What is symmetry?


is a
phenomenon, class of properties, concept
which is present in
all scientific disciplines and all kinds of the arts
Symmetry bridges
different disciplines, sciences and arts, different cultures,

The term symmetry is of ancient Greek origin. Its meaning is in close association with the related terms of asymmetry, dissymmetry, antisymmetry. Symmetry and the lack of symmetry characterise the phenomena in our natural and artificial environment, as well as our ideas about the world.

Traditional meaning of symmetry

The meaning of this term went through a fabulous transformation during its use for dozens of centuries. The proper translation of the Greek term symmetria – (from the prefix syn [common] and the noun metros [measure]) – is ‘common measure’. The Greeks interpreted this word, as the harmony of the different parts of an object, the good proportions between its constituent parts. Later this meaning was transferred to e.g., the rhythm of poems, of music, the cosmos (‘well-ordered system of the universe as contrast of chaos’). Therefore the Latin and the modern European languages used its translations like harmony, proportion until the Renaissance. In wider sense, balance, equilibrium belonged also to this family of synonyms. Some way symmetry was always related to beauty, truth and good. (These relative meanings determined its application in the arts, the sciences, and the ethics, respectively.) Symmetry was not only related to such positive values, it became even a symbol of seeking for perfection.

Common meaning of symmetry

In its everyday use symmetry is associated with its most frequent manifestations, like reflection or, in other words, mirror-symmetry, rotation (rotational symmetry), and repetition (translational symmetry). A few further geometrical appearances of symmetry belong also to this class of interpretations, like glide reflection, similitude, affine projection, perspective, topological symmetry.

All they are associated with the observation, that one performed a certain geometric operation (a transformation) on an object; and during that transformation one (or more) geometric properti(es) of that geometric object did not change (were conserved). That/those property/ies proved to be invariant under the given transformation. They are called symmetry in everyday life.

Generalised, contemporary meaning of symmetry

In generalised meaning one can speak about symmetry if

  • under any (not certainly geometric) kind of transformation (operation),
  • at least one (not certainly geometric) property
  • of the (not certainly geometric) object is left invariant (intact).

Thus we made a generalisation in 3 respects:

  • any transformation,
  • any object, and its
  • any property.

This generalised meaning of symmetry made possible to apply symmetry to materialised objects in the physical and the organic nature, to products of our mind, etc. Over geometric (morphological) symmetries, we can discuss functional symmetries and asymmetries (e.g., in the human brain), gauge symmetries (of physical phenomena); properties, like colour, tone, shadiness, weight, etc. (of artistic objects).

Asymmetry: The lack of symmetry
Dissymmetry: The observed object is symmetric in its main features, but this symmetry is slightly distorted (e.g., an arabesque ornament)
Antisymmetry: The observed object is symmetric in one of its properties, but one of its other properties changes to its opposite (e.g., a chess-board)

(G. Darvas ©)

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