György DarvasDirector, SYMMETRIONP.O. Box 994, H-1245 Budapest, HUNGARY
symmetria, ,
is 'common measure', from the prefix sym (),
common, and the noun metros (),
measure. The Greeks interpreted this word to mean the harmony
between the different parts of an object, the good proportions
between its constituent parts. Later this meaning was applied
to such things as the rhythm of poems, of music, and even the
cosmos ("well-ordered system of the universe, in contrast
with chaos"). Up until the Renaissance, Latin and the emerging
modern European languages translated "symmetry" such
as harmony and proportion. In a wider sense, the terms balance
and equilibrium also belonged to this family of synonyms. It
is not too difficult to deduce that in many ways symmetry was
always related to beauty, truth and the good. These related meanings
determined its application in the arts, the sciences, and ethics
respectively. Symmetry was not only related to such positive
values, it even became a symbol of the search for perfection,
as in the writings of Plato.The earliest surviving description of symmetry was given by
the first century B.C.E. Roman architect Vitruvius, in his Ten
Books on Architecture, which laid the foundations for the particular
meaning of the notion. His usage suggests symmetry is a general
term related to the meaning of words such as harmony, proportion,
rhythm, etc. It was not by chance that the humanists of the quattrocento
rediscovered Vitruvius's treatise for themselves. With the publication
of several translations, the term "symmetry" replaced
earlier translated versions of the word, and took its place within
modern European languages, first in Thus, after the Golden Age of Greece (fifth century B.C.E.), the paths of science and the arts crossed again in the Renaissance, only to divide once again -- for centuries -- and then to meet for a third time in the cultural melting pot of the twentieth century. This has provided fertile ground for the application of symmetry in both spheres, art and science, of human creativity, to products of two opposite cerebral hemispheres.
Figure 1).Figure 1. Reflection. Drawing by the author. A considerable proportion of readers may mention n) divisor of 360°, we return to the
original object after n transformations (rotations). In
these cases we speak of n-fold symmetry. Similar observations
can be made when rotating three-dimensional objects (Figure
2).Figure 2. Rotation. Drawing by the author. Fewer readers will mention the most frequent manifestation
of symmetry: Figure 3).Figure 3. Translation. Drawing by the author. The conservation of other geometric properties may serve as
the basis for other types of symmetry, such as similitude, affine
projection and topological symmetry. Similitude is a symmetry
transformation whereby the distances between the corresponding
points of two objects change, but the ratios between the lengths
and the angles are preserved; thus the shape of the object remains
similar to the original ( Figure 4. Similtude. Drawing by the author.
Figure
5).Figure 5. Affine projection. Drawing by the author.
Figure
6).Figure 6. Topological Symmetry. Drawing by the author.
To generalise the concept of symmetry, first, replace the geometric transformation with any kind of transformation; second, apply such transformations not only to geometric objects, but to any kind of object; third, investigate not only their geometric properties, but consider any kind of property of the objects. In other words, in a generalised interpretation of its meaning,
we can speak of symmetry (not necessarily geometric)*under any*(operation)*transformation*(not necessarily geometric)*at least one**property*(not necessarily geometric)*of the**object*
any transformation, any
object, and any of its properties.This generalised conception of symmetry makes it possible for us to apply symmetry to the materialised objects of the physical and organic world, as well as to products of the mind. In addition to geometrical (morphological) symmetries, we can now discuss functional symmetries and asymmetries (such as may occur in the human brain, for example); gauge symmetries in physical phenomena; and properties like colour, tone, light and shadow, weight, in objects of art.
Figure 7).Figure 7. Dissymmetry in an arabesque fragment from the Alhambra. Photograph by the author. Over a hundred years ago Pierre Curie, the great physicist and crystallographer, claimed, "Dissymmetry makes the phenomenon." What could he have meant? For a scientist, the subject of study should be sought where symmetry is distorted; what is there to study in a "perfect" object? Indeed, the dislocations of matter provide the most interesting phenomena for crystallographers. They provide one of the most interesting examples of dissymmetry: modern-day semiconductors, without which most of our devices would not work. The epoch-making discovery that allowed their manufacture was that very, very small amounts of contamination (that is, very small distortions of a once-perfect crystal) can completely change the electrical conductivity. While jewellers look for perfect gem crystals and never find them, scientists produce (almost) perfect artificial crystals and "distort" them, because they have learned how to exploit the advantages of this kind of "dis-perfection." The history of particle physics in the recent half-century can be considered as a discovery of symmetry-breaking.
, and is frequently applied
in frieze patterns (glide reflectionFigure 8). Other geometric symmetry
transformations can also be combined. To give but one example,
a helix can be obtained by combining translation and rotation
about the axis (Figure 9).
Figure 8(left) Glide Reflection.Drawing by the author. Figure 9 (right) Double helix by Watson and Crick.
Figure 10. Similitude and affine projection. From "Calculating Diminishing Size in Linear Perspective" © 1995, Ralph Murrell Larmann. Figure 11. One vanishing point. From "Exteriors in 1-Point Perspective" © 1995, Ralph Murrell Larmann. Similarly, we may think of aerial perspective as combining
similitude and colour change ( Figure 12. Aerial perspective. From "Atmospheric or Aerial Perspective" © 1995, Ralph Murrell Larmann.
Figure 13. Two vanishing points. From "Exteriors in 2-Point Perspective" © 1995, Ralph Murrell Larmann. Figure 14. Three vanishing points, worm's eye view. From "Exteriors in 3-Point Perspective" © 1995, Ralph Murrell Larmann. Figure 15. More vanishing points; shadow and light vanishing points. From "The Perspective of Shadows" © 1995, Ralph Murrell Larmann. In the twentieth century, such developments were taken further,
as can be seen in the work of cubists like Georges Braque and
Pablo Picasso, and futurists such as Umberto Boccioni. The application
of topological symmetry combined with similitude resulted in
new ways of seeing, and new tools for artists. Thus they were
able to give up the straight lines and fixed direction demanded
by the affine projection, and replace this by a topology, making
it possible for them to stress certain important features of
the represented object. An example of this would be M.C. Escher's
The use of two or more centres of projection, or even the
dissolution altogether of fixed centres, liberated the painter
from central perspective and opened up new vistas for artists
to provide the spectator with multiple-sided representations
of the delineated object.[ We ought to note that pictures liberated from central perspective and other constraints, appear at first not to be symmetrical at all. In the old-fashioned, everyday, meaning of the term which restricts symmetry operations to reflection, rotation, translation, and perhaps similitude and affine projection, the naive spectator would be right in assuming there is no symmetry. Having admitted combined symmetries and topological symmetry, however, our concept of symmetry has been expanded, making it possible for us to understand the view of artists who broke with conservative traditions and allowing us to find the beauty (via symmetry, that is, the harmony of details and balance between the parts) in artworks that dare to go beyond traditional composition. To validate this approach, and go even further, we should adopt the generalisation of the concept of symmetry described above. There were at least two transitional phases in which the change
of the traditional perspective concept appeared in early twentieth-century
art. The first step was the multiplication of the viewpoints
of the artist, while the vanishing point(s) were fixed. That
is, the artist would see the individual details of his or her
object from different directions. An example of this is Picasso's
Nude on a Beach,
1929.The next phase was when both the vanishing points and the
artist's viewpoints moved. For example, in Picasso's Figure 17. Pablo Picasso, Portrait of Marie-Thérèse,
1937These transformations appear multiplied in several cases.
Multiple perspective is used by cubists in horizontal perspective,
as in Braque's The roles of the horizontal and the vertical directions in
our view, and therefore in perspective representation, are not
identical. We perceive the length of straight lines of equal
length as different depending on whether they are horizontal
or vertical. Imagine a tree 10 meters high at a distance of about
300 meters in front of you; you perceive (see) the tree as being
much longer (higher), than a 10 meter long (horizontal) fence
at the same distance. Since their roles in the view cannot be
interchanged, we say that their roles are not symmetrical?that
is, not invariant under this change. Thus the perspective will
be different if we look at our object horizontally or in a vertical
direction from above. The latter, vertical, view is characteristic
of futurists works such as Boccioni's
Does a perfect symmetrical image exist at all? In an ancient
Sumerian picture previously analysed by H. Weyl in 1952 and reproduced
here in Figure 18. Dissymmetry on a Sumerian picture (H. Weyl, Symmetry, fig. 4).What are we to believe about the perfection of our world's
symmetry? The facts do not demonstrate that our world lacks symmetry,
but only that its symmetry is never perfect. If you look around
you, you cannot fail to find many symmetrical objects, both natural
and man-made. As the discussion above shows, there is symmetry,
harmony and beauty in our environment. The
world is almost perfect; there exist some distortions, admittedly,
but it retains its Symmetry is quite natural. Human beings are almsot symmetrical (who sees that our heart beats on one side, or that our motions are co-ordinated from the left hemisphere of our brain?). Flowers are symmetrical, butterflies even more so. Not perfectly, but practically symmetrical. It seems quite natural to us that a tennis court is symmetrical. It must be symmetrical on two accounts: in geometrical terms, and in moral terms, since both the physical conditions and the rules of the game for the two players should be identical, that is, symmetrical. Nevertheless, these conditions and rules were not always so self-evident. In the sixteenth century, when this game originated, Henry VIII played it in his Hampton Court palace in an asymmetrical room, which together with the partial rules provided an advantage for the king. The self-evident and natural qualities of symmetry may depend on local and temporary, physical and social conditions. Something similar may be said of the acceptance of artistic trends, i.e., the changes in the perception of beauty, harmony, and symmetry. Objects and phenomena around us show signs both of symmetry and its lack at the same time. In reality, a thing is symmetrical in one or more aspects. In other words, it conserves one or more of its properties under a particular transformation, (such as a reflection or a rotation), while it is asymmetrical in other aspects: that is, its other properties are not conserved. There is no perfect symmetry (when all properties are preserved) and no perfect asymmetry (when no single property is preserved). A very asymmetrical world would be ugly, while a very symmetrical world would be boring. This concerns not only our physical environment, but also its artistic representations. Perspective as a symmetry operation that conserves properties during the process of artistic representation, helps us to preserve this beauty in all its classical and modern forms. Therefore we should also accept the sophisticated forms of combined symmetries that appear in modern art, and which are products of a long development in multiplying and transforming the vanishing points and the artist's viewpoints, as manifestations of perspective representation. Accept it, as it is!
I Commentarii); Leon Battista
Alberti (De pictura, Della Pittura); Piero della
Francesca (De prospectiva pingendi); Luca Pacioli (Summa
de arithmetica, geometria, proportionalita, De divina
proportione); Leonardo da Vinci; and Albrecht Dürer
(Undeweysung und Messung, Vier Bücher von menschlicher
Proportion). return to text[2] See the Math
Forum webpage on perspective drawing. [3] See "Art
Studio Chalkboard: The Perspective of Shadows" and also
"Art
Studio Chalkboard: Drawing Subjects." [4] Multiple-sidedness is meant here first in geometric
terms, and secondly figuratively.
Peternák, M., N. Eross, A. Eisenstein,
and L. Beke, eds. 2000-2001. Weyl, H. 1952. Williams, K. 1999. Symmetry in architecture.
Symmetry:
Culture and Science (1990) . He is author of about 150
publications, was an invited lecturer at several institutions
from Japan to the US, and delivered lectures at many world-wide
conferences, as well as he organised a series of international
meetings and exhibitions in the field of science-art relations
and on the applications of symmetry.
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