Dimorphic

The term dimorphic refers to a 'specialised' type of tessellation, whereby the tiling unit, by way of its 'line arrangement', permits the possibility of two distinct ways of being 'stacked' as a tessellation, this being in contrast to most tiles, whereby only one such stacking is possible. Because of this feature, two distinct tessellations thus arise from a single tile. Such a property is rare, and consequently of more interest than any arbitrary tessellation. However, not until relatively recent years (1977), with an article by Heiko Harborth in the Mathematical Gazette, were such specialised tessellations studied. However, subsequently, Branko Grünbaum and G. C.  Shephard undertook the first considered approach, in the book Tilings and Patterns, albeit the background to these, and indeed the whole book, is of a decidedly advanced nature, albeit much is still readily accessible to lesser mathematicians. Further articles on the subject can also be found with George E. Martin and Anne Fontaine in the Journal of Combinatorial Theory and Mathematics Magazine.
Now, due to the above-stated rarity factor, any tile which possesses this feature is thus of more interest than any given, arbitrary tile. However, it is hard to find any such examples at all. Therefore, the addition of life-like motifs is of more interest than in normal circumstances. The example below uses a dimorphic tile apparently devised by Percy MacMahon, in New Mathematical Pastimes, of which I myself show a bird motif 'of sorts', albeit somewhat limited as regards the quality – various shortcomings can be seen here, such as the head being considerably out of proportion. However, of necessity, such matters have to be essentially accepted on this particular occasion, due to the factors detailed above. In normal circumstances, such a relatively poor quality bird motif would be deemed 'borderline' as to whether or not I would produce a 'finished' example, as here. However, due to the above detailed 'rarity' factor, such perceived shortcomings can indeed be justifiably overlooked on occasions, as with this example.

Colouration
A minimum of two colours is required for both tessellations. In retrospect, when taken as a 'paired colouring', as was intended, the colouring scheme is somewhat poor. Firstly, although the choice of four colours is consistent, this is only ideal for (a), as the motifs appear in four orientations. In contrast, (b) has the motifs in only two orientations, thereby inconsistencies occur. Furthermore, the colours chosen are different between the two. Again, inconsistencies thus occur.

References

Grünbaum, B. and G. C. Shephard. Tilings and Patterns. W. H. Freeman and Company New York, 1987, pp. 46–49. 

Harborth, H. 'Prescribed numbers of tiles and tilings'. The Mathematical Gazette 61 (1977): 296–299.

n-morphic tilings, in response to Grünbaum and Shepherd’s patch-determined article.

MacMahon, P. A. New Mathematical Pastimes. Cambridge University Press 1921 and 1930. (Reprinted by Tarquin Books 2004), p. 94.  

Martin, George E. and Anne Fontaine. 'An enneamorphic prototile'. Journal of Combinatorial Theory and  Mathematics Magazine, Series A, Vol. 37, Issue 2, p. 195.

Page History. 21 October 2025.  Page updated from Classic Sites. The conversion had left the page intact; only minor text corrections were made in Grammarly. One addition was a list of references derived from the text, which I had overlooked.
Oddly, there was no date of creation recorded, but it was early, perhaps at the beginning (2003).