Designing Tessellations

Designing Tessellations The Secrets of Interlocking Patterns by Jinny Beyer

This book, written by a quilt practitioner without a mathematical background (not that there is anything wrong in that), is somewhat awkward to assess. Broadly, it can be divided into two discrete areas, of ‘mathematical tiling’ (of non-Escher-like motifs), Chapters 1-9, and a relatively brief creating Escher-like art section, Chapters 10 and 11. Perhaps surprisingly, given the author’s non-mathematical background, the ‘mathematical tiling’ aspect 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 quality Escher-like art, with some truly dreadful examples, frequently unacceptable, and although there is the occasional respectable tessellation, the others are best described as ‘how not to’.

Chapter 10 Creating Representational Tessellations

1. House, and two variations thereof, 10.4 page 206 (also see page 222)
Unacceptable. Firstly, the following comments are applicable to all three examples (and a fourth instance on page 222). All these are is a shape with house-like elements added. These are particularly unacceptable, and they show by their inclusion that Beyer does not understand the principles of creating (good) Escher-like art. Just because a largely arbitrary tile outline here 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 misunderstanding). A cursory glance at the ‘houses’ here should be enough to see the various shortcomings. What sort of house is it that has the door half way up the building (actually its missing, although implied…)? What sort of house is it that has a pronounced ‘gouge’ at the bottom? What sort of house is it that has the side missing? How could these houses possible stand? No, no, no. These are simply unacceptable. Admittedly, one might argue that there are indeed house-like identifiable elements here, as indeed there are, with a chimney, and so such critiques are out of place. However, all that has been accomplished here is a ‘pulling’ of a chimney outline from the underlying polygon. The skill and imagination required to do this is in effect zero; anyone could do this. If this is what tessellation art is supposed to be about then I give up.

2. Enchanted Forest, page 207
Unacceptable. Tree-like in a simple way, with branches, but without the trees having a trunk…  Again, no understanding is shown; how can a ‘tree’ be portrayed without a trunk?

3. Tessellating Sue, page 208
Reasonable. 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
Reasonable. Of its type, reasonable, and indeed cannot be faulted – the hat and head are in perfect proportion. However, it is of a lower tariff of difficulty.

5. Black Spruce, page 210
Unacceptable. Presumably a tree is intended, but again is without a trunk. Again, the same failing repeating as above… See comments above, 2.

6. Sport (Horse Head and Neck), page 211
Reasonable. This does at least resemble a horse’s head and neck. Would benefit from the addition of interior detail, such as an eye and mane. However, it is of a lower tariff of difficulty.

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

8. Swan, page 213
Poor. 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 decoration details that simply flatters to deceive in terms of anatomical correctness.

9. Butterflies, page 214
Reasonable. Broadly identifiable as a butterfly, albeit with reservation. 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, from top to bottom, 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
Unacceptable. 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
Poor. A shape with butterfly-like elements added. A butterfly? Quite simply, the silhouette is nothing like a butterfly.

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

13. Bird, page 217
Unacceptable. 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)
Poor. This combines the motifs of Figures 10.10 and 10.12. See above for comments.

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

16. Houses, page 222
Unacceptable. A shape with house-like elements added. Houses? Two different ‘houses’, without a floor and side, 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 time and again in another incarnation…

17. Endangered Species (Swan and Eagle Head), page 223
Poor. 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
Poor. 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
Poor. 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 good, but overall let down by the shortcomings of the others.

Chapter 11 Metamorphosis

House Metamorphosis, page 234
Unacceptable. Although a metamorphosis is given and explained, this is in actuality the wrong way it should be undertaken, as Beyer begins with squares and ends of a ‘house’. How can one ‘develop’ something without an target goal? 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. Why include this?

Cat Metamorphosis, page 235
Unacceptable. 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…?


Well, a very 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 an exemplary series of diagrams as how not to do tessellation.


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Last updated: 29 March 2010. Revised 25 June 2012