As can readily be seen, tessellation art varies a great deal in its quality, from quite frankly execrable efforts of no worth whatsoever (of which it regrettably has to be said is frequently encountered), to examples what can be described as superlative, a highlight of the art, and, as to be expected, which are less commonly to be found, and of which this page is concerned with.
Indeed, the number of ‘superlatives’ is decidedly small, and can be said to be less than approximately 20, a ridiculously small number. Regrettably, all too often, viewers do not discriminate between execrable and superlatives, and all gradations between, resulting in tessellations of worth not being recognised for what they are. Therefore, to set the record straight, I now discuss in detail about what it is that makes for a ‘superlative’ tessellation, and not just good, or even very good, but superlative, followed by some actual instances taken from the few artists who are doing anything of worth in the subject. Simply stated, for a superlative, it’s all about the recognition of the silhouette, almost invariably of an animate, whole-bodied creature, and its articulation and definition thereof in every aspect in its outline, recognised instantly, at-a-glance, without any ambiguities as regards the type of the motif it supposedly portrays. All the elements of the creature it is supposedly portraying should be obvious, and in proportion. For example:
There’s nothing difficult about the premise, anyone should be able to understand this. Also, ‘categories of difficulty’ come into play; some instances cannot possibly be considered as superlative, and should be excluded:
Furthermore, I am not interested here in assessing as against a percentage of superlatives as against an ‘average’ quality tessellation in one’s oeuvre; all I am interested in here, plainly and simply, are those of the highest possible quality and higher degree categories, high tariff, and not lower tariffs as outlined above, just superlatives, nothing more. Unashamedly, I am elitist here, and not just interested in the ‘merely’ good, or even very good, but if you like in the broadest sense ‘extraordinary’, and so worthy of the description of superlative, of outstanding inherent quality.
Another aspect to be ideally striven for, but is largely of circumstance, is whether the motif/s adopts a ‘sensible’ orientation, which is to be favoured aesthetically. By ‘sensible’ I mean where the motif/s is portrayed in a ‘normal orientation’. For example, a human would normally be seen upright, whilst an insect would be seen from above. Anything incongruous, such as with a human figure upside down in relation to neighbouring motifs is lacking aesthetically. An instance of the latter is Escher’s horses, PD 8, where both upright and upside down ones are shown. Such instances jar on the senses, in that it does not look ‘natural’. Indeed, Escher himself for this very reason referred to instances of this type as ‘absurd’. By far the best aesthetically is where the motifs remain in their proper orientation. However, as alluded to above, such matters are largely circumstance, depending on the symmetry chosen; for example with human figures, tilings with translations or glide reflection will be ‘sensible’ (i.e. appearing in the same orientation), whilst those with rotation will not be (i.e. appearing upside down), or in Escher’s words, ‘absurd’. However, examples with ‘absurd’ orientations should not necessarily be thought ‘inferior’ per se. If circumstances dictate that the motif is upside down, than this is simply a consequence of the symmetry chosen, and not of inherent quality; it can still be a superlative. However, as an abstract argument, if two broadly alike superlative motifs are of the same quality, but one is ‘sensible’ and the other ‘absurd’, then the sensible one is obviously to be preferred. If you like, it has ‘added value’. Where this occurs below, I simply state sensible orientation, with a nod to a superior type in this context.
Aesthetics also come into play. When a composition has two or more motifs, there should be confluences, as opposites or connected, for example angels and devils and eagle and rabbit respectively. Examples are marked down if they consist of incongruous elements: A giraffe next to an ink bottle? A horse next to book? There is no reason for these dispirit elements to be associated. Also, if a composition has grossly, disproportionate sizes, this by its very nature cannot be a superlative: A giraffe with an ant the same size? A dog with a same size whale? Again, there is no reason for these to be associated. Such examples cannot be termed as superlative.
Another reason for compiling this page is to show that Escher did not do everything. All too often, as regards life-like tessellations, mathematicians only discuss Escher, to the near total exclusion of others. This annoys me tremendously. It’s almost as if Escher is the only one to be ‘allowed’ to do tessellation. Sure, Escher was the first was the first to do so (negating rudimentary examples of Puchinger and Moser), and so great kudos to him. Certainly, without Escher, I and likely all others here would not have become involved in tessellation. But is the first person to do something in their field necessarily the best? Are there not painters after anonymous cave painters? Astronomers after Hipparchus? Mathematicians after Archimedes? Of course there are, they are many eminent people in these fields. However, with tessellation, no one else is seemingly allowed or gets a look in; it’s just Escher, Escher, Escher. Therefore, to redress this imbalance, I on occasion, where the motifs are alike in premise directly compare Escher and an artist, and objectively survey their work, of which Escher is noticeably lacking. Although superlatives should not necessarily be thought of as the only test of worth, nonetheless it’s a mighty fine start; what could be more to one’s credit that a tessellation of a superlative nature, of a high tariff of difficulty?
As such, this is very much a work in progress, I intend to add more detail.
Artists featured include:
Makato Nakamura (11)
Nick Scalfittura (6)
Alain Nicolas (5)
Bruce Bilney (4)
Kurt Komada (2)
Maurits Escher (2)
Andrew Crompton (1)
Dominique Ribault (1)
Henk Wyniger (1)
Hop David (1)
The Superlatives as According to Artist
I now discuss the superlatives as according to the artist, as according to the number of examples they have. I begin with a general overview of each artist, and of how they qualify for superlatives, followed by a discussion of their superlatives.
Makoto Nakamura, arguably the world’s finest ever tessellator, living or dead, has many examples of the superlative type, more than any other artist. Indeed, many of the effects he uses are original with him, with a series of ‘trademark poses’, of which great praise should be bestowed upon his numerous and original innovations.
A typical ‘Nakamura trademark’ is of a quadruped, such as rabbits, dogs, in a running or jumping pose with both front and rear legs shown extended. Typically, quadrupeds are decidedly awkward to undertake, in that when stood upright, with typically spindly legs, such a pose makes for a most challenging tessellation, although it can indeed be overcome, but with difficulty. Nakamura looks for another solution, in which the pose is changed, and of which this offers greater scope for success (note that Escher did not realise this).
Another Nakamura trademark is of an antlered motif, such as a deer or gazelle. A moments thought will instantly serve to show how difficult such creatures is to achieve, on two counts (i) thin legs, (ii) antlers, with typically thin, jagged outlines, all of which mitigates against a acceptable tessellation, never mind a superlative.
Another Nakamura trademark the sheer variety of poses of his human figures, with these adopting a variety of poses, such as swimmers. In this particular pose he is unique, no one else does this particular pose.
1. Dogs 1
This example is particularly good. The pose, of a leaping dog, is completely natural. Indeed, the silhouette has all the dog-like elements: head, ear, stop, neck, legs, body and tail are all easily identifiable, and all are in proportion. Of note here is the superb articulation of the legs. Typically, for a dog, these are portrayed with the legs merged i. e. two front legs, two back legs. And even then, it will be found that the typical artist will compose a most awkward, contrived dog. However, Nakamura rises above this, not with two, or three, but all four legs, clearly shown. Superb.
Typically, quadrupeds are decidedly awkward to undertake, in that when stood up, with typically spindly legs, such a pose makes for a most challenging tessellation, although it can indeed be overcome. Nakamura here looks for another solution, in which the pose is changed, of which this offers scope for success.
2. Rabbit /Eagle
Rabbits/Eagles 1 is worthy of special praise, with a natural confluence (hunter/prey), with good articulation. The degree of articulation of both creatures is quite outstanding. The elements of both motifs are readily discernable, with for the rabbit, namely head, neck, long ears, body, four legs, and bob tail, whilst for the eagle, namely head, neck, wings talons, hooked beak, ‘oversize’ wings, and tail. Also, the scale of the respective creatures is true. Indeed, one can go further on this, in that it would appear that the eagle has caught the rabbit in its talons, i.e. yet another natural aspect.
3. Gazelles 2
As such the gazelle adopts Nakamura’s trademark leaping pose, of which this, in contrast to others, has even more confluence with the motif it is portraying. Also as a secondary aspect, the tessellation is sensible, in that the motifs are shown in a sensible orientation, i.e. upright.
4. Gazelles 1-2
Of praise especially is Gazelle 2, in an action pose. The degree of articulation is quite outstanding. The elements of the motif are readily discernable, with antlers, head and neck, legs portrayed leaping, as does a typical gazelle, body and tail.
Of praise especially is 1, in an action pose. The degree of articulation is quite outstanding. The elements of the motif are readily discernable, with antlers, head and neck, Legs leaping, as does a typical deer, body and tail. Admittedly, in the strictest terms, this is a little contrived, but we are talking here in the strictest of terms. To articulate so many awkward, different aspects is worthy of the greatest praise.
6. Shark and [Swimming] Woman 252
Another trademark of Nakamura is with swimmers, again unique to him. Exceptionally pleasing, both the swimming woman and shark are fully articulated, and are a natural confluence, with both motifs in the same sensible orientation. Also, the woman is essentially shown in a natural state, with only a swimming costume, thereby negating artificial aid in the form of loose clothing, thus adding further quality here.
7. Swimming 3 [Swimmer doing front crawl and backstroke]
Another delightful instance, in Nakamura’s trademark swimming pose, with considerable articulation. Another delightful aspect to this is the portrayal, with the swimmer being shown doing a front crawl and backstroke. Here, the artist shows great imagination, in effect taking advantage of the rotational symmetry to permit this to happen. In normal circumstance, and other motif would be upside down, and so not be sensible. However, here, this feature is used to overcome this; what is more natural that a swimmer doing both the front crawl and the backstroke in the same composition. Wonderful imagination.
8. Swimming 4 [Swimmer with Dolphins]
Exceptionally pleasing, both the swimming woman and a (bottle nose) dolphin are fully articulated, and in a composition with natural confluence. Also, the woman is essentially shown in a natural state, with only a swimming costume, thereby negating artificial aid in the form of loose clothing. Both motifs are in a sensible orientation.
9. Girls and Rabbits 263
This is not just very pleasing, but quite superb, with both motifs fully articulated, the rabbit adopting Nakamura trademark pose, and furthermore in an aesthetically pleasing same orientation. Although there is no obvious confluence, and a noticeably discordant scale, the sheer quality of the motifs in a sense overrides these cavils.
10. Dance 1 [Man and Woman Dancing]
The many dancing articulations here, in costume, of both the man and woman are just superb. In addition, the motifs are of a commensurate scale, of the same overall size, and all elements are in proportion. Perhaps some people might cavil at the pointed head, and flared skirt of woman, but this is cavilling at the extreme.
11. Dance 5 [Woman Dancing]
A woman dancing on here own. Again, so many articulations Perhaps some people might cavil at the sheer simplicity, but this is cavilling at the extreme.
Nick Scalfittura’s tessellations, or more exactly of the human figure, his trademark, of which he specializes to the near exclusion of other motifs, are quite superb, and indeed, I rate him most highly indeed, of the top ranking. He has many instances of themes, with confluences, that no one has done before, notably such as of a ‘restaurant’ theme, of three instances, featuring waiters and chefs and food utensils, in various complex articulations (Coffee Waiter, Chef, Wine Waiter). As a concept, such a pose, with numerous articulations, is fraught with difficulty to accomplish to an even satisfactory degree, never mind a superlative as here, of which these are the epitome of the art. These are quite superb; the quality is simply outstanding. I just drool at these. I cannot praise these highly enough. Indeed, even within a superlative category, these should be noted as superior to others here.
1. Coffee Waiter (waiter at table with coffee being served, and holding cheeseboard)
A highlight, amongst highlights, is that of the ‘coffee waiter’. The sheer number of different elements with articulation is most praiseworthy, a human figure, in a typical garb and waiter pose, with apron, not to mention heeled shoes; holding a cheeseboard aloft in one hand, and pouring coffee from a jug into a cup resting on a table with table cloth. All those articulations, and without any degree of contrive to accommodate all these features. Magnificent!
2. Chef with flambé and tureen
This other example of the restaurant theme is again so impressive. Again a human figure, the chef, of a typical rotund nature, with all the accoutrements, such as a chef’s hat, and holding a tureen in one hand, and flambéing in the other, is delightful. Everything is in proportion and scale. So many articulations without any compromise being required, superb.
3. Wine Waiter
Another highlight, with a waiter holding aloft a tray with wine and cheese, and a table cloth in the other. Again, the articulations are superb.
4. Dog Running
Here the dog adapts the leaping pose as a workaround the difficulties of it standing. All the elements are here, head, ears, neck, body, legs, and tail. The articulations here include
Again, another classic, a pose reminiscent of a gangster, sat in a chair with drink and a cigar, sharp suit and shoes, with dark glasses with a menacing pose. Again, so many articulations, all achieved without any compromise.
6. Fighting Scotsman
Another classic, of a ‘fighting Scotchman’, in a kilt, adopting a typical, aggressive boxing pose, with two raised fists in a menacing manner. Superb articulation.
Alain Nicolas has many examples of the superlative type. Furthermore, many of his two-motif tessellations are of a themed nature, in that the motifs have a natural confluence to each other, as in a bucking bronco with cowboy, a magician and rabbit (albeit the latter is of noticeably different scale). Likely, he must be purposefully striving for such confluences.
His hallmark, unique to him, is of a ‘bucking bronco’ pose, of which even a moments thought of this as an abstract concept should show how difficult this is to achieve, in that the pose is action packed, with numerous articulations to be addressed if the motif is to realisable, of which Nicolas achieves superbly. Another instance is of a karate player, with two instances.
Superb are his human figures. The articulation on the examples below is most impressive. Of note is the Running Men, page 74, Fraternity, page 18. Note how each element of the figure is articulated here, down to the hands and feet. Other examples are also pleasing, such as the Karate Man, page 17.
A drawback to seeing his entire oeuvre is that he has not published all of them on his website, but rather the website acts as a vehicle for his book, which does show all the tessellations. Just buy the book!
1. Cow-boy (Bucking Bronco and Rider 1), Page 80
A highlight amongst many highlights here, and arguably the best is that of the ‘Cow-boy’, page 80, perhaps better described as a rider and bucking bronco. Here, with a rodeo theme, of a cowboy and a ‘bucking bronco’, both motifs adopting typical poses, the rider with his arm raised, holding his hat, whilst the horse adopts the typical bucking pose, with an arched back and kicking action. The articulation of both is quite superb, and in particular the quality of the horse deserves special praise, a very difficult motif to undertake. Indeed, upon correspondence with Nicolas, he told me that he regards this one as his best work.
2. Rodeo (Bucking Bronco and Rider 2), Page 81
Another instance of the theme of Rider and Bucking Bronco. Again, this has all the merits of ‘Cow-boy’ as described above, although here with a subtlety, or innovation all of its own. Note the innovation of Rodeo 2, where the cowboy has ‘lost’ his hat, this serving to define the space between the horses’ neck and legs for the sake of better articulation. Great imagination; this is most innovative and impressive.
3. Plongeon en eau trouble (Diving Girl), Page 84
A girl adopting a diving pose, with full articulation. The verisimilitude here is quite superb, and is worthy of the highest praise. A absolute delight. Furthermore, no concession is made to clothes, and so this instance is thus of a higher degree of tariff than with a clothed human, although clothed humans should not necessarily be decried.
4. Karate, Page 17
Here the motif adopts a typical Karate pose, on bended knee, with an arm raised as if ready for combat, with very good articulation – three limbs are clearly shown, and the fourth is ‘half implied (the arm bends at the elbow, being surface decoration). That said use of, and advantage is taken, with loose clothing, but this is as close to a real life karate outfit there can be, and so the tiling retains integrity. (Contrast this with lesser artists who use clothing in a much looser way)
5. Chess Knight, page 85
Of its type, quite superb. The verisimilitude here is quite superb, and is worthy of the highest praise.
Bruce Bilney has a few superlatives in his work. By choice, he restricts his interests to favouring indigenous motifs from his homeland, Australia. His trademark is kangaroo motifs.
1. Big Game [Hunter and Prey]
A highlight of his work, with considerable innovation in many different ways, is Big Game, of a series of consistent African native animal themes. This is built on the premise of hunter and prey, with four examples of each type of animal: (i) Hunter – gazelle, impala, deer and suchlike, (ii) Prey – leopards, cheetahs, tigers, and lions, pleasingly of equal number. The composition is perfectly ‘balanced’, in that each of the rows has the motifs ‘pursuing/fleeing’ in alternate directions. The motifs are all of an ‘extended leg’ position and are of a commensurate size. All are recognisable in silhouette, despite the increases in the number of motifs, and are shown in a sensible orientation.
This is most innovative and impressive, due to the sheer number of motifs possessing inherent quality. Typically, with such an increase, the quality goes down, but not here. Indeed, this particular pose appears to be one of Bilney’s specialities, in that few tessellation artists have realised the possibility of such a device.
Of praise is TwoRoos (albeit misnamed, as it’s of only one kangaroo; the title arises from the kangaroo being in two orientations) where the articulation and definition is quite superb, with head, ears, neck, body front and back legs and tail indistinguishable from a real life motif. The verisimilitude here is quite superb, and is worthy of the highest praise. Additionally, the motifs are in a sensible, upright orientation, all of which adds value.
An ‘honourable mention’ can also be made for a like tessellation OORoos (of different symmetry), although this lacks the all articulation here (and is in a not so sensible orientation, with the motifs upright and upside down)
A particularly pleasing innovation concerns Tess-Elephants, where part of the tessellation serves as a dual function, for a better overall aesthetic appearance. Specifically, this refers to the front and rear legs, where the elephant is both upright and upside down; one leg is used twice, front and rear. This displays great imagination, and very nicely overcomes a design problem, in which the legs would not otherwise be as good as they are. (Note that this is very much of the same concept as Escher’s Bulldogs, albeit of a different nature, where the teeth also served as nails.)
An ‘honourable mention’ can also be made for a like tessellation Curly Elephants (of different symmetry), although this lacks the all articulation and innovation here (and is in a not so sensible orientation, with the motifs upright and upside down)
4. Yabbies Galore
Yabbies (Lobsters) are a type of creature most difficult to accomplish, with many spindly features, such as claws, and numerous legs, all of which generally conspire against good, not to mention excellent tessellation. Bilney admirably overcomes this.
Although Komada has done very few tessellations, he is indeed capable of superlatives, namely of rabbits and a dog.
This is superb, with all the features of a fully articulated rabbit, in a running pose.
Very pleasing indeed, with the dog in an action pose, with all elements instantly identifiable, with very good articulation. Note that three of the legs are fully articulated. Typically, with quadrupeds, even when portrayed to a good standard, the two front and rear legs merge i.e. of just two articulations. This is thus of a higher standard, or tariff than otherwise in this regard. However, it is not quite as good as Nakamura’s dogs, where all four legs are articulated. Even so, this still admirably qualifies as a superlative.
M. C. Escher
As can be seen, Escher has, perhaps somewhat surprisingly, given his reputation, relatively few instances of superlatives. As such, he regrettably didn’t really challenge himself with such instances, being content largely with lower tariff birds and fish. Indeed, without contemporaries to spur him on, I believe he largely coasted at times.
As can be seen, the squirrel adopts a typical pose, sitting up, with all aspects fully articulated. Observe the differential between the tail and body – lesser artist would of necessity merge these, Escher does not.
2. Reptile, Periodic Drawing 25
Very good indeed, with superb articulation.
Crompton has one example of the superlative type. His trademark, to the near exclusion of everyone else is of countries, of which the UK example in particular can be said to be a superlative. Many others of a very high quality indeed, with much of merit, but I consider these to be just slightly below a superlative rating, as much as I would like to include them. Possibly, this is down to style; as he favours a ‘simple’ rendering.
1. United Kingdom
A remarkable verisimilitude to the UK
Ribault one example of the superlative type.
This is quite outstanding, due to the difficulty of this particular motif, and indeed of this particular pose, of the animal rearing, requiring particularly good imagination to succeed. Indeed, very few artists have been able to compose such a fine instance, which shows how difficult this is. As such, the zebra has resemblance to a horse, and indeed one could transpose between the two here.
Of note is that Escher also showed a horse in a similar (rearing) pose, but his is by far the weaker, relying on ‘fantasy’ wings, whereas Ribault’s zebra has no need of such artifice.
Wyniger has one example of the superlative type.
Although a witch falls within a lower category of difficulty, being a fantasy with considerable leeway of outline, on occasions due to the artist’s skill such examples rise above this, as shown by Wyniger’s splendid effort. This is very pleasing indeed, with all the witch-like elements – observe the pointed hat, the prominent nose, cape, broomstick; numerous subtleties and articulations, and yet the silhouette remains compelling. That said, it is indeed of a lower tariff, and cannot be compared to the exalted instances of Nakamura and Scalfittura, but even so, it is more than deserving of being regarded as a superlative.
David has one example of the superlative type.
1. Horse and Jockey
Absolutely delightful, with considerable articulation of the horse in particular. Upon a horse of general good proportion, observe the attention paid to detail, with ears and tails, all of which combine in a compelling silhouette to give a truly stunning example of a horse. Furthermore, the rider (a logical addition) is in proportion to the horse, something of which is necessarily easily achieved, and so this yet adds more worth to the composition. As detailed above, creatures with spindly legs, such as horses here, are notoriously difficult to successfully draw, at least to some degree of respectability. However, David admirably overcomes this.
By contrast, of a more or less alike pose, consider Escher’s own ‘Horse and Rider’, Periodic Drawing 67. This has received considerable publicity and praised, adorning the cover of numerous books, and been much talked about. But is it really that good? No; it’s is grossly overrated, as I shall explain. By means of analysis let’s separate the two figures. The horse is noticeable anatomically challenged – compare the head and neck region with the body, they are noticeably disjoint. Also, the tail is entirely mere surface decoration. Also compare the scale of the two figures; the rider is way too large for the horse (or the horse too small for the rider, depending on interpretation). This is indisputable. Contrast this with David’s effort, both the horse and rider as distinct entities are anatomically correct. Also, when duly ‘assembled’, both are in true scale. Contrast David’s with Escher’s; individual elements correct, respective scale in conjunction with each other, and a distinct tail and ears. Admittedly, Escher has three of the legs articulated, as against David’s two, but David also has a tail articulation that Escher does not, and so on articulation these broadly equal each other out. So that leaves proportion comparison, where David’s wins hands down! Simply stated, David’s is far superior! And yet does David’s Horse and Rider get a mention? How unfair.
Created 3 May 2012: Makoto Nakamura, Nick Scalfittura, Alain Nicolas, Bruce Bilney. 12 October 2012: M.C. Escher, Kurt Komoda, Dominique Ribault, Henk Wyniger, Hop David.