A wall sculpture at Leeuwarden celebrating the artistic tessellations of M. C. Escher. Image credit: Bouwe Brouwer
The historical roots of tessellation are found in Mesopotamia where the Sumerians included it in their architectural buildings using clay tiles to create patterns. However, that was 4,000 years ago and the earliest documented findings of tessellations were in 1619 by Johannes Kelper. In his notes, he talked about semi-regular and regular tessellations, covering planes with polygon shapes.
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
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. |  |
Have you ever seen the floor of a church, the body of an animal such as the cheetah, or maybe, in food such as pineapples? Everything around us is made up of tessellations from soccer balls to the stones, the treads underneath the sneaker, to the paving stone in the city. Tessellations are everywhere!
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