On occasions, various people have written tutorials about the creation of life-like tessellations, of which I now examine their contributions. Note that on occasions here, the term ‘tutorial’ is stretched somewhat, in that the piece of writing is not necessarily intended as a tutorial, instead consisting loosely of thoughts about tessellation. Consequently, one could argue that I am being unfair to consider these as tutorials for review purposes, but none the less, these all contain tutorial elements, or intent, to greater or lesser degrees, and so I thus discuss. On each author, where this blurring occurs, this is duly acknowledged.

D B Sullivan’s Tutorial ‘How to Draw a Tessellation, Part 1’

Sullivan’s tutorial, of 21 pages, is very much is aimed at the complete beginner, new to tessellation, although the title is slightly misleading, in that it neglects to mention the Escher-like aspect, of which this is a prominent feature, in which the tutorial leads up to and ends on. Although not ordered as such, the tutorial is in seven stages, excluding an introduction, in which he address first principles, followed with increasing levels of relative depth, in which he explains how to create a Escher-like tessellation. Whether by accident or design, mathematical explanations are kept to a minimum, or are non-existent, likely in regard to the target audience.

Introduction
Sullivan begins with the more basic of first principles, with a definition of tessellation with everyday objects, along with advice as to pens and paper. The principle is explained as an abstract checker board, of which the idea is to change the existing sides. Reference is paid to life-like tilings, with the appeal to the imagination in forming objects and animals in the well-worn ‘what do you see in a cloud?’ premise.

Stage 1 – Let’s Get Started
Sullivan essentially then sets the framework as to how he goes about designing, with a 3 x 3 square format, and explains why this is his favoured format.

Stage 2 – Now for the Rules
Sullivan than emphasises the rules to the game with the following advice:
There is one basic rule to this game: if you change one [side], you must change them all
and illustrates this with the left and right sides of a square. This refers to the changing of the lines, in mathematical terms a translation. Essentially, this just sets out the principle involved.

Stage 3 – Playing The Tessellation Game
This stage is by far the more involved, at least in terms of length. Having stated the ‘rules to the game’ above, this is then illustrated with a very simple altered line (Line A), at the side of a square, to establish the principle, and then applies this to his 3 x 3 format, followed by a colouring in, titled as Design A. This is then followed by the same process again, but with a different line, of a more ‘complex’ nature, in terms of its elements, titled as Design B. This is then followed by a third and fourth example, Designs C and D, in which the altered line is at the top. As such, this doesn't seem strictly necessary, perhaps to reinforce an example with design B would be acceptable, but nothing more is relay required, the premise has been established. Whatever, no great harm is done beyond repetitive advice. Following this, the top and side lines are combined to form a tessellation in a combinatorial manner, Design AC, and then AD, BC, BD. Caution is advised when using certain types of lines when these are found to overlap, followed by a workaround, Design EF. Further advice is given concerning that at this stage it is not necessary to differentiate the tiles by colour. All of which is good advice.

Stage 4 – Finding the Hidden Figure
Having established the basics, Sullivan moves onto the creation of Escher-like motifs, with ‘Finding the Hidden Figure’. Here he takes a couple of the existing tessellations, Design BC and BD using the ‘what do you see in a cloud’ process as regards using the imagination for seeing animals. Good advice is given about rotating the initial presentation, as this offers a different viewpoint for the imagination, as in any one orientation may be unsuitable, whereas in another a life-like motif can spring into view. This is nicely illustrated, with upon the initial orientation not resembling anything, which when rotated Sullivan sees a ‘man with a hat, reading a book’, of which this is a fair, if not absolute resemblance to this. No matter, the point is successfully established. Even so, Sullivan continues with experimenting, turning the tile in yet another orientation, this time seeing this as a turkey, of which there is indeed a resemblance, but only rudimentary. Sullivan then looks at another of his designs, Design AD. The same procedure is then repeated, with upon an initial failure (a typical experience for all tessellators, no matter how good) with a seal motif spotted, again, somewhat rudimentary.

Stage 5 – Refining The Design
Sullivan then explains how he then refines the initial seal design, with the desire to make for a more real-life motif. Here, the existing geometric lines are now transformed into a series of gently undulating curves reminiscent of the original geometric line. Immediately, with such treatment the motif is now transformed into a recognisable as a (rudimentary) real-life seal, although still with much to be desired in terms of anatomical accuracy, but nonetheless of distinct promise. Sullivan then advises to continue with such refinements, with all the previous geometric lines replaced by a curved lined that resembled the original geometric one. Once he is happy with the standard, the seal design is shown as a proto tessellation, i.e. in sketch form.

Stage 6 - Transferring Your Design
The problem of then transferring your design to create a finished artwork is then addressed, with a tracing paper method given. As such, although this suffices, it is not conducive to accuracy. However, for the level intended, I am not too critical of this; this is not the most important aspect, being one of design.

Stage 7 – Finishing The Tessellation
After establishing the design by tracing paper, this is then shown drawn out fully, in outline, as a plane tessellation, followed by advice as to methods of colouring, in either traditional mean (paints) or contemporary (computers).

All in all, this is an exemplary tutorial, given the intended audience. The various elements to producing a life-like tessellation are broken down into a series of easily digested stages, of which Sullivan adroitly addresses, with the text and diagrams being of a clear and straightforward nature. That said, although the intention was likely to keep the tutorial as brief as possible, I do feel that some brief mention should have been made as to birds and fish are the most likely motifs to be designed. Indeed, likely this would encourage the intended audience, with a successful tessellation, where in contrast other motifs can be most difficult to achieve which might otherwise occur, and so discourage.

Andrew Crompton

Lifelike Tessellations

As such, I was in two minds as to whether consider discussing Crompton’s short paper for tutorial purposes here, in that the intent is not necessarily tutorial, but rather as part of a ‘general discussion’ on lifelike tessellations, in which he includes other aspects of the subject, with historical elements, along with discussions on more ‘advanced’ tiling per se, with isohedral and isohedral tilings. Indeed, what I term as the ‘tutorial element’ is very brief indeed, of little more than a single page. However, as the discussion does indeed touch on tutorial elements, as it includes his opinions as to the creation process, I thus include, albeit with reservation. Disregarding for analysis purposes here the above ‘general discussions’ which are not tutorial in intent, he begins with a general observation as to the difficulties involved in creating lifelike tessellation, at least of some degree of quality, of which I now examine, with my comments interspersed:

It is trivially easy to construct shapes which will tile the plane but to draw one which looks like something can be difficult. To do it successfully involves a little geometry, a little art, and a lot of willpower. The results are often more like miniature works of engineering than drawings. The process is well suited to an architect since it resembles drawing a plan in that one has to continuously refine, adjust and make compromises between competing elements.

Indeed it is trivially easy to construct shapes which tile, of which such type, with the addition of motif ‘detail’ frequently assault ones senses. As he says, the matter of drawing a truly representational motif is difficult. But this should be where the challenge lies, to turn the initial shape into something resembling the creature so titled, and not the first rudimentary thing as finished artwork. Of note is the reference to refinement of the initial figure, and any comprises arising thereof, a common situation, and an all important point, of which most tessellation artists tend to disregard. Such refinement is crucial to the quality aspect. Also he refers to the tenacity involved, again crucial. Most people settle for inferior motifs. Also of interest is his reference to architecture. Although one can see his point, such an interest is not strictly required. Escher had only minimal interest, and I myself have not studied it at all. So, do not be put off by thoughts that such a background is necessary.

A good tessellation has a slightly lunatic air, and ought to seem vaguely impossible, the important thing is that the perimeter of each tile has a double value, if this is not done properly the shapes will simply seem to overlap each other.

Again true understanding is displayed here, in that he refers to a ‘double value’, or as I prefer to describe a ‘double contour’. It’s a simple premise; one side of the line represents one thing, the other side something else, not necessarily of different creatures.

Tessellations are minimal drawings, for example the patterns of badgers shown here is formed by repeating this motif. It is difficult to imagine a badger drawn with a simpler line. This economy of expression is achieved making every point along its length have two meanings, one for each side of the line.

Again, the fundamental double contour principle outlined, with a badger tessellation. The right side line represents both front and rear, whilst the top line represents top and under.

This visual irony is the quality which brings the drawing to life. One approach to drawing lifelike tilings is to use an algorithm to create tiles at random in the hope that one will be produced which looks like something. This crude method works surprisingly well, and is good for creating tilings which interlock shapes in unusual ways.

I am somewhat uncertain as to what Crompton is referring to here. Does he mean tiles as according to a recognised tiling system, with the ‘what do you see in the cloud approach’?

The more usual method however, is to draw the thing to be tessellated then try to fit it to copies of itself using tracing paper.

I take issue that this is the ‘more usual method’. On the contrary, this is used less frequently; certainly myself and Escher did not use this, although this is not to say that it is inferior. Indeed, it may very well have advantages of its own, in that the ‘Escher way’, of distorting the sides of an existing tessellating polygon typically leads to ‘simple’ motifs, such as birds, fish, flatfish, of creatures without specific outlines, in that the outline is vague. Many of Escher's birds and fish can be mistaken for one another. On the other hand, more specific motifs, such as humans, may, but not necessarily be, better produced from an initial human figure, and ‘adapted’, as epitomized by Crompton here.

The process is made easier if it is realised that certain combinations are always impossible, for example, although four or six tiles may be fitted around a point, five never can. Although ordinary graphics software such as Freehand is useful for drawing the final pattern software for inventing the thing in the first place is generally an impediment. Some simple programmes for generating patterns automatically whilst working on a motif do exist but they obscure rather than illuminate the essential problem, which is working up and improving a simple line. This is an activity, like life drawing, in which using paper and a pencil is an advantage.

The final aspect discusses the use of computer software, in regards to its efficacies of designing. Indeed, he finds that for the initial drawing this is an impediment, and its only use is in repeating the design upon completion. I and others agree on this. Of note is that Doris Schattschneider said the same in Visions of Symmetry, being of the opinion that Escher would not have used software for designing. Certainly, this is so for most people, but subsequent to this paper Craig Kaplan’s Escherization program shows that progress is certainly being made, but this is in a different context to the one envisaged by Crompton. Also prominent here is the insistence of working up and improving a simple line, which is apt advice. Indeed, this is what it’s all about, of striving to make the motif as identifiable as possible in outline, and not just settling from some initial shape which bears not the slightest resemblance with just motif details added, an all too often failing with most people.

Although somewhat short, the essay does indeed contain much of merit as regards the principles involved in designing quality motifs. Crompton correctly identifies what should be striven for in terms of premise, and illustrates successfully in an exemplary manner with a badger, of which although a simple premise, one which would hardly stretch even a school child, a succession after succession of artists fail or seem to (or perhaps just conveniently overlook) want to realise this. As such, this is all just common sense; there is nothing abstruse in this.

It is trivially easy to construct shapes which will tile the plane but to draw one which looks like something can be difficult. To do it successfully involves a little geometry, a little art, and a lot of willpower.

All of this is very apt.

The results are often more like miniature works of engineering than drawings. The process is well suited to an architect since it resembles drawing a plan in that one has to continuously refine, adjust and make compromises between competing elements.

Of note is the reference to refinement of the initial figure, an all important point, of which most tessellation artist tend to disregard.

A good tessellation has a slightly lunatic air, and ought to seem vaguely impossible, the important thing is that the perimeter of each tile has a double value, if this is not done properly the shapes will simply seem to overlap each other.

This is then illustrated with an actual example, of a badger, with its outline, and then as a tessellation. He continues

Tessellations are minimal drawings, for example the patterns of badgers shown here is formed by repeating this motif. It is difficult to imagine a badger drawn with a simpler line. This economy of expression is achieved making every point along its length have two meanings, one for each side of the line. This visual irony is the quality which brings the drawing to life.

He then details general procedures

One approach to drawing lifelike tilings is to use an algorithm to create tiles at random in the hope that one will be produced which looks like something. This crude method works surprisingly well, and is good for creating tilings which interlock shapes in unusual ways.

Although slightly ambiguous, this seems to refer to the process as used by M.C Escher.

The more usual method however, is to draw the thing to be tessellated then try to fit it to copies of itself using tracing paper. The process is made easier if it is realised that certain combinations are always impossible, for example, although four or six tiles may be fitted around a point, five never can.

I’d take issue that this is the ‘more usual method’. On the contrary, this is used less frequently, although this is not to say that it is inferior.

Finally, he then discusses the potential of software, first as to the design, and then of the to aid in the time consuming task of showing as a tiling

Although ordinary graphics software such as Freehand is useful for drawing the final pattern software for inventing the thing in the first place is generally an impediment.

I quite agree.

Some simple programmes for generating patterns automatically whilst working on a motif do exist but they obscure rather than illuminate the essential problem, which is working up and improving a simple line. This is an activity, like life drawing, in which using paper and a pencil is an advantage.

Pleasingly, Crompton emphasises the basic aspect, which lies at the core of lifelike tessellation, namely of refining the initial curved line so that it at least results in a motif that bears some resemblance to the creature it is portraying, or aspires to be.