Snelsonian Motion

The underlying structures of the Snelsonian Motion series are based on quasi-symmetrical tessellations—(shadows from 6 and 10-dimensional space), comparing to one of the “String Theories”, which describes our reality as 10-dimensional.

The paintings utilize a "crystalline model for the atom" developed by sculptor Kenneth Snelson , and superimpose onto it curvilinear paths of sub-atomic motion resembling “Brownian Motion” in their unpredictability, as they trace a particle’s random meanderings through the structure.

R-P Motion (Richert-Penrose) | 2008 - Acrylic on Canvas 70x70in

Snelsonian Motion #3 | 2008 - Acrylic on Canvas 70x70in

Riemann Curvatures | 2008 - Acrylic on Canvas 70x70in

Snelsonion Motion #4 | 2008 - Acrylic on Canvas 70x70in

Snelsonion Motion V | 2008 - Acrylic on Canvas 70x70in

Quasi-Symmetries

The quasi-symmetry series utilizes the 5-fold symmetries that I first started investigating in the mid-60's in Drop City. Since that time, they have recurred frequently in my spatial investigations. I regard these paintings as shadowy expanses from 6-dimensional space (the structural modules having been projected from the triacontahedron on to a plane).

The primary issue being explored here is the structure of space.

Through the esthetic logic I am working with, and implicit in space (being equally extended in each dimension), is the sphere. Inherent in the sphere, by it's structural relationship to the isometric Platonic solids - such as the arrangements of evenly spaced points on their surface, are the triacontahedron and 5-fold symmetry. The non-periodic tessellation that underlies each work in this series, facilitates an interplay and interconnectedness between symmetry and asymmetry, between order and disorder. The deep structure, reflecting 5-fold symmetries, is permeated by relationships in the golden proportion. Implied by 5-fold symmetry is the golden mean.

The Golden Mean (or phi) is one of those mysterious natural numbers, like e or pi, that seem to arise out of the basic structure of reality. Unlike those abstract numbers, however, phi appears regularly in our natural world, in things that grow and unfold in steps, including living things. For instance, plants and trees grow in the phi ratio. Likewise, the nautilus shell and the sunflower spiral in the phi ratio. And in human anatomy, the spinal vertebrae are relative to each other in the golden mean. Leonardo da Vinci understood that humans are constructed in the proportion of phi and that much of the natural world is phi-based. In fact, it was recently discovered that DNA itself incorporates phi proportions in its construction. Viewed from the end of the DNA molecule (or from it's side), the proportion of the golden mean is evident in the molecular structure.

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Note: The "quasi" in quasi symmetries, means "some, but not all of the attributes" of a symmetrical system while "symmetry" refers, not simply to mirror image, but to balanced ordered systems and especially "dynamic symmetry", which is a proportioning system based on the "golden mean". "Quasi" also refers to the Quasi-crystal - a chemical solid that differs from a crystal in that it lacks a regular repeating structure. The paintings in this exhibit are based on the same structural system that led to the discovery of the quasicrystal.

Another note: The Quasi-Symmetry paintings appear flat but they actually represent volumetric space. They could be viewed from the front - or they could be viewed from the side. Either way they would look approximately the same and from both views would be permeated by the phi ratio.

It follows that implicit in 5-fold symmetry is life. Or to bring the logic (or quasi-logic) full-circle to completion: Implicit in nothingness is life.

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Phi Regions | 2010 - Acrylic on canvas, 70 x 70in

A_C Triacon

Quasi Kepler - Acrylic on canvas 70 x 70in

A_C Enneacon

Starpants - Acrylic on canvas 70 x 70in

A_C Kepler

5-Zone Series

The 5-Zone System, a series of five paintings, continues Clark’s 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 Clark Richert 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. He 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).

Quasi Schechtman | 2011 - 70" x 70" acrylic on canvas

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.

5 Zones | 2012 - 35" x 35" digital print

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.

Series Events | "Five-Zone System" RULE gallery exhibition, Nov. 2011

 Excerpted 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.

Kepler Relief | 2012 - 70" x 70" acrylic on canvas

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 | 2012 - 70" x 70", Acrylic on canvas

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. App. 10, 2660) but became popular when Martin Gardner described these in an article in Scientific American (1977).

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’)…