1. Background Principles Underlying the Prints
1.1. Metamorphoses
1.2. Counterchange
1.3. Self-Similarity
1.4. Circle Limits
1.5. Development
1.6. Tessellation Fragment
1.7. Mathematical Concepts
1.8. Compositional Ideas
1.9. Economy of Effort
1.10. Commissioned Prints
1.11. Techniques

Introduction
As such, due to the unique nature of the content and background to his tessellating-based prints, the ideas and concepts that Escher portrays are frequently not readily apparent, at least not without considerable effort being involved, of which ones interpretation may even then still be incorrect. Consequently, some sort of explanation by himself as to these matters would be warmly welcomed, essentially clarifying such matters. However, in general terms he appears to have been most reluctant to do so, essentially contenting himself with brief comments on an ad hoc basis, with the occasional article, along with more ordered comments, albeit of a generally lightweight nature, in The Graphic Work of M.C. Escher. Such ‘reticence’ to give a more detailed account is regrettable, as with the more complex examples, such as with Metamorphoses II, III and Verbum, this aspect is of even more importance due to the intricacies involved, and so consequently some point or aspect that is of the utmost importance will be at best misunderstood or at worst simply incorrect. Indeed, there is simply no substitute for these matters, and it is personally with regret that Escher did not at least give a more exact explanation of these more involved examples. Although further comments by him were prepared for a series of lectures in the United States in 1964, to be delivered in person, due to ill health he was not able to travel, and so consequently, these insights were not given. However, the text of these has been preserved, and are to be found in Het Oneindige: M. C. Escher over eigen werk, with an English translation by Karin Ford, Escher on Escher, Exploring the Infinite. Unfortunately, these are still very much of a broad nature, of generally one or two sentences of simple descriptions. Although various authorities have contributed to their understanding with their own interpretations, these are very much in a similar lightweight manner, mostly on an ad hoc basis. These vary in intensity and substance, from a few words to whole paragraphs. Consequently, to compose a unambiguous list of references to his prints is of necessity a somewhat eclectic nature. With this caveat in mind, those that are of some significance are thus duly noted below in the ‘discussed in’ reference. Surprisingly, a large number of the prints have still to be commented upon. In contrast to the previously ad hoc discussions, I here examine each one in relative detail, as according to chronology, with appropriate comment.

As such, his prints of a tessellation nature have been commented upon and analysed by various authorities, of which despite having a distinguished backgrounds in mathematics and scientific disciplines in various fields, are generally lacking in one, if not the most important aspect, namely that of having practical experience of creating such representational tessellations. Although it would indeed be foolhardy to denigrate their abilities, it is self evident that without such practical experience their comments, although of value, cannot possibly penetrate to the essence of the subject. Consequently, perhaps the people most qualified in this sense are those that do indeed undertake such matters, and although as previously mentioned, despite generally lacking a scientific background, their abilities makes them thus more uniquely qualified to give their views on his tessellation prints than otherwise, of which in the same spirit I now do. This takes the form of a few generalities concerning the types of prints in terms of their distinct aspects, followed by an analysis of the content/compositions in his tessellation-based prints, in chronological order.

1 Background Principles Underlying the Prints

1.1 Metamorphoses
A favourite device of Escher's was to utilise metamorphosis effects. As such, these were designed so that the eye could scan, generally in a long strip type of format, either arranged vertically or horizontally. When so appropriately viewed, the motif gradually becomes the background, without the viewer essentially being aware of the effect, from which the previously mentioned background then becomes another motif, resulting in a most pleasing effect on the eye. Such effects are taken to the ultimate in Metamorphosis II and III, where a stream of such changes occurs. As such, examples of this type are difficult to undertake because of a number of reasons. As this effect depends on the source tessellation having two (or more) distinct motifs, such examples are less frequent (and therefore less available) in comparison to tessellations with only a single motif, as discussed in Essay 1. Furthermore, tessellations of this type can be, but not necessarily so, of an inferior quality, essentially unsuitable. Another potential drawback concerns the tessellations aesthetics, as a situation may arise whereby a high quality has been composed, but due to the symmetry system the motifs may be, say, upside down in relation to each other, and so essentially unsuitable on account of ‘absurdity’ as Escher himself stated. Therefore, despite having possibly many tessellations at ones disposal, very few of these in practise will be found suitable. Even then, presupposing that reservoirs of suitable tessellations are indeed to hand, these are thus found as not necessarily suitable for linking disparate elements, from which of necessity tessellations that are indeed suitable then have to be specially composed, with all the difficulties thus imposed by such restrictive practises. Consequently, the number of such metamorphic type of prints is thus relatively low in number.

1.2 Counterchange
As such, counterchange is related to metamorphosis type of effects in that a ‘switchover’ takes place, albeit with one difference, in that the motifs essentially remain static in their underlying grid, whereas with metamorphosis inherent movement can be seen to take place. Although the difference between the two is not great, as a distinction between them can indeed be discerned, these are thus noted. Further, more specific details pertaining to all aspects of counterchange are discussed in Essay 1.

1.3 Self-Similarity
A somewhat lesser feature of the prints is that some are based upon the concept of self-similar tessellations, of which simply stated, these consist of tessellations where despite the proportions of the tiles remain the same, their size either increases or decreases (with implications of infinity) upon ones own interpretation. Such examples are a later development, as not until 1955 was the first of these so composed. Notably, Escher devised his own scheme, of subdividing a square into right-angled isosceles triangles, of which although inherently simple (of which anybody is surely capable of drawing), had not as yet been investigated.

1.4 Circle Limits
Another type of self-similar tessellation, not of Escher's own devising, is that of the hyperbolical plane, of a somewhat advanced mathematics at a university type of level. Escher first saw this in 1958, as a figure from ‘A Symposium on Symmetry’ by H. S. M. Coxeter, from which he then composed four prints on a ‘circle limit’ theme. Although the mathematics behind this was way beyond Escher's understanding, in simple terms he managed to adapt the underlying lines for his own purposes, of which he utilised as the framework for Circle Limits I-IV. Of these, I, III and IV are of a life-like nature, albeit the ‘fish’ motifs of I and III are noticeably of an inferior quality, whilst II consists of crosses. A question to address is to why Escher chose to utilise such relatively poor quality motifs for I and III. Possibly, although not necessarily, this was dictated by the necessity of having an ‘vague, ambiguous’ outline due to the restrictions of the reduction process. However, as IV is indeed of a high quality motif, and furthermore Dunham, in Art and Science, pages 398 shows computer drawn images of drawings 20 and 54, both of inherently better quality than with the ‘fish’ Escher utilised, from which ‘vague, ambiguous’ lines are not thus necessarily a prerequisite. Interestingly, Escher apparently created these motifs for I and III on an ad hoc basis for this from previous studies, as there is no numbered drawing preceding this, albeit subsequently, in April 1964, he did indeed complete definitive examples.

1.5 Development
As such, an ‘development’ can be seen in the prints, of which I define the terminology in a specific manner as pertaining to the growth in stages of an increasing angularity of the outline of a tessellating tile until a fully formed outline is thus arrived at. This is generally of an animal-like motif, albeit on occasions a purely geometric example can be seen, as at the near beginning of Metamorphosis III, with squares being transformed into rhombs.

Concerning the process of composing such examples, although the impression given is one of growth, in actuality decomposition takes place, of necessity. Essentially, the given tessellation is decomposed in stages back to its underlying polygon.

1.6 Tessellation Fragment
Some of the prints consist of a ‘tessellation fragment’, essentially of a self-contained block of an arbitrary number of motifs, noticeably without any rectangular border. Generally, these were composed for the purpose of inclusion in publicity material for Escher's exhibitions, and as these are thus essentially a repeat of a plane tessellation in a different format, they are of no real consequence per se.

1.7 Mathematical Concepts
Although by its very nature a tessellating print can be said to be mathematical in concept, the term can be used in a more exact manner. Occasionally, instead of the tessellation per se being the central focus, it became secondary to the abstract idea being portrayed.

1.8 Compositional Ideas
Note that as regards to his compositions, the idea came overwhelmingly not in an abstract sense per se, but first with a tessellation, from which the composition than sprung forth, with the motifs thereby naturally ‘suggesting’ some concept or idea. For example, drawing number 18, of birds and fish, of which the concept of sky and water respectively being an obvious connecting thought, resulting in the print Sky and Water I. Although it is indeed possible to begin with an abstract idea and then try to compose a tessellation with a thought in mind (of which Verbum is a notable example), such a planned effort is fraught with difficulties due to such restrictions of potential suitable motifs. By far the easiest way is to simply compose an arbitrary tessellation and then come up with a compositional idea for this subsequently. Essentially, Escher went about such matters in this way, with the creation of a tessellation followed by a composition, ‘if appropriate’ i.e. generally, but not invariably if the motifs are shown in an aesthetic orientation. Such matters as regards the aesthetics of the motifs are most important, and, as a composition invariably involves much time and study, and so consequently, it is futile to choose an inappropriate tessellation – this matter is discussed in further detail in Essay 1.

A noticeable feature of Escher's prints is that of the compositions inherent quality in terms of aesthetics – for such works, it may be reasonably have been thought that his earliest efforts in this field would be somewhat crude or lacking in quality, in effect reflecting his pioneering tessellations of 1937, which are somewhat lacking in this aspect. However, most surprisingly, this is not the case, as from the very beginning high-quality examples not only exist, but indeed abound. For example, by common consent his most popular print, Day and Night was only the third of this type. Indeed, in retrospect this was arguably the most creative period of his life, certainly at least for these tessellating based prints, as print after print followed in quick succession. Furthermore, Escher states this explicitly in a letter to Gerd Arnzt concerning extending the Metamorphosis II print in 1967, whereby he states ‘… Around 1940 my inventiveness certainly was greater than now. Shall I succeed in reaching the same level? …’

1.9 Economy of Effort
In general terms, there is effect of what I term as economy of effort underlying many of his prints, all of which effectively enhances the final product with no superfluous factors ‘hindering’ the composition – essentially there is no ‘padding’ or extraneous motifs in these instances. A typical example of the former would be Sky and Water I, whereby birds and fish are effectively ‘discarded’ as the composition proceeds, more of which I discuss as appropriate below. However, not all are of the same intrinsic quality, as the follow up, Sky and Water II displays in effect ‘extraneous’ motifs in different orientations, of which, for conceptual reasons, in simple terms is strictly unnecessary – as such the phrase ‘less is more’ is apt in this case. Again, this is discussed in more detail below.

1.10 Commissioned Prints
Although Escher undertook commissions that were generally based upon non-tessellating examples (especially of his earlier days), occasionally he undertook commissions whereby a tessellation element occurred. Indeed, it was not until 1948 that the first tessellation-based of these appeared. Furthermore, these were generally of a less extensive nature in their content than for his inherent own graphic works. However, the ones he produced are indeed interesting in their own right, as generally some ‘innovation’ is to be seen. Undoubtedly the most significant of these was for the book Regelmatige Vlakverduling (Regular Division), commissioned by the De Roos Foundation in 1957. Pleasingly, here in this essay Escher goes into considerable more depth than with his brief comments (presumably written with brevity in mind for a more general audience) than in Graphic Work. This involved an essay along with six appropriate illustrations of a substantial nature, of which Escher went to considerable time and effort to compose. Essentially, he set out in relative detail aspects of ‘regular division’ of which he wrote in relative detail about the prints illustrated (along with other pertaining aspects such as the ideal ‘viewpoints’ of the motifs), albeit still not of the genesis but more generally of the symmetry principles underlying these. Furthermore, many of these (Plates I-V) illustrate principles of counterchange, of which Escher only mentions this aspect indirectly in the text.

1.11 Techniques
As regards his techniques of the artistic aspect, namely with woodcuts, wood engravings and lithographs, this is, I consider, another matter entirely, and furthermore, as I have no experience in these matters, any comments pertaining to this would be inappropriate, and so consequently this aspect is not discussed here, albeit I will say that considerable skill is undoubtedly shown. As regards his techniques of the artistic aspect, namely with woodcuts, wood engravings and lithographs, this is, I consider, another matter entirely, and furthermore, as I have no experience in these matters, any comments pertaining to this would be inappropriate, and so consequently this aspect is not discussed here, albeit I will say that considerable skill is undoubtedly shown.

The Tessellating-Based Prints

Of these, the print title is given first, with the Bool catalogue number as given in Life and Work. This is followed by the month and year of creation along with the periodic drawing number that acts as its source, and month, year for this. The following abbreviations for book titles and essays (based on their title in English) are used:

Books

Graphic Work: Grafiek en tekeningen (The Graphic Work of M.C. Escher, 1960)

Visions of Symmetry: Visions of Symmetry. Notebooks, Periodic Drawings, and Related Work of M.C. Escher, 1990

Magic Mirror: De toverspiegel van M.C. Escher (The Magic Mirror of M.C. Escher, 1976)

Art and Science: M.C. Escher: Art and Science, 1986

Life and Work: Leven en werk van M.C. Escher (M.C. Escher, His Life and Complete Graphic Work, 1982)

Regular Division: Regelmatige vlakverderling (Regular Division of the Plane, 1958)

Zodiac: De tekens van de dierenriem (The Signs of the Zodiac,)

Escher on Escher: Het oneindige: M.C. Escher over eigen werk (Escher on Escher: Exploring the Infinite, 1989)

Magic: The Magic of M.C. Escher, 2000

World: De werelden van M.C. Escher (The World of M. C. Escher, 1971)

Essays

Approaches: Approaches to Infinity by M.C. Escher

The Work: The Work of M.C. Escher by J.L. Locher

Science and Fiction: Escher: Science and Fiction by C.H.A. Broos

Structural: Structural Sensation by G.W. Locher

Mathematical Implications: The Mathematical Implications of Escher’s Prints by H.S.M. Coxeter