Design 5 - Flexing

Under construction, subject to change, beginning 20 July 2018

This study is somewhat different from the usual practices of my own invention, as instead, it arises from correspondence with the Canadian quiltmaker Bart Braun. Our interaction beginning on 6 May 2018, who upon his asking for permission for one of my tilings to use for a quilt, then drew my attention to his quilt tessellating work on the online marketing site ‘Bonanza’ (of which previously I had not heard of). One work in particular, titled ‘Little Men’, Fig. 1, caught my attention due to its inherent quality. Upon further correspondence, and a request as to concept sketches, Fig. 2, this was sent, and other related figures on the same page, both below.

Fig. 1, Fig. 2

© Bart Braun

His Little Men was pleasing in many ways, of a realistic humanoid figure, and of a minimalist nature, of few lines. Further, surprisingly, despite the simplicity, I was unaware of the figure (and annoyed at myself for not discovering this!). However, this is not to say the figure was wholly satisfactory, as the head to body length ratio was lacking, although to a degree this could be overlooked. Ideally, the figure would be lengthened, all things being equal. Consequently, with such a promising figure, I decided to have a look at the possibilities, with the premise of a systematic analysis and finding some not previously found by Braun. Of course, all this emanates from Braun, as without his work I simply would not have devised this. To what degree this can be considered as a single Braun or Bailey study or Braun-Bailey collaboration is a moot point. Certainly, there is no active mutual designing between us per se. More to the point, I build upon Braun’s work, and then expand, considerably so. Seemingly as ever, the study expanded more than anticipated.
  Note that a basic premise of tessellating art is that of striving for improvements upon the initial design, which is rarely ideal, complete as a finished work. This aspect is all too often neglected, with the artist content with an initial design that although may be acceptable, and indeed good, loosely defined, could, with a little extra effort, be improved upon. Such strivings should be accepted with equanimity, although of course to a practical degree. For instance, as concepts, spending an hour making the figure 1% better, yes. To spend days, weeks, no. The various methodologies shown below are perhaps a little excessive, but as the figure showed such promise I consider that the time involved was justifiable.     

    Note that the study, at least of the beginning, retains the 45° orientation of the figure for the sake of consistency, although at such an angle the figure is not posed in an ideal manner, as obviously he should be upright, of which later I then adopt.

Analysis of the Little Men Figure
To begin the analysis, I show a single figure with colour coded symmetry, Fig. 3. This can be described as of a single line, reflected along the diagonal, and then translated left to right and top to bottom. A feature here is of a ‘shared line’ of a line segment. As can be seen, the resulting tile thus strictly ‘loses’ the original square vertices, and as a consequence, now requires the use of three colours for a map colouring.
Further to the analysis, it can be seen that the figure consists of six line elements, again, shown colour coded, Fig. 4. Beginning top left and then downwards, I use a rainbow colour ordering. The reason for this will become clear in the next section, where I discuss what I term as ‘flexing’.


Fig. 3, Fig. 4

Upon initial ad hoc ‘variation’ studies, I eventually came to realise that each of these lines can be systematically ‘flexed’, inside and outside the figure, giving a series of five diagrams to a given concept. Given that there are six distinct lines of the figure, this then gives 6 x 5 = 30 diagrams (in theory, as not all of the procedures are admissible), Fig. 5. Note that I have progressed systematically, beginning at the top of the head, and progressively working downwards. The core value Little Men is in the centre strip each time, from which the flexes emanate. Proceeding on the basis of a picture is worth a thousand words, the diagrams should make this clear what I am doing. As can be seen, this procedure does not always result in a permissible tiling; some of the flexes result in tanglement. Indeed, only 14 (including Little Men) are permissible, with a further * in a state of semi-permissibility, where the line to some degree directly overlaps with an existing line, but with the figure being ‘non-standard’ in the context of the sequence. Whether this should be included in the set is a moot point. Be that as it may, I show these separately. For the sake of convenience and future reference, I show all these as a distinct, numbered set. An open question is to ‘validity’ as to new instances as to originality matters. Are these sufficiently different from the source Little Men to be considered as distinct entities in their own right (admittedly with a nod to the source), and not just some insignificant change of no consequence? One could argue different ways. However, by and large, I think yes. Some contain new ideas. For instance, Row 2, No. 1 is suggestive of a girl with hair in a bun.

Fig. 5 All possibilities, leading to a set of 14 distinct Little Men variations

Flexing along the long axis, from the centre and beginning of Little Men

Continuing the flexing theme, it can be seen that the premise can also be applied to the figure in a different way, along the long axis (head to feet), instead of ‘localised’, as with Chapter 1.1, Fig. 6. Again, I largely proceed on the basis of a picture is worth a thousand words, rather than a lengthy and potentially convoluted text.

    As can be seen, there are two possibilities here, of five figures each, aside from the Little Men all new, thus giving a Set of 8.

Fig. 6a Flexing along the long axis, from centre of Little Men

Fig. 6b Flexing along the long axis, from beginning of Little Men

Fig. 6c

Fig. 6d

Fig. 6e
Fig. 6f

Fig. 7

Now, given that the Set of 14 are all topologically the same as Little Men, just as this figure can be lengthened, so can the Set of 14, as below. However, although a set of 14 is given, No. 8 can be seen to be strictly inadmissible, as lines ‘impinge’ upon the interior of the figure, although where these occur upon could arguably be removed, although then the figure is then not strictly true to the premise.

Fig. 8


Aside from the single figures, and flexing/lengthening thereof, there are other possibilities, with what I term as ‘hybrids’, Fig. 9. This is described as a tiling of two (or more) tiles, of tiles based on different elements of Little Men that won’t tile singly, but will in conjunction with another (of the same premise). A single example is shown below. However, although of interest as a concept, I see little merit per se in this. The tiling simply shows what can be described as off-shoots of Little Men of different proportions. As such, there seems little merit aesthetically in this particular combination. To me it can be described as being ‘clever’ for its own sake. Note that there are many hybrids possible, even including the Little Men figure itself, in association with the above (not shown). However, given that the concept is lacking aesthetically, I am broadly content to disregard such instances, and simply outline the possibility.

Fig. 9

Lengthening and Shortening

Now, an open question is that of trying to improve upon the initial figure in terms of proportions. As a concept, one could envisage a situation of various body proportions either lengthened or shortened. Either/or may be an advantage, although here, with a somewhat ‘compact’ figure, likely a lengthening would be advantageous. To this end, I now investigate, within a systematic premise, Figs. 10-15. Simply stated I investigate specific body parts, or regions. As can be seen, I undertake both shortening and lengthening on the figure, inside and outside, As can be seen depending on circumstances, this leads to a situation where further continuation is inadmissible, or can be continued to infinity, although where in such cases of the latter quickly become ridiculous in terms of ideal proportions of the figure, and so is then stopped after a few cycles. For each instance, I then show as 2 x 2 tilings.

Now, referring to Little Men, one incontrovertible shortcoming as when compared to a real-life figure is the head to body ratio, c. 1:3, which is notably different from a real-life figure, which is 1:7.5. Does the lengthening improve matters? As a bald statement, yes.

Fig. 10a
Fig. 10b, Fig. 10c
Fig. 10d, Fig. 10e

Fig. 10f, Fig. 10g

Fig. 10h

Fig. 11a
Fig. 11b, Fig. 11c

Fig. 11d

Fig. 12a
Fig. 12b, Fig. 12c

Fig. 13a
Fig, 13b, Fig. 13c

Core Value Changing Body Proportions

Now, the above studies, despite seemingly superficially being notably different, are, save for the lengthening, topologically of the same figure. And even of the lengthening, these remain very much of Little Men. Now, one property of Little Men is that the sides of the head and body are parallel, of both variations and lengthening, and so this can thus be described as a core value. An open question is to whether there are still yet more possibilities, for instance with a bigger/smaller head in relation to a bigger/smaller body. This is indeed so, and Little Men can be described as one of an infinity of such instances, as shown below, although these quickly become disprortionate. In practice, there are about three examples worthy, and no more. 

These instances can be described as ‘either side’ of Little Men. As can be seen, shortening the head width has a small cycle, as very quickly the lines of the procedure become inadmissible. In contrast, the head can be extended considerably more, but quickly becomes disproportionate. Be that as it may, both of the examples where the premise is made clear, an open question is to whether any of these are preferable to the initial figure, where a shortcoming is the rather large head to body ratio. Arguably, the narrower head instance is superior to the initial figure.

1 and 2

3 and 4

Other Grids (Isometric)

Another possibility for variation is that of other grid papers, aside from squared paper. An obvious next choice is that of isometric paper. To what extent does this offer new possibilities? It will be seen that it is possible to draw a square tile on isometric vertices, and then, with additional lines recreate the Little Men figure. However, this is a pointless exercise, as all I am merely doing is recreating the 90° grid instance on isometric paper! More to the point is that of a ‘broad transfer’ of the figure, with new lines of 60°. The figure below shows one such instance; there are indeed others. Further, I also show that the same procedures as with the core values of the square-based tile can also be applied to this.

1 and 2

3 and 4

Interior Detail

Finally, upon having concerned myself with matters of the outline, which is the essence of a high quality tessellation, in that in short the figure should be recognisable as in silhouette, I now address matters of interior detail. As such, although the figure can indeed stand alone as a silhouette, I consider that interior detail should be added. As detailed in my essays, this should be of a simplified nature, as exemplified by Escher. Examining Little Men, the head causes problems. An open question is to whether the figure should be shown wearing a hat, to in effect disguise rigid lines.   Would the head also be better drawn looking downwards? Should the figure have a discernible neck? What about the feet, which are somewhat ambiguous, which also serve as legs. Further, with the head to body ratio in mind, it could perhaps be better titled; Little Men suggest a full-grown adult, whereas Little Boys is perhaps a better title. Possibly, the gender title could be changed Little Women/Girls, to better reflect a more typical extravagant hairstyle. But there again, the figure is clearly clad in trousers, a more typical male attire. As usual for Escher-like tessellation, compromises are in order. Likely there is no one ideal outcome. Indeed, it is perfectly feasible, and appropriate, to have different versions, as a silhouette and interior detail, not to mention variations as detailed above.



To better aid understanding the procedures, below I show the flexing line premise, with a red coloured square for each of the six rows.

Rows 1 to 3

Rows 4 to 6

To be continued.