non-bilateral symmetry

Below is a paper I wrote last year for one of my graduate Architecture courses. It was originally entitled Vitruvius and Symmetry.


Symmetry is most commonly thought of as the geometric state in which an object is able to be divided about an axis or plane and the resulting parts will be mirror images of each other. There are other types of geometric symmetry including radial symmetry, nested symmetry, glide-reflected symmetry, translational symmetry, and group-point symmetry, but bilateral symmetry (defined above) is the most basic geometrical concept of symmetry. Merriam-Webster uses the following four definitions:
1 : balanced proportions; also : beauty of form arising from balanced proportions
2 : the property of being symmetrical; especially : correspondence in size, shape, and relative position of parts on opposite sides of a dividing line or median plane or about a center or axis
3 : a rigid motion of a geometric figure that determines a one-to-one mapping onto itself
4 : the property of remaining invariant under certain changes (as of orientation in space, of the sign of the electric charge, of parity, or of the direction of time flow) -- used of physical phenomena and of equations describing them
Definitions two through four are the most commonly used and apply primarily to geometry and science. The first definition is less scientific and definite, but equally logical and perhaps more applicable to the study of Architecture, especially that of ancient Greece. This definition lies closely with the definition Vitruvius gives. “Symmetry is the proper agreement between the members of the work itself, and the relation between the different parts and the whole general scheme, in accordance with a certain part selected as standard.” (Vitruvius, 14) He goes on to say that if a temple is properly symmetrical, one can measure any of its various parts and know from their proportion the measurements of the other parts. In this way each measurement is reliant on the others and no measurement is arbitrary. His idea of symmetry lends itself more closely with the idea of symbiosis, because the measures are not constant, but change in each application. They do, however, remain proportional to one another. In the same way a lichen is composed of a symbiotic algae and fungus that are not constant, nor do they simply mirror each other but they rely on each other and when one grows or dies the other grows or dies along with it.
The greatest example Vitruvius gives of perfect symmetry is the relationship of the members of the ideal human body. This can initially be confusing for those who are used to the geometric definition of symmetry because the ideal human form is bilaterally symmetrical, but Vitruvius never mentions this type of symmetry in his description of the human form. He does, however, go into great detail when describing the proportions of the ideal human form as well as some of the geometry that describes them. “…in the human body the central point is naturally the navel. For if a man be placed flat on his back, with his hands and feet extended, and a pair of compasses centered at his navel, the fingers and toes of his two hands and feet will touch the circumference of a circle described therefrom. And just as the human body yields a circular outline, so too a square figure may be found from it.”(Vitruvius, 73) This description of the circle and the square in human proportion illustrates the vitruvian definition of symmetry in a geometric pattern of human proportion.
In his description Vitruvius states “The other members, too, have their own proportions and it was by employing them that the famous painters and sculptors of antiquity attained to great and endless renown.” (Vitruvius, 77) Interestingly, it was in his studying of the description of human proportions that Leonardo da Vinci drew his “Vitruvian Man”, an image that he is widely known for and also a symbol of the Renaissance.

While it is easy to get caught up in the logic and the depth of description used in the description of the ideal human form, it must be said that at some point the dimensions must be arbitrary to a certain degree. Who decides what the perfect proportions are? Who is the model for the ideal man? If perfection was an unattainable form as Plato believed, then how did Vitruvius decide that the distance from the chin to the forehead should ideally be one-tenth the height of a person? Today we would most likely take a survey of the population and calculate averages and use those figures to determine the most common proportional relationships. Did Vitruvius study a few visually pleasing specimens or were the proportions a tradition that had been developed over years, or did he study the sculptures and paintings that he spoke of?
It is interesting that he never explains how those specific proportions are derived as ideal, but he does mention the controversy over the perfect number. While the debate over which number is most perfect (in this case 6 or 10) seems a bit trivial, the importance lies in the fact that the most perfect number would be the most used system of numbering, and would affect the money that was minted, the architectural proportions, et cetera.
More recently, the psychologist and mathematician Michael Leyton has written several works detailing Group Theory, Nested Symmetries, and how they describe Architecture. His definition of symmetry is the following: “…the term symmetry means indistinguishability under transformations…” (Leyton, 45) His definition fits all of the geometrical forms of symmetry and in a way is broad enough to cover the Vitruvian definition in that any transformation in one part would be translated proportionally in all parts and would be indistinguishable unless compared with another standard. Leyton also states that “Symmetry is always the absence of memory.” (44) In other words a symmetrical relationship is assumed to be constant and to have always existed in the past. This ties together well with Vitruvius’ ideas on the permanence of architecture, especially Temple architecture which he fully expected to last indefinitely.
Conversely, the statement is made that “Memory is always in the form of asymmetry.”(Leyton, 44) Through this, the author seeks to explain that the Architecture of the 20th century is filled with asymmetrical design because asymmetry is dynamic, it is a memory of the forms that it is derived from and thus denotes a change from that platonic form to its present form. Leyton summarizes in the principle “An asymmetry in the present is assumed to have been a symmetry in the past.” (44) This falls in line with the idea that when Vitruvius saw a man that was proportioned less than ideally he was still fashioned after that ideal human form, and likewise Plato when he saw an object that was less than perfect assumed that it was fashioned after the intangible ideal form that must exist for us to understand the ideal.
In Leyton’s description of nested symmetries, he details a psychological experiment in which people were presented with a rotated parallelogram. “they then reference it to a non-rotated one, which they then reference to a rectangle, which they then reference to a square.” (42) His experiment attempts to demonstrate that the average person can see in a seemingly asymmetrical object the symmetrical shape that it is derived from and the geometric operations that describe it. Similarly, Vitruvius takes the human form, with its complex relationships and proportions and shows us the shapes that it is derived from in the circle and the square. These platonic forms govern the body and make the proportions somehow easier to understand. The idea of nested symmetry is that the most complex objects can have symmetrical forms as their origin or “nested” within them.
One of the most interesting things about reading Vitruvius and Leyton side by side is that their definitions of symmetry are so different and have such different explanations yet they effectively interact with one another in theory. Leyton bases his definition of symmetry in mathematics and psychology while Vitruvius describes symmetry based on human proportion and the philosophy of ideals. Both definitions have the potential to greatly affect how we view symmetry in design and to help in our understanding of the concept of symmetry. By broadening the definition of symmetry to accept the ideas of proportion and memory we can better rationalize what makes designs successful.
In a way we can learn a great deal from both schools of thought because both are well reasoned and have a strong train of logic.






Bibliography;
Leyton, Michael. “Group Theory and Architecture”
Nexus network journal 3.2 (2001): 39-58.

Merriam Webster Online /2006-2007. 8 September 2007 http://merriamwebster.com/dictionary/symmetry

Vitruvius, “The Ten Books on Architecture” Trans. Morris Hicky Morgan.
New York: Dover Publications Inc. 1960