Symmetriis in Veritate

Symmetry is a super important concept in pretty much every discipline I’m interested in, so I think about it a lot.  Sometimes, that thought is conscious, such as centering the Escher tiling at the beginning of this post.  Other times, it’s subconscious, like looking at an image or an object and “feeling” that something is off, that there is some sort of visual tension.  Indeed, that creation or elimination of tension is the primary manifestation of symmetry in visual design, which is how most people think of symmetry in the first place.

The word “symmetry” invokes images of starfish, spirals, and perfectly nested squares within squares.  The very word stems from the Greek words “syn” and “metron”, meaning “with measure” and has been used to denote the concept of proportions, pleasing or otherwise, since the mid 16th century.

But that’s not the kind of symmetry I want to talk about…

I’ve watched many science documentaries over the years, especially about physics.  Invariably, someone in the program will be attempting to explain just what it is that physicists do when they invoke a particular phrase: “physicists look for symmetries in the universe.”  What does this mean?  It calls to mind a geometric picture of the universe, one with some kind of symmetric order, like Kelper’s astrolabe.  Kepler envisioned a cosmology in which the planets orbited the Sun on crystal spheres, each inscribed with one of the Platonic solids.

Roll for initiative, Mars.

Unfortunately for Kepler, there are only five Platonic solids (the tetrahedron, the cube, the octahedron, the dodecahedron, and the icosahedron.  TL;DR – D&D dice) and he didn’t know about two of the planets (Uranus and Neptune), not to mention the fact that the planets have elliptical orbits and the entire model is BS.  But, it was beautiful and not the most ridiculous cosmological idea we’ve had…

This idea of cosmological symmetry, though visually appealing, is not what is being referred to when one says that physicists look for symmetries in Nature, however.  I must admit, it wasn’t until a few years ago that I truly understood this idea myself.  It’s really not an idea that you encounter until you’re in physics grad school.

Symmetry in Physics

We physicists have a very specific definition for the notion of symmetry, a mathematical one.  When we say that physical system exhibits symmetry, we mean that it contains some attribute that remains constant under transformation.

Great.  What does that mean?

Well first off, we have to understand what a transformation is.  Imagine a set of points, like the four points that define a box in a two-dimensional graph.

A transformation is a mathematical process that will “map” the set of points to some other set of points in the same graph by using some rule.  The simplest example of this would be to simply add some constant number to the position of each of the points.  The result would be a transformation we refer to as translation.

Another example would be to move each point through some fixed angle around the origin of the coordinate system, a transformation called rotation.

Yet another example would be to change the spacing between the points by some fixed amount, a transformation called scaling.

There are, as you can imagine, many more transformations of increasingly esoteric nature.  Some have familiar names, such as reflection and shear.  Others are much more mathematical in nature, such as the affine transformation.  The point, however, should be clear; you have some set that is mapped into some other set using a specific mathematical rule.

Now, what does it mean for an attribute to remain constant under transformation?  Well, consider the above example of translating a box.  The process of doing this changes the position of the corners, but doesn’t change the shape of the box in any way.  Certain attributes of the box, such as its perimeter or its area, then, also must not change.  So, we would say that, for instance, the area of the box is translationally symmetric.  Note that the shape of the box doesn’t change when I rotate it, either.  So, we can also say that the area of the box is rotationally symmetric.  However, scaling the box obviously changes the area; the box becomes larger or smaller.  So, the process of scaling breaks the symmetry of the box’s area.

I should point out that this mathematical idea is what you intuitively relate to when you look at on object and observe its symmetry.  The thing that makes a sphere look symmetric is the fact that it looks the same no matter how you rotate it; another way of interpreting that with the language above is that a sphere is invariant under rotation.

This is how symmetry manifests itself in physics.  Well, sort of.  Everything I’ve mentioned up until now is mathematical, not physical.  To understand why this is important physically, I need to introduce you to one of the most interesting people in the history of physics: Emmy Noether.

Noether’s Theorem

Emmy Noether was called, by no less than Einstein, the most important woman in the history of mathematics.

A stupendous badass, if ever there was one.

High praise, to be sure, and completely warranted.  Over her life, during the tumultuous transition from the 19th to the 20th centuries, her work defined the mathematics of so-called rings and fields.  She also developed the discipline of abstract algebra.  These would go on to be the foundational concepts of much of higher mathematics and theoretical physics.  In physics, her work manifested as the celebrated Noether’s Theorem, one of the most beautiful ideas I know of.

Regardless of how much scientific training you may have had in your life, there’s one idea in physics you’ve almost certainly heard of: the conservation of energy.  It’s one of those things that has been completely appropriated by culture, much like the phrase “everything is relative”.  But what is far less known is why energy should be conserved at all.

Another example, though less commonly known unless you’ve taken an introductory physics course, is the conservation of momentum.  Momentum defines important qualities of an object’s motion in space.  It is defined as the product of an object’s mass and its velocity.  So, a slowly traveling glacier has tremendous momentum because it is very massive and a bullet has tremendous momentum because it is traveling very quickly.  When Isaac Newton wrote the Principia in the 17th century, he defined force as a change in momentum.  You have to overcome an object’s momentum to change it’s motion; you have to apply a force to speed it up or slow it down.  Strictly speaking, everything I just said pertains to linear momentum, that is momentum associated with an object’s linear motion through space, like a car driving down the road.  There is also angular momentum which is, obviously, associated with an object’s rotation.

So, we have these two huge ideas: the conservation of energy and the conservation of momentum.  There are, in fact, more conservation laws in physics, such as the conservation of electric charge.  Conservation laws are critically important in physics because we know they yield values that remain constant through the system, useful signposts when trying to understand a system’s dynamics.  Take, for example, the formation of the solar system.  If you look down on the plane in which all of the planets lie, you’ll notice an interesting thing: everything rotates in the same direction.  The Sun rotates in the same way that all the planets do, as do all the moons.  The planets and moons also all orbit in the same direction they rotate on their axes (again, almost).  Why should this be?  Because of the conservation of angular momentum.  The original cloud the solar system formed from was rotating that way, so all the pieces that formed out of it must also be rotating that way.  Now, there are some pretty important exceptions.  Venus, for instance, rotates in retrograde, opposite the other planets; on Venus, the Sun rises in the west and sets in the east.  Since the conservation of momentum tells us it shouldn’t, we can surmise that something must have happened to Venus, such as a collision, that altered its rotation.  There are several moons that break the rule as well and, similar to Venus, this tells us that something must have happened to make them do that.  So, this simple conservation law has huge predictive power and utility.

So, what about Noether’s Theorem.  What she discovered is the following: conservation laws in physics are associated with the symmetries of the physical system.  Unfortunately, the specific details of this discovery are very mathematical, but the informal idea of the theorem is easy to grasp.  If a physical system exhibits symmetry in some property, then there must be some quantity that is conserved.

Take, for example, the much-lauded conservation of energy.  It turns out that this conservation law arises from a concept known as the homogeneity of time.  Imagine swinging a pendulum, such as that on a grandfather clock.  The only thing that differentiates each swing is the time at which it occurred.  The motion of the pendulum exhibits a symmetry in that the motion is identical regardless of which swing it is.  This symmetry in time, by Noether’s theorem, must then lead a conserved quantity.  In this case, that quantity turns out to be energy.

The conservation of linear momentum arises from something called the homogeneity of space.  Remember that idea of translating a box through space?  Imagine that there is a pendulum swinging inside that box.

Now, other than the fact that the pendulum is now over there, there is no change in the way it moves.  The motion of the pendulum is identical no matter where it is.  Again, there must be a conserved quantity associated with this symmetry and that quantity is linear momentum.

The conservation of angular momentum arises from something called the isotropy of space.  Imagine the grandfather clock again.  If you were to rotate the clock so that it faced the wall rather than away from the wall, would that change the way the clock worked?  No, other than you’d be silly because you couldn’t read it.  So, another symmetry emerges related to the orientation of the object.  The idea that the system exhibits a rotational symmetry leads to the conservation of angular momentum.

I before E, except after C

From everything that I’ve said, you may have the impression that energy and momentum are always conserved.  The idea that “energy can never be created or destroyed” gets a lot of play in popular culture.  But these things are not always conserved.

Just imagine a pendulum again, like on the grandfather clock.  If you let it swing for long enough, it slows down and stops.  Why?  Well, there is friction in the mechanical parts of the clock.  There is also air resistance each time the pendulum swings.  Remember, the thing that led to the conservation of energy was the fact that the motion was symmetric with respect to the passage of time.  However, with air resistance, that’s not true.  Each swing doesn’t quite reach the same height as the previous swing.  The symmetry is broken.  If we take away the symmetry, we must also lose the conserved quantity!

What about momentum?  Well, here’s a simple experiment.  Pick up a pencil and drop it.  Initially, when it’s in your hand, it’s not moving.  Since momentum is the product of mass and velocity, if it’s not moving, it has no momentum.  But then, when you let it go, it  clearly is moving.  So, as some other point in space as it falls, it has some momentum.  Obviously, “some” is not “none”, so momentum isn’t being conserved.  What’s going on?  There is a force that is accelerating the pencil.  What this means is that the motion of the object is changing from position to position.  So, like with the clock, the symmetry is being broken, in this case by gravity.  And like with the clock, if the symmetry goes, so does the conserved quantity.

This happens with angular momentum as well.  The Earth’s rotation has been slowing down over time because of the gravitational force it feels from the Moon.  If the Earth was a perfect sphere, this wouldn’t happen.  But, it’s obviously not.  The gravitational pull it feels from the Moon changes based on the Earth’s orientation, like when the Himalayas point away from the Moon rather than towards it.  Once more, the symmetry is broken, this time the symmetry of rotation.

So, it sounds like all the conservation laws are as much BS as Kepler’s crystal spheres.


All is not lost, however.  This problem is surmounted by how we define “the system”.  Let’s revisit the falling pencil for a bit.  There, “the system” is just the pencil.  The pencil falls to the ground because of gravity accelerates it and that’s what broke the symmetry.  However, recall that gravity is a mutual attraction between two objects (something I’ve talked about before).  When the pencil falls to the Earth, the Earth also “falls” to the pencil.  Of course, it doesn’t move very much at all since the average pencil is about a billion billion billion (!!!) times less massive than the Earth, but it moves none the less.  If I change “the system” to be not just the pencil, but the pencil AND the Earth, then you will find that the amount of momentum gained by the pencil as it falls is the same that the Earth gains in the opposite direction (since it “falls” up, whatever that means).  If you add them up, you’ll get zero everywhere all the time, since they are in opposite directions.  The thing that broke the symmetry that lead to the conservation of momentum wasn’t that gravity exists, it was that when only considering the pencil, gravity was outside the system.

The same thing goes for energy.  If I expand the system to include all the sources of friction as well, then any energy the pendulum looses is accounted for by energy gained in other parts.  For instance, the energy lost to air resistance will heat up the surrounding atmosphere.  If I expand “the system” to include the surrounding atmosphere, we’re golden and all is right in the world.

For the love of god, wrap this up will you…

So, that’s how we think about symmetry.  We can define a system of some kind and derive its equation of motion.  Symmetries in those equations, such as consistency with respect to time, position, and orientation, must lead to conserved quantities such as energy and momentum.  We can then test the dynamics of the system by seeing what breaks those symmetries and what happens when we do that.

Experimentally, we can do all sorts of nifty things to break symmetry.  I’ve already talked about the effect of gravity.  Applying an electric field (like with a battery) or a magnetic field can achieve this as well.  They’re better than gravity because, try as we might, we can’t turn gravity off.  Maybe we can cool a substance down so that it freezes and removes some degree of freedom from the system.  We can reverse the direction we apply these effects, essentially running the clock backwards and seeing if the symmetric is broken by reversing time.

And by doing all of this, we can probe the fundamental nature of the physical theories we come up with to describe Nature.  We’re pretty sure that energy is conserved with “the system” is the whole universe; I mean, that’s the only time that should really make sense, right?  But what if it’s not?  Could we use that to expand “the system” to parallel universes?  Who knows.

Bottom line: symmetry has powerful ramifications for the way we view the world.  And not just because that picture is crooked.  Though, that’s pretty important too…

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