Ambiguity of Outline
Fish are motifs that are ideal for tessellation purposes, and indeed arguably of the simplest of all to achieve. This is due to their ambiguity of outline, in which a fish’s body has considerable variation, along with fins and tails. Quite simply, a fish’s body can be either long/narrow/squat and yet still remain instantly identifiable. In addition, the fins differ in their shape and number, with two, three of four all being common. Furthermore, the tail can be just about any shape or length within reason, and yet again still remain true to life, all of which any ‘field guide' book will show. Therefore (as with birds), seemingly whatever combination of the above elements is used, they all remain fish-like, with the result that there are thus simply more opportunities of ‘ambiguity of outline' and hence more tessellation examples. Indeed, it is arguable that fish motifs are the most ambiguous of all, and as such, it is simply a ‘quirk of fate' that more are not shown on the fish page category.

Escher shows numerous examples of fish, some noticeably stylised as seen in an unnatural position, as from above rather than the more naturally recognisably sideways view. Such examples are relatively easy to compose, and are not particularly praiseworthy. Undoubtedly, in portraying fish like this, he erred. Ideally, examples of this ‘Escher’ type should be avoided in favour of the 'classic' viewpoint.

Elements

Perhaps of most significance is the shape of the fish's body. Quite simply, there seems to be no ‘typical' body – any shape is possible. Consequently, such variety thus leads to considerable opportunity for such motifs.

Fins
Again, ambiguity abounds – just about any shape is acceptable.

Tail
Although the tail is somewhat more defined as according to a usual outline, considerable variety still exists.

Mouth
Although of an essentially minor nature (due to size), on occasions the mouth can used, specifically when in a gaping position. Essentially, when so open, the mouth can define another outline (of which Escher shows this with drawings 29, 72-73, 84 and 120). Undoubtedly, such instances will be uncommon, but again, it is yet another device the tessellator should be aware of when the opportunity permits.

However, although it is possible to compose a most ‘unlikely’ fish using these elements, none the less the ideal remains to be that the fish should be ‘typical', and not too outlandish.

Fish Conclusion
As the fish elements of head, body, fins, tail, all thus has considerable ambiguity of outline, it is thus possible to in effect ‘assemble’ these onto an arbitrary body and yet remain instantly identifiable as a generic fish. Consequently, this particular motif thus has an abundance of ambiguity of outline (in contrast to say, a human figure, where for instance any significant departure from 'correct proportions' would result in a ridiculous appearance). And this therefore is why fish are such a popular selection – a given outline simply has more ‘opportunity’ to be fish-like, rather than a creature of less intrinsic variety. Of course, one than has to ask if a fish motif then provides enough of an intrinsic tariff of difficulty as to be deemed ‘worthwhile’…, but that’s another matter.

Human Figures

Ambiguity of Outline

Human-like figures are noticeably less frequent than with the more commonly found birds and fish, which thus retorts the question as to why, as all three are possessed of essentially like gently curved lines, and thereby it may be thought likely that there is no intrinsic difference in frequency of occurrence between them. However, in comparison, as can be seen by observation, there is a distinct lack of such tessellation motifs, of which the reason pertains to the previously discussed ambiguity factor. Now, in this particular aspect the outline of a human figure is to be found decidedly limited as regards proportions. For example, any slight deviation from the correct proportion of the body, with say the arms being noticeably longer in proportion to the body, thus results in a ridiculous outcome, of what is self-evidently anatomically incorrect. Such an analogy is also applicable to any other body part. In short, correct proportion is a necessity, as otherwise the figure appears absurd. In contrast, as with the previously discussed birds and fish they have an abundance of ambiguity, all whilst remaining in proportion, hence their more frequent occurrences; again, there is simply more opportunity.
Therefore, due to the above constraints, the drawing of human-like motifs (if of a decent enough quality) is cause for praise, as it is a strict test of one’s tessellation capabilities to compose such figures (as against ‘easier’ birds and fish). Indeed, it is noticeable just how few such quality human figure motifs are to be found. As such, I frequently see so-called ‘human figure’ examples of gross distortions and strange protuberances that should be best kept unseen and not displayed in all their supposed glory. As such, the ‘quality question' arises, and on many occasions this aspect is neglected in the attempt for an obviously desirable choice of motif.
As an indication as to how difficult it is to compose such motifs (at least to an acceptable standard of course), even Escher himself only composed four clearly ‘unambiguous' human figures (periodic drawings Nos. 3, 4, 5 and 21); contrast this with the large number of birds and fish he did. Furthermore, these (human figures) are to a certain extent examples of the above discussed ‘compromises', as the figures are in various ways out of proportion, but not to a gross intent that would render them as unacceptable.
On occasions, it is possible to ‘enlarge’ upon the principle and introduce ‘fantasy people', with such examples being angels and devils, and as these only exist in the imagination, there is thus plenty of scope (ambiguity) in this aspect. A frequent device with these is to equip the figure with wings, of which any wing-like outline will be acceptable within reason, as this is a non-critical aspect per se. Therefore, such examples have further ‘ambiguity of outline' possibilities, and so are thus relatively easy to undertake in the context of human figures, and should be assessed accordingly (i.e. of a lower tariff).
An aspect of composing human figures that is noticeably different from all other life-like motifs is that the human figure is typically shown clothed, and not in its natural (naked) state as with other motifs. By such additions, this thus adds to the ambiguity factor, of which such matters are a decided assistance in what is a most difficult motif to achieve.
However, despite the most thorough attempts in composing such motifs that are essentially in proportion, of necessity ‘compromises' of outline are still generally in order. Therefore, such ‘disproportionate aspects' essentially have to be accepted with equanimity, as it remains most difficult (if not impossible) to compose a human figure having absolute correct proportions, and so consequently, some marginally, but not grossly inferior examples are thus ‘permissible' as a matter of course.
Now, as human figures are somewhat ‘complex' for tessellation purposes the following breaks down the various aspects that pertain to the above difficulties of this motif.

Elements:

Head
A somewhat minor aspect that specifically pertains to the head of figures that possess mirror reflection symmetry is that it will be found possible to include minor variations, if so desired, by the process of what I term as ‘symmetry breaking.' Now, with an arbitrary mirror symmetrical outline, it is natural to draw the head as seen ‘full-on,' thereby preserving the overall symmetry. As such, this is all well and good and perfectly valid. However, such a ‘view' should not in anyway be regarded as ‘cast in stone,' as the head per se is not part of the outline, which is instead formed by the hair. Therefore, this thus opens up the possibility of variations, as the head can thus be drawn in different views, as turned slightly to the left and/or right. An actual example of this can be seen on the angel of Human Figure No. 4. Because of this, there is now three possible heads, of which I describe as ‘full-on', ‘left' and ‘right,' and thereby if used in combination an appropriate tessellation consisting of motifs in three orientations should thus be chosen. Of my own examples, the most appropriate for this would be No.1, as it possesses all the necessary attributes, albeit not shown in this way.

Hair
Of considerable assistance in the composing of figures will be found to be the hair, despite this not perhaps being what is a ‘fundamental aspect' of the human body, as with, say, arms. Quite simply this aspect has considerable ‘ambiguity of outline' with different hairstyles, along with short or long, flowing hair, of which the latter can frequently be employed to advantage. This is due to the long hair, which in outline is obviously non-critical, as it is not of any great importance per se as to its waves or curls. An example of my own that illustrates this principle is shown with Human Figure No.1, where the hair is adjacent to the leg of a neighbouring figure. This being so, it is obviously of more importance that the leg is anatomically correct than the hair, and so the hair can thus be of the above ambiguity aspect.
Furthermore, as alluded to in the introduction, as the female generally has a longer, more flowing hairstyle (in contrast to the shorter hair of the male) and so this thus makes a female motif more likely than not.

Arms
As arms are of an essentially spindly nature, such aspects are most trying in attempts at adapting for tessellation purposes. (Such difficulties of a spindly nature, of a different type of motif, bird’s legs are discussed above.) Although different poses can be used, such as these being shown outstretched or bent at the elbow, in general terms these will be found to be somewhat contrived. Consequently, the difficulties in portraying arms are circumnavigated by the possibility offered up by clothing, in which the difficulties of this aspect can effectively be disguised, with the only the hand showing.

Legs
As regards legs, these have much in common with arms (along with the same problems), both being of an essentially spindly nature, albeit the leg is slightly more substantial in mass. However, it remains essentially spindly, and so difficult to incorporate in a representational tessellation. Again, as with arms, the legs can be shown bent to a degree, but care must be taken not to appear to be too contrived, with the leg at awkward angels. Whenever possible, the leg should appear natural. Essentially, the same way of circumnavigating the problem as with the arms can be employed, this time ‘obscuring' the legs either partially with a dress/skirt or completely, with trousers.

Clothing
The clothing of the figure obviously opens up considerable ‘ambiguity of outlines,' with various styles and fashions possible, of which such choice is obviously fertile ground for tessellation purposes. Furthermore, as males and females typically wear different types of clothes, some gender differences occur, all of which can be used when appropriate.

Suits and Dresses
Generalising and simplifying, the (western) male wears suits and the female dresses, of which there is an obvious difference in outline between the two garments. In short, the female apparel, with a shorter or longer hemline, and of different possible fashion possibilities has more variation or ambiguity of outline than the males clothing. In contrast; the male’s clothing, with the typically angular lines of his suit and trousers, is therefore less variable, and so possesses fewer such possibilities of ambiguity. However, where specifically a ‘geometric' tessellation is considered, the angular lines are, or can be, more suitable to the male figure, because of the wearing of the suit and trousers, which can thus be depicted as essentially straight lines. Therefore, although there are various nuances on this matter, the fact remains that as a general principle, a female motif is easier in theory to compose than a male.

Hats
When designing a human figure, a very common occurrence to be found is that frequently, indeed, more often than not, the figure will be seen to be wearing a hat, of various styles and proportions. As such, this is not due to a fixation in dressing the figure with this apparel for the sake of it, but is instead to do with obtaining a better resembling figure. There are a variety of reasons here. Typically, but not necessarily, the head will ‘terminate’ at a vertex, resulting in the figure being distorted or indeed pointed somewhat, and so, consequently, the ‘addition' of a hat is thus used to remedy what would otherwise be an inferior figure. Also, the hat can be described as ‘accommodating’ to the figure, of which as outlined is above is of a specific proportion; a hat is not. Consequently, this arises to suit the tessellation, and the inclusion is not of any importance per se beyond this. Quite simply, this refers to the previously discussed ‘ambiguity of outline', as by their very nature this can vary, as a specific outline is in general not a necessity for a hat (unless called for) and thereby thus offering up more possibilities of composing a more realistic figure than otherwise.

Of Escher's four human figure examples, Nos. 3, 4 and 21 terminate at a vertex, with Nos. 4 and 21 possessing such hats, of which, although a very small sample to survey, gives some indication of their desirability of inclusion. No. 3 has the hairstyle ‘adapted' at the vertex. Indeed, such ‘hat wearing' is a device I (and most artists) frequently use, as seen on the appropriate page. As such, it is to be encouraged if it results in a more ‘likely’ figure.

Human Conclusion
As the human elements of head, body, legs and arms, all thus has less ambiguity of outline than of other motifs (i.e. birds and fish), it not quite so easy to accomplish, and so is less frequently seen. however, in contrast to other motifs, the opportunity of using clothing exists, and so this gives a little leeway. Simply stated, it is more difficult to ‘assemble’ these elements on to an arbitrary body. Consequently, there are fewer such motifs; there is simply less opportunity. Of course, such higher tariff motifs provides more of a challenge than the easier birds and fish.

Dogs

Ambiguity of Outline
Dogs are another tessellation motif that in comparison to the previously discussed birds and fish are appreciably fewer in number, and indeed, they pale into insignificance when so compared. Again, the obvious question to pose is why that this should be so, as arguably with the previously discussed ‘ambiguity of outline', there at first consideration seems to be potential in this aspect, with different breeds possess widely different outlines. For example disparate, arbitrary dogs such as corgis, bulldogs and greyhounds, with different body sizes, legs and tails, all thus appear to have to have possibilities in this aspect. However, as examples of this motif are conspicuous by their absence (with even Escher only showing two such examples, namely No.16 and 97), such matters thereby require investigation.

Elements:

Legs
Now, perhaps the most noticeable difference between the ‘more frequent' birds (in which the leg is generally disregarded, as detailed above) and fish and ‘less frequent' dogs is that the latter has legs of a more substantial nature, that cannot simply be overlooked as with bird instances. As previously discussed, legs being of a typically spindly nature make for a most trying set of circumstance for composing representational tessellations of this, or indeed any motif. Indeed, as the legs are fundamental to this creature (this being in contrast to the bird’s leg, which is inconsequential), the animal can hardly be portrayed without it, as they make up a considerable proportion of the animal. By their very nature, legs are long and angular, of which such an aspect has essentially no ‘ambiguity of outline' – in effect, the legs must be drawn as anatomically correct, with no leeway, as otherwise the motif will appear ridiculous. Such matters are thus difficult to incorporate into a tessellation. Although it is possible to circumnavigate this difficulty by showing a dog in different poses aside from the ‘classic' sideways portrayal, such as lying down or curled up, such positions generally make for a contrived representational tessellation. Therefore, because of this, the number of dog motifs that attain the required standard as regards quality is thus very small indeed. Far too often dogs can be seen that fail the 'quality test' and so consequently are of an unworthy nature for showing, and yet they still appear, unfortunately.
Note that although the above text pertains to a dog in its own right, the material can also largely be applied to any other arbitrary quadruped, such as cats or tigers, or indeed any other animal having a similar four-legged outline, as essentially the same explanations are applicable.

Dog Conclusion
As the dog elements of head, body, legs, all thus has less ambiguity of outline than of other motifs (i.e. birds and fish), it not quite so easy to accomplish, and so is less frequently seen. Simply stated, it is more difficult to ‘assemble’ these elements on to an arbitrary body. Consequently, there are fewer such motifs; there is simply less opportunity. Of course, such higher tariff motifs provide more of a challenge than the easier birds and fish.

Escher’s Views on the Subject

Of interest would be Escher's views on the subject. Unfortunately, he said (all too) little in his essays (notably in contrast to other, unimportant aspects, of which he was much more expansive, such as ‘barber’ (p. 167-168) and ‘music’ (p. 170) digressions (and others), see Escher: The Complete Graphic Work. Broadly, he contenting himself with a brief passage in Regelmatige vlakverderling, concerning only birds and fishes; no other creature is mentioned, stating that (p. 164):

My experience has taught me that the silhouettes of birds and fishes are the most gratifying shapes of all for use in the game of dividing the plane. The silhouette of a flying bird has just the necessary angularity, while the bulges and indentations in the outline are neither too pronounced nor too subtle. In addition, it has a characteristic shape, from above and below, from the front and side. A fish is almost equally suitable; its silhouette can be used when viewed from any direction but the front.

However, although somewhat brief, this at least contains the basics, but from the viewing and examination of his works, the same conclusion could equally well be reached. What is required is a more exact, in-depth discussion, with the merits of each creature discussed separately (as above), along with as to why each ‘element' (e.g. birds, with beak, wings, body and tail) is so suitable. Of assistance in drawing motifs is in having to hand suitable ‘field guide' books, from which the different ‘elements' of the creatures can readily be compared.

On occasions, more specifically with representational tessellators of an inherently lower quality, there will be found examples of creatures in the broadest sense of what I term as of a ‘unidentifiable motif,' whereby although vaguely reminiscent of some sort of motif, this cannot be identified as essentially recognisable, such as with birds, fishes, lizards…. (Note that this does not include the ‘imaginary' creature category, which is altogether another matter.) As such, these examples are thus to be regarded as of the lowest possible quality, and indeed strictly speaking, I consider this type to be unacceptable, essentially unworthy of the dignity of the title ‘representational tessellation.' Essentially, all the tessellator is doing here is that after composing a non-representational tessellation, this is then supposedly made animate by the addition of an eye(s), along with a few vague, suggestive body markings. Such efforts are obviously inferior to the more life-like examples in silhouette, and hence essentially unacceptable. As such, examples of this type may be understandable in ones ‘early' days of attempting representational tessellation (indeed, they are hardly unavoidable as a result of experimentation, and I have many examples myself, but are designated as essentially ‘not for show').

Interestingly, Escher's examples from the ‘early years' do not include any of this type, from which this thus indicates that he was aware of the triviality and ease of such a type from the very beginning, and so he did not concern himself with such ‘unworthy' examples. Presumably, in his early days, whilst experimenting, he must have composed some examples of this type, but what became of these has not been stated. Curiously, somewhat oddly, in later years he did include some examples of this type, with drawings Nos.36, 103 and 116. (No.36 was undertaken for a specific purpose, of a ‘missing' symmetry type that Escher desired to include for the sake of completeness.)

Less frequently accomplished is a representational tessellation can be seen to be composed of two (or more) distinct motifs, such as a bird and fish or cat and dog, to give two arbitrary examples. An obvious question to ask is why this should be so. The explanation of such rarity is that in general terms, by introducing additional motifs, by whatever method you so use, the number of lines involved of necessity increases, and thereby the difficulties lay. Quite simply, by increasing the number of lines this thus results in additional difficulties, as the outlines has to represent yet another motif, and furthermore ideally all of the same quality as regards the veracity of the motifs. A further obstacle to overcome is that the motifs should ideally be in proportion to each other, as this makes for a sensible and not absurd combination. Using Escher's drawings for comparison, drawing 22, of birds and fish is ideal, as both are in proportion. In contrast, drawing 30, of fish and boats is clearly lacking, and thus can be regarded as inferior in comparison. Now, as evidenced by the lack of good-quality tessellations of even one motif, of the simpler, less numerous lines required, the introduction of additional lines, of obvious necessity for multiple motifs, is thus more difficult, albeit by no means impossible, to achieve.

However, putting the ‘practical difficulties' aside, bird and fish motifs in unison remains ideal for this, due to their own ‘ambiguities' as discussed above. Furthermore, in such a combination, if of a sufficient quality (as with drawing 22), the serendipitous contrast of the two motifs, with concepts such as ‘above and below' or ‘sky and water' in mind, lend themselves aesthetically to a superb natural composition, of which Escher took full advantage, admirably demonstrated by his print Sky and Water I. Indeed, it really is of purely fortuitous circumstances that the above motifs in combination are to be so appropriate, and lend themselves so readily to such concepts.

Another aspect of the motif is one that I shall term as familiarity of motif. This refers to the knowledge of the animal in terms of its portrayal. As such, this can be divided into two distinct types:

• Animals that is familiar in everyday life, e.g. cats and dogs

• Animals that are rarely, if ever encountered in normal life, e.g. giraffes and kangaroos

From the former group, the proportions of the animal come readily to mind, whereas with the latter, although pictures of these will have been seen, the exact proportion of the animal is not fixed in the mind. For example, is the proportion of the giraffe's neck to the length of its body the same, or is it two, three times as long? Likewise with a kangaroo - for example, is its tail length the same as of its body, or is it again, two, three times as long? As such, only an animal enthusiast with knowledge of a wide variety of animals can readily judge the veracity of resemblance of the motif to the actual animal. Therefore, in certain aspects this factor can work to the advantage of the tessellator, as faced with an unfamiliar animal in a tessellation, people will accept this at face value and not go to the trouble of comparing this with an actual picture, whereby its possible shortcomings would then be evident. Therefore, the tessellator thus effectively gains potential credit for a supposed accurate portrayal from the general lack of knowledge of such creatures. However, this of course should not be used as an excuse for possible slipshod work, in effect taking advantage of the general ignorance in such matters. Undoubtedly, the aim remains, as always, in portraying the animal in question as close as is possible to its natural appearance.

Agree/disagree? E-me.

Last updated: 23 September 2009. Revised and expanded 29 January 2013