5-Zone System

The 5-Zone System, a series of five paintings, continues my investigation of 5-fold symmetry. Each of the paintings are constructed from 5 sets of parallel lines called zones. The five zones are derived from the sets of parallel lines that appear in the shadow cast by the triacontahedron (a zonohedron). A zonohedron is a polyhedron in which the edges surrounding each face can be grouped into zones.

In 1970 I observed that the shadow of the triacontahedron produces two sets of rhombi (a wide rhombus and a narrow rhombus) that "close-pack" together to tile an infinite plane without forming repetitive patterns. I call the tiling a "quasi-pattern". This non-periodic tessellation underlies the work in this series reflecting 5-fold symmetries,  permeated by relationships in the golden proportion (phi).

The phi ratio seems to arise out of the basic structure of reality appearing regularly in the natural world, in things that grow and unfold in steps, including living things. For instance, plants and trees grow in the phi ratio. Leonardo da Vinci understood that humans are constructed in the proportion of phi.  In fact, it was recently discovered that DNA itself incorporates phi proportions in its construction.

In 1976, The physicist Roger Penrose devised matching rules to force the "wide and a narrow rhombi" into non-periodic tilings of the plane.  This anticipated the discovery in 1982 of the Quasicrystal.

The final painting in the series, "Quasi-Schechtman" is titled in honor of chemist Daniel Schechtman's 1982 discovery of the quasicrystal. In 2011 he was awarded the Nobel Prize in Chemistry. An article about Schechtman in the February 2012 issue of "Chemistry in Australia", reproduced this painting and acknowledged my use in artwork of the non-periodic tiling system in 1970.

Google panorama map of the "Five-Zone System" gallery exhibition, November 2011.
Chemistry in Australia magazine article, February 2012.

Scroll to access painting images.


  5 Zones , 35" x 35", digital print, 2012      

5 Zones, 35" x 35", digital print, 2012



  Quasi Schechtman  , 70" x 70", acrylic on canvas, 2011        

Quasi Schechtman, 70" x 70", acrylic on canvas, 2011 



oil on canvas, 70" x 70", 2011  

Exerpted from February 2011 Chemistry in Australia, QUASI-CRYSTALS OR QUASI-SCIENCE/ 2011 Nobel Prize in Chemistry, by Peter Karuso: 

... Art precedes science

To understand why this should not be possible, 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.

As an artist, Clark Richert has been working with five-fold symmetry since the 1960s 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 andinterconnectedness between symmetry and asymmetry, order and disorder – the harmony of opposites. 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 was published in 1974 in an obscure ‘Institute of Mathematics’ bulletin (B. I. Math. App10, 2660) but became popular when Martin Gardner described these in an article in Scientifc American (1977). This pattern became known a ‘Penrose tiling’ and uses only two types of tiles (the fat andthin 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’)…

  Kepler Relief  , 70" x 70", acrylic on canvas, 2012       

Kepler Relief, 70" x 70", acrylic on canvas, 2012



  Slanting In On Penrose , 70" x 70", acrylic on canvas, 2012

Slanting In On Penrose, 70" x 70", acrylic on canvas, 2012