The coloration (of which I regard black and white for the purpose of this discussion as colours as well) and contrasts of tessellations is an aspect that I consider to be of the considerable importance and yet is generally treated in a casual manner (although not, I hasten to add, by myself). Indeed, I am most critical of other people's efforts, including the few who are capable of composing quality examples, and even with Escher himself on occasions. As a rule, the tessellation per se appears to assume pre-eminence, with the subsequent coloration aspect of it very much neglected, if even considered at all. Such apparent difficulties in coloration, for such an essentially simple task of differentiating the motifs, seem to cause no end of trouble. Obviously, the tessellation per se must remain the most pre-eminent aspect, but such disregard for this ‘additional' aspect is short-sighted, as by carefully selecting an ‘appropriate' choice of colour the tessellation can thus be inherently enhanced, and not, as so often occurs, essentially ‘let down' by an inferior selection.
Escher briefly mentioned aspects of coloration in the book Symmetry Aspects of M. C. Escher's Periodic Drawings, whereby in the preface (VII), he emphasised 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.
However, although Escher does indeed (generally) use contrasting colours, the choice appears to have been essentially at whim, using arbitrary combinations of colour. Certainly, there is no investigation into a separate study per se.
Colour can now be applied in more than the 'traditional' way employed in Escher's time, namely with the electronic medium of computers. However, for truly understanding aspects of colour theory, the traditional media is, in my opinion, far superior. Therefore, being more familiar with artistic aspects than with computer coloration, the essay mainly discusses the implications involved concerning colour as with paint and brush rather than electronically, albeit the latter is indeed discussed briefly.
As regards to colour, 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, thereby clearly ‘delineating' adjacent motifs. 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 will give the best contrast effect. This aspect of aspect of contrast is known as complementary colour, as has been the subject of much conjecture. Indeed, the whole subject is rife with inexactness, with authorities (generally artists) all too often lacking in rigour, contenting themselves with generalised statements of a most lightweight nature. Firstly, it must be established exactly which type of complementary is being referred to. As such, two distinct types can be discerned:
- mixing complementary, whereby when two given colours are thus duly physically mixed, thus resulting in a grey
- visual complementary, 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 simple 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 of 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. When so compared the differences are noticeable. Consequently, the term ‘blue' must be quantified. 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, a ‘scientist artist' 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.com. (Many other so-called truisms of painting and colour theory are also examined and put to the test before being found wanting, and then duly being corrected.)
Of interest would be knowing Escher's views on this subject, of which he would almost certainly be aware of complementary colour theory. Regrettably, as he did not write anything on this subject, the only guide we have to his knowledge is the drawings. However, upon examining his examples, he seems to have essentially neglecting this aspect, as not a single example shows a complementary pairing. As a rule, he favoured a light and dark contrast without being too concerned as to which colours were used. Such apparent disregard for complementary colour I find most surprising, as the suitability of such a colouring scheme is so obvious
In general terms this particular choice of coloration is the one I prefer, of which numerous examples can be seen (and are discussed) on the site.
Variation of Colour Schemes
The above type of coloration can in broad terms be described as inherently simple, as the coloration consists of a single colour for any given tile. However, 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, such as its head, body, 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 too simplistic) single colouring scheme. 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, as it is not clear as to what colour is part of the motif, 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 then be made.
Colour rendition refers to the inherent quality of any finished motif, and as such, this aspect 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 be shown as of a mere handful of motifs or literally hundreds. Indeed, if the latter type is chosen, the ‘tedium factor' must be taken into account. Quite simply, in colouring so many motifs in an essentially mechanical way rather than being ‘creative', the potential for mistakes or slipshod work is increased, as motif after motif follows with tedium setting in. Therefore, such vagaries must be borne in mind. However, in taking an arbitrary tessellation in one of my favoured formats, namely of a ‘4 x 4' 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 respectively to complete, a ratio of 16:1. As this thus involves a considerable discrepancy in time quantities, the obvious question that arises is of how best to use the time at ones disposal. Effectively, it comes down to between a choice of a single, high quality (photo-realist) example, and sixteen (ability permitting of course!) tessellations 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 my own work.
Upon examining Escher's periodic drawings for comparison, it can be seen that he too favoured the ‘simplistic' approach in general terms, with only occasionally highly detailed rendering occurring. However, such comparisons are not strictly fair, as the tessellations were not intended to be regarded as finished works of art in their own right. Indeed, some of these are most crude, with the impression given that they are of a study nature. As such, they can be said to be a 'report' of his experiments of composing various representational motifs. Consequently, the rendition is a reflection on this.
The Computer and Coloration
Also to be taken into account is the possibilities of using the computer in coloration, as this has a notable bearing on the issue in terms of practicalities. For example, upon determining a coloration for a single motif, this can then be repeated endlessly and quickly, and so the whole tessellation can thus be coloured in a matter of moments. In contrast, using the ‘old fashioned' paint and brush, with of necessity each motif having to be coloured anew, will obviously require more time. However, although the computer can appear advantageous in this matter, the artistic finish does indeed have merits of its own. Essentially (in my opinion), a computer coloration can be said to be too perfect – its somewhat ‘soulless', whereas in contrast, a hand finish, by whatever media, has a 'charm' of its own, despite the ‘imperfectness' arising due to each motif being coloured anew. However, this should not be used as an excuse for apparent slipshod work – the aim being to replicate the colouring of each motif as alike as is humanly possible.
Black and White
As such, black and white obviously provides the greatest possible effect in terms of contrast, 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 fifteen unambiguous examples (Nos.45, 63, 73-75, 80, 97, 104, 124,126-131), along with some others which although are in intent black and white, are not quite strictly so. Perhaps somewhat surprisingly, he did not use this colouring scheme at the beginning of his studies (where applicable, as most of these require a third colour), and it was not until as late as 1941, with periodic drawing No.45, did he undertake the first usage of black and white.
However, to use such a stark contrast scheme to the exclusion of any colour would soon become tedious, if not to say demonstrably 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, of which colour usage is thus forced, although to be pedantic, black, white and grey could be so used.
Firstly, rather than giving an lengthy 'all encompassing' mathematical treatise that would be inappropriate here on this type of coloration, I instead detail the basics. (For those so interested in a more exact discussion, G.C. Shepherd delves into this subject in M.C. Escher: Art and Science 'What Escher Might Have Done', pages 111-122.)
The term perfect colouring is used by mathematicians to describe a tessellation whereby the coloration is compatible with its symmetry. For example, where the tile in a tessellation appears in four orientations, when this is then so colored with four colours. However, this will be found not to be the minimum number of colours required, of which two or three would suffice. Therefore, as regards matters of aesthetics a dichotomy exists, of which in effect a 'choice' of two coloring 'schemes' occurs:
* Perfect Coloration. Compatible with the symmetry of the tessellation, but is not compatible with the minimum number of colours.
* Non-Perfect. Compatible with the minimum number of motifs, but is not compatible the tessellation.
Therefore, when so faced with a tessellation of so how 'best' to thus colour, then any one type has both advantages and disadvantages, these being evenly balanced out as above.
Of interest is how Escher approached the possibilities in this field. Of his 137 periodic drawings, only two, Nos.20 and 118, show more colours than the minimum that is required. (On No.20 he notes that a three-colour example is possible.) Furthermore, a preparatory sketch of No.3 shows four colours. As both types of colouring are equally valid, it is perhaps somewhat surprising that Escher did not make at least two examples of any one given tessellation when appropriate.
'Map Colouring' is a term taken from mathematics, of which this refers to colouring a map such that no country has a border colour that is contiguous with that of one of the same colour. If such a clash of colours were to occur, then the countries would be less easily identifiable than otherwise could be. Consequently (political, rather than relief) maps are shown in this way. Now, such a aspect is also relevant to tessellation, as essentially what one is doing when colouring the motifs is echoing this, by colouring a map in the broader sense i.e. consisting of contiguous motifs. Indeed, such a principle has distinct merits, in that as with the different countries of the maps being readily identifiable, likewise with tessellations. However, most people involved in tessellation do not discern the subtlies behind such a rule. Indeed, instances occur where this is blatantly violated. Therefore, in the stricter sense, such examples should be regarded as inadmissible. Of course, there is nothing 'wrong' in such a colouring per se, beyond it being of a non map nature. However, as a rule, each tessellation should consist of such a feature. Indeed, of my own work, every tessellation is suitably coloured. Of interest is Escher's work. Of his 137 numbered drawings, only No. 43 (Flowers and Leaves) violates this rule. Undoubtedly, Escher thereby inherently followed such a scheme. Of interest therefore is to why the above colouration was so shown, when he could have used an additional colour. As such, this can be explained by the requirement that the leaf motifs, if they were to be of a natural colour, would of necessity have to be a single colour, namely green. Possibly, one could argue that autumnal colours could have been use, thereby preserving the map colouring. However, although this would be a solution, this is somewhat contrived, especially so for such a simple dual concept of flowers and leaves. Escher here undoubtedly opted for a colouration that was more true to life rather than slavishly following the map colouring rule. Such a colouring permits one to assess a non map colouring, and without any doubt, the individual leaves are not so readily discernible.
Last updated: 4 January 2006 (2007?) (perfect coloration added). Addition: 29 May 2007 (Map Colouring Section)