In 1967, the British mathematician Roger Penrose was interested in crafting this infinite variation. If you walked along an infinite surface with this kind of tiling, no matter how long you walked for, you would never be able to predict what comes next. However, there is such a thing as an aperiodic tiling – one which uses a small set of tile shapes that cannot form a repeating pattern, no matter how large the area. If you walked along a surface with this kind of tiling, you would be able to easily predict what comes next. Single molecular units can act as tiles and combine to build complex three-dimensional molecular tessellations properties of these tessellations such as their geometric symmetry can help scientists understand self-organised systems in biology and nanotechnology – especially as more and more complex tessellations are found.Īs of yet, all the tessellations we have looked at have been periodic that is, a tiling that has a repeating pattern within itself. Evolutionarily, this provides an advantage to the bees as less beeswax is needed overall for the same area of construction.Īt a molecular level, tessellation can also be found, especially within crystal structures. It’s more difficult to see this with squares in comparison to hexagons, however again if you construct a square with the same perimeter, the hexagon again has a greater total area as it is rounder in shape. However, you can clearly see that the hexagonal shape provides a larger area, fitting six triangles as opposed to four. The perimeter of both of the shapes is exactly the same – each is made up of six edges of the small triangles and is therefore six units in length. Take the side length of each small triangle above to be one unit. This can be easily demonstrated for hexagons and triangles. In 1999, the mathematician Thomas Hales proved the ‘honeycomb conjecture’, which demonstrated that the hexagon provided the least total perimeter out of these three options. However, this still leaves the three options for the shape – a square, triangle, or hexagonal cell.

This is because each cell is exactly the same. Honeycomb has such a regular tessellation to allow for the most efficient construction, allowing multiple bees to build different cells at one time. Within nature, tessellations tend to be regular or semi-regular. Other common types of pattern are semi-regular tessellations, which allow more than one type of regular polygon as tiles, or monohedral tessellations, which only use congruent tiles. This is because these are the only regular polygons with internal angles that are a factor of 360 degrees. From the fact that angles around a point must sum to 360 degrees, it is easy to infer mathematically that there are only three types of regular polygons which can make up these tessellations – equilateral triangles, squares, and regular hexagons. These are monohedral tilings (where every single tile is congruent) of regular polygons (all having equal sides and angles). The simplest types of tessellation are referred to as regular tessellations. As an art form, tessellation is particularly rich in mathematics, with ties to geometry, topology and group theory. They have even made their name in popular culture, from tiling puzzles and the tangram to the video game Tetris, where the aim is to create tessellations as best you can from a set of falling tetrominoes.Ī tessellation is defined a repeating pattern made of one or more shapes, without the formation of gaps or overlaps. Tessellations can be found everywhere: in the periodic arrangement of hexagons in honeycomb, stained-glass windows and mosaic tilings, and the striking artwork of M.C.

From my favourite book as a child Penrose the Mathematical Cat to the video series Doodling in Math Class by Vihart in teenage years, the beauty in this form of mathematics has always fascinated me. For me, this beauty is found in tessellation. But it is much more than this – ask a mathematician which part of maths they find the most intrinsically beautiful and you’ll receive a wide variety of answers, each very individual to the person. At first, it can just seem like they’re referring to the perfection of a nice proof when it comes to its conclusion. You will often hear mathematicians say that maths is beautiful. However, many don’t realise that this beauty is in fact mathematics. Have you ever noticed patterns in the brickwork of the pavement as you’re walking into town? Or in the wallpaper at your grandparents’ house? Or in the quilt on your bed? Most people have at some point in their lives noticed the beauty of the patterns in the world around them. “Triangles are my favourite shape, three points where two lines meet … let’s tessellate” Harriet Wood – student winner of the 2020 Teddy Rocks Maths Essay Competition Notes on honeycomb and M.C.

Geometric fish tessellation