An essentially additional aspect of tessellation possibilities is that upon having designed a motif, e.g. a bird or fish of a given outline, it is possible to vary the interior design, or more precisely the viewing somewhat, and combine these different interiors in the following tessellation. For example, a bird can be seen from above or below; a human figure can be seen from the front or rear, and so both offer different presentations, or views, and yet both retain their respective same outlines. This essay thus examines such possibilities, with the diagrams taken from my own existing tessellations, in outline form. Also, Escher’s usage (or more precisely non-usage) of such a device is examined.
As such, this feature should be looked upon as an innovation, rather than assuming any undue importance. In essence, it offers an optional extra. To examine the issue, this is addressed with an examination of the more frequently occurring tessellating motifs, namely birds and fish, along with other less frequently occurring motifs, such as human figures, lizards, and dogs, of which the lizards and dogs are taken as representatives of their ‘type’ i.e. ‘close-ground habituating’ and quadrupeds respectively, from which by extension the same argument can be applied to other animals of this type, so effectively either including or excluding all examples of such categories without needless repetition.
However, this device, if used at all, must be used appropriately, as it is not suitable in all instances. An additional aspect to take into consideration is the symmetry arrangements of the tessellation, as depending on the motif the different views may not be compatible with the tessellation, and so have to be ruled out on aesthetic grounds. For example, a human figure, with a tessellation in which the motifs appears in three orientations, would not be suitable, as the human figure has just two (front and rear) viewpoints. In this instance, if attempted, one orientation would be of the front view, another of the rear view, whilst the third has no ‘natural’ continuation, and so would have to be left ‘vacant’, or ‘incongruent’ additional front or rear views added, neither of which would be an aesthetic practice. A more successful outcome here would be where the motifs appear in two orientations, or of a multiple of this.
Birds pleasingly have many typical representations from the human perspective, and are highly suitable for such a process. Indeed, this is the most suitable motif of all. A bird can be seen from either above or below, both of which are commonly encountered, and are ‘natural’ viewpoints from a human perspective. An additional typical representation is that its wings are raised or lowered, and each in itself has additional view points, as these can be brought forward or back.
1 Above and Below
2 Wings Down – Forward and Back
3 Wings Up – Forward and Back
Unfortunately, a fish motif is less suitable for such dual viewpoints. Indeed, it can practically be ignored. As such, its most typical representation from the human perspective, as seen from the side, essentially has no such variety. One side of the fish is essentially the same as the other. Another possibility is that one could show the fish head on, with an alternative viewpoint as seen from the rear. However, this is now an untypical presentation, of which the fish is thus essentially unidentifiable in outline, or at least is not readily identified as a fish. One could also show the motif top and bottom, but again, this is another untypical representation. Although fish are occasionally typically seen from the top, a fish viewed from below is most unusual.
Furthermore, the front/rear and top/bottom views are such an unusual position of the fish that this causes its own difficulties in tessellation. A major advantage of this motif for tessellation purposes is that of its fins and tail, of which when shown sideways have noticeable leeway in outline, thus aiding the motif for tessellation design purposes. In contrast, when viewed from the front/rear and top/bottom, this aspect is negated, with the motif being essentially a narrow ellipse, and so lacks such leeway.
Therefore, this motif thus essentially lacks this ‘one outline, two different views’ possibility.
3. Human Figures
A human figure is pleasingly suitable for such a process, as it can have two views, as seen from the front and reverse. Pleasingly, both views are typical, although aesthetically, one obviously prefers the front view. As an analogy, consider when an artist paints a portrait of a person – the front view is invariably chosen. Another possible viewpoint would be top/bottom, i.e. top of head and sole of feet. However, this would be an untypical representation, with the motif essentially unrecognisable in outline. Furthermore, the top/bottom views are such an unusual position of the human figure that this causes its own difficulties in tessellation. A major advantage of this motif for tessellation purposes is that the hair, arms, legs, and clothing, of which when shown front/back have noticeable leeway in outline, thus aiding the motif for tessellation design purposes. In contrast, when viewed from the top/bottom, this aspect is negated, with the motif being essentially formless, and so lacks such leeway.
This particular example is an example of an incompatible type, as discussed above. If the plane tiling is examined, it will be seen that this tiling is based on the motifs being in three orientations. Therefore, as this motif has only two different views, this would be incompatible. Therefore, I would choose for the finished tessellation either the front or back view, and negate the ‘one outline, two different views’ in this instance. Without doubt, the front view would be selected.
4. Lizard (and other ‘close-ground habiting’ animals)
Lizards (and other ‘close ground habiting’ animals) are less suitable. Indeed, there is no opportunity for such a process. This motif lacks different typical viewpoints from the human perspective, and has only one typical representation, as seen from above. One could possibly argue a case for a sideways motif, but as one side of the motif is alike (as with the fish), this is thus not possible. Therefore, this motif is unsuitable for ‘one outline, two different views’.
By extension, other ‘close-ground habiting’ motifs, such as beetles or ants have the same shortcoming, and are likewise excluded from this possibility.
5. Dogs (and other quadruped animals)
Dogs (and other quadruped animals) are less suitable. Indeed, there is no opportunity for such a process. This motif lacks different typical viewpoints as seen from the human perspective, and has only one typical representation, as seen from the side, either standing of sat down. As explained above, such a sideways view, as with fish and lizards (albeit the latter for different reasons), this motif is unsuitable by the same argument. Although one could arguer that a front view would also be considered as typical, a side view is by far the most typical representation. Therefore, this motif is unsuitable for ‘one outline, two different views’.
By extension, other quadruped motifs, even of noticeably different scales, such as horses and tigers all have the same ‘shortcoming’, and are likewise excluded from this possibility.
Escher’s Usage of ‘One Outline, Two Different Views’
Well, this is simply described – upon examining the 137 periodic drawings, Escher did not use this process at all! Note that although No. 31 might be thought to be an (only) example, as the fish are in two placements; this is not the same as different views. So, why did he not use this? Was he not aware of the possibility? This seems unlikely, as in different drawings of his bird tessellations he shows both back (e.g. No.18) and belly (e.g. No. 29) views. Indeed, No. 28 shows both over and under views in a single tessellation, but of different outlines of birds. The next obvious step is to combine these views in a single tessellation. As such, I can offer no explanation for such an omission. Although such matters are not of undue importance, as detailed above, nonetheless this still offers innovation, and so should on occasion be used for the sake of variety, if nothing else.
As can be seen, this device has limited application. In practice, it is limited to just birds and human figure motifs. Furthermore, although a motif may be suitable per se, its tessellations symmetry may not (the human figure example discusses the symmetry aspect in detail). Indeed, both considerations should be considered in tandem. Therefore, opportunities for using this are thus limited to an extent. However, as bird motifs are such a common motif in tessellation, the possibility of using this will nonetheless frequently arise.
Created: 11 September 2009