#tilings

hello! i've had this ask sitting here for more than half a year but since i am currently taking a seminar on tessellations i finally feel semi qualified enough to talk about them :D

tessellations are also known as tilings and they're the covering of a plane or surface with the repeating pattern of a specific shape (or multiple shapes) without any gaps or overlaps. like a grid of squares or like in honeycomb which is composed by a bunch of hexagons

there are many types of tessellations and they are usually classified through their types of symmetry. the most basic group of tessellations are called the wallpaper group and they can be simply broken down into regular hexagons (like the ones in the honeycomb), squares, rectangles, equillateral triangles and parallelograms with different types of symmetries depending on things like colors.

for example the four images below (all taken from wikipedia):

regular hexagons, any type of parallelogram (which include squares, rhombi, rectangles, etc), and triangles can always form tilings, which is actually used intuitively in design, like the patterns of fabrics like the dotted pattern below, which can be classified into a symmetry of triangles arranged hexagonally

there are infinite types of tessellations that don't belong to the wallpaper group. a specific type that caused a lot of uproar in the math community recently are called aperiodic tilings (you might have seen me talk about this recently bc I was very excited lol)

aperiodic tilings are types of tilings that can't be classified into any type of grid (above you can see triangle grids, square grids etc) in any part of the tessellation. some really popular ones are penrose tilings

the one below is a penrose tiling of two rhombi (again picture taken from wikipedia)

so the interesting thing of the new aperiodic tiling that was discovered (the hat) is that it is a single tile that is aperiodic and no combination of positions of that shape can create a grid, so it is an obligate aperiodic tiling !!! like, the individual pieces of the penrose tiling above could be rearranged into a grid but the hat can't! and it's the first of its kind (the paper hasn't been peer reviewed yet or anything but still very exciting news)

there are a lot of cool examples of tessellations in art like very well known in Islamic art (that is in the tomb of I'timād-ud-Daulah, Agra taken from wikipedia) there's a set of tiles used specifically in islamic geometry called the girih tiles but I don't really know much about them to talk abt them rn

some other well known tessellations are by mc escher (pic taken from this article) and you can see the symmetry behind the patterns

there's so sosos much more about tilings that I could talk about and probably will because I have been absolutely obsessed lately :~D

Here are some links if you're interested in further reading:

Tilings and Patterns, Grünbaum & Shephard (this one is more technical but has a lot of cool examples)

Visions of Symmetry, Schattschneider

The whole wikipedia page of the wallpaper group, has a very technical part but also a lot of visual examples that have helped me understand

The Symmetries of Things, Conway (i dont have a link for this but u can dm me for the pdf)

I also like this video on aperiodic tilings

look at this little gif i made about the symmetry of some escher like tilings :)