Tessellations: From Regular Tilings to Penrose's Forbidden Symmetry

Filling the plane without gaps or overlaps

A tessellation covers a flat plane with shapes so that every point belongs to exactly one tile, with no gaps and no overlaps. The simplest case is a regular tiling: copies of one regular polygon, edge to edge. Only three regular polygons manage it — the triangle, the square and the hexagon — because the interior angles meeting at every vertex must sum to exactly 360°.

triangle: interior angle 60°   -> 6 meet at a vertex  (6 x 60 = 360)
square:   interior angle 90°   -> 4 meet at a vertex  (4 x 90 = 360)
hexagon:  interior angle 120°  -> 3 meet at a vertex  (3 x 120 = 360)
pentagon: interior angle 108°  -> 360 / 108 is not a whole number, so no
          regular pentagon tiling exists at all

Semiregular tilings and the 17 wallpaper groups

Mixing polygon types at each vertex — as long as the same sequence of polygons surrounds every vertex — gives the eight Archimedean (semiregular) tilings, such as the familiar 3.3.4.3.4 pattern of squares and triangles. Every periodic pattern in the plane, however elaborate, belongs to exactly one of 17 wallpaper groups, a full classification (proved rigorously in 1924 by George Pólya) of the ways translations, rotations, reflections and glide reflections can repeat a motif without gaps. M. C. Escher's interlocking birds and lizards are ordinary periodic tilings whose tile edges were distorted with matching translations so the boundary between neighbours becomes invisible.

Penrose tiling: order without repetition

In 1974 Roger Penrose found a set of tiles — most famously a "kite" and a "dart", or equivalently two rhombi — that tile the whole plane only aperiodically: the pattern never repeats by translation, no matter how far you tile, yet it is built from just two shapes under strict edge-matching rules. This is not a loophole in the wallpaper-group theorem; those 17 groups classify periodic patterns, and a Penrose tiling is deliberately not periodic. Enforcing the matching rules (usually drawn as coloured arcs that must line up across every shared edge) is what prevents the tiles from collapsing into one of the ordinary periodic tilings, and it forces a form of five-fold symmetry that no periodic tiling can ever have — the crystallographic restriction theorem bans 5-fold rotational symmetry from any periodic pattern, precisely because 108° does not divide 360° evenly.

Building one: the pentagrid method

N. G. de Bruijn showed in 1981 that a Penrose tiling can be generated directly, without trial and error, by drawing five families of parallel lines spaced at 36° to each other (a pentagrid) and taking the dual graph of the arrangement: every intersection of the grid lines becomes a vertex of a rhombus. Equivalently, a Penrose tiling is the shadow cast by projecting a slice of a five-dimensional cubic lattice down into two dimensions — the aperiodicity is simply what a periodic structure in 5D looks like once you view it from an irrationally-sloped 2D plane.

Quasicrystals: the physics caught up in 1982

For decades this looked like pure mathematics. Then in 1982 Dan Shechtman observed an aluminium-manganese alloy whose electron diffraction pattern showed sharp spots with ten-fold symmetry — forbidden to any ordinary crystal by the same restriction theorem that forbids 5-fold tiling symmetry. The atoms were arranged like a 3D analogue of a Penrose tiling: ordered enough to diffract sharply, but never periodic. The finding was initially dismissed, and Shechtman was awarded the Nobel Prize in Chemistry in 2011 once the field had come around; these quasicrystals are the physical proof that Penrose's abstract tiles describe a real, stable state of matter.

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