"forms must be given life and the right of individual existance." Kazimir Malevich
"I do not think God is weakly left handed" Wolfgang Pauli
The appreciation of forms and patterns dates back to the early days of civilization or even to our predecessors. Simple forms of symmetry are recognized by animals, (and even by computers). Symmetric forms appear in caves and have been used throughout the ages to express religious beliefs. Other forms like circular shapes were the basis of designs for items in practical use. Repetitions of identical units have been used in applied art (like rugs) and in early architecture. It was the Greeks, however, who defined the concept and used it in various contexts in science, architecture, the arts and in many ways of everyday life. The name itself comes from the Greek word "symmetria" or "same measure".
Our inclination to symmetry may have deep biological origins, we likely choose symmetrical mates over asymmetrical ones, as some recent studies indicate. This most probably played important role in the evolution through the--obviously sub conscientious and maybe misguided--notion that symmetrical individuals less likely carry genetic defects. Our external appearance, like the appearance of most animals, has one form of symmetry called mirror or reflection symmetry, a feature important in moving forward along a straight line between two points in space (and pursue prey). It is not surprising, therefore, that symmetry is related to somehow feeling right--to our appreciation of balance, equilibrium and, consequently, longevity. A collection of objects can also display a symmetry of a different kind if they are ordered in certain periodic fashion. Implicit in this form of symmetry is the satisfaction of having returned to origins where we have been before, whether it be in space or in time. At the same time, perhaps it is also less exciting than a disordered state because its predictability.
In mathematics symmetry is not a form or an object, it is a precisely defined mathematical method called a transformation. What is a transformation? It is a recipe of moving things around--to pick an object up and than put it back down on top of itself. If this transformation leaves the object apparently unchanged to the viewer with respect to its surroundings, or in other words, if the transformation leads back to the original position, then the object has translational symmetry.
The obvious examples of rather symmetrical objects are the circle, a work by Kazimir Malevich, the leading Russian Supremacist painter, and the sphere, called by Pythagoras the most perfect body of them all. (Figure 1.1) Rotation around the center of the circle leads to a new position for every point in the circle--going from point P to point P11, but the entire circle looks the same after the rotation. (Figure 1.2) Of course rotating around any point different from the center leads to a different position easily distinguishable from the original; the circle is not invariant under such rotation. A reflection around a line (the full line on the lower part of Figure 1.2 flips the circle over the line, placing the point P on P1. If the line does not go through the origin, the resulting circle (the blue) differs in position from the original (the red). Only the line that goes through the center leaves the circle apparently unchanged. There are, however, an infinite number of such lines, pointing in different directions.
Next consider a square, another shape painted by Kazimir Malevich. (Figure 1.3) A rotation by an arbitrary angle leads to a different position; there are only four rotations--by 90, 180, 270 and 360 degrees--which lead to a square equivalent to the original. (Figure 1.4) This shape then has a so-called four-fold symmetry. Next look at reflections. There are two ways by which the original shape can be recovered--by reflection through the two full lines indicated on the bottom (reflections around the dotted lines are regarded as equivalent to the reflections around the full lines).
What we are most accustomed to is the most straightforward--the so-called mirror symmetry. The name refers to the fact that the two sides of the object are mirror images of each other. This type of symmetry has been recognized by early civilizations, and was the dominant force of artistic expression of order in the middle ages as well as in modern art. (Figure 1.5) This is the only symmetry the figure has, in contrast to the Navajo rug. (Figure 1.6) The rug is also symmetric with respect to a reflection--in fact, we can perform such operation with respect of different axes, one horizontal one vertical, both going through the middle. But it is also invariant with respect of a 180 degree rotation (turning the rug upside down). This illustrates two-fold symmetry. The snowflake has six-fold symmetry, reflecting on a macroscopic scale the symmetry of the atomic arrangements in the snow crystal. (Figure 1.7)
Rotation and reflection are the fundamental symmetries related to a single object. There are other forms of symmetry associated with an arrangement or a collection of objects. Trees planted next to a road form a periodic array, and we also have a tendency to form such a pattern under some circumstances-soldiers marching, for example. Atoms or even electrons can form such periodic arrays; atoms in crystals have such arrangements, but forcing electrons to achieve such state is more difficult, requiring extremely low temperatures and sometimes high magnetic fields. A drawing by the Dutch artist (some say illustrator) Mauricius Escher demonstrates this point. (Figure 1.8) Such structures have so called translational symmetry. For an infinite row of identical units (the horses in this case), a displacement by the length equal to the distance between two members of the group gives back the same pattern (assuming of course that the horses are identical). Such arrangement suggests law and order. A perfect periodicity is called long range order. In general, an interaction between the members of the group, keeping them apart, is required in achieving this state.
While for a geodeter the concept of symmetry is close to what we are accustomed to and we also appreciate - although most often we are not able to sharply define--in another branch of mathematics, algebra, it may appear in a completely different and unexpected way. The following equation
| (1) |
dreamed up by the young genius Ramanuyan has, on the left hand side an integral, and on the right an infinite series. For a mathematician the above equation has a perfect mirror symmetry: the right side is equal to the left side. Such symmetry, however, can be verified only after substantial calculation.
Symmetry is associated with many of the most fundamental laws of science, whether it be physics, chemistry or biology, and it has been used repeatedly to uncover hidden relations, forms, or states of matter. For example, the correct structure of the DNA has been invented by looking at symmetry relations both between the member molecules which build up the famous double helix and within the symmetry of the entire structure. The former was a matter of chemistry, the latter a matter of analyzing the X-ray scattering performed on the DNA.
Physical laws also have the symmetries described above, and some of the principles involved seem obvious and trivial. If we perform an experiment, we assume that the identical experiment performed somewhere else under identical conditions will lead to the same result. We say that the physical laws are invariant under a translation. This sounds trivial. Somewhat less natural is the fact that an experiment is also invariant with respect to a uniform velocity. The outcome does not depend on whether the entire experimental arrangement is at rest or moving with a constant speed (with respect to an outside observer). Standing still we can throw an apple up in the air, and it will fall back into our hands. We can do the same by walking, with our arms executing the same upward motion while throwing the apple; and again it will fall back into our hands. We try it by spinning around, and it does not work; a rotation (even with a constant angular speed) leads to changes in the end result. We say that some physical laws are not invariant with respect to rotation with constant angular velocity. These symmetries have deep and fundamental consequences. Among others, they form the basis of the theory of relativity, for example.
What we are going to discuss is more trivial and transparent. It is the relation of external forces to equilibrium, and the concept of symmetry involved. The first argument of this sort has been used by Anaximander, in his elegant but incorrect statement (as described by Aristotle):
There are some who say, like Anaximander among the ancients, that the Earth stays still because of it's equilibrium. For it behooves that which is established at the center, and is equally related to the extremes, not to be borne one whit either up or down to the sides... so that it stays fixed by necessity.
To see how symmetry leads to stability let us look at a ball in a well. (Figure 1.9) The ball shown on the right, being "in the middle" of the symmetric well stays there. The forces which act on it, the gravitation and the forces exerted by the walls, cancel to zero. The ball is in equilibrium. There is no motion. The ball in the middle figure will not stay there but will roll down as the forces acting on it are asymmetric. The ball on the right is also in equilibrium; this however is rather different from the equilibrium the ball on the left enjoys.
Of course all this can be expressed in terms of mathematics and the relevant equation, Newton's fundamental law:
 | (2) |
Here "F" refers to the forces acting on the object; 
means the summation of all
available forces; "m" is the mass of the object and "a" is the acceleration. The equation expresses in the language
of mathematics what we have said before: if there is no force, or if the sum of all forces is zero, then there in no
acceleration, and the object is in equilibrium.
Of the two equilibria, one on the left and one on the right, the former is stable. Nature is going to seek out such a state. The latter is an unstable solution. Something catastrophic occurs if the exact balance is broken. These simple effects have been used repeatedly to convey a message or a feeling, as the sculpture by Newmann so clearly demonstrates. (Figure 1.10)
How do these symmetry arguments, and others, come to play a role in physics? The precise, and necessarily dry, answer was formulated by Pierre Curie (the husband of Mme Curie):
under conditions which determine a unique state of equilibrium, the symmetry of the conditions must carry over to the state of equilibrium.
In our previous example the well is symmetric and so is the gravitation with respect to a well; therefore, the ball's equilibrium must be in the symmetric, middle position. The same applies to the balance. We know since Euclid that equal weights on equal arms lead to equilibrium, the same concept as before. We can, however, balance a scale with unequal weights on unequal arms. (Figure 1.11) While this looks asymmetric there is still symmetry involved; instead of the forces the torques, in this case the mass M times the arm l, of the two sides are equal,
| M1l1 = M2l2 | (3) |
From this point of view we retain a symmetry not related to the external appearance, but a symmetry related to the physically significant parameter, a torque. Many of the ancient tools used in building monuments are based on this principle (where we have to go longer distance but with smaller force to have the same effect). This principle is also seen in many sculptures by Alexander Calder. (Figure 1.12) The fact that two, or many, smaller weights on one arm can compensate for one large weight on the other is not obvious and needs detailed construction. Supplementing our thoughts about the forces with similar thoughts about torques, these considerations are precisely the ingredients of many of the Calder mobiles. They also illustrate one more point: the shallower the well or the more balanced the forces are the easier is to move the ball away from the center. It is more responsive to external disturbances and is inclined to move about. The name "mobile" refers to these aspects of the sculptures.
The symmetry discussed above is intimately related to so-called conservation laws; by this we mean that during the motion certain physical quantities remain constant. Consider the asymmetric balance. (Figure 1.11) By raising one arm the other moves down. (Figure 1.13) There is an energy related to the weight--potential energy--given by the object's height times its weight, M x h. The potential energy associated with the weight on one arm increases; this increase is proportional to the weight times the displacement, M1 x h1. The energy associated with the other weight decreases, a decrease of M2 x h2. h1 is related to l1 as h2 is related to l2; the increase of the potential energy on one arm is exactly the same as the decrease on the other arm. In other words, the total energy change is zero. Energy is conserved during the motion of the scale.
The next painting by Malevich lacks the precision required by geometry (Of course neither the black circle not the black square painted by Malevich are perfectly symmetric; the paint has an internal and random structure and therefore a rotation leads to shifting this internal pattern. (Figure 1.14) This pattern is most certainly not an accident, it signals an artists deep suspicion toward perfect symmetry). The sides of the square are not fully parallel--surely not an accident--and our prior argument about the rotation and reflection now does not apply, we say the form is asymmetric. This concept of asymmetry however is not well define. It is merely an expression of a certain degree of deviation from a well defined, symmetric state.
As for the symmetry associated with a collection of objects, deviations from such ordered state come naturally. Chance leads to disorder and so do random forces. The birds are in an apparently random and at the same time highly dynamic state, although there is some order (called short range order) as the birds try to avoid each other. (Figure 1.15) Chance is also contributes to the removal of order, and this is elegantly expressed on the collage by Jean Arp. (Figure 1.16) We still have the remains of symmetry and order, but chance also is slowly emerging by having misplaced elements, asymmetric shapes, and rugged edges.
Next we look at a situation with two wells. (Figure 1.17) Here the ball has to make a choice. In principle, in can rest on the right or on the left of the bottom of the well. For these situations the ball is not in the middle, we say symmetry is broken. This happens more often than we think, we live in a broken symmetry world. How do we proceed from the top to the bottom on the figure?
An equation can be used to describe this scenario:
| V = Ax4-Bx2 | (4) |
where V is the shape of the well and x is the horizontal deviation from the middle of the well. (We say V(x), V is the function of the position x). A and B are constants, and their magnitude determines what shape we have. On the right hand side the first, positive term (the expression Ax4) is responsible for curving the well upwards, and the second term, with the negative sign in front of it, leads to a downward curvature. When B=0 we have a normal, U-shaped well and the ball rests in the middle. When B is larger than A we have a W shaped well, (it is easy to try this by plugging different values of the variable x into the equation with different A and B constants) with the ball either in the left or in the right position. By increasing B with respect to A, we move from the top representation in the figure to the situation shown in the bottom.
There is a sudden change when B becomes larger that A. Here, a transition occurs from the single-welled to the double-welled picture, and as far as the ball is concerned, from the symmetric to the asymmetric, so called broken symmetry, position. We call this change a phase transition. Something like this happened a few seconds after our universe was born (the so-called Big Bang), and the process of water turning into ice can be described in this fashion. From the general point of symmetry these diverse phenomena are the same.
How does the ball know which direction to go, left or right? This is determined by the minuscule differences in the potential of some random events which start to push the ball in one direction or the other.
The end result is one ball in a symmetric potential, but in an asymmetric position (say on the right), clearly in conflict with our previous argument on symmetric causes leading to symmetric results. What about the next ball? It can go to the right or to the left, and so can all the others we try. The end result will be--for random differences which determine where the balls will go--roughly half the balls in the left well, and half the balls in the right. For an infinite number of balls the difference will be vanishing--symmetry prevails not for a single ball but for a collection of balls.
The type of symmetry breaking we described in terms of simple mathematics occurs also for a collection of objects--atoms or electrons for example. A crystalline state looks like the regular row of horses. (Figure 1.8) A liquid state of the same atoms like the random state of birds. (Figure 1.15) It is a nice feature if the physical sciences (some say the unusual effectiveness of mathematics) that the transition from one state to the other-- so different as far as their symmetry, their dynamism, their predictability is concerned-- can also be described by the simple equation we discussed at length above.
Symmetry and balance are as important in the arts as in the sciences, although it is certainly less sharply defined; the great artists all seek an equilibrium and balance, more through intuition than by well-defined rules. To an educated observer such aspects are immediately evident and also are of utmost importance.
None has put is more precisely than Apollinaire, the poet and art critic, the friend (and often enemy) of nearly all the artists who lived and worked in the first years of this Century, when he commented upon the art of Picasso:
...I was tempted to cry out to you Michelangelo's words on Flemish art:"This painting is nothing but rags, huts, vegetables, shadows, trees, bridges and rivers, which they call landscapes, with a few figures here, a few figures there. And all of this, although some people consider good is done without reason or art, without symmetry or proportion...
What a joy it is therefore, to find a painter today who cares about "reason", "art", "symmetry" and "proportions."