Exerpted from February 2012
Chemistry in Australia: QUASI-CRYSTALS OR QUASI-SCIENCE
2011 Nobel Prize in Chemistry by Peter Karuso:
... Art precedes science
Consider tiling your bathroom floor? You can use triangular, square or hexagonal tiles to complete the tiling with no gaps. So it is in crystallography. Since the time of Max von Laue (Nobel Prize, 1914) who, in a single elegant experiment, proved the wave nature of X-rays and the periodic lattice structure of crystals – an ‘epoch-making discovery’ – it has been understood that periodic solids (i.e. crystals) can only form with two-, three-, four- or six-fold symmetry. This theory slowly became ‘law’ and was never challenged.
Albrecht Durer in The Painter’s Manual (1525) gives four examples of pentagon tilings, two of which were periodic. This required at least two types of rhomboidal tiles to take up the space between the pentagons. In 1619, Johannes Kepler published a systematic study of the tilings of regular polygons in Book II of his Harmonices Mundi. He noted that only tiles of three, four or six sides could be packed into a regular periodic tessellation and that all other regular shapes required three types of tiles or more to complete. However, at least a century before Durer, Arab mathematicians had determined how to produce tiling patterns with five-fold symmetry that were non-repeating – ‘quasiperiodic’. This tiling, known as girih graces many Islamic buildings such as the Darb-i Imam shrine in Iran, built in 1453. There are also examples of quasiperiodicity in modern art.
Quasi-Schechtman
2011, Acrylic on canvas, 70” x 70”
As an artist, Clark Richert has been working with five-fold symmetry since the1960s when he started building domes inspired by Buckminster Fuller at the experimental artists’ community Drop City. His domes were inspired by zonahedra such as triacontrahedra. In 1970 he discovered that the shadow cast by a three-dimensional rhombic triacontahedron produced a two dimensional pattern of two rhombi (‘fat’ and ‘skinny’ diamonds) that would tile together to fill a plane nonperiodically.
That year he produced several art projects based on the system, for example a poster entitled Tree of life. The non-periodic tessellation that underlies these works displays an interplay and interconnectedness between symmetry and asymmetry, order and disorder – the harmony of opposites.
Slanting In On Penrose
2011, Acrylic on canvas, 70” x 70”
Since the 1970s,Clark has continued to work with five-fold symmetry in his art and currently has an exhibition entitled ‘Five-Zone System’ at the Rule Gallery in Denver, Colorado. An example is his most recent painting entitled ‘Quasi-Shechtman’, is dedicated to this year’s Nobel Prize winner and graces the cover of this issue.
In the 1970s, Sir Roger Penrose, an Oxford University mathematical physicist, inspired by someone’s letterhead paper that displayed a pentagon surrounded by five more pentagons nestled inside a larger pentagon, decided to try and create a tiling pattern with five-fold symmetry with the minimum number of tiles.
This pattern became known a ‘Penrose tiling’ and uses only two types of tiles (the fat and thin rhomboids seen by Clark Richert in 1970), and comes closest to Durer’s and Kepler’s dream of five-fold symmetry tessellation. Notably, like the girih patterns of 12th–15th century Arab tiling, Penrose tiling was aperiodic – it never repeated itself but yet was predictable. Penrose patented his invention in 1976 (‘Set of tiles for covering a surface’)
This was published in 1974 in an obscure ‘Institute of Mathematics’ bulletin (B. I. Math. App. 10, 2660) but became popular when Martin Gardner described these in an article in Scientific American (1977).