Aesthetics of Symmetry


A Treatise on the Aesthetics of Symmetry



I was walking along Commonwealth Drive one day in Boston admiring a wrought iron fence when I became introspective, curious as to why I appraised the fence as attractive. It had something to do with the inherent symmetrical design. With this in mind, I scrutinized other items that I traveled past exhibiting symmetry. Some were quite provocative, others were nominally patterned, and still others were rather tedious and boring. The conclusions I depict in this essay are anything but scientific.

Even if you concur one hundred percent with everything I describe, be careful when applying this to your own designs. At times the theories that we use to constrain ourselves during design come back to bite us in our rear ends. These ideas have the potential to capture your visual design processes and hold it hostage, incarcerating all of your designs to a central structural consistency. Bad plan. Please apply what you learn from this treatise where its application fits, but don't force all of your work toward satisfying these observations.

Finally, aesthetics also transpire below, on top of, and beyond symmetry. Some of the most exquisite art and craft that I've seen had nothing to with symmetry at all.

  Geometric Symmetry: Quick Review  

Symmetry has several variants, but the basic symmetrical operations are Translation, Rotation, and Reflection. Translation is simply the movement in a linear direction of a design; here is an example of a translation:


The design above has been translated four times. A rotation is the movement in a circular direction of a design; here is an example of a rotation:


The design above has been rotated six times. Finally a reflection is the inversion of a design across a line; here is an example of a reflection:


The design above has been reflected across both horizontal and vertical lines. Symmetrical operations can be combined; here is an example of a translation combined with a reflection (this is called a Glide Reflection):


Here is an example of a translation combined with a rotation (a Glide Rotation):


  What is Boring  

Symmetrical operations certainly add some elegance to an otherwise drab design. You can quite easily, however, have too much of a good thing. The following design, for example, is something that I would consider boring:


Too many repeats of the primary simple design bores us. What constitutes "too many" or "too few" repeats is a matter of taste and personal opinion. It also has to do with the complexity of the simple design, or motif. Furthermore, it might be affected by the symmetry operation:


The above example is similar to the boring example with too many repeats, however since the symmetrical operation is a Glide Rotation instead of a Translation, it is considerably more interesting.

  Grading Operations  

Can we find some arithmetic rules to show whether a symmetrical design will be visually appealing? Let's put together an equation where a higher rating indicates complete randomness (or a symmetry that is too challenging to discern), and a lower rating shows boredom. We'll call this rating the Interest level. Clearly, the more complicated the motif, the higher the rating should be. So we will want higher motif complexity to increase the number. More repeats makes for a less interesting pattern, so we will want higher repeats to decrease the number.

I therefore propose an equation for determining the Interest level of a particular symmetrical design: I = M/sn, where "I" is the Interest level, "s" is a number for the symmetry operation, "M" is a number for the complexity of the underlying motif, and "n" is the number of times that the motif is repeated. After considerable trial and error I have arrived at some appropriate values for s, the factor for the type of symmetry operation:

      Translation . . . . . 5
Reflection . . . . . 2.5
Rotation . . . . . . . 1
Glide Reflection . . 1
Glide Rotation . . . 1
  Repeated Motif  

Before we ponder some examples and contemplate the value of their Interest level, allow me to digress to discuss the M value for the complexity of the motif. I propose a simple measure of complexity for the motif: count the number of distinct elements. Here are some sample motifs, with the corresponding count of their "M":



Note that the concept of a "distinct element" is a bit subjective: it depends on what you perceive as an element. In some designs this is not readily apparent... does an arc count as it's own element, or is it part of some larger visual portion? Here you will need to use your visual design skills as a license to interpretation.

A motif can have its own symmetry beyond that of the overall design. In the example below, although the overall pattern exhibits Rotation symmetry, the motif itself has a design with Reflection symmetry:


In the case such as above where the Motif show signs of symmetry, the Interest level equation should properly be I = M/ysn, where "y", the additional divisor, represents the factor for the type of symmetry operation within the motif itself.

  Ratings Guide  
    Having explained everything that goes into the calculation, here is how I interpret the computed Interest level, I:  
      0-1.5 . . . . Boring
1.5-2.5 . . . Patterned
2.5-4 . . . . Interestingly Patterned
4-6 . . . . . Complexly Patterned
6-9 . . . . . Disturbingly Patterned
>9 . . . . . . Random
    These are not hard and fast rankings and are intended only as a general guideline. When you review the examples below, see if you agree in your artist's heart and your subjective imagination with the calculated ratings and interpretations.  

The page that is linked below has examples of various symmetrical designs, along with the computed Interest level and the factors that contributed to that rating:

  Visual Angle Constraints  

Although the above examples are appropriate for computer-based displays, design in real-life usually extends well beyond 600 by 800 pixel resolution. Because the human eye has both limited angles of visual perception and visual acuity, you need to consider the observant range when designing for larger areas. For example, a long-running fence may have several hundred repeats of a motif, even though the eyes of any given observer may only be seeing twenty of the repeats in a single glance.

A more subtle relevance is whether a design element should be considered as a motif within a larger symmetrical design, or as an individual item within a motif. This confusion can arise because the central eight degrees of the visual field are scrutinized at a higher level of detail than the surrounding background. When standing next to a large wall painting of the following design, each element is likely to be viewed as a motif in the overall design:


In other words, on a computer monitor this has an Interest level of 8/(2.5)(1)(1) = 3.2, but the Interest level standing next to a large wall painting of the same design would be 14/(2.5)(8)(1) = 0.7. Hence, be aware of how your designs are viewed in the context of the visual angles distinguished by the observer.

  Creative Ideas  

This treatise has discussed the evaluation of an existing design; if you are creative then you can use your imagination and talents to build interesting motifs from scratch. The page below flips the Interest Level formula around and demonstrates how to develop effective motifs with interesting symmetry in mind as the target.

  Offsite References  

You might enjoy these other sites dealing with the use of symmetry in design.

A scholarly review from the National University of Singapore.

An explanation of the 17 types of tiling symmetry, with historical examples.

An extensive scholarly book about symmetry and ornamentation, heavily mathematical, published online by Slavik V. Jablan.

An annotated list of Internet links related to art.