Introduction

An essentially additional aspect of tessellation possibilities is that of the serration of certain lines, undertaken for the purposes of improvement of motif, this being dependent on circumstance. This mostly and arguably more naturally, applies to birds and to a lesser extent other motifs, such as fish, horses, bugs and wild boars. More exactly upon having designed a motif, e.g. a bird of a given outline, it is possible, albeit dependent on circumstance, to introduce this feature. This essay examines the issues underlying this feature. Also, I examine Escher’s usage, my own, and Makoto Nakamura’s usage of such a device, of which the latter uses this feature the most extensively.

As such, this feature should be looked upon as an innovation, rather than assuming any undue importance, as it is of a secondary aspect to the tessellation. Broadly, it arises as a by product, in that having designed the tessellation, it is then possible to add this feature. In essence, it offers an optional extra if so desired. However, this device, if used at all, must be used appropriately, as it is not suitable in all instances, due to the symmetry arrangement of the tessellation, as this may not be compatible with the tessellation, and so have to be ruled out. To examine the issue, this is addressed with an examination of the suitable motifs.

When creating tessellations, the primary aim is, or should be, to have the outline of motif as recognisable as possible to the creature it is purporting to represent. Wherever possible, if the tessellation will admit it, fine detail, such as serrations, e.g. as occur with a bird’s wing should be added, thus aiding recognition of the motif.

Motifs
I now examine some motifs where this feature is relevant, at this point without reference to tessellation matters; in effect I establish the concept:

Bird

Background

Upon examining a real life bird, it can be seen that the wing is formed of a series of serrations, as shown below

Insert pic

This feature can also be seen to be of the tail as well.

Insert pic

Fish
Fish are somewhat of a less suitable motif for this feature. Upon examining fish, it can be seen that the fins and tail are formed of a series of serrations. However, in contrast to a bird, these are decidedly more subtle, and in practise are they are not readily seen. Indeed, one could argue that for such stylised motifs as preferred by Escher, such a feature is inappropriate, in that it is too detailed for a stylised motif. However, depending on circumstance, I consider that it can still be justified. Furthermore, Escher's fish serrations are conspicuous for their excess, of a type that no real-life fish possesses. Also, see Makoto Nakamura’s fish No.11.
Therefore, when a tessellation of a fish has a border line where:
(a) fins are on either side
(b) tail are on either side
(c) or the fins and tail in combination
It is thus possible to introduce this concept on the appropriate occasion.
Unfortunately, I have no illustration, as I have not used this for a fish.

Horses
Background

Although a horse may seem at prima facie an unsuitable motif, as it main outline has no natural serrations, it is possible to contrive a situation where it does, namely with its tail and mane flowing.
Unfortunately, I do not have any examples to show, but numerous examples can be seen in the work of Makoto Nakamura. However, as may be imagined, examples of this feature are decidedly rare per se, as not only the motif per se is most difficult to accomplish, the mane must also meet at the same border line.

Bugs
The serration effect comes into its own with bug-like creatures, as their legs, a prominent feature of the motif, consist of a series of segmented serrations. Unfortunately, I have nothing to show of my own. For examples of this, see Escher's No.54, and Makoto Nakamura’s.

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Usage by Other People

M.C. Escher’s Usage of Serration

As can be seen by an examination of the 137 periodic drawings, Escher, where the possibility existed, frequently used this feature, of approximately on half the occasions.

32, Fish, 1940 (with reservation)
39, Bug, 1941
54, Two Insects, 1942
84, Bird/Fish, 1951
87, Two Birds, 1952
88, Sea Horse, 1952
89, Fish, 1953 (top right)
90, Two Fish, 1953
92, Two Birds, 1954
93, Fish, 1954
Note that No.32 has faux serration

Escher could have used this on:
9, Birds, 1937-38
29, Birds and Fish, 1939
31, Fish, 1940
120, Birds and Fish, 1964
A13, Fish, 1958

Although it may appear that other of his tessellation are suitable, this is not so. For example, No.106 has a upper rear wing line that shares a border with the head and beak. Therefore, if the wing was serrated, this would affect the head, giving a most unlikely appearance, to the overall detriment of the tessellation. Therefore, in this instance, serration would not be selected for usage. Therefore, the opportunities for this feature are somewhat limited.

Makoto Nakamura
Nakamura’s work show considerable usage of this device, and furthermore with a wide variety of creatures, including birds, birds/flowers cats/birds, chickens, fish, horses, Pegasus, and wild boars. Interestingly, as a rule, if such a feature is possible, Nakamura uses this.

Bird 1
Bird 2

Bird 3
Bird 4
Bird 7
Bird 8
Bird 22
Bird 27
Bird 29
Chicken 1
Chicken 2
Birds/Flowers 1
Birds/Flowers 3
Birds/Flowers 5
Cats/Birds 3
Fish 11
Horses 1
Horses 2
Horses 3
Horses 4
Horse/Birds 2
Pegasus 2
Wild Boars 1
Wild Boars 2
Wild Boars 3

David Bailey
As such, this feature has had limited use in my own work, perhaps less than it should. Some examples are shown below

Figure 1

It is thus possible to introduce this concept on the appropriate occasion, as shown, with a semi-neat study, Figure 1, right, a bird with no serrations; and left, of a serrated motif of the bird at right. In this instance, it is of types (a) and (c).
Therefore, when a tessellation of birds has a border line where:
(a) wings are on either side
(b) tail are on either side
(c) or of the wings and the tail in combination
one can use this feature

The above diagram shows some experiments undertaken on a geometric bird, of type (a) where wings meet.

Above I show some further examples, of a more stylised nature, of a geometric type of motif. Firstly, I show various experiments that are undertaken in the pursuit of a best type. Note that as I have identical birds, so must the birds with serration to be consistent. To achieve this, the serrated line must be 180° rotational. Again, the purpose is one of discerning the best serration, rather than simply settling for the first, as so many other people seem to do. As I state in my other essays, quality of motif is everything!
Below this I show a wireframe tessellation, in which I show a simplified best.

experiments

Best serration, as a semi-neat wireframe

As such, tessellations pay lip service to this feature, in that such a feature is not generally applicable, although desirable. To workaround this, they put in place a series of lines, of which therefore this feature is implied. Below I show a diagram where this feature is implied.

Summary
The serration feature has less opportunity of being used, as it depends on the motifs suitability and further to that its symmetry arrangement of line having to be suitable as well. Generally, therefore, the opportunity for the addition of such a feature is not possible. However, wherever possible, I consider this an ideal feature to use, and where this situation arises, I thus use. Of note is that both Escher and Nakamura both used this to a noticeable extent, albeit they have not set out their reasons. However, it is implicit that this is advantageous, as why else use? Therefore, if the opportunity arises, this makes for a pleasing tessellation, as the motif is made more life-like, the perennial goal of the tessellator. Although both the serration and non-serration examples are in a generalised sense worthy, I consider that a slight advantage is obtained from the serration type, and, as stated elsewhere, the accumulation of a small advantage should be added where possible.

Agree/disagree? E-me.

Created: 14 September 2009