Symmetry

Georges Hattab
Jun 6, 2019 · 10 min read
Thunderbird VII. 2010. Courtesy of Susan Point.

“If we could construct a microscope of sufficient power, we should be able, by the help of such an instrument, to resolve the molecular constellations of every little terrestrial milky way, exactly as our first rate telescopes resolve the celestial nebulae and separate double and triple stars. Were our sight sufficiently penetrating we should behold what now appear mere confused heaps of matter, arranged in groups of admirable symmetry”
All the Year Round by Edwin D. Babbitt [1]

As a quality of an object, symmetry defines an invariance to any of various transformations including: reflection, rotation, or scaling. This precise definition is given in mathematics.

In everyday life, symmetry often refers to a sense of harmony or beauty. It may be why crystals, sea stars, and many other things and living organisms are appealing. Indeed, symmetry is due to a balance of proportions or to the placement of exact parts facing each other or around an axis.

Types

There are two symmetry types: plane and axis. Plane symmetry corresponds to a mirror effect. Provided a cube is set in front of a mirror, there exists an exact correspondence of points in the mirror. Axis, or rotational, symmetry is the property a certain shape has when it looks the same after a rotation or a turn. Both are present in nature and across different scientific disciplines. In the animal kingdom, two taxonomic ranks are a fitting illustration to these two symmetry types: bilateria (plane) and radiata (axis).

Nature

We are a good example of a bilateria mammal, with bilateral symmetry, i.e. with a head (anterior), a tail (posterior), a back (dorsal), and a belly (ventral); hence, a plane of symmetry. The radiata is a historical taxonomic rank that was used to classify animals with radially symmetric body plans. Example organisms are sea stars, urchins, and plants [2].

Figure 1 depicts a triangle, a form common in many crystals, and when equilateral, has three points at the angles and three at the sides. Fig. 2 is an hexagon, common in crystallization, and has twice as many possible axes of symmetry. Fig. 3 has many more possible axes than Fig. 2, while figures of the animalcules (microzoa), 4 and 5, are but an example of countless millions of amazing minute skeletons of animals out of which whole mountains are sometimes built. They have a much larger number of possible radial axes of symmetry. For instance, the circle itself being a figure which is defined as being composed of an infinite number of straight lines, which are equidistant from the same center.

Figures 6 and 7 illustrate two leaves. In 6, the leaf branches out from the bottom to the top while in 7, it branches out in many directions and meets at a general center. Figure 8 depicts a shell of Echinus which forms a little dome-shaped animal with a variety of lines of forms having an axis symmetry passing through the apex. Figure 9 illustrates few of the radiating lines of the asterias, which has a flower-like center [3].

Science

From the fractal-like growth of plants to the crescent-shaped patterns that form as wind blows sand in the dunes of the Namib Desert, geometry enables the understanding and modeling of natural patterns. Symmetry is observed from the nanoscale to higher levels of organization. A good example is the modeling of snow crystals as seen in Fig. 2.

Another interesting example is organic chemistry. The chemical properties of a molecule aresomehow related to mathematical symmetry. Indeed, highly symmetrical molecules are hydrophobic (i.e. ‘water-fearing’ or failing to mix with water), while, on the contrary, hydrophilic molecules have a unique direction, or that is to say polarity. For example, the water molecule, dihydrogen oxide (H2O) or H-O-H, is a polar molecule. It has a bent shape where the electronic trends of its two O-H bonds can add together. In contrast, non-polar molecules have high symmetry. For example, dry ice (CO2) or carbon dioxide or O=C=O is completely linear and has its electronic trends going in opposite directions from the central carbon atom.

(Left) Dry ice or CO2 molecule. (Right) Water molecule or H2O.

For CO2, these electronic trends are cancelled, because of the shape and the symmetry of the molecule. As symmetry affects the polarity, the solubility of substances is affected. Another example is the organic and hexagonal compound benzene; as shown below in different representations.

Four equivalent ways of representing the structural formula of benzene where C is a carbon atom, H is a hydrogen atom, and a line is a chemical bond.

For related reasons, symmetry also affects other properties: melting point, vapor pressure, boiling point, crystal diffraction, etc. The inherent symmetry of individual molecules is called point-group symmetry. The elements of such a symmetry are rotation, reflection, and inversion of all atoms in the molecule, in three dimensions or 3D about a fixed central point without any net change to the molecule. Indeed, a principle aspect of our universe may be understood through the optics of symmetry. However, even with technical expertise the psychology of symmetry opens the door to some interesting discussions.

Aesthetics

Symmetry and beauty are often claimed to be linked, particularly by mathematicians and scientists. Yet, the predictability of symmetry is less attractive than the more dynamic asymmetry. However, an excess merely results in chaos. From a mathematical point of view, symmetry is undoubtedly aesthetically satisfying (e.g. tessellations). In spite of that, famous minds such as the philosopher Kant and art historian Gombrich disagreed and regarded symmetry too harsh, too rigid, or qualified it as banal ‘as it holds no surprise’ [4,5]. This led to a historical struggle, in which symmetry and asymmetry were and still are regarded as two opponents of equal power. The formless chaos, on which we impose our ideas, and the all too formed monotony, which we brighten up by new accents. In a recent study that analyzed the fractal dimension of inkblots devised by Hermann Rorschach, researchers found that pareidolia, or the psychological effect of observing human faces in inanimate objects, was attributed to three factors: the shading caused by variations in the ink’s opacity in the regions enclosed by the boundaries, the boundary’s fractal structure, and the plane (left-right) symmetry [6,7].

Courtesy of Hermann Rorschach, Rorschach-Test. Verlag HansHuber, Hogrefe AG

Other metaphors have been formulated by psychologist Rudolf Arnheim: ‘Symmetry means rest and tie, asymmetry means movement and detachment. Order and law here, arbitrariness and chance there; stiffness and compulsion here, liveliness, play, and freedom there’. In this duality, Arnheim defined a ladder on which: ‘every woman’s style, every individual, and every artwork finds its own particular place’.

Weyl described this tension as being ‘like life itself, inclined to mitigate, to loosen, to modify, even to break strict symmetry’. This seems to be true with the social, biological, and physical worlds; where asymmetry is at all levels. However, there’s an argument that symmetry forms the foundation on which asymmetry can be built. A good example of reconciliation is design, where symmetry achieves function and asymmetry provides character.

Design

‘Even in asymmetric designs one feels symmetry as the norm from which one deviates under the influence of non-formal character’, as Weyl puts it. A fun example that illustrates this thought is the study of faces by Bruno Munari (excerpt below) [8].

Courtesy of Bruno Munari. Design as Art.

Symmetry is an obvious feature of good, practical, and effective design. For instance, a chair or a table exhibit a symmetry that enables it to stand and support a function. The same goes for other objects created via product design. The readily distinguishable chair designs made by Alvar Aalto are notable. In the figure below, left-right plane symmetry enabled the illustrator to present side views of each design [9].

Chronology of furniture design. Courtesy of Alvar Aalto.

A little asymmetry is also necessary, irregularity implies changes and as Jan Tschichold puts it: “Asymmetry is the rhythmic expression of functional design. In addition to being more logical, asymmetry has the advantage that its complete appearance is far more optically effective than symmetry” [10].

However, even without technical experience, we can still be drawn in the world of symmetry and understand some of the foundations of the universe through relevant art works; such as architectural tiles, or in art using the ideas expressed by M. C. Escher.

Architecture

Organic forms and lines in Gaudì buildings, such as the church of the Sagrada Familia in Barcelona, have an underlying geometry based on vertical and horizontal structures. On one hand, symmetry can be found in Byzantium and early Renaissance. In the 16th century, the symmetry of Byzantium was displayed in the ornate artistic and architectural style which developed in the Byzantine Empire and spread to Italy, Russia, and elsewhere. On another hand, the asymmetry of the High Renaissance and the Baroque period, in the 17th and 18th centuries, is exemplified by the palace of Versailles and by the work of Wren in England [11].

Courtesy of Le Vau and Le Notre, 1669. Versailles, France.

Moreover, if we look around ourselves we may find all kinds of ornamentations in form of repetitive patterns: floor tilings, wallpaper designs, ornamental brickwork, or yet more patterns on our clothing. Some of these repetitive patterns are historic ornamental art and have been reported across different cultures (c.f. Geometry). This method of tiling shapes in a repeated pattern without gaps or overlapping is called a tessellation.

Tessellation

For two-dimensional tilings of a surface, there are 17 different plane symmetry groups, cataloged by the Hungarian mathematician G. Polya in 1924. Escher wrote, in regard to regular tessellation, that “it is the richest source of inspiration that I’ve ever tapped, and it has by no means dried up yet” [12]. The term regular refers to filling a surface with identical monohedral or related objects. The example below shows Zellige terracotta tiles in Marrakech, forming edge-to-edge, regular and other tessellations.

Islamic mosaic — ceramic tile tessellations — in Marrakech, Morocco. Courtesy of Ian Alexander, 2001.

While the aforementioned molecules are three-dimensional objects, and tiling patterns are generally on a flat surface, it is possible to illustrate some symmetry relationships with tessellations. Moreover, in contrast to the aforementioned point-group symmetry, the elements of tessellation symmetry are rotation, reflection, glide reflection, and translation of a side of a polygon to one other side, to create a tile that can exactly cover the surface [12].

Artistic patterns can inspire and interest those who examine chemistry or mathematics, while less complex tilings with the same symmetries can clarify some relationships of molecules. For at least half a century, people have realized that tessellations demonstrate two-dimensional analogs of the three-dimensional symmetries of solid molecules, following the principles of crystallography [12,13,14].

Selection of X-ray crystallography fabric prints from the Science Museum Collection. Courtesy of Helen Megaw

The Continuum

Schotter or Gravel Stones, Courtesy of Georg Nees.

Studies have shown that small deviations from symmetry can have strong effects on aesthetic evaluation and perceived visual complexity [7]. In the left figure, Nees represents the fading of a continuum of strict symmetries made up of squares into an asymmetrical arrangement. As the location and the angle of the squares is jittered progressively, less symmetry is perceived as one moves down through the image.

The original symmetries are still somewhat discernable but new possibilities and relationships are also opening up. This is reminiscent of the arguments of Stuart Kauffman, who suggests that the evolution of life — that statistically most unlikely event — could neither occur in the rigid, frozen, ordered world of ice crystals, nor in the booming, Boltzman-like confusion of an ideal gas, but perhaps where ice is melting to water, where there is fluidity and change, but order is not lost to noise as soon as it is formed. Life evolved, he suggests, ‘at the edge of chaos’, and intriguingly that area is also the most interesting and pleasurable.

  1. Babbitt ED. The principles of light and color. Kessinger Publishing; 1878.
  2. McManus IC. Symmetry and asymmetry in aesthetics and the arts. European Review. 2005 Oct;13(S2):157–80.
  3. Mandelbrot BB. The fractal geometry of nature. San Francisco: W.H. Freeman and Co; 1982.
  4. Gombrich EH. The Sense of Order, 2nd edn., Phaidon.
  5. Schummer J. Aesthetics of chemical products. HYLE–International Journal for Philosophy of Chemistry. 2003;9(1):73–104.
  6. Liu J, Li J, Feng L, Li L, Tian J, Lee K. Seeing Jesus in toast: neural and behavioral correlates of face pareidolia. Cortex. 2014 Apr 1;53:60–77.
  7. Taylor RP, Martin TP, Montgomery RD, Smith JH, Micolich AP, Boydston C, Scannell BC, Fairbanks MS, Spehar B. Seeing shapes in seemingly random spatial patterns: Fractal analysis of Rorschach inkblots. PloS one. 2017 Feb 14;12(2):e0171289.
  8. Munari B. Design as art. Penguin UK; 2008 Sep 25.
  9. Pallasmaa J. Alvar Aalto Furniture. Helsinki: Museum of Finnish Architecture 180 PP. ISBN 951–9229–33–7. 279; 1984.
  10. Tschichold J. The new typography: A handbook for modern designers. Univ of California Press; 1998.
  11. Wölfflin H, Murray L, Murray P. Classic art: An introduction to the Italian Renaissance. Phaidon; 1964.
  12. Martin BD, Sarhangi R. Symmetry, Chemistry, and Escher’s Tiles. InBridges: Mathematical Connections in Art, Music, and Science 1998 (pp. 245–254). Bridges Conference.
  13. M. C. Eschers brother, Beer, a teacher of crystallography, proposed in 1948 to write a book on this subject, with his brother illustrating [2]. Their project never materialized, though Escher later helped to produce such a work: Symmetry Aspects of M. C. Eschers Periodic Drawings, Carolina H. MacGillavry, Utrecht: Oosthoek, 1965. Reprinted as Fantasy and Symmetry, 1976.
  14. Milburn G. International Tables for X-ray Crystallography. Structural Science. 1983.
  15. Fuess H. GH Stout, LH Jensen: X‐Ray Structure Determination, A Practical Guide, John Wiley & Sons, New York, Chichester 1989. ISBN 0‐471‐60711‐8. Berichte der Bunsengesellschaft für physikalische Chemie. 1991 Jan;95(1):104–5.
  16. Gartus A, Leder H. The small step toward asymmetry: Aesthetic judgment of broken symmetries. i-Perception. 2013 Aug;4(5):361–4.

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