Since the time of Escher, computers have increasingly been used in the drawing and design of tessellations, of both the mathematical e.g. of regular tessellations and representational types, as of Escher type, albeit here I restrict the discussion to the representational ones.
Consequently, the possibilities of this tool open up new vistas, in both the designing i.e. the actual creation and the subsequent showing of the repetition of motifs in an tessellation array, such as a 4 x 4 arrangement. As such, although by no means an expert in this field (indeed, my knowledge is most limited) in the applications at least of computers to tessellation, it would be amiss of me not to give my thoughts and opinions on the subject, of which the following thus discusses.
Designing on the Computer (Escherization)
The design stage as regards the representational aspect appears to me to be most limited in extent. For example, an expert in computers (and in computer graphics), Craig S. Kaplan, uses his Escherization method to compose examples, but these are noticeably lacking in quality when compared to Escher and his few successors of note. In addition, the method he uses is not a matter of what I would term as 'design', as he in effect takes an arbitrary shape and instructs the computer to adapt this into a tessellating shape. However, as it is still relatively early days with the computer (especially so as regards to tessellation), so improvements in such matters must be expected in due course. However, quite when, or indeed if this will occur is speculative. As such, the Holy Grail for a computer designed example would be a realistic, ‘complete' animal, of which even further refinements would include different breeds if appropriate. If a program can be written to accomplish this, any need for human effort would be made redundant. Indeed, progress is being made. However, such an ideal is far from even remotely being accomplished, indeed, if it ever will. An analogy perhaps can be drawn with that of a different field, music. Although it is possible to reduce musical notation to a mathematical analysis, and thereby with the potential to examine 'all possibilities', there appears to be no shortage of new music being composed, of whatever genre. A possible similar situation may also arise with tessellation – the variety of motifs, of differing poses, of differing symmetry systems may prove impractical to unite to a computer program.
In recent years various specific tessellation software packages have arisen, with those aimed at the representational type including Tesselmania!, Tesselmania Deluxe and Tessellation Exploration (2001) by Kevin D. Lee and TESS by Jeff Tupper. Examples of tessellations produced by these programs, of both Lee and Tupper and other tessellation artists can be seen on the CD-ROM from M.C. Escher's Legacy and the website www.peda.com/tess respectively. However, these to me appear somewhat limiting in their inherent capability and the extent of the software if a degree of quality of the motif is assumed (as stated in Essay 6, Categories - Assessing the Inherent Quality of Motifs, as I regard this the most important factor in designing). Indeed, Lee's software inadvertently illustrates this. As can be seen from the above, this is now into its third incarnation, and yet the tessellations produced by people using this (including Lee, who would be most aware of its capabilities), show tessellations that generally remain of an inferior standard, albeit admittedly the occasional one of relative quality is indeed shown. Generally, the motifs are of my lower categories, 3 to 5, as discussed in Essay 6. Furthermore, there is no discernable difference in quality of the motifs shown despite additions of features (such as a full complement of the 28 Heesch types and perfect colouring) between the first and last versions.
• Crudeness of Drawing Instruments
• Rubber Stamp Tool
As such, the main drawback is the relative crudeness of the drawing implements. The degree of 'fine control' that can be obtained from traditional media such as a pencil is missing. As such, I actively dislike this feature, consisting of generic motifs, such as eyes and mouths for adding interior detail. This instrument results in the motifs appearing very much cartoon-like, an inferior category of motif. On the odd occasion a quality motif appears it is thus effectively downgraded by its usage. Although this tool is well intended, being easy and simple and is thus quick to apply, and apparently aimed for those who lack talent in drawing, such matters are not conducive to the overriding concern of quality.
Although I am somewhat critical of the software as regards the designing of the motifs, it does nonetheless have merits. Some very pleasing features include:
• The Drawing of the Tessellation
Upon designing a single motif, this then can be repeated with ease, whereby the repetitive nature of repeating the motif is automatically produced by the program. In contrast, when so drawn by hand, this is a most time consuming feature.
• Instantaneous Change
Where a change is made to one side of the tile this is then automatically applied to the other. Again, this is advantageous than by hand, as the change can be seen immediately. However, in practise, this is not a significant time saver, as when doing so by hand, the change only takes a few seconds to apply.
• The Morph ('Animate') Feature
Essentially, this feature can be used to show the development, from the underlying polygon to the complete motif. Alternatively, its 'undevelopment' can also be shown. This is a pleasing feature as it offers a 'non-static' viewing of the development from beginnings to end. The 'traditional' way of showing development is of the type as exemplified by Escher's print Metamorphosis I, whereby rhombs develop into a Chinese Boy, with examples of my own shown on the Development page.
However, although the above features are praiseworthy, and indeed a welcome addition to building on tessellations, they still remain very much on the periphery as regards the actual design. As such, part of the reason for this is that people lack the understanding of tessellation as outlined in my essays.
Of interest is speculating that if such software had been available in Escher's time, would he have used it to design his motifs? As such, I believe the answer would be no. The biggest drawback to this is the degree of control that when designing is absent. Put simply, when using ‘pencil and paper', this can be regarded as a direct extension of one's thoughts, with the implements of design under full control. In contrast, when using computer input devices, such as mice or graphics tablets, these effectively are additional implements that come between the designer and the drawing (of whatever medium). Quite simply, no device is yet (indeed, if it ever will be) as manipulative or dextrous as a hand can be. As such, there is simply no comparison in the degree of control exercised by hand and the computer equivalent tools.
Graphics Programs - Adobe Photoshop
Other ways of designing include the use of general graphics programs that are not specifically aimed at tessellation, such as Adobe Photoshop. However, as with the discussion above, the same difficulties in matters of ‘dexterity' reoccur. Therefore, this appears more useful for the task of repeating the motifs (although I have yet to use this in this way) rather than their actual creation. Certainly, if the problem of ‘dexterity' can be overcome, then the various image-editing facilities offer possibilities that are simply impractical by the traditional media. Of considerable implication is that offered by the colouring of the tessellation. Upon opening up an image in Photoshop there is a wealth of possibilities of fundamentally changing the coloration, not only in its coloration but in the number of examples - one could quite easily have, say, a hundred variations. Consequently, colouring can thus become trivial, albeit used in a more considered way then variations that would be impractical by hand (due to the time that would be involved) thus become a practical proposition. Other effects that can be used include the various filters. Again, these offer up new possibilities that would be impractical by hand. However, as a rule, these are not of undue importance, the effects being essentially trivial.
Best Use of Computers
Perhaps the best use of computers at present is in the purely mechanical task of replicating and colouring the motifs. Essentially, the 'tedium factor' of drawing numerous motifs by hand can thus be eradicated. Quite simply, this factor must be borne in mind, as if an inordinate number of motifs are contemplated, the tedium of continuous repetition, of drawing the motifs and their colouring, as fatigue can set in, leading to an inferior quality of finish, no matter how dedicated one is to the task at hand. Using the computer in such a mechanical way has three distinct advantages over traditional media such as pen, paper and paint:
Time. Quite simply, a single repeating motif, once designed, can be repeated ad infinitum, with the minimum of time expanded. Such a tool has obvious advantages in terms of time saved, negating such repetitive matters of that of necessity require a new drawing of the motif each time.
More motifs. Considerably more motifs can be shown than otherwise would be the case. Simply by using the 'cut and paste' tool, vast numbers of motifs can be shown (an example of this is in The Magic of M.C. Escher, using drawings No.66 and 130 (cover and endpaper), that simply would be impractical by the ‘old fashioned' method by hand.
Coloration. This can be achieved considerably quicker using a 'paint fill' tool. Furthermore, variations of the colouring using more colours than the minimum thus become a practical proposition.
So, what can be drawn from this? Although in the modern age it may be thought that some degree of basic knowledge of computers would be highly desirable, if indeed not essential in undertaking representational tessellations, such matters are demonstrably not of necessity. Escher himself, for obvious reasons, never had access to a computer, and yet this did not hinder himself in producing quality examples. Indeed, perhaps to give encouragement to people who believe this aid to be a necessary requirement, all the drawings here have been designed using the ‘old fashioned' pen and paper (and coloured using paints). Therefore, demonstrably, the field is still open to anyone who has the desire to attempt such things – a computer is thus not essential. In short, if one cannot design quality tessellations by 'pen and paper', a computer will confer no advantage - essentially, it is a matter of understanding tessellation matters, and is thus not simply a matter of the 'computer power' or software at one's disposal.
Last updated: 11 January 2006 (substantial additions to previous 14 November 2005 text)