Birds 1

Birds 1 consist of birds of a geometric nature concerning their outline, of one motif (1-8) and two motifs  (9-12), all based upon an underlying square.

The background of this tessellation is particularly intriguing, although not in relation to the bird motif, which I devised. Interestingly, Escher studied this pattern but did not recognise its potential in terms of representation, specifically the bird motif. The tessellation originates from an illustrated article by George Pólya titled Über die Analogie der Kristallsymmetrie in der Ebene, published in the crystallography journal Zeitschrift für Kristallographie in 1924. In this article, Pólya presents 17 diagrams representing different plane symmetry groups, four of which he introduced specifically for this purpose. One of these, denoted as C4, serves as the basis for the tessellation discussed here. (Both the relevant diagram and Escher’s studies of it can be found in Visions of Symmetry, pages 23 and 25.) Given its simplicity, it would be surprising if this were its first recorded instance. If anyone has encountered a similar tile from an earlier source, I would be very interested in hearing the details.

Escher’s interest in this particular tessellation began in 1936 when he embarked on his first serious studies of tessellations. His research focused on examining various articles—specifically their diagrams, as he found the accompanying text too complex—obtained from journals provided by his half-brother, B. G. Escher, a geologist (Schattschneider provides a list of these sources, page 337). Among them, Pólya’s article stood out, prompting Escher to conduct a thorough study of its diagrams. However, despite his meticulous analysis, he overlooked the potential for a bird motif.

It is remarkable how well this outline lends itself to a bird motif. Surprisingly, since Pólya's apparent inception of this pattern, no one else seems to have recognised its representational potential—until I did in 1989 in sketch form, followed by a defining work in 1992 and 1993. This particular tessellation is a favourite of mine, as it features a bird motif with a geometric outline that, though simple, results in a design of exceptional quality. All elements of the bird are clearly recognisable—head, body, two wings, and tail. There is nothing contrived or awkward within the parameters of a geometric bird. I believe its quality is self-evident, even at a casual glance.

Another aspect of this tile, unrelated to its historical background, is its order-4 rotational symmetry. This symmetry allows for variations in the placement of the motif. The example here showcases a rotational arrangement in which all four orientations of the bird motif are used—what I term ‘fulfilling its potential.’ This differs from a translation, which would display only a single orientation.

Colouration

A minimum of two colours is required. The tessellation is illustrated in its simplest form using a two-colour complementary scheme of red and green. However, a four-colour scheme would likely be more effective, as it would emphasise each of the four orientations.

This tessellation is particularly notable due to its background, as it has a connection to Escher. The tile—though not the tessellation or the motif—appears alongside Escher's study of Pólya's illustrations (Schattschneider page 25), denoted as C4. However, it is not one of Pólya's original designs but rather a creation of Escher’s, possibly an experiment using the same lines arranged differently. Escher’s exact intent with this variation remains unclear. Nevertheless, he once again missed the opportunity to introduce a bird motif, as demonstrated in the example above.

An interesting aspect of this motif is its ability to be positioned in more than one way within a single tile, due to mirror symmetry in this case. Consequently, each tile has two distinct placements. The example above illustrates the simplest of these two orientations.

Colouration

A minimum of two colours is required. The chosen colouration is somewhat more elaborate than a conventional two-tone contrast. A basic two-colour complementary scheme—blue and orange—reflects the two orientations of the motif. Additionally, for the sake of innovation, the birds' heads have been given a third colour, brown. The colouring employs a technique in watercolour known as ‘wet in wet,’ which allows the colours to blend seamlessly.

Although the bird may initially appear somewhat unusual, with an odd, pointed protuberance extending from its head, this can be interpreted as a highly stylised crest—similar to those possessed by certain bird species.

Colouration

A minimum of two colours is required. A basic complementary scheme of red and green has been chosen, with the head and neck areas rendered in a third colour, yellow, for added variety.

As with No. 3, although the bird may initially appear somewhat unusual, with an odd, pointed protuberance extending from its head, this can be interpreted as a highly stylised crest—similar to those possessed by certain bird species.
Colouration. A minimum of two colours is required. A basic blue and orange complementary colouring scheme is chosen, of ‘one bird, one colour’. 

The background of this tessellation is particularly noteworthy—though not in relation to the bird motif, which I devised. It serves as an example of a hypersymmetric tessellation, a subject on which Branko Grünbaum has written in Congressus Numerantium. Although I do not claim expertise in this area, Escher appears to have been familiar with the outline, as his drawing 127 seems to be a variation. Doris Schattschneider briefly discusses such hypersymmetric tiles in Visions of Symmetry (page 317).

Colouration. A minimum of two colours is required. The most distinctive feature of this tessellation is its colouration, which utilises different shades of a single colour—blue. Although I previously stated that a minimum of two colours is necessary, while this example appears to use only one, the four distinct shades of blue can be regarded as separate colours in their own right. Consequently, this qualifies as a 'four-colouring.'

Due to the lack of strong contrast typical of distinct colours, this colouration scheme is used sparingly. It serves as a variation or novelty on the more conventional schemes that rely on definitive contrasts.

A feature of this design is that the bird is depicted in both 'under' (magenta, green) and 'over' (blue, yellow) views. Of course, one could equally as well have two tessellations of the distinct views.

Colouration. A minimum of two colours is required. Although two colours would be sufficient, four colours have been selected to highlight the motif's four distinct orientations, rather than using the minimum required.

Colouration. A minimum of three colours is required, although four are selected for reasons of better portraying the symmetry. Although three colours would have sufficed, such a colouration would be 'inelegant' as according to the demands of the composition's symmetry, this possessing order 4 rotation. Consequently, to emphasise the four orientations of the motif, four colours are thus used.
Most of note here concerns the colouring scheme, of a more restrictive type, whereby a basic two colours of blue and brown are chosen, these essentially being of a contrasting nature. To this is added a light and dark version of each colour, thereby resulting in four colours for all four orientations of the motif. 

A tiling based on the union of four regular hexagons. The symmetry permits four different placements of the bird; left, right, upright, and upside down, athough the simplest is shown, a translation.
Colouration. A minimum of three colours is required. An arbitrary choice of red, yellow and green is used. 

This example is based on a well-known tessellating geometric pattern, sometimes referred to as a ‘pinwheel.’ Because of its recognisability, finding a motif for this shape is of greater interest than for a random tessellation. A bird motif is indeed possible, as demonstrated here; however, its design leaves much to be desired. The design sits at the edge of what could be deemed an acceptable representation, as the bird appears somewhat disproportionate and awkward in places due to the angular nature of the outline. For example, the wings are not in alignment. Nevertheless, given the appeal of establishing a motif for such a familiar tile as the pinwheel, certain shortcomings in quality can occasionally be overlooked, which this example aptly illustrates. 

As the tile possesses order 4 symmetry, this allows the motif to be positioned in four distinct ways, thus maximising what I term as its ‘potential’. Although a translational motif could be shown, i.e. all motifs in the same orientation, this would leave other possibilities unused, hence the choice.

As such a motif of considerable quality, as the indentations of the tile, albeit of a simple nature, prove admirably suitable for a bird motif, as shown.

This example shows a somewhat rare example of the underlying placement of the tiles, as they do not meet edge to edge.
As to the composition itself, consisting of two distinct tiles, it will be seen that both have order 4 rotational symmetry, and so consequently this thus opens up plenty of scope for the placement of the motifs, of which after numerous studies the following was selected. The birds differ in inherent quality. The more ‘squat' bird can be seen to be not of the highest quality, as arguably it can be described as ‘overweight,' and not essentially streamlined, as a real-life bird would be. The other bird is more true-to-life. However, seemingly as ever with tessellations, compromises are inevitably in order, and as such although obviously not ideal, nonetheless I still consider that such an example, of the above rare type of tessellation, is thus in the circumstances justifiable for inclusion.
Colouration. Each of the two types of bird is coloured with two colours, with red and yellow for the 'squat', and blue and green for the 'standard' bird.

This and the following tessellation (No.11) are interrelated. The two examples essentially show different placements of the bird motifs, as both tiles possess order 4 rotation symmetry, and so consequently there are four possible placements of the (bird) motif in any one tile. The examples above and below show examples with a high degree of symmetry, with No. 10 showing a single placement (translation) and No.11 showing two placements (order 2 rotation).

The birds differ in inherent quality. The more ‘squat' bird can be seen to be not of the highest quality, as arguably it can be described as ‘overweight,' and not essentially streamlined, as a real-life bird would be. The other bird is more true-to-life. However, seemingly as ever with tessellations, compromises are inevitably in order, and as such although obviously not ideal, nonetheless I still consider that such an example, of the above rare type of tessellation, is thus in the circumstances justifiable for inclusion.

Colouration. Complementary colours of red and green are chosen for maximum contrast.

As the tile possesses order 2 symmetry, this allows the motif to be positioned in two distinct ways, thus maximising what I term as its ‘potential’. Although a translational motif could be shown (as with No. 10), i.e. all motifs in the same orientation, this would leave other possibilities unused, hence the choice.

First as this example possesses some subtleties that are not readily apparent at first glance, the reader may first want to test their ‘appreciation' of this tessellation by examining it in more detail before reading the following text.

This example is somewhat more complex in concept than with most of my tessellation, as it is of a type that I term as ‘Mother and Young', a term that without further explanation would not be at all obvious as to its intricacies. Now, upon observing the tessellation, in simple terms it can be described as of a less frequent larger bird along with more frequent smaller birds, from which the concept of ‘mother and young' might thus be apparent, albeit the true value is somewhat more involved than such a simple description, as subtleties lie behind this. Now, if the larger and smaller birds are examined (the latter in groups of four) it will be seen that in both cases the outline is the same. From this, the interpretation is that by subdividing this larger tile into four identical smaller ones, the notion of ‘mother and young' thus more directly arises, as the mother bird has thus literally given birth to the (four) smaller ones. A tessellation of just the mother bird is No. 6.

As such, a most pleasing tessellation as the concept behind it is most subtle and rare, and is therefore of more interest than any other ‘arbitrary' tessellation. A further example of the use of this tiling outline, involving the larger bird and neglecting the ‘subdividing' aspect, is shown as * , and *.

Colouration. A minimum of three colours is required, from which a 'simple' colouring is given, with the 'mother' bird in blue and the 'young' in red and green. 

28 May 2025. Minor revision of existing text from Classic Sites.
9 June 2025. Black and white birds were added upon revision of the page. Previously in Classic Sites, they were placed under 'Black and White'. Also, birds consisting of two motifs were added (previously, the page consisted of one motif).
10 June 2025. Added text to 'complete' the page where necessary.