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The mathematical art of repeating patterns



A wall tile at the Alhambra, in Granada, Spain. (Photo: Wikimedia Commons, (CC BY-SA 3.0))

From patterned wallpaper to decorative mosaics, paving can be found all around us. The mathematical art of creating repeating patterns dates back to 4000 BCE, when the Sumerians used clay tiles to decorate their homes and temples. Since then, virtually every other civilization throughout history has adopted tilings in art and architecture.

Read on to learn about the history of paving and how the complex theory is still used today.

What is a paving?

A tessellation occurs when a geometric shape (or tile) repeats over and over again, covering a 2D or 3D surface without any gaps or overlaps. There are different styles of tiling depending on the shapes used.

The word “paving” comes from the Latin tessera which means a small stone similar to a tile. Tessera was used to make tessellata, that is to say the mosaics and tiles that decorate the ancient Roman buildings.

Types of paving

Paving art

An example of a semi-regular tessellation (Photo: Wikimedia Commons, (CC BY-SA 3.0))

There are many types of paving, all of which can be classified as those that repeat, are not periodic, quasi-periodic, and those who are fractals. The most common configurations are regular paving and semi-regular paving.

Ordinary periodic tiling is to create a repeating pattern from polygonal shapes, each meeting summit to summit (the point of intersection of three or more adjacent tiles). A checkerboard is the simplest example: it consists of square tiles in two contrasting colors (usually black and white) that can be repeated over and over again.

Semi-regular tessellations occur when two or more types of regular polygons are arranged so that each vertex point is the same. Each vertex is surrounded by the same polygons arranged in the same recurring order. (See the example above.)

Now that we’ve covered the basic math of tilings, read on to learn more about their use throughout history.

Tessellations in ancient Islamic art and architecture

Paving art

Tessellations of ceramic tiles in Marrakech, Morocco (Photo: Wikimedia Commons, (CC BY-SA 3.0))

While the Sumerians of the 5th and 6th BC Egyptians, Persians, Romans, Greeks, Arabs, Japanese, Chinese, and Moors all adopted repeating patterns in their decorative arts.

Perhaps the most famous style of paving can be found in Islamic art and architecture. Islamic religious art is generally characterized by the absence of figures and other living beings. This is because many Muslims believe that the creation of living forms is solely the work of God. Therefore, they adopted the abstract characteristics of tessellation and used colorful geometric tiles to create non-figurative patterns.

One of the most famous examples of Islamic tessellation art is found in the Alhambra, a huge palace located in Granada, Spain. It was built by the Muslim Moors in the 14th century and became the royal residence and court of Mohammed ibn Yusuf Ben Nasr. Inside the fortress, the walls are adorned with countless colorful tiles in geometric formations.

In Islam, using tilings to decorate surfaces and is called zillij. This style of mosaic tile is made from individually hand-chiseled pieces into a plaster base. You can still find zillij installations in Morocco and other predominantly Islamic countries, on the walls and floors of mosques, houses, public places and tombs.

Paving art

Tiling work in the Moroccan pavilion in the global showcase of the EPCOT Center. (Photo: Wikimedia Commons, (CC BY 2.0))

The tessellation in art history

When we compare the methodical theories of mathematics and science to the free thought process of artistic creation, it is easy to forget that the three disciplines are often linked. Many artists have focused on creating art that follows certain mathematical rules.

MC Escher: the father of modern paving

The mathematical theory of tilings has also had an influence on the art world. Perhaps the most famous artist for using geometric grids in his work is MC Escher. Also called the “father of modern paving,” the Dutch artist created irregular and interlocking tiles in the shape of animals and other natural objects.

Although he had no formal mathematical training, Escher was concerned with precision and a natural understanding of geometry. With his woodcuts, lithographs, black prints and drawings, he just wanted to achieve balance, harmony and perfection. He has already been quoted as saying, “Order is the repetition of units. Chaos is multiplicity without rhythm.

While Escher’s work is often associated with the Op Art movement, other artists used mathematical tilings in a more practical and decorative way.

The models of Koloman Moser


Lithograph “Die Reciproken Tänzerinnen” (The reciprocal dancers) by Koloman Moser, 1901 (Photo: Wikimedia Commons, public domain)

Austrian artist Koloman Moser is known for its Art Nouveau-style patterns which follow the principles of tessellation. His influential and elaborate designs, often featuring natural patterns and geometric shapes, have been used as textiles for fashion and interiors, wallpaper, posters, furniture, jewelry and even postage stamps.

Along with Gustav Klimt, Moser was one of the founders of The Vienna Secession. The avant-garde movement was a counter-foot to the conservative artistic institutions of the Austrian capital at the end of the 19th century. The artists promoted a contemporary approach to art and each developed their own styles which are still widely celebrated today.

Here are some examples of how contemporary artists use tilings in their art.

Matthew Chilean’s paper sculptures

Matthew Shlian

Read more: Paper Artist Engineers Incredible Fully Handmade Relief Sculptures

Tessellation Surreal Landscape Drawings by Tim Stokes

Hand-printed designs by JeongSu

Paper tessellations by Ekaterina Lukasheva

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