The imperfection of strontium titanate

When you squeeze some crystals, you distort their lattice of atoms just enough to separate a pair of charged particles and that in turn gives rise to a voltage. Such materials are called piezoelectric crystals. Not all crystals are piezoelectric because the property depends on what the arrangement of atoms in the lattice is.

For example, the atoms of strontium, titanium and oxygen are arranged in a cubic structure to form strontium titanate (SrTiO3) such that each molecule displays a mirror symmetry through its centre. That is, if you placed a mirror passing through the molecule’s centre, the object plus its reflection would show the molecule as it actually is. Such molecules are said to be centrosymmetric, and centrosymmetric crystals aren’t piezoelectric.

In fact, strontium titanate isn’t ferroelectric or pyroelectric either – an external electric field can’t reverse their polarisation nor do they produce a voltage when they’re heated or cooled – for the same reason. Its crystal lattice is just too symmetrical.

The strontium titanate lattice. Oxygen atoms are red, titanium cations are blue and strontium cations are green.

However, scientists haven’t been deterred by this limitation (such as it is) because its perfect symmetry indicates that messing with the symmetry can introduce new properties in the material. There are also natural limits to the lattice itself. A cut and polished diamond looks beautiful because, at its surface, the crystal lattice ends and the air begins – arbitrarily stopping the repetitive pattern of carbon atoms.

An infinite diamond that occupies all points in the universe might look good on paper but it wouldn’t be nearly as resplendent because only the symmetry-breaking at the surface allows light to enter the crystal and bounce around. Similarly, centrosymmetric strontium titanate might be a natural wonder, so to speak, but the centrosymmetry also keeps it from being useful (despite its various unusual properties; e.g. it was the first insulator found to be a superconductor at low temperatures, in 1967).

Tausonite, a naturally occurring mineral form of strontium titanate. Credit: Materialscientist/Wikimedia Commons, CC BY-SA 3.0

So does strontium titanate exhibit pyro- or piezoelectricity on its surface? Surprisingly, while this seems like a fairly straightforward question to ask, it hasn’t been straightforward to answer.

A part of the problem is the definition of a surface. Obviously, the surface of any object refers to the object’s topmost or outermost layer. But when you’re talking about, say, a small electric current originating from the material, it’s difficult to imagine how you could check if the current originated from the bulk of the material or just the surface.

Researchers from the US, Denmark and Israel recently reported resolving this problem using concepts from thermodynamics 101. If the surface of strontium titanate is pyroelectric, the presence of electric currents should co-exist with heat. So if a bit of heat is applied and taken away, the material should begin cooling (or thermalising) and the electric currents should also dissipate. The faster the material cools, the faster the currents dissipate, and the faster the currents dissipate, the lower the depth to which the material is pyroelectric.

In effect, the researchers induced pyroelectricity and then tracked how quickly it vanished to infer how deeply inside the material it existed.

Both the bulk and the surface are composed of the same atoms, but the atomic lattice on the surface also has a bit of surface tension. Materials scientists have already calculated how deeply this tension penetrates the surface of strontium titanate, so the question was also whether the pyroelectric behaviour was contained in this region or went beyond, into the rest of the bulk.

The team sandwiched a slab of strontium titanate between two electrodes, at room temperature. At the crystal-electrode interface, which is a meeting of two surfaces, opposing charged particles on either side gather and neutralise themselves. But when an infrared laser is shined on the ensemble (as shown above), the surface of strontium titanate heats up and develops a voltage, which in turn draws the charges at its surface away from the interface. The charges in the electrode are then left without a partner so they flow through a wire connected to the other electrode and create a current.

The laser is turned off and the strontium titanate’s surface begins to cool. Its voltage drops and allows the charged particles to move away from each other, and some of them move towards the surface to once again neutralise oppositely charged particles from the other side. This process stops the current. So measuring how quickly the current drops off gives away how quickly the voltage vanishes, which gives away how much of the material’s volume developed a voltage due to the pyroelectric effect.

The penetration depth the group measured was in line with the calculations based on surface tension: about 1.2 nm. To be sure the effect didn’t involve the bulk, the researchers repeated the experiment with a thin layer of silica (the major component of sand) on top of the strontium titanate surface, and there was no electric current when the laser was on or off.

In fact, according to a report in Nature, the team also took various precautions to ensure any electric effects originated only from the surface, and due to effects intrinsic to the material itself.

… they checked that the direction of the heat-induced current does not depend on the orientation of the crystal, ruling out a bulk effect; and that the local heating produced by the laser is very small…, which means that the strain gradients induced by thermal expansion are insignificant. Other experiments and data analysis were carried out to exclude the possibility that the induced current is due to molecules … adsorbed to the surface, charges trapped by lattice defects, excitation of free electrons induced by light, or the thermoelectric Seebeck effect (which generates currents in semiconductors that contain temperature gradients).

Now we know strontium titanate is pyroelectric, and piezoelectric, on its surface at room temperature – but this is not all we know. During their experiments (with different samples of the crystal), the researchers spotted something odd:

The pyroelectric coefficient – a measure of the strength of the material’s pyroelectricity – was constant between 193 K and 225 K (–80.15º C to –48.15º C) but dropped sharply above 225 K and vanished above 380 K. The researchers note in their paper, published on September 18, that others have previously reported that the strontium titanate lattice near the surface changes from a cubic to a tetragonal structure at around 150 K, and that a similar transformation could be happening at 225 K.

In other words, the surface pyroelectric effect wasn’t just producing a voltage but could in fact be altering the relative arrangement of atoms itself. What the precise mechanism of action could be we don’t know – nor any other features that might arise in the material as a result. The researchers hope future studies can resolve these questions.

The symmetry incarnations

This post was originally published on October 6, 2012. I recently rediscovered it and decided to republish it with a few updates.

Geometric symmetry in nature is often a sign of unperturbedness, as if nothing has interfered with a natural process and that its effects at each step are simply scaled-up or scaled-down versions of each other. For this reason, symmetry is aesthetically pleasing, and often beautiful. Consider, for instance, faces. Symmetry of facial features about the central vertical axis is often translated as the face being beautiful, not just by humans but also monkeys.

This is only one example of one of the many forms of symmetry’s manifestation. When it involves geometric features, it’s a case of geometrical symmetry. When a process occurs similarly both forward and backward in time, it is temporal symmetry. If two entities that don’t seem geometrically congruent at first sight rotate, move or scale with similar effects on their forms, it is transformational symmetry. A similar definition applies to all theoretical models, musical progressions, knowledge and many other fields besides.

Symmetry-breaking

One of the first (postulated) instances of symmetry is said to have occurred during the Big Bang, when the universe was born. A sea of particles was perturbed 13.75 billion years ago by a high-temperature event, setting up anecdotal ripples in their system, eventually breaking their distribution in such a way that some particles got mass, some charge, some spin, some all of them, and some none of them. This event is known as electroweak symmetry-breaking. Because of the asymmetric properties of the resultant particles, matter as we know it was conceived.

Many invented scientific systems exhibit symmetry in that they allow for the conception of symmetry in the things they make possible. A good example is mathematics. On the real-number line, 0 marks the median. On either sides of 0, 1 and -1 are equidistant from 0; 5,000 and -5,000 are equidistant from 0; possibly, ∞ and -∞ are equidistant from 0. Numerically speaking, 1 marks the same amount of something that -1 marks on the other side of 0. Characterless functions built on this system also behave symmetrically on either sides of 0.

To many people, symmetry evokes the image of an object that, when cut in half along a specific axis, results in two objects that are mirror-images of each other. Cover one side of your face and place the other side against a mirror, and what a person hopes to see is the other side of the face – despite it being a reflection. Interestingly, this technique was used by neuroscientist V.S. Ramachandran to “cure” the pain of amputees when they tried to move a limb that wasn’t there).

An illustration of V.S. Ramachandran's mirror-box technique: Lynn Boulanger, an occupational therapy assistant and certified hand therapist, uses mirror therapy to help address phantom pain for Marine Cpl. Anthony McDaniel. Caption and credit: US Navy
An illustration of V.S. Ramachandran’s mirror-box technique: Lynn Boulanger, an occupational therapy assistant and certified hand therapist, uses mirror therapy to help address phantom pain for Marine Cpl. Anthony McDaniel. Caption and credit: US Navy

Natural symmetry

Symmetry at its best, however, is observed in nature. Consider germination: when a seed grows into a small plant and then into a tree, the seed doesn’t experiment with designs. The plant is not designed differently from the small tree, and the small tree is not designed differently from the big tree. If a leaf is given to sprout from the node richest in minerals on the stem, then it will. If a branch is given to sprout from the node richest in minerals on the trunk, then it will. So is mineral-deposition in the arbor symmetric? It should be if their transportation out of the soil and into the tree is radially symmetric. And so forth…

At times, repeated gusts of wind may push the tree to lean one way or another, shadowing the leaves from against the force and keeping them form shedding off. The symmetry is then broken, but no matter. The sprouting of branches from branches, and branches from those branches, and leaves from those branches, all follow the same pattern. This tendency to display an internal symmetry is characterised as fractalisation. A well-known example of a fractal geometry is the Mandelbrot set, shown below.

An illustration of recursive self-similarity in Mandelbrot set. Credit: Cuddlyable3/Wikimedia Commons
An illustration of recursive self-similarity in Mandelbrot set. Credit: Cuddlyable3/Wikimedia Commons

If you want to interact with a Mandelbrot set, check out this magnificent visualisation by Paul Neave (defunct now 😦 ). You can keep zooming in, but at each step, you’ll only see more and more Mandelbrot sets. This set is one of a few exceptional sets that are geometric fractals.

Meta-geometry and Mulliken symbols

It seems like geometric symmetry is the most ubiquitous and accessible example to us. Let’s take it one step further and look at the meta-geometry at play when one symmetrical shape is given an extra dimension. For instance, a circle exists in two dimensions; its three-dimensional correspondent is the sphere. Through such an up-scaling, we are ensuring that all the properties of a circle in two dimensions stay intact in three dimensions, and then we are observing what the three-dimensional shape is.

A circle, thus, becomes a sphere. A square becomes a cube. A triangle becomes a tetrahedron. In each case, the 3D shape is said to have been generated by a 2D shape, and each 2D shape is said to be the degenerate of the 3D shape. Further, such a relationship holds between corresponding shapes across many dimensions, with doubly and triply degenerate surfaces also having been defined.

Credit: Vitaly Ostrosablin/Wikimedia Commons, CC BY-SA 3.0
The three-dimensional cube generates the four-dimensional hypercube, a.k.a. a tesseract. Credit: Vitaly Ostrosablin/Wikimedia Commons, CC BY-SA 3.0

Obviously, there are different kinds of degeneracy, 10 of which the physicist Robert S. Mulliken identified and laid out. These symbols are important because each one defines a degree of freedom that nature possesses while creating entities and this includes symmetrical entities as well. So if a natural phenomenon is symmetrical in n dimensions, then the only way it can be symmetrical in n+1 dimensions also is by transforming through one or many of the degrees of freedom defined by Mulliken.


Symbol Property
A symmetric with respect to rotation around the principal rotational axis (one dimensional representations)
B anti-symmetric with respect to rotation around the principal rotational axis (one dimensional representations)
E degenerate
subscript 1 symmetric with respect to a vertical mirror plane perpendicular to the principal axis
subscript 2 anti-symmetric with respect to a vertical mirror plane perpendicular to the principal axis
subscript g symmetric with respect to a center of symmetry
subscript u anti-symmetric with respect to a center of symmetry
prime (‘) symmetric with respect to a mirror plane horizontal to the principal rotational axis
double prime (”) anti-symmetric with respect to a mirror plane horizontal to the principal rotational axis

Source: LMU Munich


Apart from regulating the perpetuation of symmetry across dimensions, the Mulliken symbols also hint at nature wanting to keep things simple and straightforward. The symbols don’t behave differently for processes moving in different directions, through different dimensions, in different time-periods or in the presence of other objects, etc. The preservation of symmetry by nature is not coincidental. Rather, it is very well-defined.

Anastomosis

Now, if nature desires symmetry, if it is not a haphazard occurrence but one that is well orchestrated if given a chance to be, why don’t we see symmetry everywhere? Why is natural symmetry broken? One answer to this is that it is broken only insofar as it attempts to preserves other symmetries that we cannot observe with the naked eye.

For example, symmetry in the natural order is exemplified by a geological process called anastomosis. This property, commonly of quartz crystals in metamorphic regions of Earth’s crust, allows for mineral veins to form that lead to shearing stresses between layers of rock, resulting in fracturing and faulting. In other terms, geological anastomosis allows materials to be displaced from one location and become deposited in another, offsetting large-scale symmetry in favour of the prosperity of microstructures.

More generally, anastomosis is defined as the splitting of a stream of anything only to reunify sometime later. It sounds simple but it is an exceedingly versatile phenomenon, if only because it happens in a variety of environments and for a variety of purposes. For example, consider Gilbreath’s conjecture. It states that each series of prime numbers to which the forward difference operator – i.e. successive difference between numbers – has been applied always starts with 1. To illustrate:

2 3 5 7 11 13 17 19 23 29 … (prime numbers)

Applying the operator once: 1 2 2 4 2 4 2 4 6 …
Applying the operator twice: 1 0 2 2 2 2 2 2 …
Applying the operator thrice: 1 2 0 0 0 0 0 …
Applying the operator for the fourth time: 1 2 0 0 0 0 0 …

And so forth.

If each line of numbers were to be plotted on a graph, moving upwards each time the operator is applied, then a pattern for the zeros emerges, shown below.

The forest of stunted trees, used to gain more insights into Gilbreath's conjecture. Credit: David Eppstein/Wikimedia Commons
The forest of stunted trees, used to gain more insights into Gilbreath’s conjecture. Credit: David Eppstein/Wikimedia Commons

This pattern is called the forest of stunted trees, as if it were an area populated by growing trees with clearings that are always well-bounded triangles. The numbers from one sequence to the next are anastomosing, parting ways only to come close together after every five lines.

Another example is the vein skeleton on a hydrangea leaf. Both the stunted trees and the hydrangea veins patterns can be simulated using the rule-90 simple cellular automaton that uses the exclusive-or (XOR) function.

Bud and leaves of Hydrangea macrophylla. Credit: Alvesgaspar/Wikimedia Commons, CC BY-SA 3.0
Bud and leaves of Hydrangea macrophylla. Credit: Alvesgaspar/Wikimedia Commons, CC BY-SA 3.0

Nambu-Goldstone bosons

While anastomosis may not have a direct relation with symmetry and only a tenuous one with fractals, its presence indicates a source of perturbation in the system. Why else would the streamlined flow of something split off and then have the tributaries unify, unless possibly to reach out to richer lands? Anastomosis is a sign of the system acquiring a new degree of freedom. By splitting a stream with x degrees of freedom into two new streams each with x degrees of freedom, there are now more avenues through which change can occur.

Particle physics simplifies this scenario by assigning all forces and amounts of energy a particle. Thus, a force is said to be acting when a force-carrying particle is being exchanged between two bodies. Since each degree of freedom also implies a new force acting on the system, it wins itself a particle from a class of particles called the Nambu-Goldstone (NG) bosons. Named for Yoichiro Nambu and Jeffrey Goldstone, the presence of n NG bosons in a system means that, broadly speaking, the system has n degrees of freedom.

How and when an NG boson is introduced into a system is not yet well-understood. In fact, it was only recently that a theoretical physicist, named Haruki Watanabe, developed a mathematical model that could predict the number of degrees of freedom a complex system could have given the presence of a certain number of NG bosons. At the most fundamental level, it is understood that when symmetry breaks, an NG boson is born.

The asymmetry of symmetry

That is, when asymmetry is introduced in a system, so is a degree of freedom. This seems intuitive. But at the same time, you would think the reverse is also true: that when an asymmetric system is made symmetric, it loses a degree of freedom. However, this isn’t always the case because it could violate the third law of thermodynamics (specifically, the Lewis-Randall version of its statement).

Therefore, there is an inherent irreversibility, an asymmetry of the system itself: it works fine one way, it doesn’t work fine another. This is just like the split-off streams, but this time, they are unable to reunify properly. Of course, there is the possibility of partial unification: in the case of the hydrangea leaf, symmetry is not restored upon anastomosis but there is, evidently, an asymptotic attempt.

However, it is possible that in some special frames, such as in outer space, where the influence of gravitational forces is very weak, the restoration of symmetry may be complete. Even though the third law of thermodynamics is still applicable here, it comes into effect only with the transfer of energy into or out of the system. In the absence of gravity and other retarding factors, such as distribution of minerals in the soil for acquisition, etc., it is theoretically possible for symmetry to be broken and reestablished without any transfer of energy.

The simplest example of this is of a water droplet floating around. If a small globule of water breaks away from a bigger one, the bigger one becomes spherical quickly. When the seditious droplet joins with another globule, that globule also quickly reestablishes its spherical shape.

Thermodynamically speaking, there is mass transfer but at (almost) 100% efficiency, resulting in no additional degrees of freedom. Also, the force at play that establishes sphericality is surface tension, through which a water body seeks to occupy the shape that has the lowest volume for the correspondingly highest surface area. Notice how this shape – the sphere – is incidentally also the one with the most axes of symmetry and the fewest redundant degrees of freedom? Manufacturing such spheres is very hard.

An omnipotent impetus

Perhaps the explanation of the roles symmetry assumes seems regressive: every consequence of it is no consequence but itself all over again (i.e., self-symmetry – and it happened again). Indeed, why would nature deviate from itself? And as it recreates itself with different resources, it lends itself and its characteristics to different forms.

A mountain will be a mountain to its smallest constituents, and an electron will be an electron no matter how many of them you bring together at a location (except when quasiparticles show up). But put together mountains and you have ranges, sub-surface tectonic consequences, a reshaping of volcanic activity because of changes in the crust’s thickness, and a long-lasting alteration of wind and irrigation patterns. Bring together an unusual number of electrons to make up a high-density charge, and you have a high-temperature, high-voltage volume from which violent, permeating discharges of particles could occur – i.e., lightning.

Why should stars, music, light, radioactivity, politics, engineering or knowledge be any different?

The Symmetry Incarnations – Part I

Symmetry in nature is a sign of unperturbedness. It means nothing has interfered with a natural process, and that its effects at each step are simply scaled-up or scaled-down versions of each other. For this reason, symmetry is aesthetically pleasing, and often beautiful. Consider, for instance, faces. Symmetry of facial features about the central vertical axis is often translated as the face being beautiful – not just by humans but also monkeys.

However, this is just an example of one of the many forms of symmetry’s manifestation. When it involves geometric features, it’s a case of geometrical symmetry. When a process occurs similarly both forward and backward in time, it is temporal symmetry. If two entities that don’t seem geometrically congruent at first sight rotate, move or scale with similar effects on their forms, it is transformational symmetry. A similar definition applies to all theoretical models, musical progressions, knowledge, and many other fields besides.

Symmetry-breaking

One of the first (postulated) instances of symmetry is said to have occurred during the Big Bang, when the observable universe was born. A sea of particles was perturbed 13.75 billion years ago by a high-temperature event, setting up anecdotal ripples in their system, eventually breaking their distribution in such a way that some particles got mass, some charge, some a spin, some all of them, and some none of them. In physics, this event is called spontaneous, or electroweak, symmetry-breaking. Because of the asymmetric properties of the resultant particles, matter as we know it was conceived.

Many invented scientific systems exhibit symmetry in that they allow for the conception of symmetry in the things they make possible. A good example is mathematics – yes, mathematics! On the real-number line, 0 marks the median. On either sides of 0, 1 and -1 are equidistant from 0, 5,000 and -5,000 are equidistant from 0; possibly, ∞ and -∞ are equidistant from 0. Numerically speaking, 1 marks the same amount of something that -1 marks on the other side of 0. Not just that, but also characterless functions built on this system also behave symmetrically on either sides of 0.

To many people, symmetry evokes an image of an object that, when cut in half along a specific axis, results in two objects that are mirror-images of each other. Cover one side of your face and place the other side against a mirror, and what a person hopes to see is the other side of the face – despite it being a reflection (interestingly, this technique was used by neuroscientist V.S. Ramachandran to “cure” the pain of amputees when they tried to move a limb that wasn’t there). Like this, there are symmetric tables, chairs, bottles, houses, trees (although uncommon), basic geometric shapes, etc.

A demonstration of V.S. Ramachandran’s mirror-technique

Natural symmetry

Symmetry at its best, however, is observed in nature. Consider germination: when a seed grows into a small plant and then into a tree, the seed doesn’t experiment with designs. The plant is not designed differently from the small tree, and the small tree is not designed differently from the big tree. If a leaf is given to sprout from the mineral-richest node on the stem then it will; if a branch is given to sprout from the mineral-richest node on the trunk then it will. So, is mineral-deposition in the arbor symmetric? It should be if their transportation out of the soil and into the tree is radially symmetric. And so forth…

At times, repeated gusts of wind may push the tree to lean one way or another, shadowing the leaves from against the force and keeping them form shedding off. The symmetry is then broken, but no matter. The sprouting of branches from branches, and branches from those branches, and leaves from those branches all follow the same pattern. This tendency to display an internal symmetry is characterized as fractalization. A well-known example of a fractal geometry is the Mandelbrot set, shown below.

If you want to interact with a Mandelbrot set, check out this magnificent visualization by Paul Neave. You can keep zooming in, but at each step, you’ll only see more and more Mandelbrot sets. Unfortunately, this set is one of a few exceptional sets that are geometric fractals.

Meta-geometry & Mulliken symbols

Now, it seems like geometric symmetry is the most ubiquitous and accessible example to us. Let’s take it one step further and look at the “meta-geometry” at play when one symmetrical shape is given an extra dimension. For instance, a circle exists in two dimensions; its three-dimensional correspondent is the sphere. Through such an up-scaling, we’re ensuring that all the properties of a circle in two dimensions stay intact in three dimensions, and then we’re observing what the three-dimensional shape is.

A circle, thus, becomes a sphere; a square becomes a cube; a triangle becomes a tetrahedron (For those interested in higher-order geometry, the tesseract, or hypercube, may be of special interest!). In each case, the 3D shape is said to have been generated by a 2D shape, and each 2D shape is said to be the degenerate of the 3D shape. Further, such a relationship holds between corresponding shapes across many dimensions, with doubly and triply degenerate surfaces also having been defined.

The tesseract (a.k.a. hypercube)

Obviously, there are different kinds of degeneracy, 10 of which the physicist Robert S. Mulliken identified and laid out. These symbols are important because each one defines a degree of freedom that nature possesses while creating entities, and this includes symmetrical entities as well. In other words, if a natural phenomenon is symmetrical in n dimensions, then the only way it can be symmetrical in n+1 dimensions also is by transforming through one or many of the degrees of freedom defined by Mulliken.

Robert S. Mulliken (1896-1986)

Apart from regulating the perpetuation of symmetry across dimensions, the Mulliken symbols also hint at nature wanting to keep things simple and straightforward. The symbols don’t behave differently for processes moving in different directions, through different dimensions, in different time-periods or in the presence of other objects, etc. The preservation of symmetry by nature is not a coincidental design; rather, it’s very well-defined.

Anastomosis

Now, if that’s the case – if symmetry is held desirable by nature, if it is not a haphazard occurrence but one that is well orchestrated if given a chance to be – why don’t we see symmetry everywhere? Why is natural symmetry broken? Is all of the asymmetry that we’re seeing today the consequence of that electro-weak symmetry-breaking phenomenon? It can’t be because natural symmetry is still prevalent. Is it then implied that what symmetry we’re observing today exists in the “loopholes” of that symmetry-breaking? Or is it all part of the natural order of things, a restoration of past capabilities?

One of the earliest symptoms of symmetry-breaking was the appearance of the Higgs mechanism, which gave mass to some particles but not some others. The hunt for it’s residual particle, the Higgs boson, was spearheaded by the Large Hadron Collider (LHC) at CERN.

The last point – of natural order – is allegorical with, as well as is exemplified by, a geological process called anastomosis. This property, commonly of quartz crystals in metamorphic regions of Earth’s crust, allows for mineral veins to form that lead to shearing stresses between layers of rock, resulting in fracturing and faulting. Philosophically speaking, geological anastomosis allows for the displacement of materials from one location and their deposition in another, thereby offsetting large-scale symmetry in favor of the prosperity of microstructures.

Anastomosis, in a general context, is defined as the splitting of a stream of anything only to rejoin sometime later. It sounds really simple but it is a phenomenon exceedingly versatile, if only because it happens in a variety of environments and for an equally large variety of purposes. For example, consider Gilbreath’s conjecture. It states that each series of prime numbers to which the forward difference operator has been applied always starts with 1. To illustrate:

2 3 5 7 11 13 17 19 23 29 … (prime numbers)

Applying the operator once: 1 2 2 4 2 4 2 4 6 … (successive differences between numbers)
Applying the operator twice: 1 0 2 2 2 2 2 2 …
Applying the operator thrice: 1 2 0 0 0 0 0 …
Applying the operator for the fourth time: 1 2 0 0 0 0 0 …

And so forth.

If each line of numbers were to be plotted on a graph, moving upwards each time the operator is applied, then a pattern for the zeros emerges, shown below.

This pattern is called that of the stunted trees, as if it were a forest populated by growing trees with clearings that are always well-bounded triangles. The numbers from one sequence to the next are anastomosing, only to come close together after every five lines! Another example is the vein skeleton on a hydrangea leaf. Both the stunted trees and the hydrangea veins patterns can be simulated using the rule-90 simple cellular automaton that uses the exclusive-or (XOR) function.

Nambu-Goldstone bosons

Now, what does this have to do with symmetry, you ask? While anastomosis may not have a direct relation with symmetry and only a tenuous one with fractals, its presence indicates a source of perturbation in the system. Why else would the streamlined flow of something split off and then have the tributaries unify, unless possibly to reach out to richer lands? Either way, anastomosis is a sign of the system acquiring a new degree of freedom. By splitting a stream with x degrees of freedom into two new streams each with x degrees of freedom, there are now more avenues through which change can occur.

Water entrainment in an estuary is an example of a natural asymptote or, in other words, a system’s “yearning” for symmetry

Particle physics simplies this scenario by assigning all forces and amounts of energy a particle. Thus, a force is said to be acting when a force-carrying particle is being exchanged between two bodies. Since each degree of freedom also implies a new force acting on the system, it wins itself a particle, actually a class of particles called the Nambu-Goldstone (NG) bosons. Named for Yoichiro Nambu and Jeffrey Goldstone, the particle’s existence’s hypothesizers, the presence of n NG bosons in a system means that, broadly speaking, the system has n degrees of freedom.

Jeffrey Goldstone (L) & Yoichiro Nambu

How and when an NG boson is introduced into a system is not yet a well-understood phenomenon theoretically, let alone experimentally! In fact, it was only recently that a mathematical model was developed by a theoretical physicist at UCal-Berkeley, Haruki Watanabe, capable of predicting how many degrees of freedom a complex system could have given the presence of a certain number of NG bosons. However, at the most basic level, it is understood that when symmetry breaks, an NG boson is born!

The asymmetry of symmetry

In other words, when asymmetry is introduced in a system, so is a degree of freedom. This seems only intuitive. At the same time, you’d think the axiom is also true: that when an asymmetric system is made symmetric, it loses a degree of freedom – but is this always true? I don’t think so because, then, it would violate the third law of thermodynamics (specifically, the Lewis-Randall version of its statement). Therefore, there is an inherent irreversibility, an asymmetry of the system itself: it works fine one way, it doesn’t work fine another – just like the split-off streams, but this time, being unable to reunify properly. Of course, there is the possibility of partial unification: in the case of the hydrangea leaf, symmetry is not restored upon anastomosis but there is, evidently, an asymptotic attempt.

Each piece of a broken mirror-glass reflects an object entirely, shedding all pretensions of continuity. The most intriguing mathematical analogue of this phenomenon is the Banach-Tarski paradox, which, simply put, takes symmetry to another level.

However, it is possible that in some special frames, such as in outer space, where the influence of gravitational forces is weak if not entirely absent, the restoration of symmetry may be complete. Even though the third law of thermodynamics is still applicable here, it comes into effect only with the transfer of energy into or out of the system. In the absence of gravity (and, thus, friction), and other retarding factors, such as distribution of minerals in the soil for acquisition, etc., symmetry may be broken and reestablished without any transfer of energy.

The simplest example of this is of a water droplet floating around. If a small globule of water breaks away from a bigger one, the bigger one becomes spherical quickly; when the seditious droplet joins with another globule, that globule also reestablishes its spherical shape. Thermodynamically speaking, there is mass transfer, but at (almost) 100% efficiency, resulting in no additional degrees of freedom. Also, the force at play that establishes sphericality is surface tension, through which a water body seeks to occupy the shape that has the lowest volume for the correspondingly highest surface area (notice how the shape is incidentally also the one with the most axes of symmetry, or, put another way, no redundant degrees of freedom? Creating such spheres is hard!).

A godless, omnipotent impetus

Perhaps the explanation of the roles symmetry assumes seems regressive: every consequence of it is no consequence but itself all over again (self-symmetry – there, it happened again). This only seems like a natural consequence of anything that is… well, naturally conceived. Why would nature deviate from itself? Nature, it seems, isn’t a deity in that it doesn’t create. It only recreates itself with different resources, lending itself and its characteristics to different forms.

A mountain will be a mountain to its smallest constituents, and an electron will be an electron no matter how many of them you bring together at a location. But put together mountains and you have ranges, sub-surface tectonic consequences, a reshaping of volcanic activity because of changes in the crust’s thickness, and a long-lasting alteration of wind and irrigation patterns. Bring together a unusual number of electrons to make up a high-density charge, and you have a high-temperature, high-voltage volume from which violent, permeating discharges of particles could occur – i.e., lightning. Why should stars, music, light, radioactivity, politics, manufacturing or knowledge be any different?

With this concludes the introduction to symmetry. Yes, there is more, much more…

xkcd #849