Symmetry means indistiguishability. The degree of symmetry means a level of indistinguishability. For example, on Figure 1 A we can permute (swap) any yellow point (pixel) with any other and no change will occur. The picture will remain symmetrical (invariant) to such changes. Swapping the points does not make difference. On the contrary Figure 1B will change thoroughly after arbitrary replacement of a larger set of points. Only for a subset of such replacements (e.g. dark dots with other identical dark dots, or yellow dots with identical yellow dots) will the picture remain the same. Swapping of points does make difference. So the panel B has a broken symmetry with respect to panel A. One can see that it is the loss, or breaking, of symmetry that reveals a presence of a structure. Panel A possesses a structure. Structure means distinguishability. The symmetry breaking is what happens when a structure is created out of unstructured homogeneous field.
Figure 1. A. Every yellow point (pixel) in the field can be replaced by any other and nothing will change, the field will remain the same. When one replaces (transforms, varies, changes) the components of the object and the object still remains unchanged (invariant) we say that it is symmetrical with respects to those transformations. The original and the transformed picture will be indistinguishable. B. In this case arbitrary replacing the points in the field will result in lost of the structure and the meaning of the picture. The original and the transformed picture will be distinguishable. Because the transformations (point replacements) will change the object, we say that it is not symmetrical with respect to those transformations. Its symmetry is broken.
Symmetry, symmetry breaking and Information
Imagine you are headed to visit your friend. Slowly you’ve got surrounded by a dense fog (Figure 2 A). You rotate your neck and the whole body to orient yourself, in vain. Everything looks the same. Every direction is indistinguishable from any other. Looking around does not make difference. You are lost in the symmetry. You feel uneasy because no relevant information exists to guide your decisions and actions. Your decisions-actions are stuck. Suddenly, a small patch of less dense fog occurs and the symmetry breaks down (Figure 2 B). A structure occurs. The structured ambient light now informs your decisions and actions.
Figure 2. A. A dense fog is symmetrical with respect to your actions and perception. You can look around you (i.e. rotate yourself), walk forward or backward, left or right (i.e. translate yourself), but everything looks the same. The symmetry does not allow you to guide your actions. No preferred direction of walking exists. You cannot decide where to go. B. As the fog becomes less dense, the symmetry gets broken and a surrounding structure occurs. Now, the structure of the ambient light can guide your decisions and actions. One direction of walking becomes privileged by your decision and action system.
Information is linked to symmetry. High level of symmetry (dense fog) enables no information. Lower levels of symmetry, mean existence of structure and enable information. This information specifies decisions to be made and actions to be undertaken. In similar vein, on Figure 1 B the structure of bananas, their shape and position specify how they can be reached and grasped.
Consider you are on open sea. You can distinguish only the up and down direction. Up is the blue sky and down is the water. This up-down distinguishability (broken symmetry) is due to the gravity gradient or force (see the text bellow). However, the horizon line is indistinguishable wherever you look. You won’t be able to tell east from west and north from south just by looking around yourself (there is rotational symmetry of the horizon line). Where ever you look the sky touches the sea at the horizon. Everything looks the same. During daytime only the position of the sun breaks this rotational symmetry and helps in identifying the directions of our actions. During the night it is the star structures (constellations) we use to orient our actions. A not-so-direct visual means to orient on open sea is the compass because it uses the broken (nort-south) spatial symmetry that the Earth magnetic field creates. It is this broken symmetry that induces structure with respect to rotations around our own body vertical axis. This structure is information bearer that guides our actions. It also guides actions of many sea- and air- faring creatures, like whales and birds.
Scale symmetry in geometry
The scale symmetry keeps indistinguishable the shape of figures and objects while their size changes. If we multiply the radius of the circle by some factor, then the size of the circle will increase but the shape will remain a circle. This is called scale symmetry. The same is valid for any geometric object, curve, or field.
Figure 2. Scale symmetry. Multiplying the radius of the circle A by some number does not change the shape of the circles B and C.
Symmetry and symmetry breaking (rearrangement) in geometry.
The circle (Figure 3A) has the largest symmetry of two-dimensional figures. Analogously the sphere has the largest degree of symmetry in three-dimensional objects. Why? Because they stay the same for any transformation we subject them to. For example, for any (in fact infinitesimal) degree of rotation the circle or the sphere will stay the same. For any straight line (axis) that passes through their center and divides them on two halves, those halves will look the same. Circles and spheres contain infinite number of mirror symmetries.
Figure 3. A. One can rotate the circle for arbitrarily small angles without noticing difference. It has a continuous rotational symmetry. The circle stays the same (invariant) for any size of rotation. In similar vein there is infinite number of axes for which the circle will have two equal halves (mirror symmetry). B and C the hexagon and triangle have a discrete symmetry. They do not remain the same for continuous rotations. They have a broken symmetry with respect to the circle. The triangle has a broken symmetry with respect to the circle and the hexagon.
That is not the case with the hexagon (Figure 3 B). Only for certain set of rotations (in fact multiples of 60o) will the hexagon look the same. Also the set of mirror symmetries is smaller than the one of the circle. It has 6 mirror reflection symmetries. The triangle has even less symmetries than the hexagon. We say that the hexagon has broken (lower order) symmetry in comparison with the circle and the triangle has broken or lower order symmetry than the hexagon and the circle.
While in the case of circle (and sphere) there are no preferred rotations and mirror reflections, in figures and bodies with broken symmetries there are, as we have seen above. Preferences are connected with broken symmetries. In fact atomic and molecular structures arise as a consequence of the broken spherical symmetry of the spatial distribution of, so called, electron clouds
Crystals in nature are well known macroscopic continuations of the previously mentioned realizations of the symmetry breaking of the maximal spherical symmetry at atomic and molecular levels of reality.
The symmetry of undisturbed water under external sound driving breaks and wave patterns emerge: These patterns contain complex symmetries, but of lower order than the fully symmetric one that belongs to the undisturbed water. They are unstable and the stable state is the fully symmetric (i.e. the undisturbed) one because as the external drive fades out the pattern fades out too. The undisturbed, symmetric state is the attractor of the water when the external driving is switched-off.
Permutation symmetry and symmetry breaking in language, logic and music
Consider the following two cases:
a. Statements such as: X is a brother of Y and Y is a brother of X; has the same correct meaning, because ‘being brother’ is a symmetric relation if X and Y are males. If someone who is male is a brother to me it means that I am necessarily a brother to him; however statements such as:
b. X is mother to Y and Y is a mother of X do not both have the same correct meaning, because ‘being a mother’ is assymetric relation. If X is a mother to Y, then Y can be a son or a daughter to X, but not a mother.
So, in the case (a) we can swap (permute) the places of X and Y and the meaning will remain the same. However, in the case (b) this cannot be done. Hence, in the first case we say the statement contains a permutation symmetry (indistinguishability of meaning) with respect to transforming the places of variables X and Y and in the second case that symmetry is broken.
c. Musical melody is a pattern with broken symmetry with respect to the permutations of notes and their timing. However, the melody is symmetrical (invariant) with respect to the energy (loudness) of the sound. The melody remains the same if you play it very quiet, moderately loud or very loud (of course, with some limits at the extremes). The information does not depend on energy.
Synergies and symmetry
Synergies in biological systems are phenomena when components of the system work together (synergize) in such a way that they reciprocally compensate each other in order to maintain the intended performance of the system.
For example, there may be an infinite number of combinations of forces that muscles and tendons of two fingers can exert with the performance of 10 Newtons being constant. For example 10 = 5+5, but also 10 = 6+4 or 10 = 4+6 or 10 = 7.3 +2.7 (Newtons).
Another example is keeping the body mass constant by compensation in consuming and spending the same amount of calories.
So, always when the sum is kept constant by different values of the summands there is a symmetry with respect to some additive group of numbers.
Conservation of physical properties and symmetries
We all know that in systems that conserve energy the total energy E may be transformed from potential V to kinetic K energy and vice versa, so that: E = K + V. In collisions of systems that conserve the energy momentums of separate colliding bodies change, but their total sum remains the same. These conservation laws are associated to time and space translation symmetries . Also the conservation of angular momentum is associated with space rotation symmetries. In general the conservation (invariance) of some other physical properties (e.g. charge) and conservation laws of physics is connected with certain symmetries.
The creation of elementary physical forces in the early Universe
In physics symmetries and their breaking play a major role. Among the other the creation of elementary physical forces (the strong and weak nucler forces, the electromagnetic force and the gravity) in the very early and early Universe , have been separated as distinguishable forces in a series of phase transitions in which a cascade of spontaneous symmetry breakings occured driven by decreasing temperature in quickly expanding Universe. In this process the initial undistinguishable state of interactions gave a way to separate types of forces that today in this state of low energy shape the processess on elementary physical level. Also, the emergence of rest mass of elementary particles, the emergence of nucleons such as protons and neutrons as bound states of quarks and other important moments in early Universe is due to special types of a spontaneous symmetry breaking events. These spontaneous symmetry breaking processess are responsible for our physical world and Universe to become as we observe it now.
The gravity gradient brakes the directional symmetry
The gravity brakes the rotational symmetry of the local space. We can choose to move freely left-right forward-backward, but not in up-down direction. If we jump up, we will quickly be pulled down counter to our wish. The breaking of symmetry, and the preferred direction (up-down), is induced by the gradient of the gravitational potential of our planet. In order to restore our freedom of choice (and consequently, the symmetry) we will have to apply large amounts of energy to counteract that gradient. That is why the direction of flow of rivers, waterfalls and rain is directed up-down. Structured motions with preferred directions are a consequence of broken symmetry.
However, the energy from the Sun succeeds to counteract this gradient by providing larger freedom of motion (symmetry) to water molecules by evaporation.
Symmetry and symmetry breaking in chemical, biological, psychological and social levels of reality
In the previous text, we briefly discussed why symmetry breaking is important for creation of the elementary natural forces and structures like protons and neutrons. However, symmetry and symmetry breaking give rise to much more structures and phenomena than just physical ones. As we said at the beginning of this chapter, homogeneous states are states of symmetry (see Figure 1 A and explanations therein). In chemically homogeneous systems, at some critical point of the changing context, arise symmetry breaking, and consequently dynamic phenomena, such as oscillations, in which two chemical products concentrations periodically change or produce spatial waves.
In biological systems the weakly broken bilateral symmetry of the body is a well known fact. Also, the morphogenesis is a process in which through symmetry breaking some homogeneous state of cell population starts to take a definite shape. Color coat patterns in animals also arises as a consequence of spontaneous symmetry breaking of the previous homogeneous state of pigment distribution. Decision making, formation of action or, indeed, adoption of any criterion, value, preference, attitude at psychological or sociological level is a symmetry breaking bifurcation, arising from the previous unstructured state. Because structures arise from symmetry breaking events and structures mean information we see how symmetry breaking is responsible for creating information at every level of reality.
Symmetry and symmetry breaking in sports and physical activities
1. Consider two or more athletes running with the same speed. Although they change the distance from the start or the end point, their interpersonal distance does not change. It remains invariant with respect to the changes of distance from the start or end point of the run. The interpersonal distance possesses symmetry with respect to these transformations. The same is valid if all athletes simultaneously start to accelerate. However, the symmetry will brake if some of athletes change the speed and others not. Then a re-structuring of the running formation will occur.
2. Consider the problem of sports scouting and the performance analytics in sports. If it does not make difference which player will play on which position then there is symmetry with respect to swapping (permutation) the roles of players. We all know that this is not the case. Different roles need players with different properties and skills. Living beings are not identical. In this sense, the permutational symmetry is broken and leads to the need of gathering much more information about the player’s properties than if there was symmetry with respect to their roles on the field. Note that if players were identical the personal information about each of them would be not important. The information for one person would be enough. Players’ properties as constraints make an important context within which the game unfolds. This is why scouting and performance analytics is important in sports and other human performance sectors.
3. Further, during the match, the changing positions of players do not elicit one and same action. Different changes of the structure of players bring about different decisions and consequently actions on the field. The environmental structure and its change matters to our decisions and actions. They are not invariant (symmetrical) with respect to the changes in the environement.
3. Consider a collective sport situation in which all your team mates are well marked by opponents and you are in a ball possession. Each team mate’s action is fully compensated by the opponent’s actions. You see the same environment wherever you look. All your teammates are well marked. This is a symmetrical state of the system. No passing opportunity emerges. Hence, no game starts. The situation is stuck. Then, one of your team mates succeeds to demarcate. A fluctuation (deviation) from the symmetric state occurs. You pass the ball to her/him. The symmetry is broken and the game starts, all over again. The game dwells on fluctuations and broken symmetries.
But the previous symmetry may just rearrange itself into two (or more) equally attractive symmetrical states. Two players simultaneously demarcate and your decision may be again stuck, at least for a moment – an instance of Buridan’s donkey problem in sports. In symmetric cases the system usually switches to one of the new born states due to any random event. So, although there are two equally attractive states, it selects the behavioral solution spontaneously, one would say, by chance, relaxing in one of the equally attractive states. This is called a spontaneous symmetry breaking because there is no clearly discernible constraint there that forces the system to prefer one of the solutions more than the other. This kind of bifurcation is connected to what is referred to as a second order phase transition or a pitchfork bifurcation.
However, usually there exists a bias in the informational constraints. Such events are connected to what is referred to as a first order phase transition or a saddle-node bifurcation. In this case asymmetry is induced into the system by some asymmetry control parameter. This kind of symmetry breaking is called forced symmetry breaking, because some kind of bias forces the system to contain asymmetric behavioral solutions. In sports settings an example of this kind of symmetry breaking would emerge when one of the team mates has a better position to score a point than the other one. The difference of the positions of the two team mates induces a bias which constrains the decision of the one possessing the ball, so s/he passes the ball to the one who is better positioned. The diagonal stance in martial arts also may play a role of symmetry braking parameter, i.e. a bias inducing constraint, forcing either left or right handed punches to be more used, respectively.
In forced symmetry breaking systems an interesting effect arises – a hysterezis. It would arise in the previous example if the player who is in possession of the ball continues to pass the ball to one of the team mates although the other one attains a better position to score. In one moment the player who is in ball possession decides to pass it to the player who is better positioned for a score. But, now he becomes stuck by passing to this team mate although the first one becomes better positioned for scoring. This type of inertia or memory is typical for systems with forced symmetry breaking. As we said earlier, this phenomenon cannot be found in pitchfork bifurcations, i.e. the second order phase transitions.
Symmetry and symmetry breaking… in general
Any artwork is breaking of the symmetry, homogeneity and randomness and creation of patterned spatio-temporal structures. It creates information. It is the pattern formation through symmetry breaking that unifies the humanity with the divinity.
In this sense all natural and human made processes, structures and functional products may be considered as coming from a same source. This mechanism is in function from the initial moments of our Universe when elementary forces were created through symmetry breaking, to the physical and chemical patterns, to the defense and attack patterns in sport matches, narrative patterns in novels, musical and architectural multileveled patterns. Natural, psychological and social phenomena become one.
Robert Hristovski 19.08.2020