Then in 1891, Yevgraf Fyodorov, a crystallographer, set off a chain of mathematical events when he announced that every tessellation contained seventeen different sets of isometries. This proved to be a startling discovery, enticing people’s interest of what a tessellation is, it’s role in mathematics, and how many kinds of tessellations are there.
What is a Tessellation?
A tessellation on a flat surface is the tiling of a plane using more than one geometric shape to create it. It doesn’t contain any gaps or overlaps. In Mathematics, the concept of tessellations is studied as an independent subject. It tries to figure out a way to arrange the tiles on a plane without leaving any gaps or overlaps. In order to fill in the plane, people must follow a specific set of rules such as that all corners should meet up with no corner of a single tile to align along the edge of a different tile.
However, following the rules isn’t true for some tessellations, notably the tessellation designed using bonded brickwood. Regular tessellations made up using a triangle, hexagon, and triangle follow this rule whereas semi-regular tessellations are created using the same kind of polygon. If you want to view artwork using the concept of tessellations, head over to the page of M.C. Escher to see how he made tessellations using asymmetrical interlocking tiles, shaping them up like objects or animals. In addition to regular and semi-regular tessellations, there are seven other kinds of tessellations.
The Seven Types of Tessellations
1. A Normal Tiling
Each tile is topologically equal to a disk, two intersecting tiles are connected by a single or empty set, and all of the tiles are evenly bordered.
2. A Monohedral Tiling
Each tile is compatible with each other, meaning that it has one prototile.
3. The Hirschhorn Tiling
The tiling contains an irregular pentagon.
4. An Isohedral Tiling
Every tile belongs to one transitivity class, which means that they all are duplicates of the same prototile that belongs to the symmetry group.
5. Penrose Tilings
It is made up of two quadrilaterals, both different from each other, creating non-periodic designs.
6. Aperiodic Tilings
It creates patterns that can’t tessellate sporadically.
7. Dirichlet or Voronoi Tilings
Every tile is defined as a pair of points nearest to only one of the points of the detached pair of defining points.