Designing Tessellations The Secrets of Interlocking Patterns by Jinny Beyer

This book, written by a quilt practitioner without a mathematical background, is somewhat awkward to assess. Broadly, it can be divided into two discrete areas, of ‘mathematical tiling’ (non-Escher-like motifs) and Escher-like art. Perhaps surprisingly, given the author’s non-mathematical background, the mathematical tiling is reasonably strong, giving some interesting ways of composing tessellating tiles. Indeed, a measure of its quality is that I fully intend to study this is depth at some date. However, in contrast, the ‘Escher art’ sections, Chapter 10, Creating Representational Tessellations and Chapter 11, Metamorphosis, where Beyer shows her own Escher-like tessellations (occasionally with other people included), of which we may expect some ‘artistic insight’, are particularly weak, to put it mildly. These show a lack of understanding of the issues of creating, with some truly dreadful examples, and although there is the occasional respectable tessellation, the others are best described as how not to.

Critiques

Chapter 10 Creating Representational Tessellations

1. House, and two variations thereof, page 206
Firstly, the following comments are applicable to all three examples. A shape with house-like elements added. These are particularly poor, and they show by their inclusion that Beyer does not understand Escher art. Just because a tile has the addition of a few house-like elements to the interior (such as a window) does not mean that the tessellation is in any way worthy (a common failing of understanding). A cursory glance at the diagrams should be enough to see the shortcomings. What sort of house is it that has the door half way up the building (actually its missing, though implied…)? What sort of house is it that has a pronounced gouge at the bottom? How could these houses possible stand? No, no, no. These are simply unacceptable.

2. Enchanted Forest, page 207.
Tree-like in a simple way, but without the trees having a trunk…

3. Tessellating Sue, page 208
Although not shown as a tessellation as such, Beyer includes some possible interiors for the tile in diagrammatic form, namely of a cat and fish. Both are quite reasonable, broadly identifiable.

4. Head - Girl with Bonnet, by Carole Nicholas, page 209
Of its type, reasonable, and indeed cannot be faulted – the hat and head are in perfect proportion.

5. Black Spruce, page 210
Presumably a tree is intended, but this is without a trunk…

6. Sport (Horse Head), page 211
A reasonable tessellation, as it does at least resemble a horses head. Would benefit from the addition of interior detail.

7. Opposites Attract (Bird Heads), by Danielle Brower-Noble, page 212
A shape with bird-like head elements added...

8. Swan, page 213
A shape with swan-like elements added. Again, when in silhouette, this is unrecognisable as a swan. Although titled as a ‘swan’, in reality there is nothing swan-like about this. The neck and head in the real animal are held separate, with the head in a noticeably higher position. In contrast, in Beyer’s example the head, neck and body merge. Again, this is essentially a tile with the addition of a few swan-like details that simply flatters to deceive in terms of anatomical correctness.

9. Butterflies, page 214
A reasonable tessellation. Although at first glance these may be thought to be of true worth, as once again a plausible butterfly outline is shown, in reality this is not so and is anatomically incorrect. Simply stated, a real life butterfly has wings that taper, rather than of ‘equal halves’ as here. Furthermore, this is probably not all Beyer’s own work, as it would appear to be based on Escher’s own Butterfly (No.12), and so is not truly original. (Note that Escher’s own butterfly has anatomical inaccuracy).

10. Crab, page 215
A shape with crab-like elements added. A crab? Again, an essentially arbitrarily outline with a few crab-like features added. Unworthy.

11. Butterflies, page 216
A shape with butterfly-like elements added. A butterfly? Quite simply, the silhouette is nothing like a butterfly.

12. Frogs, page 216
A shape with frog-like elements added. Frogs? Not really, but better than some of Beyer’s efforts…

13. Bird, page 217
A shape with bird-like elements added. Oh dear. What sort of bird is it that has no body and tail? Worthless.

14. Two Interlocking Butterfly motifs, page 218 (top)
This combines the motifs of Figures 10.10 and 10.12. See above for comments.

15. Three Butterflies, page 218 (bottom)
This combines the motifs of Figures 10.10 and 10.12, and has a further addition, of arguably the most unbutterfly example of the lot…

16. Houses, page 222
A shape with house-like elements added. Houses? Two different ‘houses’, of a similar obscure nature to those on page 206, to which the same negative comments apply. What can one say when the same inferior type is repeated in another incarnation…

17. Endangered Species (Swan and Eagle Head), page 223
A shape with swan and eagle-like elements added, by subdividing a tile. Again, the swan is similar in nature to that of page 213, with the same shortcomings. The eagle head is essentially unrecognisable…

18. Birds and Fish, page 226
A somewhat odd, incongruous composition, of three motifs, of two distinct fish and one bird that is obviously unsatisfactory in terms of its aesthetics. Additionally, the fish are anatomically incorrect, as although the fish is seen from above, the tail is at right angles, as if seen from the side. Essentially, the fish has two viewpoints. Pleasingly, at least the bird is acceptable.

19. Butterflies, Fish, and Birds, page 227
Again, the same shortcomings of the fish above arise. The butterfly is nothing like a butterfly, even by Beyer’s standards….Again, the bird is acceptable.

Chapter 11 Metamorphosis

House Metamorphosis, page 234
Although a metamorphosis is given and explained, this is in actuality the wrong way it should be undertaken, as Beyer begins with squares with the ending of a house. How can one ‘develop’ something without an initial source? See my ‘How Escher Did…’ section for the best process i.e. a deform (to give the impression of a development), rather than a creation. As detailed above, the house is worthless.

Cat Metamorphosis, page 235

Where to begin…? A metamorphosis of sorts, with an uneven tempo, unfortunately. (Four identical fully formed ‘cats’ are preceded by two tiles, and then one, and another one….)
A cat? Where…? No. A shape with cat head-like elements is all that is here. This is just a shape with the occasional cat-like feature, noticeably the pointed ears, and that’s just about it…What can one say…?

Summary

Well, a sorry state of affairs indeed. Just about every tessellation has shortcomings, with most of them unworthy of being in print. The motifs are generally of poor quality, and most would fail the ‘silhouette test’ of inherent quality. There is absolutely no evidence whatsoever that Beyer has any insight to Escher-like representational tessellation matters, in either their composing or understanding the underlying issues, and even worse, the impression given is that Beyer believes that these are ‘good’. … All in all, this can only be described as a exemplary series of diagrams as how not to do tessellation.