Aperiodic Monotiles

A brief background. In March 2023, four researchers, David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss, announced the proof that the tile discovered by David Smith is an aperiodic monotile, i.e., a solution to the einstein problem, a problem that seeks the existence of any single shape aperiodic tile. In May 2023, the same authors published a chiral aperiodic monotile with similar but stronger constraints. The two tiles, both of 13 sides, as discovered and named by the above, are titled the 'Hat' and 'Spectre'.

An obvious idea is to 'Escherise' the tiles with representational motifs, such as birds and fish (or other motifs), as often used by Escher. The tiles differ in their leeway in this regard. The Hat cannot be deformed in any other way, while the Spectre has limited possibilities. The latter can have curved sides, albeit only the same line throughout. Therefore, there are strict restrictions on the Escher-like artist. Typically, in a generic tessellation, there is more leeway in lines, not to mention in the choice of tiles, and so these are easier to achieve (albeit most people still struggle). Therefore, any Escher-like motif that has a degree of artistic integrity for the aperiodic monotiles (naturally ruling out 'abominations') is cause for praise. The tiles themselves do not particularly resemble animals, and so the teasing out of a motif is cause for praise. Specifically, I have found three bird motifs and two dogs, which I now detail below. In total, there are seven artworks. Wherever possible, for each of the works I strive to be innovative in the use of colour wherever possible for the sake of variety. Each entry is discussed in two parts; a general overview and the colouration. Although of a secondary aspect, colour can add value, hence my careful consideration. Without explanation, an otherwise subtlety may be missed.

The works below are all hand-drawn, watercolours, with a black pigment pen outline, on Bockingford 16" x 10 or A3 watercolour paper.

Dog 1

The first of two dog heads. This work was judged the better of the two, albeit slightly, but enough to establish a difference in quality. A dog head is clearly identifiable in profile, with a chin, stop, top of head, and ears. Colouration. A basic four-colour scheme is chosen. Red, yellow, green, and blue. No attempt is made in terms of colour innovation.

Dog 2

The second of two dog heads, this here judged the slightly inferior of the two, but enough to establish a noticeability difference in quality. Nonetheless, although judged 'inferior', it still has much artistic integrity and is a worthy inclusion.

Colouration: A basic four-colour scheme is chosen. Red, yellow, green, and blue. No attempt is made in terms of colour innovation.

Two Dog Heads in Combination

A hat tiling featuring the two distinct (albeit alike) dogs above (1 and 2), suitably combined in an innovative composition in different ways. It will be seen that in an 'upright' hat (being asymmetrical), both the left and right-hand sides will permit a slightly different dog head. Therefore, an obvious (and exciting) possibility is combining both views in a single composition (of course, both heads will also tile separately) to add interest. For this, use is made of the possibility of using both unreflected and reflected tiles for the dogs (hence the disparity in the distribution of the motifs, as there are naturally more unreflected tiles than reflected).

Further, in keeping with the dog theme, the composition is self-contained within a (stylised) dog bone outline, just as an amusing 'novelty' presentation, rather than the more usually seen square or rectangular formats.

Overall, this composition is my favourite in the series. It displays great imagination. And incidentally, was awarded an 'honourable mention' in the international Einstein Mad Hat competition.

Colouration: A two-colour scheme is chosen. Two (complementary) colours (orange and blue, for maximum contrast) are used, given the two dogs and two types of tile (as above), to reinforce the point and not the use of more colours (as with four or six colours commonly seen, of which the subtlety would then be lost at a casual glance). Of course, there is a trade-off here in terms of distinction, as the tiles are not map-coloured, but on occasion, such concerns can be overridden in terms of the demands of the innovative composition, which I think this fully justifies.

Birds 1

Arguably the best of the bird series, in that each of the main elements of a bird is distinct, i.e. head, body, tail, and two wings, and are broadly in proportion. For such a restricted tile, this is no mean feat.

Colouration: A six-colour scheme is chosen. Given that the tile appears in six orientations (including its reflection), an obvious thought is to use the six colours of the colour wheel, namely red, orange, yellow, green, blue, and violet, as a unifying concept. Of course, there is a trade-off here in terms of distinction, as adjacent motifs of the same orientation have the same colour, and so the normally 'ideal' map colouring aspect is lost (a four-colouring is also possible, showing distinct tiles, but at the loss of 'same orientations', including reflections, as here). However, on occasion, such concerns can be overridden in terms of the demands of the greater colouring scheme, which I think this fully justifies.

Birds 2

Arguably, the bird is a little contrived and was only completed due to the 'special' nature of the tiling. Although all the elements are on view i.e. head, body, tail, and two wings, the bird is overall 'clunky'. It is only borderline acceptable (to my high standards of veracity).

Colouration: A broad six-colour scheme is chosen. Each bird is of two colours, with the colours blending between the head and body around the colour wheel; red, orange, yellow, green, blue, red, violet, e.g. yellow body to green head.

Birds 3

A bird tiling of the hat. In terms of innovation, of which little is possible here, the birds are shown within a hat outline, to as it were 'emphasise' the tile shape, which is thus made clear at a glance. The composition is centred on a bird (in red) in its most natural position, flying 'upright', echoing the orientation of the greater tile outline, which gives a nice unifying touch to the composition.

Colouration. A six-colour scheme is chosen. Given that the tile appears in six orientations (including its reflection), an obvious thought is to use the six colours of the colour wheel, namely red, orange, yellow, green, blue, and violet as a unifying concept. Of course, there is a trade-off here in terms of distinction, as adjacent motifs of the same orientation have the same colour, and so the normally 'ideal' map colouring aspect is lost (a four-colouring is also possible, showing distinct tiles, but at the loss of 'same orientations', including reflections, as here). However, on occasion, such concerns can be overridden in terms of the demands of the greater colouring scheme, which I think this fully justifies.