions of Symmetry, page 111 shows that he refined a given outline.)
Critiques of Tessellations
1. Head – Horse. Anonymous. Page 137
A shape with eyes. Cartoon-like, with conflicting perspectives, with the head seen in profile, with the eye straight on. Furthermore, a horses head in profile is not like this. Note the difficulty in identifying the motif here.
2. Head – Witch, by Steve Dawson. Page 188
A shape with witch-like elements added. Loosely, yes, but of the easiest tessellation type (head).
3. Fish with wings and an elephant-like trunk, by Steve Dawson. Page 188
A shape with fish, wing, elephant-like elements added. Somewhat far-fetched…
4. Head - Dog, by Tracy Steszyn. Page 189
A shape with cartoon like dog-like elements added.
5. Head – Woman, by Christina Jams. Page 189
A shape with woman-like head elements added. The body is formless…
6. Head – ‘Mutant Beast with open mouth’, by Steve Dawson. Page 190
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. Head – Mutant Elephant Head with human like features, by Steve Dawson. Page 190
A shape with mutant elephant-like elements added. Various uncertainties arise - does these posses horns or is this horn like appearance actually part of the neck? An emphasised outline would negate this ambiguity.
8. ‘Hockey Players of unidentifiable creature’, by Steve Dawson. Page 191
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.
9. Head – Dog. Anonymous. Page 194
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 vague. However, this matter aside, the detail does indeed resemble a dogs head.
10. Head – Eagle, by Lyda Kobylansky. Page 195
A shape with eagle-like elements added.
11. Head – Horse, by Henry Furmanowicz. Page 195
A reasonable attempt at a horse head, as it does indeed bare a likeness, albeit let down by antler-like protrusions above the head.
12. Head - Cartoon Dog with bow tie, by Lyda Kobylansky. Page 196
A shape with dog-head-like elements added. Why should a dog be wearing a bow tie…?
13. Head – Pirate, by Nick Zannella. Page 196
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. Head – Cartoon, by Steve Dawson. Page 197
A shape with human-like cartoon elements added.
15. Bird, by Henry Furmanowicz. Page 197
In contrast to most of the examples, 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…
16. Head - Rabbit Head and ‘Bowtie creature’, by Steve Dawson. Page 200
A shape with head-like elements added. 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. Anonymous. Page 204
Loosely bird-like, especially so concerning the head.
18. Head – ‘Cartoon Moose’, by Steve Dawson. Page 205
A shape with moose-like elements added, As a whimsy, reasonable.
19. Head – Spiv, by Steve Dawson. Page 205
A shape with spiv-like elements added. As a whimsy, reasonable.
20. ‘Fighter’. Anonymous. Page 208
A shape with fight-like elements added. A lack of defining outline renders this most inconvenient.
21. Bird, by Steve Dawson. Page 209
Loosely bird-like. The detail on the wings displays a lack of anatomical knowledge.
22. Head – Dog? ‘Tisha’, by Stephen Makris. Page 212
A shape with cartoon like dog-like elements added. As a whimsy, reasonable.
23. Heads - Duck, Bird, Mutant Horse, Goblin, Dragon, and Cuckoo, by Steve Dawson. Page 213
Firstly, as this is in two parts, with single motifs and a tessellation, I discuss separately.
Single motifs:
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, 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, Anonymous. Page 216
A shape with bird-like elements added. At least the beak is portrayed accurately...
25. Head – Clown, by Jill Britton. Page 219
A shape with cartoon-like clown-like elements added. Of its type, quite pleasing, with no obvious shortcoming per se.
26. Head – Devil, by Steve Dawson. Page 222
A shape with cartoon-like devil-like elements added. Again, of its type, 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 Anonymous. Page 225
Fish-like in outline, of a reasonable fish, albeit the upper fins are not streamlined.
28. Dogs by Anonymous. Page 229
This has likeness to the St Bernard tessellation of Joseph Teeters. A reasonable tessellation of a sitting dog.
29. Owl, by Steve Dawson. Page 230
A shape with owl-like elements added. 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
A shape with sea lion-like elements added.
30. Birds. Anonymous. Page 233
Reasonably good.
31. Heads – ‘Pig-like nose and ear with human-like features of eyes and mouth’, by Sheila Le Blanc. Page 233
A shape with cartoon-like pig-like snout and human elements added. What can I say…
Summary
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 requiring a simple format suitable for printing in a largely uniform manner. Even so, this should not negate the inherent quality aspect.
To the fore are examples of the lowest type of quality, a type classed as ‘heads’, with a total of 20 of the 32 tessellations shown, in which as I have discussed elsewhere (Essay 6, Categories of Difficulty), is the easiest to compose, of which no real skill is required. Even within this basic category, there are many undeserving examples here.
Most of the examples can be described derogatorily as ‘shapes with eyes’, with all that the artist has done is to add animal-like elements to a outline that does not bear any resemblance to the creature it is portraying. Ironically, on page 234 Britton give 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 people listen….