The problem of covering a flat surface — a subset of the Euclidean plane or the whole plane itself — using some fixed geometric shapes and with no overlaps is probably one of the oldest in mathematics. Such a covering is called a tiling, or also a tessellation or a mosaic.
https://arxiv.org/pdf/2310.18950
Although it might look strange, mathematics also studies the placement of tiles on a floor! If the floor is an infinite plane, we have a tiling of the plane. A tiling is a way to cover the plane using nonoverlapping copies from a small set of geometric shapes.
In a periodic tessellation there is a region that repeats itself by translations. Mathematicians discovered that there are exactly 17 different kinds of periodic tessellations (the well known tessellations that cover the Alhambra walls in Granada contain examples of each of the existing 17 structures).
On the contrary the Penrose tessellation is a nonperiodic way to fill the plane, therefore there is no region that continually repeats itself. In a sense one could say that the Penrose tessellation is very varied, never repeating itself. To construct it, we use two forms: darts (dark colored tiles) and kites (light colored tiles). The fascinating thing is that the Penrose tessellation is not only non-periodic, but there is no way to place the tiles in a periodic way: like the pieces of a jigsaw puzzle, the way to fit darts and kites invariably leads to a non-periodic tessellation of the plane.
https://penrose.dmf.unicatt.it/
To understand what makes Penrose tilings so special, we first need to explore the idea of tiling itself. Usually, when you tile a surface, you use simple shapes like squares or triangles, and the pattern repeats periodically, like a checkerboard or bathroom tiles. These traditional tilings are periodic, meaning that if you slide or shift them by a certain distance, the entire pattern aligns with itself again.
Penrose tilings break this rule. They are non-periodic, meaning that no matter how you shift or rotate the pattern. It never repeats itself exactly. Yet, despite the lack of periodicity, Penrose tilings maintain a kind of order. It’s as if they are carefully balanced between chaos and predictability — a structure that is endlessly complex but still follows precise mathematical rules.
There are different versions of Penrose tilings, but the most famous involves two shapes: a fat rhombus (a squashed square) and a skinny rhombus (a squashed diamond). Penrose discovered that by using these two tiles, and applying a specific set of rules for how they must fit together, you could cover a plane in an arrangement that never repeats, no matter how far you extend it.
https://medium.com/@prmj2187/the-puzzle-that-never-ends-penrose-tilings-explained-9ca9822690ec
https://e.math.cornell.edu/people/mann/classes/chicago/penrose%20reading.pdf
Tilings Encyclopedia
Aperiodic
There are disputations among the experts how to define “aperiodic”. One possibility is to use it synonymously with nonperiodic. This is somehow a waste of this term. Others refer to an “aperiodic tiling” as one, which is created by an aperiodic set of tiles. This is unsatisfactory since this is rather a property of the set of prototiles than the tiling itself. Another definition is: A tiling is aperiodic, if its hull contains no periodic tiling. Personally, I like the latter definition (DF). Then a sequence …aaaaabaaaaaaa…. is not aperiodic (since its hull contains the periodic sequence aaaaaaaaaaaaa….), but the Fibonacci sequence is aperiodic.
Nonperiodic
A tiling T is called nonperiodic, if from T+x=T it follows that x=0. In other words, if no translation (other than the trivial one) maps the tiling to itself. In the theory of nonperiodic tilings usually the repetitive ones are the objects to be investigated. Usual constructions for repetitive nonperiodic tilings are substitutions, cut and project methods, and matching rules. Another well-studied class of nonperiodic (non-repetitive) tilings are random tilings, which can also be viewed as being generated by matching rules.
https://tilings.math.uni-bielefeld.de/
