The Tessellations of Jinny Beyer
Designing Tessellations. The Secrets of Interlocking Patterns
(No website of tessellations)
Jinny Beyer’s tessellations are of the opposite end to the spectrum of the other people detailed here. These are, in the strictest sense, essentially worthless (as I will demonstrate below). As such, these seem to be very much secondary to her main interest, in quilting. Normally, I would not deign to discuss or include such examples, but in this instance I have decided to include these as a single example of decidedly inferior tessellations of the type to avoid doing, by means of contrast with the superior examples of the other people as detailed.
As such, the inclusion of Beyer’s work is open to merit here, in that her book is mostly concerned with the creation of tessellations per se, rather than life-like tessellations. Therefore, one could query the appropriateness of her work for this discussion. Indeed, the creating Escher-like chapter is only a relatively small part of the book, a single chapter out of eleven. I could easily as have chosen other people. For example, Patrick Murphy, in Modern Mathematics Made Simple; and Chris De Cordova, in the Tessellations File Escher-like efforts leaves me speechless! Other, like poor examples can be found on the web, with many children’s efforts, all of which I could have chosen to comment upon. However, what arouses my ire and separates Beyer and the above is that her tessellations are shown in a sumptuously produced book, with the impression that her tessellations are given from a position of authority on the subject. Further ‘credence’ to this is giving by the author stating ‘Thank you … to Doris Schattsneider for [her] advice’ (hopefully this was for the other, non-Escher chapters). Aside from this, external factors, such as the books’ review on Amazon, where it is given 5-star reviews, thus further contributes to this impression. Therefore, on the face of it, this is a book of undoubted substance and worth. This being so, one has a right to expect the contents to live up to its production and billing, which sadly is not the case. The tessellations are quite frankly execrable, and show a complete lack of understanding of the issues. This sorry state of affairs simply cannot be allowed to remain unchallenged. Therefore, I thus select this book (or more precisely, the relevant chapter, Chapter 10 ‘Creating Representational Tessellations’) as the one source to discuss of as inferior standards in tessellation.
On only three of the nine determining aspects of ability and understanding of the issues does she score (and these are of the lesser important by far art aspects), with criteria as listed in the introduction:
(1) X The inherent quality of the motif (silhouette)
(2) X Showing the whole motif (excluding ‘heads’)
(3) X The number of tessellations in the body of her work
(4) X Variety of motifs
(5) X A tendency to the more difficult to achieve motifs
(6) ✓Coloured or shaded tessellations (excluding wireframes)
(7) ✓Map colouring of tessellations
(8) ✓Degree of finished rendering
(9) X Borderline
In more detail:
(1) Recognisable in Silhouette
The aspect of recognisably of the motif, as seen in silhouette is fundamental to the premise of a quality tessellation. Here Beyer shows little understanding of this issue, with many examples unrecognisable. I struggle to find a single example that I am entirely happy with. Of these, the only one of worth is Sport (a horse head), but this is of the lower category of difficulty of motif here, a head only. At a pinch, one could also include Tessellating Sue (a cat). Those aside, nothing else. This is simply not good enough, and so I fail her here.
Lesser artists, like Beyer, struggle with this concept (inexplicably so, given its simple premise), and fail to recognise its importance and unfortunately delude themselves as to equating interior life–like interior detail with a recognisable exterior outline.
(2) Shows the Complete Motif
Beyer’s tessellations are mostly of ‘complete’, whole body motifs, albeit some supposedly ‘whole’ motifs lack fundamental parts. These include birds without bodies, houses without floors, trees without trunks, and so one cannot truly say that this is a complete motif. A few incomplete head examples are also included, such as an eagle head, a horses head. Although the majority of hers are indeed of the whole body, in percentage terms the ratio to ‘whole body’ and ‘parts’ is too high, and so I thus fail her here.
Lesser artists, like Bayer, frequently do not understand the difference between the two types, and undertake such ‘head’ only examples (on account of their less challenging aspect), and unfortunately delude themselves as to equating these with the more challenging whole body motif. Contrast this with Bailey, Bilney, Crompton, Escher, and Nakamura.
(3) The Number of Tessellations
Beyer shows 20 examples, which is far below the arbitrary benchmark figure of 50 and above as regards the highest standards. Escher (137) and Nakamura (268) show considerably more. Given such a low number, one might expect that these would be of quality, on the basis of quality, not quantity. However, this is not so, as these do not show true worth, as detailed above. Such paucity is unacceptable, and so I thus fail her here.
Typically, lesser artists, like Beyer, will show a large number of inferior examples, and consider that such large numbers outweighs quality. Beyer, thankfully, does not at least present even more…
(4) Variety of Motifs
Beyer shows just 10 different motifs, which is noticeably below Bilney (39), Escher (32), Nakamura (39) and Crompton (25). Very few of the motifs are of interest as regards inherent interest, such as the ‘rarity’ aspect, of creatures not commonly seen (which is Bilney’s speciality). Such few numbers in variety are unacceptable, and so I thus fail her here.
Lesser artists, like Beyer, frequently shy away from undertaking variety, showing simpler to achieve birds and fish (I exclude her from this) to the exclusion of variety, and unfortunately delude themselves as to equating these with the more praiseworthy variety of motifs.
(5) Challenging Motifs – Human Figures
The more challenging and interesting human figure is ignored in its entirety. Aside from this, to perhaps try and ‘rectify’ this omission, with other examples of a challenging motif these are not shown (contrast this with Bilney). Therefore again, I fail her tessellations in this category.
Generally artists, as with Beyer, for other reasons than the desire not to repeat themselves, shy away from such challenging motif examples (on account of their degree of difficulty).
(6) Coloured or Shaded tessellations
All of Beyer’s tessellations are shown coloured (like Bailey, Bilney, Crompton, Escher, and Nakamura). No inferior wireframe examples are shown. Therefore, I pass her tessellations of this aspect. However, this should not be thought to be a case for undue celebration, as this point is not concerned with the all important design aspect, but rather of presentation, and is very simple to achieve, requiring no further skills beyond colouring in.
Lesser artists frequently show wireframe examples, for no good reason (from which one can only conclude is that they do not understand the issues).
(7) Map Colouring of Tessellations
Most of Beyer’s tessellations are shown with map colouring, the exceptions being Birds (page 217) and Endangered Species. Therefore, I pass her tessellations of this aspect. However, again, as above, this is not a cause of undue celebration, in that one simply uses contrasting colours, requiring no further skills beyond colouring in.
Lesser artists frequently disregard this aspect, for no good reason (from which one can only conclude is that they do not understand the issues). Perhaps surprisingly, of note is that other leading tessellators, such as Nakamura and Bilney also disregard this aspect on more occasions than I would like.
(8) Finished Rendering
Assessing the rendering is a little difficult, as it is not clear as to who exactly produced these. The book credits Kandy Petersen as the illustrator. Therefore, although it would appear strictly that Beyer did not do these per se, they were likely executed supplied from drawings under her watch. Therefore, below I refer to ‘her’ tessellations, in the sense of as by Beyer.
Beyer’s style, as regards finish, is of a consistent approach, with just the right amount of detail, with little if any variation of note. These are drawn and rendered by computer. Generally, she uses single ‘strong’ colours without any pretence of three-dimensional shading. As a rule, generally a simplistic finish is to be preferred (as here), as too much detail i.e. a photorealistic rendering hinders a clear interpretation of the motif. Therefore, I pass her tessellations of this aspect.
Lesser artists frequently disregard this aspect, with a rendering that is too real, too photorealistic, in that although well intentioned, such rendering is to the overall detriment of identifying the motif/s of the tessellation.
Beyer shuns the use of a decided borderline, with the outlines of the same thickness as of the interior detail. Normally, I would prefer a decided borderline in such circumstances However, the use of self-defining colouration, with ‘strong’ colours, as here, does not necessarily require a borderline. Even so, in such instances, I consider a borderline ideal, but not a necessity. Therefore, I give her a weak failure of this aspect.
However, the omission of a borderline cannot be said to be a fault as such, in that the inclusion or exclusion is down to personal choice, depending on the circumstances of the tessellation. Undoubtedly, this is secondary to the tessellation itself, and so of less importance to other, more fundamental issues, as detailed above.
Beyer’s tessellations have little or nothing to recommend them. Throughout, her tessellations have been examined and found to be wanting in the all important design aspects, criteria 1-5. Only in the lesser important art aspects, 6-8, is there a satisfactory outcome.
Unfortunately, I cannot say a single positive word as to her tessellations. Even her best tessellation, Sport, is merely of a lower difficulty head type.
Unfortunately, shortcomings, if not outright failings abound in her work. On all of the major design issues as detailed above (criteria 1-5), the tessellation falls woefully, if not embarrassingly, short.
Particularly poor are her ‘house’ tessellations, of which she shows no less than six examples, of the same premise, that bear not the slightest resemblance to a house! These show, by their quality and number, that the author does not understand the issues of quality Escher-like art. When these are tested, as coloured in as a silhouette, the resulting shapes are unrecognisable as houses. Just because the tile has the addition of a few house-like interior elements (such as a window) does not mean that the tessellation is worthy! (A common failing of understanding.) A cursory glance at the tessellations should be enough to see the shortcomings – what sort of house is it that has the door half way up the building? What sort of house is it that has a pronounced gouge at the bottom? What sort of a house is it shaped like a hexagon? No, no, no. These are simply unacceptable. And then, if this was not bad enough, to compound matters, she then shows another five! The impression given here is that Beyer thinks that these are good, and so shows further examples of the same kind! What is the point?
Another decidedly weak point is the ‘butterfly’ motifs. Although at first glance these may be thought to of true worth, as a apparently plausible butterfly is shown, in reality they are not so as they are anatomically incorrect. Simply stated, a real-life butterfly has wings that taper noticeably from top to bottom, rather than of ‘equal halves’ as here. Furthermore, this is unlikely to be of here own work, and is probably based on Escher’s own Butterfly (No.12), and so is not truly original. (Note that Escher’s own butterfly has anatomical inaccuracy). That said, as stylised butterflies, than a case can be made for these as regards ‘worthiness’ in the broader sense. However, against this is the relative ease of creating such motifs, of which these are decidedly simple to undertake. Simply stated, one or two examples of this kind is enough. However, Beyer again repeats herself, and shows no less than four further examples. Again, the impression is given that these are thus more worthy. What is the point? One could have, say, 10 or more, with ease, of the same kind of butterfly.
Further poor examples are Black Spruce (trees, page 210), a tree without a trunk, and Bird (page 217) a bird without a body or tail. The thinking here runs along the lines of ‘Oh, I can’t accommodate a trunk, I’ll just disregard it; Oh, I can’t accommodate a body and tail’, I’ll just disregard it’. Such an attitude is just not good enough. A tree without a trunk? A bird without a body and tail? No, these are just arbitrary shapes with minor tree and bird detail added. What is the point?
Perhaps at first glance Swans (page 213) may be thought good, in that a swan is indeed recognisable. However the swan detail is merely surface embellishment. Test this yourself – take a copy, and colour the tile as a silhouette, and the ‘swan’ promptly disappears, replaced by a tile that resembles no such motif. Again, people with lesser experience to tessellation will be taken in by apparent ‘quality’.
Perhaps the ‘multi-motif’ examples may be thought to be worthy? On the face of it these are indeed recognisable as distinct creatures. However, again, this is deceiving, as various shortcomings permeate these. Let’s look more closely at two examples: 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 (again, this is likely borrowed from Escher, see page 235). Essentially, the fish has two viewpoints. Pleasingly, at least the bird is acceptable, but one out of three is hardly worthy. 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 low standards. Again, the bird is acceptable. As an entity each time, this is just not good enough.
As detailed above, the exclusion of the more difficult to achieve human figures is somewhat to the detriment of the oeuvre here, as ideally she would show at least one. However, she is not alone in this, in that other respectable tessellators, such as Crompton and Bilney show very few. However, there is a world of difference in their other respective efforts!
No innovation of idea is shown, no specific ‘mathematical’ tilings, such as with Penrose tilings, or isohedral types or others (see Crompton). In her defence, very few artists concern themselves with this latter aspect, and so criticism here is perhaps unwarranted.
Beyer is a very poor tessellator indeed, and is unworthy of the status of the name. Throughout, as demonstrably shown, shortcomings, if not downright failings per se are very much to the fore, with the work essentially worthless. As detailed above, she does not understand the various issues underpinning the composing of inherent quality of tessellations. I struggle to praise a single one. How was this chapter allowed to be in the book?
Is she better than Escher? Definitely not! I’d put her not just slightly below Escher, but below nearly every one else in the field, schoolchildren included. The examinations below should show this:
Nearly all fail the ‘silhouette test’ (contrast with Escher)
Less number of tessellations than Escher (and furthermore of grossly inferior quality), 20 against 137 (albeit both are padded somewhat by unacceptable examples, Escher’s with Flatfish; Beyer’s with Houses)
Less variety of motifs than Escher, 10 against 32. (Furthermore, the 10 are barely discernable as recognisable creatures)
Less challenging motifs (i.e. human figures) than Escher, 0 against 4 (contrast with Nakamura’s 70)
Do you care to agree/disagree? Does anybody want to fight her corner? Email me.