1. Escher's Beginnings
Of interest is to how Escher began his studies, by what route, from what sources, from which one can then more readily comprehend the difficulties he faced. Furthermore, by such means one can thus, in theory at least, build upon his own efforts by the creation of new representational tessellations, having seen the background. First, it should be noted that his interest in creating representational tessellations was essentially a personal one; as such, an idea that had no real precedent of note. Although tessellation per se has a long history, apparently nobody thought of the idea of making tilings (or adding decoration) in the form of some recognisable figurative motif before Escher. That said, there can be seen to be forerunners, with the work in the Art Nouveau period of Erwin Puchinger and Koloman Moser in 1899 (although Escher was wholly unaware of Moser until as late as 1964, and likely Puchinger entirely), this cannot be considered in the same manner, their tessellations in concept, number and quality paling in comparison. As regards Escher, he essentially worked independently, with ‘artistic intuition’, as he had no real knowledge of any mathematics underlying tessellation principles. The first such example that pertains to tessellation was undertaken in 1920 or 1921, albeit to be exact these are not tessellations in the normal sense as essentially two broadly human-like motifs are ‘adapted’ to the confines of a rhombus, which was then repeated as a tilling pattern, as shown on p. 7, Visions. Although that said this idea is not new, as earlier examples of this type can be found, for instance as wallpaper patterns. However, Escher's effort is noteworthy in that a conscious desire to fill as much as is possible of the polygon is to the fore. This is in contrast to the others, where essentially the motif is placed within the confines of the polygon without undue attention paid to the filling out aspect. In contrast, Escher's efforts can be said to be intermediary between ‘placement’ and tessellation. However, the motifs are somewhat inelegant, and so presumably on account of their relative poor quality Escher did not proceed further with this idea, although in the same year he briefly returned to a polygon of a non-repeating nature (essentially triangular) filling with a human figure as a woodcut (cat. 85). Interestingly, the same type of hand can be seen on both examples.
1.1 The First Bona Fide Tessellation, January-March 1922
Upon returning to more orthodox (landscape) works, he once more returned to the problem of plane tiling with recognisable motifs in January, February or March 1922 (the exact month is uncertain), whilst still a student at Harlem, of what is indisputably his first ‘proper’ tessellation, of Eight Heads, these being shown in profile, four female, four male, albeit the black male figures are somewhat contrived. Interestingly, these are not of the type with which he later proceeded (whole bodied creatures), as they (heads) represent only a partial element of the whole figure, and so lack challenge. As is self evident, these are again of a most poor quality in terms of their inherent aesthetics, with furthermore half of the motifs upside down in relation to each other, which Escher probably did not like (and of which of such examples he spoke out about later). However, although possessing notable shortcomings, the principle of a life-like tessellation is indeed firmly established. Again, presumably due to the inherent poor quality, upon a initial effort Escher did not continue further with such examples of this type, returning to his more orthodox studies of a landscape nature.
1.2 First Visit to the Alhambra and La Mezquita, October 1922
Later in the autumn of the same year, he embarked upon a sea voyage, from Holland to Spain, of which this journey was to have most serious consequences concerning his future artistic career. Upon arrival, he visited many cities, presumably with no thought of tessellation necessarily in mind. Most notable of these places were Granada, with the Alhambra. Evidently, he was much impressed with the mosaics, making a detailed copy of an arabesque pattern, of 20 October 1922, as shown on p. 9, Visions. Upon examining the drawing, it is evident that this displays relative complexity. Indeed, considerable credit should be given to Escher in being able to both draw and thus understand its construction. As such, considering Escher's essentially non-mathematical background, the accomplishment of such a fine copy is most surprising. However, this once again did not lead to an inherent change in his work. A change of abode, from Holland to Italy then occurred. From his new base, Escher then made frequent additional journeys, visiting many churches and buildings including those at Ravello, Amalfi, of which by accident or design contained repeating patterns in the form of mosaics. Examples of the drawings, some made by Jetta (although the exact attribution is lost), are shown on p. 13, Visions. Unfortunately, these are not dated, but were most likely from 1924-1933. Interestingly, the study of these seem to have reinvigorated his interest in plane filling, as he once again attempted some life-like motifs, albeit of barely recognizable creatures, of, as he terms them, ‘lions’ and ‘bats’. Periodic drawings of these appear as numbers 1 and 2 of the numbered sequence. These led to examples printed on satin and silk, of 1926, shown on p. 11, Visions. However, once again, this was very much an essentially isolated piece of study. Indeed, he essentially then once more returned to his favoured landscapes.
1.3 Second Visit to the Alhambra and La Mezquita, May 1935
The year 1935 was significant for Escher, as the family left Rome for Switzerland, this caused by the deteriorating political situation in Italy. However, he continued in his desire for travel, with journeys to many Mediterranean ports, whereupon on the 23 May he once more visited the Alhambra. Presumably, with many more years of experience of creating tiling patterns, albeit still of a most sporadic nature, he was freshly inspired by the many mosaics there, of which he made considerably more sketches than on his previous visit in 1922, a selection of which are shown on p 17, Visions. Again, these were in association with Jetta, of which the same lack of exact attribution arises. This was followed on the 30 May by a visit to the mosque La Mezquita in Córdoba, where again he once more made sketches of the mosaics, these also shown on p. 17. The next destination was Seville, with the Alcazar, also containing Moorish decorative art. Leaving by boat on the 7 June, Escher then spent two more weeks on the return voyage to Trieste, adding to his collection of sketches. From September 1936 to March 1937, Escher then used the sketches of the non-tessellating nature for the series of woodcut prints for the Adria company as previously agreed. Interestingly, along with this he also began to undertake more extensive efforts of composing representational tessellations, the first exact one dated being No.3, of October 1936, followed quickly by a whole host of others. Presumably, Escher must have studied his tessellations from the Alhambra and Córdoba as regards how the line arrangements tessellate before then using this knowledge to create his own. Interestingly, Escher was not content with ‘merely’ having composed a tessellation, as he immediately set about putting these to ‘use’, as a composition, in the form of a print (Metamorphosis I, of May 1937), with the tessellation providing the backdrop. This thus confirms his later view that these drawings were not works of art in their own right, as can be intimated by their generally low-quality rendition, but were intended as necessary ‘precursors’ for the print.
1.4 The Mathematical Breakthrough, October 1937
The tessellations and Metamorphosis I print was subsequently shown and discussed in October 1937 at Escher's parents home at the Hague, with also being in attendance his half brother B. G. Escher, a professor of geology, who from a crystallographic viewpoint recognised their significance. This shortly (1 November) led him to compile a list of sources for Escher to reference (of which a full list of these is given on p. 337, Visions). Upon studying the various articles in these crystallographic journals, or more precisely the diagrams, Escher found two in particular most useful. These were by George Pólya and F. Haag, namely in the journal Zeitschrift für Kristallographie, with ‘Über die Analogie der Kristallsymmetrie in der Ebene’ of 1924 (the diagrams shown on p. 23) and ‘Die regelmässigen Planteilungen und Punktsyteme’, of 1923 (‘Regular Divisions of the Plane and Point Systems’) respectively. These articles contained tessellating diagrams, of an essentially simple nature, which Escher studied with vigour, reproducing the diagrams (as shown on pp. 24-29, Visions). Pólya’s contained representative samples of the 17 plane symmetry groups, and from this Escher then used these as the ‘scaffold’ for periodic drawings, with No.17 (Eagle) and No. 21 (Imp). As an aside, Escher missed some further examples more directly based on these tiles – see my Birds 1, No.1 (which uses Pólya’s C4) and No.2 (based upon Escher's ‘offshoot’ sketch of C4, p. 25). Furthermore, C3, D4 and D3 all have bird-like possibilities.
Essentially, at this point Escher must then have realised that tessellations were of a mathematical nature, from which having thus been pointed in the right direction he could then proceed on firm foundations, this being in contrast to the previous essentially ad hoc studies. Interestingly, somewhat paradoxically, the earliest tessellations (arbitrarily stated as Nos. 1-11) are amongst the most difficult of his to understand in terms of the underlying construction, with apparent underlying tessellating polygons frequently (but not invariably) bearing little, if any, obvious connection to the motif. Indeed, although it may have been thought that he would have begun with ‘straightforward, simple’ process of opposite translations of squares and rectangles, this appears not to have been the case, as the first example of this type does not occur until July 1941, with drawing 38. After this initial outpouring, further studies then led to a succession of drawings and prints of outstanding originality in remarkably short time.
1.5 Summary
It is thus evident that at first Escher composed representational tessellations with an absolute minimum amount of knowledge as to the mathematical procedure. Indeed, at the very beginning he had nothing in the way of references or sources to refer to, and so consequently of necessity thus stumbled around. However, upon finding the Alhambra and La Mezquita mosaics, more specifically on his second visit, his latent interest was reawakened, from which he then set out to copy these and then having thus understood their constructions, he was then in a position to apply these principles to his own ideas. Although these remained somewhat crude, progress was definitely being made. Undoubtedly, the introduction by B. G. Escher as to their relation concerning mathematics was a pivotal moment. As such, it is debatable as to whether or not Escher would have continued with the relatively limited means at his disposal without this fresh input. However, once the mathematical door was opened, Escher entered it with considerable fervour. Therefore, from this it can be seen that no ‘special’ abilities per se are needed to compose representational tessellations – essentially the field is open to anyone; Escher freely admitted that he was no mathematician. Furthermore, since Escher’s time, it is now consequently much easier to attempt this, due to a plethora of popular books and articles available on the subject.
2 Background Details Concerning the Drawings
This section discusses the various background details concerning the drawings, in short everything except the drawing itself, with the drawings being discussed individually in the following section.
2.1 Text on the Drawings - Sequenced Numbers, Number of Motifs and Symmetry System, Place of Origin and Date and Improvement
As regards the text on the drawings, this takes various forms, of different periods, sometimes added retrospectively, all of which causes difficulty as regards a true chronology. Without fail, all the drawings bear text, either all or some of the following:
2.1.1 Sequenced Numbers
Escher added numbers (usually at the top) to the drawings generally following their chronological sequence, albeit on occasions there is a discrepancy between the given sequence and chronology. A further addition to the numbering, pertaining specifically to drawings 18, 20-22, 25, 56, 59, 63, 66-67, 71-72, 80, 91 and 96 can be seen in the form of a pencilled ‘X’ next to the number. Quite what the purpose or significance is of this notation remains unknown. Another pencilled X (which is circled) is occasionally on the drawings from the Alhambra of 1936 (p. 17). Whether this is related to the above drawings is likewise unknown. Can anyone shed some light on the ‘X notation’?
2.1.2 Number of Motifs and Symmetry Type
Escher added the number of motifs (if there was more than one) and symmetry type and classification of the tessellation system used (of Escher's own devising) as a group, placed at the lower left of the drawing. However, this text was not always contemporary with the drawing, with the symmetry type and classification text being added in 1941-1942, when Escher had set out in his notebooks the classification system to his preceding examples, of which he then went back and pasted over now outmoded classification of drawings 1-33 with the newer findings.
2.1.3 Place of Origin and Date
Escher invariably added the places of his reside (perhaps somewhat unnecessarily and unusually) and date, with the month and year, placed at lower right. Can anyone shed some light as to reasons why he added his residence?
2.1.4 Improved Drawings
In contrast to the above text that always appears on the drawings, on occasions the drawings also possess the term ‘improved’, accompanied by the date of this occurrence, this being generally added at the lower right. This refers to essentially a re-working of an existing drawing on the given date, in which he ‘improved’ the drawing with further colouration and rendition and/or emphasising the motifs outlines in black ink. However, this happened on relatively few examples, of which the purpose of this action was for their inclusion in the book Symmetry Aspects of M. C. Escher's Periodic Drawings, by Carolina H. MacGillavry. For the book, she selected some of Escher's tessellations to illustrate crystallographic principles, of which Escher was presumably unsatisfied with the inherent quality of colouration and rendition of the mooted tessellations, and hence decided to improve.
Consequently, as such additional matters effectively obliterate the original; this thus results in some of the drawings appearing in a somewhat disjoint manner with contemporaries. However, occasionally Escher left portions of the underlying drawing showing, from which it is thus possible to see it essentially as it once was. As such, this addition, even if it is indeed of an ‘improvement’ nature, does not find favour with myself, as it destroys (or at least spoils) the intrinsic appearance as regards earliest chronology.
2.1.5 Instructional Text
Aside from the above main body of text, on occasion further additions of a more specialised nature can be seen that does not pertain to the drawings themselves. Escher added these for the purposes of guiding the photographer of his drawings for The Graphic Work of M. C. Escher, of 1959. Consequently, these are of no interest in terms of the drawings themselves.
2.2 Titles
Of the drawings, Escher did not give titles to any of them, and although most are self-explanatory, on occasions the motifs are ambiguous, thereby causing difficulties for reference purposes. Upon noting that Schattschneider has added titles of her own, in my own discussion that follows these are thus retained for reasons of consistency.
2.3 Preparatory Studies
A major disadvantage in commenting upon the drawings is that, with few exceptions, the all-important preparatory studies as to their genesis remains unpublished. Consequently, for the most part, how easily or with considerable difficulty these came into being remains speculative. Essentially only fragments are to be found, such as in Visions, with a considered sequence from the very first line to a definitive nature ready for a polished drawing not shown. Where preparatory examples do occur, even of a minor nature, these are referred to as ‘Preparatory Studies’, followed by their source.
2.4 Paper
As tessellations are of a repeating mathematical nature with an underlying geometrical grid, Escher naturally used quadrille paper as being the most appropriate. This was of generally the same overall size throughout the years, with paper measuring approximately 14¼�? x 10½�? being the most favoured. However, quadrille paper is generally of inherently lower quality, typically of a lightweight nature, it being designed for pen work (with lines) and not the application of colour per se. When watercolour is applied, this type of paper will buckle, resulting in the unwanted effect of pooling, this being notable in many of the drawings by a ‘blotchy’ appearance.
Interestingly, he retained quadrille paper despite the tessellations on occasions having an apparently unsuitable underlying symmetry, such as those based on equilateral triangles (which cannot be drawn on the intersections directly). Indeed, it is possible that all of them were undertaken using quadrille – upon examining them in Visions and Mind Play (these being of the best reproductions), only a handful cannot be determined one way or the other as to be certain of this matter. Certainly, such usage of inappropriate grids is not ideal, albeit in practise not a major hindrance. Apparently, the reason for this, at least during the war years, was that isometric paper was expensive or barely obtainable, of which of necessity when Escher did acquire such paper this was covered liberally with unrelated diagrams – see Visions, p. 144.
2.5 Escher's Views on the Drawings
As such, Escher looked upon his tessellations not as finished works of art per se, but as ‘reservoirs’ (his own words) or sources for future graphic works. (In contrast, the present day trend is to regard a tessellation as a work of art in its own right, not necessarily having been composed for a ‘picture story’ composition as visualised by Escher.) Indeed, he inadvertently ‘emphasises’ this by the rendition of the drawing, these frequently lacking in that quality, especially of the earlier years. Consequently, Escher’s approach to the drawings must be considered when assessing the inherent quality of each tessellation, especially so for these aspects, as they were purposefully neglected in the stricter sense.
Although he was apparently very pleased with the drawings, they were not exhibited in the normal sense, as a more ‘orthodox’ work of art would be.
2.6 Purposeful Tessellation
Although overwhelmingly Escher composed tessellations as essentially according to whim in terms of choices of motif and symmetry, on occasions he purposefully undertook examples whereby of necessity he was restricted in such matters. These include:
• Metamorphoses II and III.
With tessellations to suit the purposes of linking the motifs.
• Missing Symmetries
Upon composing a notebook in 1941-1942 in which he set out his symmetry systems, he noticed that examples of ‘missing’ symmetry types that had as yet to have an illustrative example, from which he then composed examples, with drawings 36-39, this primarily being of 1942. A noticeable feature of these is that many are of an inferior quality, with motifs generally of an imaginary or fanciful nature. Oddly, many of these are of a superior rendition belying their inherent quality,
• Caroline MacGillavry
In 1960, Carolina H. MacGillavry was commissioned to write a book (published in 1965) on crystallography by the International Union of Crystallographers, Symmetry Aspects of M. C. Escher's Periodic Drawings of which to illustrate the various crystallographic principles involved used 42 of Escher's tessellations. For this, she requested further ‘missing symmetries’ of examples that Escher had not done. As such, as examples of these ‘forced’ types are thus more difficult to compose, with self-imposed restrictions thus resulting in fewer potential possibilities, and although this is not necessarily an impediment to quality, the tessellations so composed for MacGillavry are noticeably poorer in comparison to his ‘normal’ standard.
Furthermore, Escher ‘improved’ upon some earlier drawings for this book, by reworking these, in which the rendition is considerably better.
• Tessellations for Pre-conceived Concepts
Occasionally, rather than the tessellation providing the source for a following print, the reverse happened, and with a concept or composition in mind, Escher then specially designed tessellations for this exact purpose. As such, these are the most difficult to design, due to the severest of restrictions. These include:
•Verbum
Arguably, this was the most difficult of the pre-conceived concept examples that Escher undertook, on a number of counts. Firstly, upon having outlined the idea, of carefully balanced creatures in terms of their appropriateness to air land and sea, specific combinations then had to be composed, i.e. birds, birds; birds, fish; fish, fish; frog, fish; frog, frog and frog, bird. Secondly, due to underlying structure, these had to be of a specific polygon, of equilateral triangle arranged as rhombuses. As such, a most trying set of circumstances, a true test of ones capabilities, of which Escher admirably rises to the occasion. Of the drawings en masse, only drawing 52 is noticeably lacking, with the frog in particular of poor quality. Somewhat strangely, despite being composed for a ‘series of six’, the drawings are not coloured in a consistent manner, with various combinations being used.
2.7 Colouration and Rendition
Concerning the colouration, Escher apparently selected arbitrary contrasting colours without too much concern for exact contrasts of the type known as complementary, of which this aspect per se is discussed more thoroughly in Essay 1. Essentially, at least for the ‘early’ years, Escher invariably used what I term as ‘flat’, simple colouring, with no apparent favoured colouring combination, although on the ‘early’ examples red, blue, and white are to the fore.
Concerning the usage of white, this was achieved by leaving the paper used, and where this occurs, in the text below of the 137 drawings I state this as ‘white (unstated)’.
Interestingly, there is no attempt at portraying three-dimensions with these ‘early’ drawings, with examples of this type occurring much later, beginning with drawing 30, of March 1940. Generally, Escher was not too concerned with this aspect, with only on rare occasions ‘considered’ rendition occurring (for example with drawing No.45 of angels and devils), albeit from this point onwards this matter is addressed, sporadically, presumably for reasons of practicality (as discussed in detail in Essay 1).
Another matter concerning colouration is the number of colours required to colour any given tessellation as according to contiguous regions being of a contrasting nature. As such, there are various subtleties to this, and indeed the subject of which can be most involved, essentially a study in its own right. However, without the necessity of such an involved study being required, it will be found that for any given tessellation, in which a ‘minimum’ colouring is strived for, and according to the symmetry demands of the tiles themselves, two colours are usually sufficient, with occasionally three being required. However, it will be found that such apparent general rules, although sufficient, do not necessarily result in the ‘only’ or ‘best’ colouring solution. This aspect can be seen in one of Escher's earliest drawings, No.3, where he uses three colours, this being more compatible with the minimum number of colours rather than the symmetry of the composition, which requires four (the use of two colours, in this instance, are incompatible). Therefore, examples of this type have no ideal single colouring solution, as both types have pros and cons. The minimum three colouring, although ideal in terms of the smallest number of colours to be used has the shortcoming that the colouring is unaesthetic in terms of symmetry, whilst the use of four colours, while aesthetic, results in an additional, strictly unnecessary extra colour being used. Therefore, in situations such as this, ideally there would be two examples, showing both types of colouration.
2.8 Grid Lines
Frequently the drawings can be seen to possess grid lines, most noticeably with the examples from the early years. Somewhat curiously, these are more often than not shown without any apparent bearing of the drawings themselves at the most expected places, namely the vertices. Indeed, they can appear to have been arbitrarily placed - most strange. Presumably, at one time these must have played a part in the original framework, but due to Escher's design process, in which the original vertices have been ‘moved’, these have subsequently become redundant.
2.9 The 1941-1942 Notebooks and Tessellating Systems
After an initial period of study (1936-1940), in which Escher composed tessellations in an essentially ad hoc manner, he decided to be more systematic in his approach to tessellation theory, composing a notebook in which he set out his symmetry systems. This took the form of both text and illustration, of a ‘visual mathematical’ nature, of considerable depth (of which Schattschneider devotes a whole chapter in Visions, pages 53-104, reproducing the whole series of drawings and diagrams). From his cataloguing of the numbered tessellations (of which he added retrospectively to the drawings preceding this date), he found that some of the symmetry types had yet to appear as a finished, numbered drawing, upon which he then specially composed tessellations for these ‘missing’ symmetries, illustrated by No.36-39. However, in general terms, these so composed can be seen to be lacking in inherent quality in comparison with his other, ‘non-forced’ tessellations.
2.10 Transfer of Definitive Drawing
Of interest is to how Escher, upon having composed a definitive motif, then ‘transferred’ this to the finished state, i.e. of numbered drawing. Although this aspect is not of the greatest importance, as such matters are essentially of a mechanical nature, it nonetheless remains a relevant area of interest. Although Visions, page 113, discusses this, the description given is not of the greatest clarity, of which a series of illustrations would have been more informative. However, from the text, it is apparent that he used a single motif as a ‘master copy’, with punched holes for registration marks at the vertices, from which he then repeated this in the definitive drawing, although the specifics of transferral as regards interior details are not discussed. If the above interpretation is indeed correct, it can self evidently be seen (due to the 137 drawings) that this method suffices, albeit not one that I myself favour, as by its nature, the greatest accuracy is impractical by this route. (The transfer method of my own, using graph paper, involving drawing out of the motifs fully (including all interior details) is discussed in Essay 1.)
2.11 Format
Of interest is the type of format that Escher used for the drawings, and although this is not of the greatest importance, it nonetheless remains of interest. The most noticeable feature of this is that without exception, all the drawings have a square or rectangular format, with an ‘unstated’ arbitrary border. Indeed, this format never changed throughout the course of his lifetime. Curiously, he never showed his tessellation as an arbitrary ‘patch’ or ‘grouping’ as frequently shown by myself and others, of which this particular format, with typically fewer numbers of motifs, has an advantages in terms of the time required to draw the tessellation. Presumably, he desired to show the repetitive, infinitive nature without any possible uncertainty caused by a self-contained unit mentioned above.
3 The 137 Drawings
This section is wholly concerned with the drawings themselves, in which I comment on these in relative detail. Rather than a somewhat lengthy and daunting single page discussion, these are arbitrarily placed into five sections of thirty drawings (1-30, 31-60, 61-90, 91-120 and 121-137). Each drawing’s discussion is broken down into three distinct parts:
• Background details
(i) The number and title (as given by Escher and Schattschneider respectively)
(ii) The place of origin and date
(iii) The media used – watercolour, ink, pencil
(iv) If discussed elsewhere, this is duly noted
(v) The source for the preparatory drawings (if applicable)
(vi) The concept drawing, in relation to the graphic print (if applicable)
(vii) Related or other works, in which the drawing was subsequently used, along with the Bool catalogue number (if applicable)
• Comments concerning the drawing itself
This concentrates, albeit not exclusively, on the merits of the quality of the motif(s), and not, as with MacGillavry and Schattschneider, who concerned themselves, admittedly simplifying (especially MacGillavry), mostly on the symmetry aspects per se. Even so, Schattschneider’s discussion undoubtedly contains much insight of background matters. All references to page numbers concern Visions of Symmetry. As such, ideally only people like myself, who have indeed composed representational tessellations, are thus suitably qualified for such matters in the stricter sense.
• Comments concerning the colouration and rendition of the drawing
These related aspects are discussed as a broad statement in three elements:
(i) Flat, in which the colouring is shown in a flat, single colour in a simple manner, with no attempt at portraying three dimensions.
(ii) Minimum, in which the minimum number of colours are used as to be consistent with Escher's rule of contrasts. Where a non-minimum usage of colour has occurred, this is stated instead.
(iii) The number of colours used, along with a description as to their colour, e.g. blue, red.
(iv) The rendition (if applicable).