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# What is a Tessellation?

Mary McMahon
Mary McMahon

A tessellation is a tiled pattern created by repeating a shape over and over again, with no overlaps or gaps. A classic example of a tessellation is a tile floor in which the floor is covered in square tiles. Tessellations appear in numerous works of art in addition to architecture, and they are also of mathematical interest. These patterns crop up in a variety of settings, and once people start looking for tessellations, they tend to start seeing them everywhere, including in nature.

Tessellations are basically mosaic patterns which are made with a repeating polygonal shape. They can be used to tile a flat plane, or a sculpted surface. In all cases, the tessellation can theoretically be repeated infinitely, with the pattern remaining consistent and the shapes retaining their positions in relation to each other. Certain shapes will not tessellate, or cannot tessellate infinitely because the pattern eventually reaches a point where shapes start to interlock or gaps form.

In regular tessellations, also known as periodic tessellations, a single shape is used to tessellate. Only equilateral triangles, squares, and hexagons can be used in a regular tessellation. Semi-regular or non-periodic versions have two or more shapes. The art of M. C. Escher often includes non-periodic tessellation as a stylistic element, sometimes with very complex shapes, such as interlocking animals. This type of tessellation is also used in geometry and other math classes to introduce students to a number of concepts.

The mathematics background of the tessellation may explain why it is such a popular design element. Many recurring themes in artwork can be described mathematically, suggesting that there is a universal appeal in mathematically-bounded and described concepts. From the cobblestoned streets of Paris to the complex tessellated designs of Islamic art, tessellation can be seen everywhere, in a variety of levels of complexity. Like art, math can be a universal language which may be understood by anyone, and it is interesting to trace commonalities in radically different styles of artwork which can be linked to mathematical concepts.

Exploring tessellation can help children learn about shapes and basic math, and these patterns can make interesting, fun, or engaging projects for students. Students can play with ideas like seeing how many colors they need to ensure that shapes of the same color will not touch, and they can also experiment with visual illusions created with specific shapes and colors in a tessellation.

Mary McMahon

Ever since she began contributing to the site several years ago, Mary has embraced the exciting challenge of being a WiseGEEK researcher and writer. Mary has a liberal arts degree from Goddard College and spends her free time reading, cooking, and exploring the great outdoors.

Mary McMahon

Ever since she began contributing to the site several years ago, Mary has embraced the exciting challenge of being a WiseGEEK researcher and writer. Mary has a liberal arts degree from Goddard College and spends her free time reading, cooking, and exploring the great outdoors.

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OeKc05

I got my daughter a couple of wooden jigsaw puzzles featuring animal tessellations. She had a lot of fun learning how they fit together, and they looked really cool once they were totally finished.

One featured an elephant tessellation. Two shades of brown were used, and a light brown elephant would lock trunks with a dark brown one, so my daughter always knew which ones to fit together.

We also have a lizard tessellation puzzle. Three of the lizards are arranged so that they touch noses in the center, while the other three touch tails.

seag47

@Oceana - While it would be cool to have a unique tiling tessellation, I actually prefer the old-fashioned kind found in the cafes and diners of years past. I like the black and white squares that have been tilted to look like diamonds and have been alternated to play with your eyes.

Staring at them for any length of time will make your eyes do strange things. It’s sort of an optical illusion, yet it is so simple at the same time.

I have my kitchen tiled like this, and to me, it really brightens up the place. I think it helps my eyes and my mind wake up in the morning.

Oceana

I saw several examples of tessellation in my daughter’s geometry book last semester. Many of the examples were quite creative, and I never would have been able to think them up on my own.

There were several examples where triangles and squares had been used together. They formed curving, twisting shapes that I wouldn’t have imagined in a tessellation, but there were no gaps between the shapes, so technically, they were tessellations.

I think that if I were going to use tile in my house, I would lay it out in the shape of one of these creative tessellations. It would be something that not everyone has in their home.

shell4life

A honeycomb is one of the tessellation shapes that occurs in nature. If you can get close enough to a beehive to look at one without getting stung, it is really interesting to observe.

I don’t understand how the bees make the hexagons so perfectly. They fit together so that there are no gaps, making a tessellation.

I don’t know exactly how they go about building one, but they must have an excellent mathematical instinct. Human artists struggle to achieve something resembling the natural tessellation that bees are capable of making.

anon113119

yeah, good article, but more information can be added, e.g. mathematical concepts involved in tessellation, reflection, transition, rotation, etc.

anon69324