Birds 6

A dedicated section of what I term the 'Pólya tile', specifically of C4, 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 bird 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. 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. 

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.

Colouration Studies. A dedicated colour arrangement study, with colour as the main premise. Specifically, this refers to what I term as 'partition colouration', in which the bird motif is divided into a series of regions (in this instance, four - head, body, wings, and tail), and then coloured using four colours (with the map-colouring condition). The tessellation, of order 4 rotational symmetry, was especially chosen to complement this possibility. The challenge then is to colour this in all possible distinct ways, of which there are 5 examples, with 3 distinct types, as according to the vertex arrangements, with colour distribution ratios of 1, 1, 1, 1 (No.1 and 2), 2, 1, 1 (No. 3 and 4), and 2, 2 (No. 5).

All five instances were exhibited at the 2008 Bridges Mathematical Art conference, in Leeuwarden, the Netherlands in 2008. 

Black and White

6 June 2025. New dedicated page on the Pólya tile. Although previously (Classic Sites), all were shown, this was not as a dedicated page.  No. 1 was on a geomatic bird page, the black and white was as that premise, and the 'Set of 5' were on a dedicated page on the Bridges 2008 Art exhibit.