When we look at a mirror, every object in the real world has its exact mirror image, and the two worlds taken together, the real and it's mirror image, have a perfect mirror, or reflection symmetry. Such symmetry also exists in the real world, where every shape or side reflected through an imagined mirror (or more precisely a reflection plane) has its counterpart. Humans and most animals (and practically anything which is designed to move-- like ships, cars and airplanes) have an external appearance which has this feature, and they are divided into left and right sides, as so clearly expressed by the well known icon of Leonardo da Vinci (Fig.2.1). In other words, we, as well as most living creatures, are bilateral-- left reflecting right and right reflecting left. We possess what we call bilateral, reflection, or mirror symmetry.
We recognize symmetry as something essential for survival. Bilateral symmetry is needed to move along a straight line easily, and it utilizes both sides of our body to their most effective capacities. Our symmetric external appearance prevails in spite of the asymmetry of our internal organs and even the asymmetry built into us at a molecular level. We have a subconscious attraction to symmetrical partners, as several recent studies indicate. It is argued, based on studies on animals and most recently also on humans, that there is a clear female preference for symmetrical males, and that this is the by-product of selection for male recognition. The reason for this is not entirely clear. It has been argued by biologists that it is due to the sub-conscious supposition that asymmetry is related to a defect which can be transmitted to the next generation. However, artificial networks-- which are obviously not concerned about reproduction-- also have a preference towards symmetrical images as a by-product of image recognition.
At the same time, full mirror symmetry appears sometimes lifeless. Absent are the small variations (small mistakes and weaknesses) toward which we also feel an attraction. Such so-called fluctuating asymmetry may also be an important factor in evolution.
The mirror symmetry we possess is broken when we start to move. At that point, either our left or our right leg is in the forward position, and the other is in the backward position. The same symmetry-breaking occurs in animals with four legs. This is certainly based upon economy. The most effective and least energy-consuming way of moving forward is by pushing ourselves against the ground-- not leaving it while we move. Flying is a different matter. There, the symmetric dynamic solution prevails, and wings move up and down in unison, not breaking the symmetry in this so-called dynamic state.
Because it is so easily recognizable, it is not surprising that mirror symmetry appeared in the early art of most civilizations and was the dominant form used to express (mainly religious) harmony. It prevailed throughout the Middle Ages, before more sophisticated concepts in symmetry and geometry emerged. The visual effect created the impression of balance, and mirror symmetry was used extensively to draw attention to something essential, at the center of attention (Figs. 2.2 & 2.3). The concept of mirror symmetry-- with usually god in the center of the painting or fresco-- also permeated the development of medieval iconography, modifying the tradition as set by the Bible and Liturgy. In some instances there are two good shepherds instead of one, or two magi instead of three, dividing the painting into left and right sides. Animals like two-headed eagles were created, again seeking mirror symmetry and balance.
Most medieval cathedrals have, or more precisely were designed to have, such symmetry (Fig.2.4). Asymmetry was the result of extended building periods which often left portions unfinished, or completed using a different style. The same concept is true for many other forms of early architecture, including public buildings and castles. In most of the medieval Cathedrals the two towers are different, not by design but by financial restrictions. A somewhat different and beautiful example is the Chambord Chateau in France's valley of Loire (Fig.2.5). There, the symmetric design, so apparent at first sight, is slightly broken by small additions built throughout the centuries. Though the result of chance more than design, these small asymmetries have an essential role from the aesthetic point of view. Once we recognize these distinctions-- these minor differences-- they serve as new focal points; they break the monotony and bring life to the building.
Cities were also often designed to have mirror symmetry, guiding us toward the focal point, usually a church or cathedral, as the aerial view of the Vatican so clearly demonstrates (Fig.2.6).
This principle is reflected not only in frescoes and the building and layout of cities, but also in music. The notion of 'counterpoint,' one or more independent melodies added above or below a given melody, was invented during the Middle Ages. It was then developed further to its full extent in the music of Johann Sebastian Bach, called appropriately Contrapunctus (Fig. 2.7).
It would seem trivial to postulate that our physical world is also symmetric, since every atom and molecule which builds the inorganic and living world should have a mirror counterpart. Crystals built up of molecules usually reflect their symmetry; we therefore expect to have crystals which also have reflection symmetry. However, it was surprising to find that this is not always the case. In 1848, the French chemist Louis Pasteur crystallized a salt called tartaric acid (Fig.2.8). Two crystal forms, both asymmetric, were obtained and they were mirror images of each other. The reason for this is clear today; the acid molecules are asymmetric, and can be either left- or right-handed; this is called chirality. Crystals, built up exclusively by molecules which are one but not the other of these variations, reflect this asymmetry. Many molecules, particularly those which are building blocks of living organisms, are likewise asymmetric.
The existance of asymmetric molecules is not surprising or unexpected. Many of them are built of hundreds and hundreds of atoms in an intricate arrangement. This by itself is not broken symmetry. While each individual molecule is asymmetric, in general when we prepare such molecules there exists an equal number of left- and right-handed versions, and thus when crystals are grown an equal number of left- and right-handed crystals. For each individual crystal we can find its counterpart in nature, and the full collection of these crystals still has the complete mirror symmetry.
The existance of asymmetric molecules in nature is more a rule than an exception. What is remarkable, however, is that living organisms use molecules showing only one form, the left- or right-handed but not the other. In our body we find only right-handed glucose (Fig.2.9) produced by fruits or vegetables. The left-handed form, while not occurring in nature, can be made artificially. This form, however, can not be used by us and passes through our system untouched. The reason for this is simple, but rather surprising. All amino acids in living creatures are left-handed and never right-handed. Other molecules, such as nucleic acids, occur only in right-handed forms. We can utilize only forms which are compatible with our molecular arrangement, and we can build into our system only molecules which "fit in," and this depends on their symmetry.
One also can readily see this asymmetry by conducting a simple optical experiment. Light is the radiation of an electric field which can be made to rotate right or rotate left while propagating. When it bounces off the surface, its properties depend on the properties of the surface in relation to the properties of the light. Asymmetric molecules lead to different reflections of right or left rotating light. We can take a photograph using such left or right rotating light (Fig.2.10). The fact that an insect looks different in the two cases immediately gives evidence of the asymmetry of the molecules which form the wings.
We are asymmetric also and on a much larger scale, although in most cases not in our external appearance. Our heart (and many of our other organs), is not at the center of our body. In this regard we are asymmetric. However, we are not asymmetric in the same sense as were the acid crystals mentioned earlier. If we were, this would mean that the human population would consist of the same number of people with their heart on the left side as those with their heart on the right side. Most of us have our heart on the left hand side, with only a few exceptions; thus the human population as a whole breaks such mirror symmetry. Such asymmetry abounds in the living world, and in some cases this concept is readily apparent even in external appearance. Snails are predominantly coiled to the right when viewed from above.
The reason for this fundamental asymmetry, for this chirality of life at the molecular and also at the macroscopic level is not understood, and it is subject to much speculation. We understand how asymmetry is inherited from generation to generation-- through the DNA and also through the cytoplasm of the mother. However, we do not know how this asymmetry started. It is speculated that it may be related to weak asymmetries which exist in our physical world.
Mirror symmetry may be a subtle attribute, and the external appearance can be misleading. Arguing on the basis of this may lead to many surprises. We have discussed the scale before, and the principles involved in creating a perfect balance. In order to have balance, which is in equilibrium, not tilting to the left and not tilting to the right, the scale, together with the weights, has to be perfectly symmetric.
Now, let's look at another physical experiment (Fig.2.11). If we make a loop of electrical wire and place a compass at the center, the entire arrangement, just like the balance discussed before, is perfectly symmetric with respect to the plane defined by the loop. We can arrange the compass in such a way that in the absence of electric current the compass is also in the plane as shown (we can do this by pointing it in a direction where the Earth's magnetic field is along the direction of the compass). Will the compass move when we start the current in the loop? Certainly not, we would argue, as the entire arrangement is perfectly symmetric. If there is any tendency to move, it should be with the same likelihood to the left as to the right, and (just like the famous donkey which, being put exactly halfway between two haystacks, could not make up its mind and was starved to death), the compass will stay put. As it turns out, this is not the case, and the compass is displaced to the position indicated on the figure. When this was discovered by Oersted more than 150 years ago, it was an astonishing surprise. By now, however, we understand what was going on. We were deceived in judging symmetry according to its external appearance. The electric current induces (invisibly) the magnetic field which encircles the wire in which the current flows. If we place an imaginary mirror through the center of the wire, we see that the magnetic field does not have mirror symmetry. Such symmetry would require that the current flow in the opposite direction at the two sides of the mirror, instead of in the same direction-- as is this case. Therefore, mirror symmetry is broken by the current flow, and this is the reason for the displaced compass. We can relax-- our universe is still symmetric, but we have to be aware of invisible factors which may cloud our judgment of what appears to be 'right.'
About half a century ago things again looked odd, as though nature was a little asymmetric, even when certain predicted effects, such as that discussed above, were taken into account. This led two young physicists, Lee and Young, to propose that nature may be weakly asymmetric. Experiments which they conducted revealed that this was indeed the case. Here I quote Feynmann:
We take a radioactive disintegration in which, for instance, an electron and a neutrino are emitted - an example, which we have talked about before, is the disintegration of a neutron into a proton, an electron and an anti-neutrino, and there are many radioactivities in which the charge of the nucleus increases by one and an electron comes out.The thing that is interesting is that if you measure the spin - electrons are spinning as they come out - you find that they are spinning to the left (as seen from behind - i.e. if they are going south, they turn in the same direction as does the earth). It has a definite significance, that the electron when it comes out of the disintegration is always turning one way, it has a left-hand thread. It is as though in the beta-decay the gun that was shooting out the electron were a rifled gun. There are two ways to rifle a gun; there is the direction 'out,' and you have the choice whether you turn it left or right as you go out. The experiment shows that the electron comes from a rifled gun, rifled to the left. Using this fact, therefore, we could ring up our Martian and say, 'Listen, take a radioactive stuff, a neutron, and look at the electron which comes from such a beta-decay. If the electron is going up as it comes out, the direction of its spin is into the body from the back on the left side. That defines left. That is where the heart goes.' Therefore it is possible to tell right from left, and thus the law that the world is symmetrical for left and right has collapsed.
Our fascination with mirror symmetry is evident in many ways. We create symmetric icons for objects such as the heart, which has a clearly asymmetric appearance (Fig.2.12). Discoveries which implied, and finally in some instances confirmed, that nature is asymmetric were met and accepted with great reluctance. Additionally, we greet full symmetry in life, and particularly in art with reluctance as well. We are therefore attracted to the small deviations from symmetry that make our lives more interesting.