An open question in tessellation is to whether or not to portray the (finished) motifs with or without a decided border line, and furthermore, if a borderline is used, how thick should this be? Such an approach can be criticised in terms of attempting to portray reality, as no real-life motif has a defining surrounding line. However, as regards art matters, with tessellation this is a unique situation, and so is thus allowable in this circumstance.
So, is this feature advantageous or not, or indeed, is it of any real importance? The aim of this essay is to address this issue, and along with illustrations with and without border lines try and establish the matter. In essence, this is related to the aspect of discerning the motifs, in that does adding a border line aid this feature? On the face of it, this would seem a straightforward matter to examine, as in theory one could compare two ‘identical’ tessellations, with and without a borderline, and thus decide the matter in a few moments. However, various complications ensue, militating against a straightforward comparison:
The analysis is undertaken in three parts:
Part 1 begins in a generalised sense by examining the usage of such a feature in Escher’s and another expert in tessellation, Makoto Nakamura; this accompanied by other people’s work, and seeing what can be gleaned from this overall survey.
Part 2 then examines different types of tessellation finish, such as wireframes, photorealistic, orientation, and colour, all of which impact upon the borderline aspect as regards discernment of motif.
Part 1 – General Usage
Perhaps an obvious starting point for discussion would be an examination of Escher’s 137 periodic drawings (these are discussed in greater detail later on). Upon viewing Escher's work, his own works shows examples with design lines, and with without a borderline, in roughly equal ratios. As regards the thickness of line where applicable, this varies in thickness, from, in relative terms, very thin to very thick. Throughout his work, taken as a chronology sequence, these aspects intermingle, and so it would appear he had no firm favourite choice. Annoyingly, he did not write on the subject.
Nakamura generally does not use a border line, with the motifs defined mostly by colour. Occasional use of a design line is seen. Where a border line line is used, this is relatively thin.
Other People’s Work
When looking at other people’s work, one first has to address the quality issue. Far too often, one will see a motif that bears no resemblance to a creature, a type of which I disparagingly refer to as a ‘shape with eyes’. Should such examples be considered here? To include such examples alongside Escher’s and Nakamura’s better quality work would be to distort an true assessment. Instead, I introduce a degree of quality control, in which I assess examples from Andrew Crompton’s list of 50+ people currently active who show a degree of skill. Even so, many are unworthy of inclusion. No matter, the list at least permits a rudimentary assessment. Broadly, all types of ‘finish’ are readily seen, and so I conclude that in general people do not really consider this issue, with the design of the motif taking the more importance role (of which admittedly this is obviously the most important). However, I consider this aspect, of border lining, if not of fundamental importance, then of very high importance indeed, and should be considered and implemented as appropriate.
Part 2 – Different Aspects – The Theory
Therefore, to ascertain the efficacy of such possibilities, I now examine different types of tessellation, with two tessellations side by side, with and without the feature. Although one tessellation may be thought similar to another, this is not so, as different ‘presentations’ are possible:
A frequent presentation is that of a ‘wireframe’ tessellation, with the motifs left in an uncoloured state. However, although such an instance finds favour with many people, I actively dislike this mode. As such, I consider examples of this type as unfinished. Without colour or outlining, the motifs are generally difficult to discern, and the more orientations of the motif, the more difficult it is to discern.
Of interest is how many of this type Escher showed – none. Nakamura, with over 250+ tessellations - none. Myself, likewise, none. So, this should be some guidance as to the intrinsic worth of this type…
Occasionally examples of this type are shown (in contrast to the more frequently encountered cartoon or stylized type), invariably undertaken with a computer (I know of no hand-drawn photorealistic tessellations, albeit such a technique is indeed used in ‘normal’ art. The reason for such omission is the amount of time required, which renders this impractical by hand.)
For an example of such types, see Craig Kaplan’s thesis Escherization, page 144 (where he shows a dog’s head)
A drawback for purposes of discernment is that there is a myriad of tones and colours, thus rendering the motif inconvenient to see when repeated as a tessellation. Therefore, examples of this type thereby need the border line (as in Kaplan’s example, presumably purposeful).
3. Orientation of Motifs
Broadly, the more numerous the orientations of the motifs, the more difficult it is to discern. For example, in a translation, the motif appears in just one orientation. In contrast, with a rotation, the motif can appear in up to six orientations.
4. The Inherent Quality of the Tessellation Itself
Broadly, the better the quality of the tessellation the easier it will be to discern - an inferior example will not be so easily discernable. As such, the more 'true to life' the tessellation is (i.e. it resembles a real-life creature), than the more easily identifiable it will be. All too often, as discussed in Essay 6, this aspect is not addressed - a vague, distorted, uncertain creature is most difficult to discern (and in any case should be regarded as simply unacceptable).
Just by the simple process of adding colour the discernment of the motif is made much easier. Furthermore, on can have a sub-choices, such as two or more colours when appropriate, as with ‘exotic’ creatures, such as tropical fish, or birds of paradise, with many gaily colourations. However, such multi-colours causes difficulty in discernment, with one colour generally being favoured. As such with multi-colours, this should be seen as a occasional innovation, rather than standard practice.
Generally, tessellations are shown in two formats, either as a block (such as one of my own favourites, a 4 x 4 array) or patch of individual motifs per se, or as a ‘picture frame’ outline, such as a rectangle or square. Although this feature is perhaps of lesser importance as regards discernment, nonetheless the format should be considered. As regards discernment, I consider that arrays (such as 4 x 4) or patches slightly aid the process. Broadly, the eye has an exterior outline to fixate upon, rather than as a window. However, I frequently use a square or rectangular format, and I have no objection whatsoever to finished work in the format. Interestingly, both Escher and Nakamura solely use this type of presentation for their tessellations. However, note that in Nakamura’s instance, this is by default, as he uses a repeating unit to fill a rectangle. In Escher's prints he occasionally used patches, with Reptiles, 1942, Fish. c.1942, and Fish 1943 (the examples are shown on page 271 of Visions of Symmetry).
7. The Type of Motif
Although it may be thought that one motif may be very much alike another as a concept per se, this is not so. For instance, some creatures are naturally of one colour throughout, or can be of many colours. For example, birds and fish are generally of one colour (although some admittedly can consist of many colours). A human figure, can when clothed, consists of multiple colouring, with hair, flesh and different coloured garments. Essentially, the more colours, the more difficult the motif will be too discern.
8. Traditional or Modern Media - Paint or Computer Coloured
Essentially, there are two choices of media in executing the tessellation, in a traditional sense, with artist paint materials, or the more modern, with a computer. Each of these has their advantages and disadvantages, as discussed in Essay 10 The Computer and its Application to Tessellation. The merits of paint are discussed in Essay Colouration and Contrasts of Motif. In matters of border line, both have possibilities, albeit with traditional media then some are more ideal than others. Additionally, the media may be unsuited to a border line that should be of an uniform thickness. For example, disregarding the merits per se of oil paint, to apply a line of uniform thickness is a practical impossibility. In contrast, when using watercolour, then a pen can easily be overlaid, and so there is no such problem.
As such, watercolour is ideal, as upon application and a suitable period of drying time then delineation can then be applied. In contrast, other paint media, such as oil paint is wholly unsuitable for such purposes. Consequently, the appropriate choice should be made. Typically when using traditional media such as watercolour, the paint itself is meant to be used in a generally 'light' manner, with transparent washes permitting the paper surface to still be seen - to apply colour opaquely is in this instance 'incorrect'. Furthermore, traditional media is not conducive to a truly sharp border outline, whereas the computer is. However, the difference is negligible, of which for practical purposes can be ignored.
In contrast, when coloured by computer then an opaque colouring is indeed 'permissible', and so the motifs can indeed be 'defined' by colour alone. A outline can thus be added if desired.
Comment on the Different Aspects
So, as can be seen, there are many different aspects to take into consideration. Ideally, I would like to examine the above aspects with a single, and thus consistent motif, and so thereby making for a straightforward assessment of the implications. However, in practise, this is most difficult, and although I could indeed do so, the time involved would be disproportionate as to needs, as I have to hand suitable examples, albeit of different motifs. Nonetheless, it is indeed possible to have a relevant examination of the issues, of which I now show in diagrammatic form.
Here I show an example of a wireframe tessellation, with and without a borderline. The format is of a 4 x 4 block, of a translation:
As such, by using just lines (i.e. a wireframe) to delineate the motif, the individual elements of the motif, such as wings, body, and tail are not so easy to pick out at a glance. Essentially, the aspects merge into a homogenous mass, and so the eye of necessity has to 'search' the whole composition for a complete motif, and so 'struggles' to find a whole motif. Indeed, even for a high quality motif, as here, the task is relatively difficult. When viewing a typical 'beginners' tessellation, of lower inherent quality, then the matter is even more difficult with examples of this type. Furthermore, this instance shows the motifs in just one orientation – consider how difficult it would be when more orientations are involved. Just by the simple addition of a border line assists in discerning the motif, as the outline is instantly apparent. As a type (wireframe), with or without a border line, I now exclude this from further consideration.
Colour (With and without a borderline)
To the above diagrams/types I now add colour, with the simplest colouring scheme, each bird being of one colour, with contiguous motifs of another colour. In contrast to the wireframe examples, the motifs are now obviously defined primarily by colour rather than line. To this I now add a border line
Fig.** (a) left Finite block 4 x 4 line of unit thickness and (b) right heavier outline
Here the motifs are essentially defined by colour alone, with the delineation (interior) intrinsically insignificant. Each bird motif is readily identifiable, with no 'searching' required. Of interest is to compare the respective types. Although fig.**(a) motifs can be discerned with relative ease, the addition (b) of a border line is even better. Furthermore, the addition of a black line helps to intensify the colours even more, and therefore both aspects aids the eye in ‘easy’ viewing. Essentially the use of colour greatly assists in defining individual motifs no matter whether outlined or not.
Some (inadvertent) examples of Escher's can be used for comparison purposes, for example No.15, Lizards and 21, Imp, as on the same sheet ‘before' and ‘after' border lining can be seen (albeit this is admittedly not ideal, as the ‘before and ‘after' are disproportionate in extent). From this, the improvement is, or should be, self-evident. Without any doubt, such border lining noticeably ‘permits' what I term as ‘readily viewing', as in effect the eye easily sees the motifs at-a-glance, without having to, as it were, ‘struggle'.
Concerning my own examples, the addition of colour in conjunction with a (black) border line is frequently to be seen, and as a general rule invariably employed. (Indeed, I have no example of 'finished work' of the wireframe type). However, on rare occasions, examples can be seen whereby the favoured delineating line is not to be found, this 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 being delineated by 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 day. 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 as regards chronology can be seen in Escher's work, as above, with additions of approximately twenty-five years later, thus rendering considerable uncertainties in this matter.
Coloration itself has a bearing, as any one motif may be coloured with one, two or more colours. Essentially, the more colours used the more difficult it is to discern the motif. The examples above have all been of the 'one motif, one colour' type colouring scheme. Below is shown an example of a tessellation that of necessity has to be coloured in more than one colour to remain true to life, namely with a human figure. Although it is true that Escher coloured his human figures in a single colour, such a 'simple' approach is obviously not ideal.
By simply adding colour this results in an inprovement, albeit the motif is still not noticeably defined, due to the 'patchwork' nature of the colouration. In addition, the further orientations cause difficulties. In contrast, when delineated right, this is now an noticeable inprovement. In examples of this type then the advantages of delineation are even more apparent.
Orientation of Motifs
This example has the motifs in different orientations. Again, the addition of colour greatly assists the discernability of the motif, of which, due to the extra orientations is in this instance even more ideal than with the previously discussed translation example. Immediately, the motifs are discernable, with the minimum of effort on the part of the viewer.
3. The Degree of Thickness of Outline
Upon having established the efficacy of the principle of emphasising the outline, a relevant question to ask is to what degree of thickness this line should be.
Of Escher's examples, various degrees of thickness can be seen, from ‘negligible’ to ‘considerable', in relative terms, as the differences is one of mere millimetres. As such, different line thickness intermingled throughout the years, from which it can be assumed that he had no single, favoured type. As such, the ideal to aim for is that of what I term as ‘noticeable', but not to an ‘excessive’ degree. An ideal example, using Escher's drawings, would be No.108, Bird, where the interior detail is clearly finer than the outline, with the outline being ‘noticeable', but not excessively so. None of Escher's drawings has a line that is excessively so, albeit No.81, Bat/Bird/Bee/Butterfly is arguably so, albeit this has been ‘artificially' emphasised' to improve the representational aspect of the motifs.
As such, an exact size cannot be given, as this depends on the scale of the drawing, as obviously a motif, say, twice the size of another would obviously require a proportionate line. However, for any arbitrary drawing, the ideal to aim for is that of what I term as ‘noticeable', but not to an ‘excessive’ degree (with too thick a line). Indeed, if an 'excessive' line is shown, then this affects the design itself of the motif, as it impinges on the detail i.e. 'spills over' to the motif. As the tessellation is designed with a negligible line in mind i.e. of pencil thickness, then too great an addition to this is obviously uncalled for. The example below shows such an both’ ideal’ and 'excessive' instances.
For my own work, with a typical example having a underlying square of 40 mm, these are generally outlined with a 0.7mm or 0.5mm Staedtler Marsgraphic pigment liner pen, with the interior detail of 0.1mm or 0.3mm. An advantage of this is that the ink dries relatively quickly, thereby minimising any likelihood of smudging occurring, which as may be envisioned, is infuriating after the time consuming process of drawing and colouring.
Variation of the Colour of Delineating Line
A variation on this idea of an emphasised (thicker) line is that in place of having the surrounding delineating line denoting the motif in black; other colours can be used, generally of a 'neutral' nature, such as white or sepia. Such a feature should be regarded as a 'novelty', or innovation, in itself of no significance.
Escher made occasional use of this, as exemplified by periodic drawings Nos. 72, Fish/Boat, 79, Butterflies, and 94, Fish (albeit to be precise a white space was used). Additionally, this same idea applies to a themed series, Nos.111, Flying Fish/Bird; 112, Flying Fish/Boat; 113, Fish/Boat; and 114, Fish/Frog (where here he used a sepia outline). However, as can be seen by the few times Escher employed this, he obviously did not favour this particular variation, possibly due to practical reasons, as it is far simpler do draw an opaque black outline, and it serves it purpose just as well.
A white ink is not really advisable on practical considerations, as the white line is not truly opaque to be considered a 'pure' white. Presumably, this was the reason, for Escher using ‘white space' to represent a white line. Although I have occasionally experimented with an interior colour of the same intrinsic colour, such usages are very few and far between. However, instances where a white line is used in my own work has not yet arisen.