Introduction
Although not central to the creation process, of interest is to examine one's background and then how one arrives at the subject of tessellation, and by so doing seeing if any specific route is advantageous in producing representational tessellations. Essentially, it can be said that one arrives at the topic of tessellation as an offshoot from another discipline, as it is not a core subject per se as with, say, mathematics or art.
A convenient way of seeing different people's backgrounds are given on Andrew Crompton's website, in which he has a section showing representative examples of all the (47) people who are contemporary in the field, mostly, but not always, with brief details of their background, some of these being multifaceted. However, there is a notable discrepancy in their respective abilities (in quality, as some quite frankly are unworthy of inclusion, and quantity, with some showing very few examples), albeit as a general guide to background the details are still valid. From this, it can broadly be stated that one arrives from either a scientific or artistic discipline. Now, as can be seen by the practitioners of the art, it is noticeable that a wide diversity of subjects of the two basic disciplines can be discerned, of which the following thus discusses:
Artistic Subjects:
Art
Number of people: 9 (Adrian Fisher, Robert F. Kauffmann, Makato Nakamura, Royce B. McClure, Jim McNeil, Geoff Pike, Herb Robb, Christine Shearer, Henk Wyniger)
Clearly, this category, with nine representatives, is by far the most popular field, perhaps surprisingly so, with tessellations indisputably being the domain of a mathematical nature. As such, I consider that an interest in art, and not necessarily ability per se (to whatever degree) to be of far more importance than that of mathematical ability, for reasons of which I discuss below.
In general terms, most people who arrive at tessellations do so via this way, despite an essentially nonmathematical background (if at all). Perhaps somewhat surprisingly given their background, they undoubtedly produce the most worthwhile examples. Why should this be so? Simplifying, art can be said to be about imagination, of which although it would be crass of me to state that mathematicians lack this, none the less they do seem to be lacking, at least as applied to representational tessellations.
Art can assist in tessellation in two distinct ways, with ‘drawing ability' (i.e. the drawing of the motifs) and also with the essentially secondary aspect of coloration, both of which the artist would be more familiar with through his every day experience of painting and drawing. However, I would not say that an interest in art per se is essential, although as representational tessellations are obviously of a visionary nature, the interest is obviously relevant. However, this is not to say that anybody with an interest in art will be capable of producing worthwhile efforts. Indeed, despite there being many artists the world over, the number of such artistically inclined people interested in tessellation is relatively few. Generalising, artists can be divided into two camps, those that can be described as of a ‘free spirit', and consequently having no mathematical interest per se, whilst the other type can be described as of a more ‘thoughtful' nature, essentially being not satisfied with a ‘simple' portrayal of the world around them. Therefore, the latter are the types of people who thus possess the potential for producing representational tessellations. Of significance is the fact that Escher himself came to tessellation from an artistic background, having had a childhood interest in drawing. Interestingly, his more mature art in the 1920s (with the exception of 192627 where he briefly studied tessellation) cannot be said to be as a forerunner of what was to come. Essentially, his work was concerned with the graphic field, consisting of portraits and landscapes, a very common theme throughout art in general. What is perhaps somewhat unusual about the landscapes is the viewpoint, typically as seen form a high vantage point. Although other artists have shown an interest in the same thing, these were very much secondary to their art. In contrast, Escher seems to have had a decided preoccupation with this viewpoint. Furthermore, each of these is carefully rendered, with a highly detailed, realistic finish. Certainly, they are very good in a technical sense, and furthermore somewhat unusual (due to the viewpoint), but these possess nothing by the way of inherent originality.
Personal
As regards my own background, if I have to pin a label, this would be the one I choose to put myself in. My interest in art began in my early twenties, with pencil drawings of conventional subjects such as landscapes and animals. However, very quickly, a ‘simple' portrayal did not satisfy me, of which I then tried to add complexities, drawing objects as shown in a mirror. From this I experimented with ‘effects', attempting ‘double images', as made famous by Salvador Dali (whom Escher also admired). This was then followed by a surreal period in general. From this, I then ‘progressed' to op art, which can be said to be mathematical in nature. Consequently, such an interest can thus be said to lead to a more orderly approach in thought, rather than the typically less structured world of art. Therefore, finding myself in an mathematical domain, an interest in tessellation art then is but a short step, but still not inevitable. Such a step occurred with myself purely by chance. 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 Reader's Digest magazine, of March 1981. This contained a brief article on Escher titled ‘The Artist Who Aims to Tease' by Greg Keeton (incidentally, does anyone have background detail as to who Keeton is), pages 3741, containing a handful of prints of Escher's. Amongst these was Day and Night, which intrigued me, 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 was essentially put aside. Not until much later did not turn my attention to tessellations in a practical sense, in 1986. As stated above, although finding myself in a mathematical domain with op art, there is still no direct connection to tessellation. However, I recall that having seen a general discussion of mathematical art in Sources of Mathematical Discovery, this containing tessellations amongst other matters, this thus aroused my interest in the subject once again, this time in a practical sense. From this, I could then proceed on relative firm foundations, at least for someone with no mathematical background of any real substance.
From the above therefore, it can be seen that I have a distinct lack of official training – no artistic qualifications, and certainly no art school background, more or less as with Escher. Consequently, demonstrably, such an ‘official' artistic background is not essential. Certainly though, it remains desirable, or at least to have some degree of interest in the subject.
Architecture
Number of people: 3 (Andrew Crompton, Jorge Molgora, Sanjay Sundram)
As such, despite appearing as disparate subjects, another relation from an arts field to tessellation appears to be architecture. 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 practitioner (not on Crompton's site) is William Huff, has also produced many interesting examples related to tessellation, with parquet deformations.
Interestingly, Escher was 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 in favour of the decorative arts.
My own interest in this field is decidedly limited, with only a passing interest, if at all.
Science Subjects
Mathematics
Number of people: 3 (Doly Garcia, Klara Gelencser, Robert Ingalls)
Perhaps surprisingly (and pleasingly for those of limited mathematical ability, including myself), despite tessellations having an obvious underlying connection to mathematics, mathematical ability per se is not a requisite to producing representational tessellations. Indeed, ability in this field is almost of an unimportant matter, albeit of course an interest the subject is. For instance, numerous experts, professors of mathematics, who have an interest in tessellation, such as Branko Grünbaum, G.C. Shephard and Doris Schattschneider, all of whom have made significant contributions to tessellation theory, have to my knowledge not produced any lifelike examples. Another leading light, the mathematical physicist Roger Penrose, who is able to visualise in multiple dimensions, has at least made some attempts with his nonperiodic tessellations, with chickenlike motifs. However, it must be admitted, these leave a lot to be desired in terms of their inherent quality. Also, there are 'lesser' mathematicians, such as Patrick Murphy and Bruno Ernst (arbitrarily chosen, for reasons of awareness), with mathematical degrees in the subject, who although are of a lower standard than the above luminaries, they are by any means still of a considerably advanced standard in comparison with the general populace. Murphy in Modern Mathematics, pages 194205, dedicates a chapter to tessellation, with a discourse of basic elements of tessellation per se, intermingled with attempts at representational tessellations. Similarly, Ernst does more or less likewise in Escher The Complete Graphic Work, pages 135154. However, although both are splendid mathematicians, far superior to Escher and myself, they can quite plainly be seen to be lacking in their attempt at an representational aspect. Indeed, despite Ernst having had the considerable benefit of meeting and discussing tessellations with Escher, he still could not produce anything of note. Admittedly, all the above mathematicians may be more concerned with the mathematical aspect rather than the artistic, but the fact remains, despite their undoubted knowledge; they seem incapable of producing quality examples. Demonstrably, ‘intelligence' or exalted mathematical ability is self evidently not a significant factor.
Essentially, the mathematics underlying tessellation is generally (but not always) of the most basic nature, for example being aware of principles of translation, reflection and glide reflection, the elements of which are not especially difficult even for the nonmathematician to understand. Furthermore, ‘assistance' is available by being aware of ‘general rules' of mathematics without necessarily having or requiring the knowledge to prove such matters, such as knowing that any triangle or quadrilateral will tessellate. Again, this is most simple to follow if one is lacking in understanding the underlying reasons.
Of interest is in knowing Escher's mathematical ability. Escher frequently proclaimed his lack of ability in the subject, of which his schooldays was apparently unexceptional, from which I would place his achievements in approximately the middle of the general populace, neither exceptionally poor nor good, just ‘midrange'. Upon leaving school, he apparently did not pursue the subject, as he concentrated on his art. Consequently, having essentially forgotten his school day mathematics, he essentially began his mathematical studies as regards tessellations at the relatively late age of 38 (this neglecting some ‘minor' tessellation studies of 1922 and 1926 or 1927 that although are of interest are essentially inconsequential). For this, he had to use his own intuition, essentially of necessity, as there would have been no books on the subject. Subsequently, he obtained various mathematical articles containing tessellation diagrams, which proved useful to his studies. As such, his mathematical work (tessellation) can be seen to be quite plainly ‘basic' in the strictest sense. Nothing of this work is of abstract or advanced nature, with for instance equations or algebra, just basic geometry.
As regards my own mathematical ability, this is decidedly lacking, the only qualification being from my schooldays, a CSE 3. As a rough estimate, this would place me in approximately the lowest 10% of the general populace. Indeed, it may be a good deal lower. As with Escher, I had a pause in my mathematics from my schooldays, starting again at the age of 27. In contrast to Escher, although there was available to me more books on the subject (invariably general textbooks, which I did not understand), of necessity I initially stumbled in the dark, having to think things through for myself. The biggest advantage I had over Escher was that I had was that of viewing his own tessellations, the first book of note being The Graphic Work of M.C. Escher. Essentially, I could build upon his pioneering work having seen the possibilities in the field. In contrast, of necessity, he had to start from scratch. Nothing in my background would have predisposed me to such an interest in tessellation studies, and undoubtedly, I would not even have begun an interest without having seen his work.
Demonstrably, as both Escher and I are thus quite plainly lacking in mathematical ability, but obviously not in the art of producing representational tessellations, this thus shows clearly that ability in mathematics is not strictly necessary. Indeed, arguably there can be something said for relative ignorance in the subject.
In contrast to most scientific disciplines, say chemistry or physics, whereby the 'basics' of the subject have been established for hundreds of years, with which there is no likelihood for original work by the newcomer, with originality in the field having been essentially closed to all bar the ‘expert professor', mathematics is somewhat different. Quite simply, as it is such a vast subject, no one can possibly hope to grasp all its aspects. Consequently, originality is still a possibility, of which for some the charm of discovery, of doing something previously unknown accounts for much of its attractions. As such, there is a whole realm of possibilities in the type known as recreational mathematics, of which the amateur, no matter how lowly his or her ability is, can indeed contribute. Geometry is a case in point, with many contributions made by those with an interest in mathematics despite not possessing exalted ability. Indeed, Escher admirably demonstrates this. Consequently, the subject thus provides an inherent attraction.
Mathematics can be said to be of a broad church, subdivided into distinct sections. Most of these have no bearing on tessellation, such as algebra or group theory, to give two arbitrary examples, and so consequently, study of these is strictly unnecessary. Therefore, by simply concentrating on one small aspect i.e. tessellation, one is thus not distracted by ‘inconsequential' matters that have no implications on the subject, and so more time can be given to such specifics, rather than attempting a more broader understanding of mathematics per se. Essentially both Escher and myself accept our limitations, and thus concentrate on what we do best.
Crystallography
Number of people: 1 (Imameddin Amiraslan)
Another avenue to tessellation appears to be crystallography, albeit in this field such an interest lies very much secondary to mathematics and art, it being of a somewhat specialised scientific nature. Some eminent crystallographers have indeed taken an interest, perhaps the most noteworthy being Caroline H. MacGillavry, a professor of crystallography. MacGillavry did much to popularise his works with the book Symmetry Aspects of M.C. Escher's Periodic Drawings, this being specifically aimed at addressing crystallographic matters with the aid of Escher's drawings, and additionally met with Escher.
Interestingly, Escher's halfbrother, B. G. Escher also took an interest in this subject 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 the above people having the advantage in both theoretical knowledge and indeed in personal contact, the latter of which must be greatly advantageous, no representational tessellations of any kind or worth has emanated from the above people. Again, another (scientific) discipline where the practitioners lack the ability of representational tessellations.
Upon having looked at various crystallography books, relations to tessellation can indeed be seen. However, such books are typically of a somewhat advanced, complex nature for a layperson in such matters, and so in consequence are of limited use, if at all. Indeed, no instance of a crystallographic diagram has led to a representational tessellation of my own devising.
Computer Science
Number of people: 5 (Sam Brade, Chris X Edwards, Craig S. Kaplan, Robert F. Kauffmann, Peter van Rooij)
The above term is used somewhat loosely to describe the above people in this field. Some are considerably more advanced than others, but rather than having further subdivisions, for reasons of convenience I thus group together all the ‘computer practitioners' under that classification.
A relatively new field is that of the computer, of which this can be seen to have implications for tessellation. As such, this can be separated into two distinct parts, namely that of the design itself and the subsequent drawing and repetition of this as a tessellation. Perhaps of most note in this field is Craig S. Kaplan, being an expert in both computer graphics and in mathematics, in which he uses the potential of a computer for composing representational tessellations with his ‘Escherisation' process that simply would be impractical to using the old fashioned pencil and paper. Such matters have implications over the ‘traditional' method, as human design would not then be required. However, such a situation is not immediately foreseeable. Consequently, the computer is still lagging far behind the human mind in these matters, and as yet it shows no sign of making the human aspect redundant.
Aspects of the computer are discussed more fully in Essay 10, The Computer and its Application to Tessellation.
Of note is the fact that the computer was not available in Escher's time, but as can be seen by the quality of his tessellations as when compared with modern efforts, any designed/composed by computer pale in comparison. Likewise, as with Escher, my efforts are the product of my thoughts. Although I have indeed made some attempts by computer, it is still far more practical to design by hand. Demonstrably, a computer is not a prerequisite in their design.
Psychology
Number of people: 0
Although as a category there is not a single representative from Crompton's listing, as psychology does indeed has a bearing on tessellation, it would be inappropriate of me to ignore this subject altogether. Consequently, it is thus included.
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. Escher apparently studied various articles of these matters, as documented by Marianne L. Teuber in an article in Scientific American of July 1974. 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. However, it is safe to assume that Escher had an interest in the subject to some degree.
Escher used two different aspects of figure and ground, apparently, of his own devising. Examples can be seen in the prints Sun and Moon and Development I. The former shows alternate views, whilst the latter, in which the background develops in intensity, from a grey border to an increasingly clearly defined black and white. Upon investigating various psychology books and articles, albeit by no means exhaustively, as yet I have still to find direct evidence of some previously established principle of figure and ground containing details of the type that Escher used in his prints. As such, his usage appears to be of his own devising. If anyone does indeed have relevant sources, I would be most interested in receiving details.
Summary
So, what can be gleaned from the above? What is it that gives so few people the ability to produce representational tessellations? Having analysed every likely background, all have been shown to be individually unnecessary. Certainly, it is advantageous to have a background in the core subjects of mathematics and/or art, as both of these underpin representational tessellations, but is not essential in any way, as demonstrated. More exactly, an interest in the subjects is (but not necessarily of a advanced nature as regards mathematics). Consequently, without some degree of interest in at least one of these it is hardly possible to begin, as the concept would not come to mind. However, what I consider as by far the most important factor, that almost indefinable quality that few people possess, the ‘X factor', is something that is not readily categorised, of which I simply state as the potential for shape recognition, of which I discuss in Essay 2, Abilities and the Art of Shape Recognition.
Agree/disagree? Eme.
