The necessary and sufficient condition for an irregular hexagon to be able to tessellate the plane (see previous article ) is that it has central symmetry, as in the chamfered hexagonal tessellation of the figure. In this case, the irregular tiles also have axial symmetry, an unnecessary condition, and are combined with regular tiles; but it is obvious that the plane can be tessellated without the small regular hexagons.
In recent weeks we have talked about tiles and Penrose , so it is inexcusable to join both terms and dedicate a few paragraphs to Penrose tiles.
All the tessellations that we have dealt with recently, as well as the vast majority of those that we can see in mosaics and tiling of all kinds, are periodic, which means that we can delimit in them a region that paves the plane by translation, that is, moving it without subjecting it to twists or symmetries (an informal way of saying it is that the same basic design is maintained throughout the entire tessellation) .
The regular polygons that can tessellate the plane – the equilateral triangle, the square and the regular hexagon – can only do it periodically; but with equal rhombuses, for example, we can perform both periodic tessellations – the typical harlequinade – and aperiodic (can you draw one?).
A peculiar type of aperiodic tessellations are groupings of tiles that form copies of themselves on a larger scale, that Solomon W. Golomb called reptiles (contraction of repetitive tiles , repetitive tiles), as we saw a few months ago when talking about polyominos.
For a long time it was thought that all the tiles that could give rise to aperiodic tessellations, could also be rearranged in periodic configurations, as in the case of rhombuses or reptiles ; But since the seventies of the last century, sets of tiles have been discovered that can only give rise to aperiodic tessellations, such as the six tiles obtained by Raphael M. Robinson from the square, or the six by Robert Ammann, which we see in the figure, also from the square.
But the one who has advanced the most in this field is Roger Penrose, recent Nobel Prize winner in physics, who in 1973 discovered a set of six tesserae that impose aperiodic tessellation. In 1974 he reduced them to four, and later reduced them to two.
There are two pairs of these binary Penrose tiles: one made up of two rhombuses with equal sides, but different angles (can you calculate them?), And another made up of two Quadrilaterals with axial symmetry, one concave and the other convex, obtained by partitioning the less elongated rhombus of the previous pair, and which John Conway called dart and kite (dart and kite). In order for the tessellation to which they can give rise to be necessarily aperiodic, certain restrictions must be imposed; otherwise, it is clear that the pair of rhombuses can be used for periodic tessellation, and that a dart and a kite can be coupled by reconstructing the rhombus from which they are derived, with which periodic tessellation is equally obvious. Constraints can be materialized, for example, by coloring the sides and allowing only the joining of sides of the same color, or by adding protrusions and recesses that limit the coupling forms, as seen in the figure.
Aperiodic tessellations could look like a mere mathematical fun with no connection to the real world; But the discovery of quasicrystals in the mid-eighties of the last century (for which Dan Shechtman received the Nobel Prize in chemistry), showed that in nature ordered structures are formed, but not periodic, which was a real revolution in the field of crystallography. But that's another article.
Carlo Frabetti is a writer and mathematician, a member of the New York Academy of Sciences. He has published more than 50 popular science works for adults, children and young people, including 'Damn physics', 'Damn maths' or 'The great game'. He was a screenwriter for 'La bola de cristal'.
You can follow MATERIA on Facebook , Twitter e Instagram , or sign up here to receive our weekly newsletter .