Tiles & tessellations: past & present

by Sheila A. Menzies
Tile Heritage Foundation

Many of us have our ‘tile eyes’ focused—we are drawn to what attracts us. For me, in looking at tile installations I have always been ‘drawn in’ by the observation of geometric patterns within patterns—sometimes they seem infinite... the closer you look the more there is to observe in the tessellations. I love to muse about the intertwining of math and art.

Tiling is an ancient art with purposeful decoration as well as functional application. Many tile makers in the modern world are drawn by the same rhythm of patterning in their work from the Lows of Chelsea, Massachusetts in the late 19th century to contemporary artists like Hank Saxe & Cynthia Patterson of Taos, New Mexico. (Fig.1 image here)

A tessellation or tiling of the plane is a pattern of figures that fills an area with no overlaps or gaps. Tessellations are seen throughout art history from ancient Islamic architecture to modern art and tile installations. Many different types of tessellations exist. One of these tessellated patterns is aperiodic. (See http://paulbourke.net/texture_colour/nonperiodic/ for a full discussion.)

Penrose tilings were considered discovered by mathematician Sir Roger Penrose (UK, 1974). By using two different polygons (known as kites & darts) these tilings are the most famous example of tessellations that create aperiodic patterns. (Fig 2 image here)

Penrose tiles allow a two-dimensional area to be filled in five-fold symmetry, using two shapes based on phi (which is the Golden Ratio, or Number - also known as the Fibonacci Code). The ratio of the two types of tiles in the resulting patterns is always phi - which I think is fascinating! See http://www.goldennumber.net/penrose.htm ( Fig.3 image here)

However, in 2005 a young physicist, Peter J. Lu, in observing the complex 15th century tile patterns found in a madrasa (Islamic school) in Uzbekistan, found that these tile patterns on Islamic buildings were the identical geometric forms identified by Penrose as the foundation elements for elaborate, non-repeating patterns. Lu’s research shows that these patterns had been applied long before Penrose’s discovery, in fact 500 years earlier! For a full discussion visit  http://www.saudiaramcoworld.com/issue/200905/the.tiles.of.infinity.htm 

This ‘tiling’ story becomes even more fascinating today. Daniel Shechtman of the Israel Institute of Technology in Haifa, Israel was awarded the Nobel Prize in Chemistry 2011 for the first experimental observation of what came to be known as quasicrystals (1984): non-repeating regular patterns of atoms once thought to be non-existent in nature. They appear like 3D Penrose tilings. His research, in part through observation of patterning of Islamic tiling designs, assisted his understanding of what quasicrystals would look like at the atomic level. The observation of the existence of quasicrystals has changed the understanding of solid matter. Tiles and tile makers’ use of geometric designs can make it observable for all of us. Find details at http://www.nobelprize.org/nobel_prizes/chemistry/laureates/2011/press.html