Essay 1

Concerning the process of creating �representational tessellations', as defined on the home page, as so very few people seem to be capable of composing such tessellations, a pertinent question to ask therefore is what sort of skills or abilities are required in order to undertake such matters. Therefore, the following essay thus sets out to answer this question, divided into distinct sections or aspects. As such, this follows Escher's �primary' example (although he did indeed compose other essays), in which he set out his own thoughts on the matter, first published in the Dutch book Regelmatige vlakverderling of 1958 (The Regular Division of the Plane), of which an English translation is shown in Escher The Complete Graphic Work, pages 155-173. Explanations as to techniques and methods are not readily described in words, as the artist essentially has to leave the familiar pictorial field for the literary one, and, as Escher himself pointed out,

It is outside my scope to use letter symbols, but in this instance I am forced to use them�

Indeed, Escher's attempt unfortunately has notable shortcomings, as nowhere do we see text or pictures pertaining to the all-important genesis aspect of his tessellations, with the examples that are shown being essentially in their perfected state. However, although I am somewhat critical of this particular part of his essay, nonetheless it still contains many points of interest that are most illuminating, and well worth the study.

As such, despite text being a necessary requirement to clarify the process, the phrase 'a picture is worth a thousand words' comes to mind, and is, in this particular example of my own, is most appropriate, and so consequently the text here is thus liberally illustrated with the above quotation in mind. As I am in a similar background situation to Escher myself in this matter, and therefore, despite the utmost care in trying to set out my techniques with clarity, hopefully due allowance will thus be made for any shortcomings of what follows. Note that the following methodology is based upon the �old-fashioned' way of essentially utilising �pen and paper' and nothing more by way of assistance. Although computers are indeed being used increasingly, certainly at least with the drawing and subsequent colouring of the tessellation, and also to a lesser degree with the actual designing, as yet I myself have still to be convinced in their efficacy of design, and so consequently I have not really began in this matter. Therefore, hopefully, the quality tessellations here may inspire the reader to compose examples of their own, who had previously believed that the subject has no more quality real-life examples to discover, or who thought that a computer was is in any way essential to the task in hand. Indeed, such thoughts of the former are understandable given some websites of tessellation that I myself have seen.

Now, the answer to the above question has various a disparate aspects to it, of which the following sets out the various ways of creating representational tessellations. What follows is how I go about the creative process with my own tessellations, of which the material is separated into two distinct parts, placed as two distinct essays. Essay 1 consists of essentially peripheral matters, in which I detail various aspects pertaining to aspects that lead up to the creation process, which, although are indeed of importance, are not central to the actual task of creation. In contrast, Essay 2 is solely concerned with the actual design process.

Although not central to the creation process, of interest is to how one arrives at the subject of tessellation. Now, as can be seen by the practitioners of the art, it is noticeable that a wide diversity of subjects can lead to an interest in tessellation, of which the following thus discusses:

As such, my own first �proper' introduction to tessellation (neglecting my schooldays mathematics lessons on the subject, which, although I can distinctly recall, I did not proceed further with) was through just browsing at random (in approximately 1983) a Readers Digest magazine, of March 1981. This contained a brief article on Escher titled �The Artist Who Aims to Tease' by Greg Keeton, pages 37-41, the article containing some prints of Escher's, most notably Day and Night , from which I then wondered how he did this. However, due to not understanding how to go about such matters, this thus remained at an interest level only, and essentially put aside. Not until much later did not turn my attention to tessellations in a practical sense, in 1987. Before this, I had an interest in art, of a mostly surreal nature, favouring the work of Salvador Dali (whom Escher also greatly admired), and laterally of op art. Consequently, I then gradually drifted into the world of mathematics, before finally beginning my tessellation studies, albeit still somewhat of a fragmentary nature, in 1987.

Without doubt, an interest in art is highly desirable. However, this is not to say that anybody with an interest in the subject will be capable of producing worthwhile efforts in this field. As such, despite there being many artists the world over, the number of artistically inclined people interested in the subject is relatively small, and as a consequence of having such a very narrow interest base, the associated numbers producing tessellations thus results in a mere handful of so inclined people. However, in general terms, most people who do indeed arrive at tessellations via this way, despite an essentially non-mathematical background, perhaps somewhat surprisingly produce the most worthwhile examples. Incidentally, as detailed above, this is the route I took.

Obviously, an interest in mathematics is, again, highly desirable, albeit not necessarily of an advanced or indeed of even a basic nature. Escher admirably illustrates evidence of this, as he himself essentially picked up the rudiments required, despite his many protestations as to his lack of ability in this field, having not undertaken any mathematical studies since his schooldays. Not until considerably later, at the relatively late age of 38 did he essentially begin his interest in tessellations (neglecting some minor studies of 1922 and 1926 or 1927). However, despite tessellations being primarily in the domain of mathematics, so few mathematicians seem to be capable of producing any worthwhile representational examples. Indeed, a typical example of the shortcomings of mathematicians in this field, arbitrarily chosen, can be seen in Modern Mathematics by Patrick Murphy, pages 194-205. Although he is a splendid mathematician, far superior to myself, he can quite plainly be seen to be lacking in the representational aspect.

As such, despite appearing as disparate subjects, another relation to tessellation appears to be architecture. Indeed, Escher was more than interested in the subject, apparently intending to make it his profession, attending the Haarlem School of Architecture and Decorative Arts (1919), although he quickly abandoned architecture aspect in favour of the decorative arts. Although no direct channel to tessellation can be found in architecture, its practise can be said to lead to symmetry aspects, from which tessellation can thus be seen as a near relation. A contemporary practitioner of like manner is Andrew Crompton of Manchester, England who has produced many fine examples, with further details on the links page.

Another avenue to tessellation appears to be crystallography, albeit in this field such an interest lies very much secondary to mathematics and art. However, some eminent crystallographers have been personally associated with Escher, notably Caroline H. MacGillavry, who did much to popularise his works, notably with the book Symmetry Aspects of M.C. Escher's Periodic Drawings. Interestingly, his half-brother B. G. Escher also took an interest in this in conjunction with his post as Professor of Geology, in which he taught crystallography (and also played a pivotal role in Escher's progress, introducing him to crystallographic journals containing tessellating diagrams). However, despite having the above advantages in both theoretical knowledge and indeed in personal contact, of which needless to say must be greatly advantageous, no representational tessellations of any kind or worth has emanated from the above people, or from their successors as far as I know.

Despite not possessing an obvious connection to tessellation, aspects of psychology do indeed have a bearing on the subject, of which this interacts with tessellation in a mostly subsidiary way, notably with the aspect of figure and ground , essentially to be considered as an optional �extra' upon the completion of a tessellation. Various articles of these matters were apparently studied by Escher, latterly of which an article in Scientific American of July 1974 by Marianne L. Teuber sets out the various sources that were essentially available to Escher, albeit much of what Teuber relates was subsequently rebutted by his son, George, in a following letter to the journal.

So, upon having outlined the background to ones interest in tessellation, a relevant question to ask is what sort of abilities are thus in order to produce representational tessellations. Now, it may very well be thought that people from any of the above scientific backgrounds would be at a great advantage as when compared with the artistically inclined, as they would be inherently aware of the rules underlying tessellation. However, despite many eminent professors having taken an interest, none can be seen to have produced anything that can be said to even approach Escher's in quality, or furthermore of noticeably inferior examples. Indeed, all the quality examples of tessellation since the time of Escher have been composed by the artistically inclined, and not consequently from the much more likely scientific field. Now, why should this be so? What, in essence, makes somebody a representational tessellator? As such, I believe the foremost requirement is to be one of which I describe as shape recognition. More precisely, this refers to the ability to �see' or recognise some sort of creature or inanimate object in an arbitrary shape. As such, this is can be found at the most basic levels of children's games, such as looking at clouds in the sky and then by using ones imagination trying to �see' such things. A similar occurrence is looking at the moon and then trying to �see' motifs, most noticeably illustrated by the familiar �man in the moon'. Furthermore, Escher also played likewise games, adding various motifs to paint on the wall in his bathroom to amuse his children, as recalled by his son George in M.C. Escher: Art and Science . Indeed, despite being such a simple pastime, with the thought in mind that such matters are too frivolous or childish for the scientifically inclined, such pastimes even occupied Leonardo da Vinci, who amused himself in this way. (A quotation on this matter by da Vinci is in the book Escher The Complete Graphic Work , page 160.) However, as the above �shape recognition' games have referred to shapes that are not in any way per se to do with tessellation, one may thus feel justified in asking as to how all this pertains to tessellation matters. Well, quite simply, such ability in �shape recognising' can in effect be �transferred' to tessellation, as when faced with an given, arbitrary tessellation such ability can thus be duly applied. However, such ability seems to be lacking in most people, or perhaps is lying dormant, awaiting a prompt. As to my own abilities in this matter, I do indeed distinctly recall that at a young age, approximately 10-15 years of age (unfortunately, I cannot be precise), I was indeed aware of such an ability. However, this was of an isolated example, of which that until becoming interested in tessellation I had no use for. Specifically upon observing a map of the British Isles, I thus recognised such a shape. Can you use your imagination here and see a bird, fish�? As such, once pointed out it appears so obvious, and is noticeable at each view of the map (answer given below). Indeed, as the map occurs every day with the weather reports, millions of people have had the opportunity to see this, but have not apparently recognised the shape I have noticed.

Of interest is just how soon should one expect life-like tessellations of an at least �reasonable' standard to be produced. Upon examining Escher's tessellations and background to this matter, a �probationary period' of May�June 1936 can be discerned, whereby a relatively intense period of study was undertaken. Essentially, this involved copying patterns from the Alhambra and the mezquita in Cordoba (from recent visits to these places). Upon returning home and showing these to his brother Beer, who recognised their connection to crystallography, and this was then followed by an active search of crystallographic journals by Beer in seeking out existing tessellations, of which most notably an article by P�lya was influential, with a direct study of the diagrams by Escher himself being undertaken. From this, his first �proper' tessellations then emerged (negating a brief period of study which produced a mere handful of tessellations of 1922 and 1926 or 1927, the latter essentially �abandoned' and subsequently numbered 1 and 2) quickly followed, resulting in a whole host of tessellations in the style of which he has become renowned. Indeed, these �early years' resulted in a veritable outpouring of periodic drawings, of which with subsequent years, for a variety of reasons, he was never able to match in such quantities. Essentially, it can be said that he developed the �knack' of life-like tessellation very quickly, and indeed high-quality examples emerged remarkably quickly, such as with periodic drawings No.18 and 20 (which later were utilised for his renowned prints Day and Night and Sky and Water I). Therefore Escher's �probationary period' can be seen to be most short, a matter of a mere seven months. Such a �pause' and subsequent study echoes my own studies, whereby in 1986 I merely �dabbled,' lacking any real mathematical understanding worthy of the name, before in July 1987 I made a more concerted effort. However, it was not until February 1988 that anything of true, original tessellations emerged, albeit still mostly rudimentary, although �promising' in their nature. Indeed, one of my favourite motifs, of Girl 1, No.1 dates from this period. Likewise as with Escher, this marked a watershed, with my own �outpouring' of tessellations quickly following, albeit still not in a generally finished state as with his numbered drawings.

Therefore, from all this, it can be seen that essentially one either intuitively understands tessellations or not at a relatively early stage in ones �tessellation career.' Essentially, if the early efforts are �encouraging' this thus naturally results in further study, whereas if no improvements are initially forthcoming (as Escher found in 1922), ones enthusiasm is naturally lessened, with the likelihood of pursuing such matters, at best, being put aside, or more probably abandoned in favour of other pursuits. However, pleasingly, age is not an apparent factor per se, as Escher himself began his own studies at the relatively late age of 38.

Furthermore, upon establishing tessellation of a reasonable standard, it can be seen that progression as to inherent quality does not continue with the passing of the years, as may have been thought, but essentially remains static. Certainly, �refinements' do indeed take place, but there is no arrow-like continuum. For instance, utilising Escher's examples, a periodic drawing from the 1930s could quite easily be mistaken for one of the 1960s. As such, this is not due to a lack of ability, but is rather a consequence of the specialised nature of representational tessellations in which the underlying symmetry �forces' outlines that at times leaves a lot to be desired as regards representation. Therefore, this simply has to be accepted as a matter of course, albeit this should not be used as an excuse for slip-shod work that is quite plainly of an unacceptable standard.

Concerning the motif, without doubt birds and fish are the easiest types of creatures to utilise for tessellations, with numerous examples, not only with Escher but from his successors as well, with any other type of motif very much noticeably lacking in such numbers. Now, an obvious query to ask is why that these two specific motifs should be so admirably suited, as a priori there is nothing unique about their respective outlines � why should not other motifs, say human-like figures or dogs, to give two disparate examples, all consisting of the same essentially curved lines not be equally so suitable? As such, this has a lot to do with what I term as �ambiguity of outline' in which for an arbitrarily given outline birds and fish are to be found the most suitable. The following thus sets out to answer this question in relative detail by discussing various aspects pertaining to this matter with a variety of motifs in turn.

Simply stated, the above motif, in contrast to any other type of creature (exempting fish), can be seen to have a certain inherent amount of what I term as ambiguity of outline , of which such an aspect is of fundamental importance. By �ambiguity', this term is thus applied to the various �parts' or �elements' that go into the making of an arbitrary bird. For instance, a bird with a short, medium or long wingspan it is still instantly recognisable. A likewise argument can also be put forward for other regions of a bird. For example, the tail region, with types such as swallow or fan tails, and also similarly with the length of the body � with noticeable differences between, say, disparate birds such as a sparrow and albatross, all of which any �field guide' book will confirm. As a result, seemingly whatever combination is utilised, the outline remains bird-like. From this, there is thus literally an abundance of such �ambiguities of outline', with which, in comparison with any other type of creature, disregarding fish, any other motifs simply do not possess. Therefore, when applied to tessellation, such �ambiguity of outline' simply gives more opportunities for this particular motif.

A further aspect to this choice of motif is that it permits the drawing of two distinct views, applicable when the motif has its wings outstretched, with the bird taking the form of an approximate cross, and indeed this can be most rudimentary. From this, a �under' and �over' viewpoint is thus possible, as shown in diagrammatic form below. As a consequence, this aspect thus adds to the possibilities for any given bird tessellation that possesses such a cross-like outline, as the tessellation can now be any or all of three types, namely of �under', �over' or �under and over' in combination.

In general terms, such �ambiguities of outline' also apply equally well to fish � quite simply, a fishes body can be either long and narrow or squat and yet still remain true to life. In addition, the fin and tail can be just about any shape or length, and yet again remain true to life, as any �field guide' book will show. Therefore, as with birds, seemingly whatever combination is utilised, they all remain fish-like, with the result that there are thus simply more opportunities of �ambiguity of outline'. 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.

Motifs consisting of human-like figures are noticeably infrequent in comparison with, say, birds or fishes, which thus retorts the question as to why, as all three are possessed of essentially curved lines, and thereby it may be thought that there is no intrinsic difference between them. However, 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. 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 looking outcome. Such an analogy is also applicable to any other body part. In short, correct proportion is a necessity. In contrast, as the previously discussed birds and fish have an abundance of ambiguity, all whilst remaining in proportion, hence their more frequent occurrences. Now, due to the above practical difficulties, the drawing of human-like motifs (if of a decent enough quality) is cause for praise, as it is a strict test of ones tessellation capabilities to compose such figures. 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 an indication as to how difficult this is, even Escher himself only composed four clearly �unambiguous' figures (periodic drawings No.3, 4, 5 and 21) and all these 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.

On occasions, it is possible to introduce �fantasy people', with such examples being angels and devils, and as these only exist in the imagination, there is plenty of scope 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' and so are thus relatively easy to undertake in the context of human figures.

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.

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. An additional aspect, at least of the present age, is the different type of clothing for both male and female. Generalising and simplifying, the female wears dresses, and the male suits, of which there is an obvious difference in outline between the two garments. In short, the female apparel, with a shorter or longer hemline, along with different possible fashion possibilities, thereby has more variation or ambiguity of outline, whilst in contrast, the males clothing, with the 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 indeed 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.

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 utilised 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.

Another �ambiguity of outline' is also to be found in hair and hairstyles, with short or long, flowing hair as appropriate for the tessellation, of which the outline in this aspect 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. Furthermore, in contrast to the shorter hair of the male, the female generally has a longer, more flowing hairstyle, and so this thus makes a female motif more likely than not.

When designing the motif, a very common occurrence to be found is that frequently the motifs will be seen to be wearing a hat, of various proportions and styles. As such, this is not due to a fixation in dressing the figure with this apparel, but is instead to do with obtaining a better resembling figure. Essentially 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 examples, No.4 and 21 possess such hats, of which, although a very small sample to survey, gives some indication of their necessity of inclusion. Indeed, this is a device I frequently use myself, as seen on the appropriate page.

Another aspect pertaining to the human figure is that an additional feature is sometimes possible, essentially of an optional nature, is of what I term as a �two for the price of one' type principal. Quite simply, the most common view of humans is when seen �full on' as in face to face. However, another view, equally valid, although less so in an aesthetic sense, is the back view. Now, as the outline of a human figure is appropriate to both views, such a possibility arises of combining these in a single tessellation, hence �two for the price of one'. Of course, in accommodating both of these, the symmetry of the tessellation must of necessity be taken into account. For example, a tessellation where the motifs all side by side or are in two orientations would be ideal. Less so would be an example whereby the motif appears in three orientations, as it is obviously inappropriate for this particular instance. A typical example of the latter can be seen on human figure No.1.

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 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, dogs are tessellation motifs that in comparison to the previously discussed birds and fish are of 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, with disparate, arbitrary dogs such as corgis, bulldogs, greyhounds all thus appearing to have to have possibilities. However, as they are conspicuous by their absence, with even Escher only showing two such examples, namely No.16 and 97, such matters thereby require investigation. Now, perhaps the most noticeable difference between the �successful' birds and fish and �unsuccessful' dogs is that the latter has legs, of which they make up a considerable proportion of the animal, and indeed here is where the problems lie. 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. Therefore, because of this, the number of dog motifs that attain the required standard is thus very small indeed.

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, and therefore such specific text would largely be redundant.

On occasions, more specifically with representational tessellation websites of an inherently lower quality, will be found examples of what I term as of a �unidentifiable motif,' whereby although vaguely reminiscent of some sort of creature, 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 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, 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, I have many examples myself, but are kept as essentially �not for show'). Interestingly, even Escher himself included a few of these amongst his numbered drawings (No.**) albeit these are very few and far between, and furthermore these are not amongst his earliest efforts as may have been reasonably thought. This thus indicates that he was aware of the triviality of such a type from the very beginning and so he did not concern himself with these.

Most frequently, representational tessellations can be seen to be composed of a single motif, which, as according to the symmetry arrangement of its outline appears in a number of ways, either of a translational, glide reflection or rotational nature. Much less frequent is a tessellation consisting 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 utilise, 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. 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, the natural 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 these concepts, 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.

An aspect of tessellation concerning examples of representational tessellations is one of which I term as aesthetic and non-aesthetic , of which such a classification has a bearing of which tessellation are assessed as of a lesser or better quality, along with their potential application to counterchange, of which this aspect is discussed in greater detail below. Now, the above two terms are thus utilised to describe tessellations that pertain to motifs that are normally most recognisable in a sideways position, such as fish or bids, and not downward, as for example a lizard. From this, the above criterion is thus applied in two simple ways as according to the orientation of motifs, of which I utilise Escher's periodic drawings for illustrative purposes. Now, where the motifs(s) all remain upright, as for example, No.17, of birds, this is thus defined as aesthetic, as this is a sensible orientation, and in contrast to this, any example with motifs that are both upright and upside down, which is a non-sensible orientation, as for example No.8, of horses, is thus non-aesthetic. As such, the above descriptions are thereby applied arbitrarily whatever the underlying tessellation system is, and therefore examples can occur whereby the motif itself is quite superb or alternatively very poor, and yet can still be described as aesthetic or non-aesthetic.

Therefore, as well as permitting an assessment of the tessellation, an actual application of this pertains to counterchange, whereby the choice of the type of tessellation is critical. Now, to give an arbitrary example, of a lineal counterchange, of which I thus utilise the above examples of Escher's for illustrative purposes. For ease of argument, I thus here discuss counterchange matters in a broad sense, without any pretence to the actual process. Now, the birds of No.17 are ideal for this purpose, as upon a successful counterchange having been undertaken, the motifs have remained in a sensible orientation. In contrast, the horses of No.8, is less than ideal, as upon a successful counterchange the motif is now upright and upside down � an absurd situation, as why should this horse appear in such an unnatural orientation? To be pedantic, there are more subtleties involved here, as ideally, for such a lineal strip, two distinct motifs should be utilised, but the argument is kept here as simple as possible to show the general principal of the application.

Another aspect of this concerns what I term as a � typical representation' or �viewpoint' of the creatures outline. By this, I refer to a �normal' view of the creature, whereby from its outline alone it is instantly recognisable. Such a feature is highly desirable, as it thus permits recognition at a glance, which when applied to tessellation, with creatures that at times are not too distinct due to symmetry restrictions coming into effect, is a noticeable asset in identification of the motif. For demonstrative purposes, utilising a human figure, a typical representation would be as seen full on, from which such an outline and what it would thus represent would be obvious. In contrast, a non-typical representation would be from the ground level up, with an outline that would be not so obvious as with the more normal view as above. Now, when the above is thus applied to tessellations, frequently the motifs outline, due to the inherent symmetry �restrictions' necessarily involved, can be seen to leave a lot to be desired, from which such appropriate typical representation can only assist in matters of unambiguous determination of motifs. Such analogies also apply to other motifs. For example, any quadruped animal is more easily recognisable as when seen from the side, rather than, for example, from ground level upwards. In contrast to this, as small creatures such as insects would thus normally be seen at a bird's eye viewpoint, from which their outline would thus becomes recognisable. In addition, Escher specifically wrote on the above matters in his previously mentioned essay, page 164.

The contrast and colouration of tessellations is an aspect that I consider is generally treated in a casual manner (although not by myself). As such, the tessellation per se assumes pre-eminence, with the subsequent contrast and colouration of it very much neglected, if even considered at all. Obviously, the tessellation per se must remain the most pre-eminent aspect, but such disregard for this aspects is shortsighted, as by carefully selecting an �appropriate' choice the tessellations can thus be inherently enhanced, and not, as so often occurs, essentially �let down' by an inferior selection.

Further evidence of the importance of this matter is shown by Escher in the book Symmetry Aspects of M. C. Escher's Periodic Drawings, whereby in the preface, (VII) he emphasised (albeit briefly) this point.

Another important question is shade contrast. For the Moors it was natural to compose their tiled surfaces with mutually contrasting, different-coloured pieces of majolica. Likewise, I myself have always used contrasting shades as a simple necessity, as a logical means of visualising the adjacent components of my patterns.

As such, black and white obviously provides the greatest possible effect, resulting in the tessellation motif/s being of the utmost clarity and starkness. Indeed, I frequently use this contrast effect myself, as my examples show.

Black and white was also a fairly popular colouring scheme of Escher's, as he shows fourteen unambiguous examples (all shown in the book Visions of Symmetry), along with some others which although are in intent black and white, are not quite strictly so. Perhaps somewhat surprisingly, he did not utilise this colouring scheme upon the beginning of his studies, and it was not until 1941, with periodic drawing No.45, did he undertake the first such usage of black and white.

However, to utilise such a stark contrast scheme to the exclusion of any colour would soon become tedious, if not to say lacking in imagination. Indeed, as on occasions an arbitrary tessellation requires a minimum of three colours, due to its symmetry, and so of consequence such simplicity of contrast is thus insufficient. However, although the above example gives a plausible reasoning for the use of colour, such necessity should not serve as the real reason per se for reasons of which I discuss in the next section.

As regards to colour contrast, with a great deal of choice possible, the crux of the matter is to select colours that have the maximum amount of contrast, in effect the colour equivalent of black and white. Quite simply, colour per se is an aspect that deservedly has merits in its own right, the usage of which undoubtedly gives extra refinement to the tessellation. Therefore, upon contemplating which colours to use, an obvious question to ask is which give the best contrast effect. This aspect of aspect of contrast is stated as complementary colour, as has been the subject of much conjecture. Indeed, the whole subject is rife with inexactness, with authors (generally artists) all too often lacking in rigour. Firstly, it must be established exactly which type of complementary is being referred to. As such, two distinct types can be discerned, mixing, whereby when two given colours are duly mixed result in a grey, or visual , whereby the colours give the most contrast as seen by the eye. All too often, the above distinctions are not stated, thereby leading to confusion as to which type is referred to. Neglecting this, when complementaries are indeed stated, with presumably of the visual type in mind, combinations such as blue/orange, red/green and yellow/violet are given. However, such colour descriptions are woefully inadequate, at best, an approximation. Quite simply, more precision is in order, as such a simple description as to colours is too vague, essentially being an abstract concept. For example, blue can conceivably be one of a multitude of pigments, such as a violetish blue as ultramarine (pigment blue 27), or as a greenish blue such as cobalt blue (pigment blue 28), not to mention other blue pigments that are available. Such inexact choices of colour apply to the other simplistic colour descriptions above. Quite simply, such matters have persisted, and without doubt, an unsatisfactory state of affairs that has been allowed to remain for far too long. However, thankfully the matter has finally been resolved due to the efforts of Bruce MacEvoy, who has undertaken some original work in the field, and has consequently thereby established some specifics as against the previous uncertainties. Furthermore, the information is readily accessible at a popular level, with a list of the complementaries given, published as an online article www.handprint. Many other aspects of painting and colour theory are also examined and put to the test before being found wanting, and thus being corrected.

In general terms this particular choice of colouration is the one I prefer, of which numerous examples can be seen on the site. Of interest is Escher's choice in this matter, and upon examining his examples, although many of his drawings do indeed show forethought in this matter, it cannot be said that he undertook systematic studies per se of the subject.

The above type of colouration can in broad terms be described as inherently simple , as the colouration consists of a single colour for any tile. As such, it is possible to introduce a certain amount of variety by colouring specific areas of the motif in different colours. For example, a bird motif can be divided up into many distinct parts as according to its body, such as the head, wings or tail areas, either individually or in combination as one sees fit. Such variation can be, at times, be a most welcome change from the usual (arguably simplistic) colouring schemes above. However, if overdone to an excessive degree with a multitude of colours, the result is that the individual motifs can be most difficult to perceive, which is obviously to the detriment of the tessellation. Therefore, when such �multiple colouring' is thus contemplated, it is essential to undertake colour studies beforehand, thereby permitting an assessment as to the merits or otherwise in this matter, upon which a definitive selection can than be made.

Colour rendition refers to the inherent quality of any finished motif, and as such has a bearing on any finished work. Such rendition can range from a straightforward application of either one or two shades of watercolour, up to a highly detailed, photo realistic finish, with, taking a bird motif for demonstrative purposes, individual feathers being noticeable. Therefore, when faced with these two extremes, the question arises as to which of these is most appropriate for tessellation purposes. Again, such matters depend upon circumstances, as a tessellation can consist of a mere handful or literally hundreds of motifs. Therefore, such vagaries must be borne in mind. However, in taking an arbitrary tessellation in one of my favoured formats, namely of a �4x4' arrangement which thus consists of sixteen motifs, I now examine the merits in terms of time of the two renditions. Concerning the �one or two-shade' type, a single motif could be finished in a matter of moments, let us say for the sake of argument, one minute. Now, as regards the �photo-realistic' type, this is obviously the most time consuming, of which for a single motif could easily take, say, an hour to finish. Therefore, upon applying when such renditions to the whole composition, simple calculation shows that we have examples taking sixteen minutes and sixteen hours to complete, a ratio of 16 to 1, respectively. As this thus involves a considerable discrepancy in time quantities, the obvious question that arises is of how best to utilise the time at ones disposal. Effectively, it comes down to between a choice of either a single, high quality (photo-realist) example, and sixteen of simpler, one or two-shades colourings. Now, as no one has an infinite amount of time available, such �excesses' of the former cannot thus be justified in a practical sense, therefore the �simpler' example is the only realistic choice, of which such a rendering permeates throughout this website. Upon examining Escher's periodic drawings for comparison, it can be seen that he too favoured the �simplistic' approach in general terms, albeit he did not regard these as finished works of art in their own right.

This term refers to tessellations that are typically of a more complex nature, created by advanced mathematicians, and are, in general terms, beyond the understanding of most people (including myself). However, despite lacking �theoretical knowledge,' as to their composition, it is still possible to utilise these for representational tessellation, and by so doing, gives one kudos by essentially default. In this field, depending upon the circumstances, the addition of a motif is, or can be, most difficult to achieve, and so the finding of such motifs, if of a sufficient quality, is more praiseworthy than with an arbitrary tessellation.

More specifically, examples of the above include the Penrose non-periodic tiles, of which due to their unique arrangement provide a stern challenge, as, in simple terms, they do not share the same �placement' of contour lines as with normal tessellations. Other examples include dimorphic tessellations, of which I discuss in relative detail on the appropriate page. Further advanced examples can be found in abundance in Tilings and Patterns, of which despite being of a decidedly academic nature, much remains accessible to lesser mathematicians. Indeed, it should be remembered that Escher himself was of a similar nature, making comments that such �hocus-pocus' text (as he put it) was beyond him, but all the same, he still made use of the diagrams.

Now, putting aside the mathematics and design side, I now propose to add a few words as to the choice and selection of media, and furthermore written as appropriate for tessellation. This takes the form of a few �generalities' that can be applied over a wide range of subjects, followed by more text pertaining to distinct �specifics.' The following is therefore primarily aimed for those of a scientific nature, with the assumption that those who are so inclined are unfamiliar with such matters per se, as it is outside of their field. What follows is a brief overview of the subject, as I see no real need to repeat advice/technique that can be found in any one of a myriad of such art books, from which the advice here can thus be followed up upon if so desired.

For the complete beginner, if possible, by far the best method of finding out how to use the various medias is to have some lessons, and not simply muddling through � much time can be saved by having an introduction as to their correct usage.

The description of paints for the utilisation of artists can be divided into three distinct groupings, namely artists, students and general. Those for the artist, as the name suggests, are of the highest quality, utilising pigments that can be (but not always) difficult to normally obtain, and/or require extensive manufacturing procedures, of which the price can be seen to reflect this. So-called �student' paints are of a slightly lower standard, with the more expensive pigments replaced (or omitted) by lesser qualities, and where this occurs such paints are generally described as hues. However, such a range is still more than appropriate for a beginner to painting. The �general' description refers to paints that in any store that essentially sells the paints with other, non-related products. Such materials are of a noticeably lower quality, of a very limited range, with the pigments having been extended with a lot of �filler', resulting in the paints possessing a noticeably duller colour, with again the (lower) price reflecting this, as essentially intended or marketed for children. Quite simply, such a quality is wholly unsuitable, even for a complete beginner, and not worth purchasing.

As the pigment of paint is obtained from a variety of substances, these can naturally vary in their permanence when the finished painting is exposed to light in normal, everyday situations, such as when displayed in a frame. Furthermore, the various medias have different consequences in this matter; as for example, a watercolour has a relatively thin film of paint, whereas an oil painting is noticeably more substantial. Therefore, for any arbitrary pigment, there is obviously more risk of fading to occur in the watercolour with this particular comparison. Furthermore, some pigments, no matter their media, are more prone to fade than with others, on occasions alarmingly quickly, sometimes in a matter of days or weeks, a typical (and all too frequently available) example being Alizarin Crimson (pigment red 88). Therefore, permanency of the paint should also be a factor to take into account, albeit such matters should not immediately concern the beginner, at least not with high quality, finished examples. More in-depth information on this aspect is available in a variety of books and from the manufacturers themselves, of which pleasingly in recent years such questions of permanency has been addressed, with the omission of fugitive pigments in favour of more lightfast ones.

As such, the choice of paper for the selected medium is most important, and it is not sufficient or advisable to utilise any arbitrary paper. Quite simply, there are available, in various sizes and formats, specially produced papers for the express purpose of each individual media, for example pencil, pastel, chalk and watercolour to name but few. All have different qualities due to their manufacturing process, but remain more than suitable, especially for the beginner.

As such, a wide variety of paints abounds, and so the appropriate choice/s are obviously of some considerable importance, as this is the bedrock of the application of colour and therefore discerned before undertaking a finished example. Having thus decided upon a favoured choice, such matters can then essentially be put aside for good. Now, upon contemplating this matter, various media are available, such as oils, acrylics, watercolours, coloured pencils, and inks. Which, if any, are ideal for tessellation purposes? As such, the choice of paints is of vital importance to the subject of tessellations per se , these being in their own way unique, consisting of repeating motifs, which is in contrast to almost any other subject matter, such as a portrait or landscape for instance. Now, upon considering the matter, the colouration of such relatively large numbers of motifs thus makes the choice of paint of importance. Essentially, a medium that is quick to apply and dries quickly, ideally in a matter of a few minutes, if this aspect is applicable to the paint, so that neighbouring motifs can be coloured nearly contemporaneously is obviously highly desirable. As such, each medium has strengths and weaknesses, of which the following discusses individually.

Quite simply, oil paint has few merits with many disadvantages. As it is of a slow drying medium (typically of many days, and in a technical sense of months) it is not therefore possible to colour neighbouring regions (motifs) of different colours, as they would blend at their respective borders, thereby rendering the motifs indistinct. However, if possessed enough of patience, oil colour could indeed be utilised, whereby upon having completed the colouration of the motifs of one colour, and then left to suitably dry, from which the other colour could subsequently be applied. Indeed, despite the above drawbacks, oil colour does have the potential for a most photo-realistic finish, as, for instance, any high quality portrait shows. However, as detailed in the �colour rendition' text, the ramifications behind such a choice of finish must be borne in mind. Therefore, in matters of practicality, oil paints leave a lot to be desired, and so this then effectively rules out the usage of oil paints.

As such, acrylics can be obtained in two distinct types, namely in an impasto, as with oil paints, or in a liquid manner , as with watercolours. Such distinctions must be borne in mind when purchasing. The most notable feature of this paint is that it has been formulated to dry very quickly indeed, in a matter of minutes, or even sooner, depending on the thickness of the paint layer. Therefore, in contrast to oil colours, such a medium has practical potential for tessellation purposes. However, despite such advantages, I still refrain from the impasto type of paint specifically. Quite simply, I dislike the use of such material, finding it inappropriate for my purposes. As such, the �semi-liquid' type is very much like watercolour in its consistency, and indeed its usage is synonymous with this media, incidentally of which I favour. However, even so, this media is used very sparingly, and indeed perhaps upon reflection unfairly so, and as such, I have still to choose this for any finished work. In general terms I prefer to utilise watercolour, which is practically the same when suitability added to with water.

Again, a distinction must be made here, as inks can be obtained for artistic purposes in two distinct types, namely in an opaque, with black or as translucent with any coloured example. Concerning the former, this can, and indeed is utilised in my work for such appropriate tessellations, along with the addition of white for discerning markings (such as eyes or feather lines, simply shown as small dashes and other delineations). However, although such an black and white contrast is theoretically ideal (as previously discussed), in practise I find that the black ink tends to dry unevenly (whichever brand is utilised), resulting in a �non-uniform black', albeit this effect is only visible when examined close-up. Furthermore, the white ink, when applied, is never quite opaque, resulting in lines that are not quite as distinct as they would ideally be. Therefore, because of these perceived drawbacks I tend not to utilise this particular contrast choice as much as I would like. As regards coloured inks, I find these to be noticeably lacking in quality for colouration purposes of the whole motif. As such, the ink is specially formulated to be more suited for line work, and not for colouring large areas of paper, and so consequently, this media is thus effectively ruled out.

Coloured pencils, which despite having perhaps a reputation of an inferior choice of media due to their being thought of as only suitable for children, in their higher quality examples they are indeed of possible usage for �serious artwork.' Indeed, before my interest in tessellation, this has been a medium I have favoured in my surreal artwork where colour was concerned, and as such upon being familiar with its inherent strengths and weaknesses, I duly applied it to my �early' (1987-1990) examples of tessellation.

As regards their application, there is a multiplicity of ways of using them, ranging from a light, and airy pastel-like drawing to a more �intense' approach, with the pencil being applied in a more forceful manner. As such, I tend to favour the latter, this �style' having evolved from the former. However, such a painterly approach is arguably not strictly what they were intended for, as by their very nature, being a tool for line drawing, such application may be thought incorrect. Indeed, when utilised in such a painterly manner, it is obvious that their application is limited to relatively small work, as it would be impractical to cover a large area with effectively a point source. Therefore, if such usage is indeed contemplated, the number of motifs is best kept relatively low, of which a �4x4' format that I favour being of a practical size for this purpose.

The choice of paper is another aspect of coloured pencils that is of importance, as quite simply such a decision has a bearing on the finished work. This I discuss more fully in a general sense in the paper section, from which I simply state here that the type known as Ingres is to be found ideal, in either its paper or board incarnation.

Watercolour is, without any shadow of doubt, my favoured medium for the colouration of the motifs. Quite simply, its advantages are legion, with any perceived drawbacks of no real consequence. Now, the obvious question to ask is why that this should be so. The mediums possible biggest advantage is that large scale and numerous (arbitrarily stated) motifs are �practical', in the sense that when applied by the brush a large area is quickly covered � contrast this with a point source, such as a pencil, of which a considerable longer time would be consequently arise. Although oil paint can be said to have the same possibilities in this way, it also has inherent impracticalities, of which I have detailed above. Also, watercolour can be seen to favour a flat, even colouring (known as washes), of which such an application I particularly favour � such examples abound on this website. Furthermore, watercolour also has variations in its applications, such as the technique known as �wet in wet', whereby this involves letting two colours run together on the watercolour paper, whereupon a blending of sorts occurs. As such, this technique can be a bit �hit or miss� with the results, due to various factors, such as water amount, pigments and the choice of paper all being involved. An example using such a technique is Birds (1) No.2. Yet another advantage is that as it dries very quickly, in a matter of moments, the whole composition can be finished in a single sitting � again, contrast this with oils. In addition, as this is such a popular medium in general terms, with other artists per se, a relatively larger number of distinct pigments are made available by the manufacturers than would occur otherwise normally occur, thereby permitting a greater choice of colours. However, despite the above advantages, watercolour does have a few limitations, in that it is not really advisable to overlay colours, as this essentially �stirs-up' the underlying colour, resulting in a non-uniform colour, albeit with due care and attention this can be alleviated. Another perceived drawback is that of their permanency when exposed to light. However, if not exposed to sunlight in a excessive manner, and along with selecting pigments that are inherently permanent, such matters are not of overly significance.

The paper that I overwhelmingly use for watercolour is known as Bockingford, which although not of the absolute highest quality, possesses a very pleasing surface upon which to work, and furthermore it is readily available throughout any art supply store. On occasions, I do indeed utilise others, for the sake of variety.

Therefore, with numerous advantages and only minor drawbacks, this is the medium I wholeheartedly recommend for you to use.

Therefore, upon having discussed various background materials pertaining to tessellation, I now turn my attention to the actual process of their design, from beginning to the end i.e. from the very first line to the definitive representational tessellation. Of interest would be an examination of how Escher himself went about his own tessellations, the seeing of which would be most illuminating. However, despite numerous books on Escher and his tessellations, surprisingly such knowledge is not readily available, and so it is thus remains that it is still not generally known as to how he went about this. Perhaps the best source for his "preparatory drawings" is the book Visions of Symmetry, albeit only some tantalising examples are given (notably of the �horseman' tessellation) � see page 111. Indeed, although the book starts on this very premise, with the beginning line of the preface (and is again addressed on page 106), the question is still left unanswered. Therefore, such isolated examples cannot thus be given as a full, in-depth study of his design process. This being so, in an attempt to clarify such matters, I thus show here how I go about the whole process. Admittedly, this is still somewhat of a concise nature, as it is beyond the scope of this web page to show what can be at times a most lengthy discussion.

As such, it is possible to compose life-like tessellations in three distinct ways:

(i) By taking an existing, arbitrary tessellation, with straight lines being favoured, and then utilising the outline for the addition of a life-like motif, typically of either birds or fishes. Alternatively, as a variation of this idea, one can experiment with replacing existing lines (generally straight) with arcs as according to the existing symmetry. Furthermore, such examples, by their very nature, do not require any additional study. Of my own efforts that are undertaken in this manner, many examples can be seen whereby I simply utilise a given or self-constructed geometric outline (such as with Birds 1 and Fish 1), and indeed I consider this type a speciality of mine. Quite simply, most people involved in tessellations do not seem to have realised the rich possibilities in this area. Indeed, even Escher himself can be included in this � despite studying such geometric outlines (more specifically of P�lya's article), of his 137 numbered periodic drawings (all of which are shown in Visions of Symmetry), only one such example of this type can be found, namely No.127.

(ii) By taking an arbitrary, given tessellating polygon, such as an equilateral triangle, square, hexagon, kite � one can thus experiment with adding lines as according to known rules of tessellation symmetry by which the resulting shape is known to tessellate. This I thus describe as an essentially "freehand" method. Examples of such a technique are to be seen in numerous books, an arbitrary example being Modern Mathematics Made Simple by Patrick Murphy, pages 194-205, albeit the representational tessellations shown (by the author, a mathematician) leave a lot to be desired.

(iii) By taking a non-tessellating motif, one can then "convert" this into a tessellating shape, albeit of course the original motif is thus something of a "compromise�, as the original line is of necessity distorted, due to its adaptation of the tessellation unit. This particular method is utilised less frequently than with the above, with consequently such examples being harder to find. An example this technique is shown in the book Creating Escher-Type Drawings by E. R. Ranucci and J. L. Teeters, pages 140-142. Further refinements of this idea, computer aided, are shown by Craig S. Kaplan's �Escherisation' process.

Of the above three ways, Escher utilised (ii) to the exclusion of all others. Personally, I favour (i) and (ii), with an approximately even split of usage. To all intents and purposes, I ignore (iii). However, this is not to say that any one method is better than any other, albeit due to the inherent mathematical background to (i) and (ii), these are, I consider, more elegant. Some high-quality tessellations utilising (iii) can indeed be seen on Andrew Crompton's site, of which I gather that this is his preferred way of working.

This initial stage, depending upon circumstances, can be the most time-consuming aspect, as it is here where the motifs quality is determined i.e. its resemblance to usually some animal-like form, and so frequently many trials are thus required � and all without any guarantee of success. Therefore, despite having spent potentially many hours on a tessellation, on occasions one has to finally admit defeat and thus essentially abandon it.

On occasions, depending upon the symmetry of the tessellation, and concerning almost exclusively to the geometrical aspect, it is sometimes possible to have an "additional" placement/s of the �interior motif'. More specifically, this aspect refers to a tessellation tile that possesses reflection or rotational symmetry, of which an example of the former is shown below in its �study state'. Therefore, such a possibility thus opens up further opportunities, in effect an added bonus to the original tessellation. However, such opportunities are, in my own case frequently left as possibilities � depending upon circumstances, such studies can result in such a multiplicity of potential tessellations that it is simply impractical to undertake such high quality finished examples for every one.

In broad terms, tessellations are usually shown in their �finished state' as one of two distinct formats: Firstly, of an �all over' style, with an arbitrary frame that delineates the boundary, generally of a rectangular outline, although others, such as triangles are indeed possible. Secondly, one consisting of an arbitrary number of motifs, with no such boundary, of which their number are usually determined by their underlying symmetry, such as a square, of which 3x3, 4x4 and 5x5 examples are typical for this polygon. Now, as such, there are no rights or wrongs in this matter, the choice of format being a personal decision. However, there are practical implications involved, for reason of which I now outline. Now, in considering the two types, it can be seen that the �all over' example, by its very nature, is more likely to consist of a larger number of motifs than with the pre-determined �arbitrary' example. From this, it will thus be obvious that the more motifs that are drawn, the longer it will take to subsequently add colour. As such, this is not a matter that is to be to be taken lightly, as considerable amounts of time can thus be involved, aspects of which I discuss under �colour rendition'. Therefore, as a matter of general policy, I thus favour the �arbitrary number' type, solely because of the time factor (and not one of aesthetics), as the �boundary type' takes far too long to do in a practical sense. Although examples of both types can indeed be seen, it is noticeable that the �arbitrary number' is favoured, with examples of �all over' very few and far between, generally undertaken in my �early' days, when such matters were not thought through.

Now, upon having determined all such possible placements for the motif, followed by the choice of which example/s to utilise, the next task is to begin actual preparations for the eventual finished example. Such a task takes the form of drawing out onto graph paper (utilised for the sake of accuracy) the previously determined choice, with all the lines that are intended to be shown as on the finished example. An example of this is shown below.

Here I take the previously completed graph paper drawing and attach both graph and watercolour paper together with removable tape, for the purpose of tracing through the example on graph paper. Now, as watercolour paper is opaque, it is thus of necessity to place the above attach papers onto a translucent surface, such as a light box or a window pane. (I prefer the latter, as it gives the maximum of light to see the example on graph paper.) When thus suitably attached, the underlying drawing is then traced through, in pencil, as shown below.

Upon separating the pencilled drawing on watercolour paper from the underlying graph paper, I am now in a position to add colour. The choice of the colouring scheme, having thus selected beforehand, with the previous experiments in this field having been assessed for their validity. The reason for such preparations is so that no wastage in both time and in relatively costly watercolour paper will occur, as well as permitting that ones attention is solely focussed on the actual application of colour, with no �last minute' changes of colour or technique required.

Now, when the addition of the colours has been completed and left thoroughly to dry (preferably overnight), the final stage is thus set for the delineation of the outlines by the inking-in of the individual motifs. Without doubt, this aspect is conspicuously absent with most other people involved in tessellation; with upon having added colour, the motif outline is then basically left alone i.e. non-delineated. However, with my own work I consider this aspect a necessity, as only by doing so are the motifs clearly and unambiguously delineated. Indeed, the more complex or involved the colouring scheme is, the more this is required, as without this the individual motifs, as a general principle, are not readily discernable. Such a �finishing- touch' is, in my opinion, a noticeable asset in this matter. Numerous examples in which Escher utilised this finishing touch as well can be seen in his periodic drawings, albeit such a concept did not immediately appear in his work. Upon examining the periodic drawings, it can be seen that considerable uncertainties pertain in this matter, as it is not always clear if such delineation was contemporary with the tessellation or was indeed added subsequently. On occasions, Escher went back to old works and �improved them', with colour and delineation, noting the date on the original work when so done, for the eventual purpose of a forthcoming book Graphic Work of M. C. Escher . For example, the first such instance of delineation, of No.15, can clearly be seen to be an addition, of 1963. Therefore, despite on occasions as having been delineated it is feasible that instances occurred whereby no such recording was made, and so assigning a definitive beginning of such matters is fraught with difficulty. Indeed, it is plausible that Escher did not in fact begin such delineation until 1963. Whatever, the above �chronology uncertainties' are really of academic interest only.

Now, returning to delineation per se, the decisive test of its merits is that of directly comparing �before' and �after' examples, thereby determining if such an aspect does indeed result in an improvement. Some (inadvertent) example of Escher's can be seen in some of his periodic drawings, for example No.15 and 21, from which the improvement is, or should be, self-evident. Without any doubt, such delineation noticeably �permits' what I term as �readily viewing', as in effect the eye unambiguously sees the motifs at-a-glance without having to, as it were, �struggle'. A variation on this idea is that in place of having the surrounding delineating line denoting the motif in black, a thin, narrow space is instead �surrounding' or delineating the motif, as exemplified by Escher's periodic drawings No. 72, 79 and 94, and in effect, this same idea applies to 111-114. However, as can be seen by the few times Escher employed this, he obviously did not favour this particular variation, possibly due to technical reasons. Concerning my own examples, the addition of such a black delineating line is frequently to be seen, and as a general rule invariably employed. On rare occasions, examples can be seen whereby no such line is to be found, thus effectively going against my own rule in this matter. The reason for such an apparent contradiction is that upon beginning my studies I did not immediately realise the importance of this aspect, being content with the motifs colour contrast only. Now, although I would very much prefer in retrospect to have such examples more properly finished, as in the above manner, upon due reflection, I refrain from the temptation of so doing. Quite simply, I consider such subsequent addition �inappropriate', as the tessellation is inherently of another period, with a difference of possibly many years. As such, I much prefer to let �old' work remain �as intended' for their own time. If such improvements are indeed in order, I much prefer to entirely re-do the entire tessellation. Some idea of the potential confusion in this matter can be seen in Escher's work, with additions of approximately twenty-five years later, thus rendering considerable uncertainties in this matter, as discussed above.

Upon the completion of this final stage, the tessellation is now in a finished state, and as such should be regarded as a work of art in its own right, resulting in the creation of a final, definitive tessellation of the various intricacies and ramifications of ones investigations. Indeed, frequently such a single choice is simply inappropriate, thus necessitating additional definitive examples. Interestingly, Escher was not essentially content with this, as he regarded such examples as sources for further work and not as finished examples per se. Indeed, whether by accident or design his colour rendering is most perfunctory, especially so of his earlier works, and so thus reinforces his �assessment' of his own work. In addition, these are all, without exception, shown on squared paper and not the more appropriate quality paper such as specially designed for watercolour. However, subsequent to Escher, the general opinion is that such examples are in indeed to be regarded as �acceptable' as works in their own right. Therefore, although complete by this criteria, if so desired it is indeed possible to �continue' and in effect follow Escher's advice of � do something� ', albeit he essentially refers to the motifs in this matter, and not to the person here. As such, there are two distinct ways of going about this, or indeed in combination with each other, namely with counterchange or development, both of which Escher frequently employed, of which I discuss below.

As such, the term counterchange is perhaps not one that people are too familiar with, even with those who are interested in tessellation, and so to clarify this matter as applied to tessellation I thus simply define a counterchange as

Even so, despite this, the definition still lacks clarity, as such a concept that is more readily viewed than described, of which the saying �a picture is worth a thousand words' is most appropriate. Therefore, the reader is referred to the counterchange page, where the examples will make clear such matters in an instance rather than with any amount of lengthy words.

As such, counterchange can essentially be regarded as an additional aspect to a finished tessellation, of which such a feature can be regarded as an optional extra and is certainly not to be regarded as �compulsory'. Indeed, some tessellations are wholly unsuitable in an aesthetic sense, for reasons of which have been discussed elsewhere in Processes. Essentially, upon the completion of an �aesthetic' finished tessellation, it is then possible to continue or �build' upon this with a counterchange, in essence following Escher's advice in the above book � . . . do something, come out of there and show me what you are capable of!' . Indeed, examples of counterchange were readily utilised by Escher in his prints, in various formats, most notably with Day and Night, Sky and Water 1 and Verbum, in which the motifs are all eventually �released' from their original tessellation, of which in both concept and execution all three are quite superb, all of which are exemplary examples in this field. Indeed, no praise is high enough for these, essentially the epitome of the genre, of which such work was perhaps somewhat surprisingly amongst his earliest efforts (1938 twice, 1942 respectively) and not, as one may have reasonably assumed, the product of vast experience. Some further counterchange examples of Escher's, undertaken for illustrative purposes, albeit of lesser inherent quality of these matters, in which the motifs are in contrast �retained' in their tessellation outline, are shown in his Regelmatige vlakverderling essay, translated in the book Escher The Complete Graphic Work, plates II-V, pages 155-173.

Now, the �intricacies and technicalities' of counterchange are somewhat involved, albeit still readily understandable for those of a non-mathematical background, and indeed such matters could be the subject of a whole book. However, most regrettably, very little, if any, information is available on this matter, with the background to Escher's examples being unpublished. Furthermore, despite the above-mentioned examples in the book, he does not go into specific detail. Therefore, upon trying and failing to locate any information on the subject, to understand such matters I had to essentially analyse and recreate Escher's examples, from which after due study I could then apply the theory behind these as appropriate to my own examples.

Now, a counterchange can, in theory, be applied to any tessellation, albeit some are more suitable than others, due to two factors, namely of the motif(s) themselves, birds, fish etc and their inherent symmetry arrangement, of which the motifs orientation are taken into consideration, more of which below. Another aspect to take into consideration is the format, of which various types are possible, such as lineal, hexagonal etc. are possible, the only limit essentially being ones imagination, all of which I discuss.

As such, the choice motif is of the utmost importance, and indeed cannot be overstated, as by so choosing an appropriate tessellation that is based upon an ideal motif(s) the counterchange will thus be greatly more effective in an aesthetic sense. Now, as a general principle, a tessellation of two (and not one) distinct motifs is ideal, as by so doing, the process of counterchange is then made more obvious by the subsequent �switchover' or transposition' of motifs, and ideally they should also have some connection or relation to each other in a �real-world' sense. An ideal example of this would be birds and fish (of which, purely by a quirk of fate, such a combination of motifs is relatively common in tessellation). In contrast, a tessellation of disparate motifs such as, say, a bird and horse that obviously has no such connection to each other, and thereby the utilisation of such motifs would therefore be �inappropriate' as according to the above.

Again this is of the utmost importance, as by so choosing a tessellation based upon the ideal symmetry arrangement the counterchange will thus appear correct in its aesthetics. For example, let us consider a arbitrary tessellation of the type as favoured above, of birds and fish, of which we have two distinct example, namely of a bird and fish, both of the same upright orientation, and also another, of bird and fish, this time with the fish motif upside down in relation to the bird. Now, when each of these is thus subjected to the counterchange process, in whatever format, the former will result in both of the motifs remaining in the same upright orientation, whilst the latter will have an upside down motif. Consequently, these can thus be described as �ideal' and �absurd' respectively, from which it is thus self evident as to which of the two symmetry arrangements is to be preferred.

As such, the format refers to the description or overall shape of the composition, such as a lineal strip of any convenient unit thickness (the most frequent being of vertical or horizontal orientations), or of a geometrical shape such as a parallelogram, square, and triangle, hexagon �or of any other. Furthermore, the selection of the most appropriate format is not arbitrary, as both the type of motif along with their underlying polygon influences the selection. To give an arbitrary example, two motifs, say of the bird and fish motifs, arbitrarily based upon a square. This would thus naturally favour a square format, viewed in an up and down direction, for reasons as stated above. Alternatively, a vertical strip could be utilised, of the same premise.

Of some importance is the tempo of the counterchange, and more specifically this refers to how �quick' or �slow' in relative terms as to how the counterchange proceeds in its course. Quite simply, there is an arbitrarily ideal, essentially between �quick' and �slow' whereby the counterchange can be said to be of the optimum duration, neither taking an absurdly short time or an apparent eternity to complete. Examples of the �right balance' are to be seen throughout Escher's prints, of which the three mentioned above are exemplary of the art. However, on occasions Escher did indeed utilise the �quick' type, notably with a commissioned work of a new year's greeting card for L. and K. Asselbergs, which then apparently spurred on further examples of the type in quick succession. As such, the judgement of these as to their aesthetics is somewhat arbitrary, and so can thus be said to be of a personal choice. Indeed, the quick examples are refreshing in their own right, having an immediacy that is lacking in the more grandiose examples, and despite having a personal preference for the latter type; these are, in their own way, of equal merit.

As such, the term development is utilised in the sense that from an initial start from an arbitrary tessellating polygon, the lines of this take on a more and more life-like motif, with upon �completion' the tessellation is thus shown fully developed, with the motifs being �retained' in the grid, hence development. Escher utilised many examples of this technique, such as with elements of Metamorphoses I and II, Development I. Again, these are all of a very high quality. However, although the impression given with these is indeed of a development, in actuality this is somewhat of a misnomer, as the process to create this effect is actually the reverse � essentially the motif is undeveloped back to its underlying polygon. However, the interpretation is certainly intended to consist of a �start to finish' nature. As a variation on the theme, instead of the motifs being �retained' in their grids, it is also possible to �release' them, from which a more realistic bird can be obtained. Again, Escher frequently utilised this, perhaps most notably with Day and Night (arguably his best print in terms of 'tessellation aesthetics') and Liberation. Furthermore, the two types of prints detailed here involve some of his most popular works, of which, in general terms, I much prefer in comparison to his later work of non-tessellating 'spatial constructions'.

As such, on occasions it is possible to combine both counterchange and development in a single print, essentially a tour de force of ones skills in tessellation. An example of this by Escher is of Verbum, an exemplary example of these two possibilities, and one of my own is Counterchange No.4. As such, due to inherent difficulties that are necessarily involved, such a composition can involve a lot of preliminary work, requiring numerous studies to compile the best composition. Therefore, when such a composition is thus so sought, it will be found that only high quality, appropriate motifs should be utilised, as such �investment' in both time and study is not justified for a tessellation of an inherently inferior quality.