# Can a symmetry be "broken down" (to a lower symmetry)?

Background

Symmetries are a key concept in physics, and describe the invariance of a system under certain operations (for example, rotation). Breaking a symmetry refers to modifying the system in a way so that it is no longer invariant under the same operation.

A symmetry can be completely broken, or only partially broken with a lower symmetry remaining. For example: a circle always looks the same, regardless of the angle you rotate it by. A square also has a rotational symmetry, but you have to rotate by multiples of 90 degrees to get the same picture. Hence, the symmetry of the square is lower.

Question

In the scientific literature, people use the expression "to break down a symmetry" when referring to a reduction from high symmetry (circle) to a lower symmetry (square) as compared to a complete breaking (irregular shape). They might write

The original continuous symmetry was broken down to a symmetry under 90 degree rotations.

In my opinion, the expression "to break down" is not appropriate, because it means to separate into its parts or to destroy completely rather than to reduce to a smaller part. I would instead write

The original continuous symmetry was reduced to a symmetry under 90 degree rotations.

Does the use of "to break down" with this meaning deviate from the usual meaning of this phrase in other contexts?

Here are some examples, suggesting that the use of "to break down" in the sense of reducing to a smaller part is at least common in the field:

http://arxiv.org/pdf/0708.2115 (native speaker, authority in his field)

http://arxiv.org/pdf/1312.2934

http://www.science.uva.nl/~bais/broksym.pdf

http://en.wikipedia.org/wiki/W%E2%80%B2_and_Z%E2%80%B2_bosons

• If people often use it that way, and other people understand what they mean, then yes, it can be used that way. More to the point, the meaning of break down is not as tightly limited as you suggest. Commented Aug 20, 2014 at 22:16
• Thanks. I have removed "often" because on second thought it is not such a common expression. I agree with your view, but I would still be interested if "to break down" is used in this sense in other contexts. Commented Aug 20, 2014 at 22:28
• I, for one, completely agree with you- I'd choose "reduce" over "break down" for the situation you describe. Or perhaps I'd say the when a symmetry is broken it is either broken completely (no symmetry remains) or partially ( a reduced level of symmetry remains) HOWEVER, if the convention among domain experts in your field is to "break down symmetries" you ought to follow right along.
– Jim
Commented Aug 20, 2014 at 23:21
• I have a feeling that it's an accidental usage. Physicists are used to referring to broken symmetries, so broken is psychologically linked to symmetry, and in some cases the word slips into less appropriate contexts. They then compensate by adding down. Commented Aug 21, 2014 at 4:28
• @Jim I will watch out in the future when reading papers to find out if native speakers use the term as well. Commented Aug 21, 2014 at 6:26

I must confess I have never encountered the usage of breaking down symmetry in the meaning that you state that you have encountered.

In all my education, the following have exclusive meanings
• breaking symmetry
• breaking down symmetry
• reducing/increasing symmetry

The basis of symmetry is orthogonality, or at least presumed orthogonality. Orthogonality means having a set of mutually independent aspects, where the set of all aspects considered are in complete and non-overlapping coverage of a defined realm/universe/field.

Orthogonality is also hierarchical in nature. A tree relationship without incestual parentage, rather than a network relationship, ensures mutual independence of all the aspect nodes in that tree.

For example, consider the following orthogonality

• Translational Direction: 3 dimensions of directions
• Rotational Direction
• Distance
• Matter/energy

Rotation and Distance are mutually independent because a zero distance-sized dot could rotate all it wants without traversing any translational distance.

That dot could experience compaction/rarefaction of density without rotating or traversing any distance.

Of all the orthogonalities, Direction is most familiar to us. Direction in our human consciousness is resolvable to three mutually independent directions, with complete and non-overlapping coverage of all possible directions.

These orthogonalities interact with each other

• Matter/Energy rotating
• Distance traversing in translational direction.

Because they are mutually independent, and they collectively completely covers the realm that we define, we are able to use cartesian coordinates to document the states and activities in such a realm.

Symmetry is when the characteristics of an entity is not differentiable when viewed thro two or more orthogonalities, where the orthogonalities are not in any level of parental relationship.

The study of orthogonality and symmetry of an entity or collection of entities is necessary when attempting to analyse their possible states and distribution of those states.

For example,

On a perfectly square table with red, green, blue and black legs at its corners, how many ways could you seat
• a man
• a man and a woman
• a man, a woman and a boy
• a man, a woman, a boy and a girl

I will not answer this elementary mathematical question. However, I shall illustrate symmetry.

• The number of permutations will be reduced by half if we have a diagonal pair of legs painted blue and the other diagonal pair painted red. That is because there would be two pairs of indifferentiable table sides. Their symmetry makes them indifferentiable

• What if all for legs had the same colour.

• What if the table had no legs, and the table and dinner guests were floating in the absence of gravity in outer space

### Breaking down the symmetry

is to breakdown the symmetry into their orthogonal components. It is the analysis of the degrees of freedom of symmetry.

Where the orthogonality would be the legs of the table, the dinner guests, etc.

We analyse the principal components of variation.

### Breaking the symmetry

Instead of colourless legs, we decided to paint one leg red. That would break one of the components of symmetry.

### Reducing/increasing symmetry

By using a perfectly square table instead of a rectangular table, we increase the symmetry. Vice versa.

• I actually think this question should have been asked in the Maths forum. Commented Aug 21, 2014 at 11:05

You might watch this video, a TED Talk by physicist Garret Lisi, about higher dimensional geometry and how patterns that have some sense of balance are related to symmetry (see starting at 8:30 if you want to jump right into it.)

For a non-physics example, Sven Yargs illustrated his answer with several examples: In one, Herbert Spencer referred to his predilection for following a pattern of balance in his opinions as symmetry. When he failed to follow this pattern of balance, he could say the symmetry was broken ("symmetry broke down").

Symmetry breaks down when an otherwise expected pattern of balance ceases to hold up (see video at 10:30).

What a physicist might discover, when evaluating a theory of symmetry, is that a complicated model of symmetry can be simplified, and the corresponding geometric models of symmetry that they might use to illustrate these symmetries (analogous to your circle and square) break down into something simpler, ceasing to follow the more complex pattern of the original theory.

If you are asking whether, in this context, break down means simplify to fewer dimensions, I don't think that is necessarily so. Break down means to break or to fail, and when the expected pattern ceases to exist then the original symmetry is broken. Another symmetry of lower (or higher) order would not necessarily emerge, so no, the words broken down, by themselves, should not be taken to imply this. But to say the symmetry broke down into some other symmetry says both that the original symmetry broke and something replaced it.

• Thanks. The video indeed suggests that a symmetry can be broken down, which is the answer to the title of the question. However, the example refers to the complete breaking of a symmetry. I have slightly changed the original question to include the aspect I was most interested in. Commented Aug 21, 2014 at 7:02
• I think I see. I'll add a little more to my answer. Commented Aug 21, 2014 at 7:11

From a quick Google Books search for "symmetry breaks down" and "symmetry broke down," I get the impression that these phrases have had a life of their own in logic, biology, and even political planning independent of (and perhaps antedating) their meaning in physics.

The earliest instance that a quick Google Books search for "symmetry breaks down" and "symmetry broke down" finds is from E. B. McCormick, "The Last Words of Herbert Spencer," in The Westminster Review (June 1903):

It is permissible to suppose that Mr. Spencer has not fully thought out all the difficulties of his position, and has been misled by his predilection for symmetry and completeness in his opinions. This symmetry once before, as he regretfully confessed, broke down when he attempted to deduce from evolution an ethical formulary which should introduce into the moral sphere the order and consistency of the physical processes.

Another early instance is from John D. Rogers, A Historical Geography of the British Colonies, vol. 5, part 5: Newfoundland (1911):

In 1607 a pure West-English colony was established at Monhegan Island off the coast of Maine, by Captain George Popham, brother of Sir John Popham, a Somersetshire man, and by Ralegh Gilbert, Sir Humphrey's youngest son. It failed, and the southern settlement, which afterwards monopolized the title of Virginia, succeeded. New efforts were made to colonize New England by Sir Ferdinando Gorges, who was a kinsman of Sir W. Ralegh and a Somersetshire man. Then this artificial symmetry broke down.

And from Theodore de Laguna, "The Nature of Space II. The Empirical Basis of Geometry," in The Journal of Philosophy (August 3, 1922):

It will be observed that up to this point our account of distances and of lengths has been perfectly symmetrical. At this point the symmetry breaks down.

Other instances from the first half of the twentieth century include this from Science News Letter (1939) [combined snippets]:

We expect right eyes to be like left eyes in color shape, and movement. The right side of the mouth is like the left, the right ear like the left and the hands and feet like each other anatomically.

Yet every once in a while, the general pattern of symmetry breaks down in some detail. Occasionally we see a girl with a beautiful blue eye on one side of her face, but when she turns we find the second eye of an entirely different hue, perhaps hazel or even brown.

And from S. Bhagavantam, Scattering of Light and the Raman Effect (1940) [combined snippets]:

The most important modification is in respect of the degenerate modes. They continue to be degenerate in the crystal as well, only when the crystal possesses the same order of symmetry as the free ion of the molecule. If the symmetry breaks down, the degeneracy also breaks down and each degenerate mode will split and give rise to the appropriate number of distinct modes. For example, the SO4 ion, having a tetrahedral symmetry in the free state, has two triply degenerate and one doubly degenerate normal modes. When a monoclinic CaSO4,2H2O crystal (gypsum) or an orthorhombic CaSO4 crystal (anhydrite) is built out of them, each one of the degenerate modes splits into three or two distinct normal modes in the crystal, as the case may be.

As later search results suggest, though the phrase begins to appear more frequently in the context of physics in the second half of the twentieth century, it also appears in discussions of linguistics, adaptive behavior, art interpretation, and other areas of study. As a result, even if you can persuade physicists to use the term only in the sense that you consider appropriate, you'll have to deal with seepage of the term from other disciplines, which will create a constant temptation for physicists to use the term more loosely and carelessly. After all, in those fields, theoretical opposition to the notion that symmetries can be broken down is probably nonexistent.

• Just to clarify, and in accordance with your wonderful research: I am happy to accept that a symmetry can be "broken down" (although for me the meaning would be the same as simply "broken"). I just had my doubts that "to break down" is the equivalent of "to reduce to a lower symmetry". Your quote involving facial symmetry seems to suggest that it is indeed acceptable. Commented Aug 21, 2014 at 6:23

TL;DR

This usage is uncommon but by no means unknown, and has been applied by a wide variety of people in an equally wide variety of situations. And, in the right context, yes, it does make sense.

The most compelling uses of it (to my ear) are when an entity is perceived to be "built up" upon a more primitive entity, and when it's possible for that superstructure to "break", and fall away, leaving the whole thing in a previous, lesser, state. In other words, when the reduction is discrete rather than continuous, and in particular when the loss is significant.

On to the evidence, then.

Applications of the Idiom

The phrase "break down to a lower level" meaning "to be reduced" is not common, but it does enjoy some currency in the wild; it doesn't appear to be isolated to physics or indeed to any particular community.

For example, here is an apartment posting on Craigslist:

Our new move-in special is six weeks of free rent which breaks down to a lower rent over 14 months

and a similar use appears in Forbes:

While the average price is a little higher than in Atlanta, it does include two more concerts and breaks down to a lower average price per show at \$126.77

One could argue that "breaks down to" in the preceding passages is intended to be interpreted "can be analyzed as", but it's debatable.

But even then, we do see evidence, in the wild, of unambiguous applications the "break down"="reduce" idiom, like this absolutely spot-on quote from a popular science book aimed at the general public:

Mass can also appear from a dimension reduction: the idea that a universe that is based on ten or eleven dimensions "breaks down" to a lower-dimensional space of only four evident dimensions.

If that skirts too close to the "physicists use language weirdly, even when talking to laymen" line for you, here's a couple of identical usages by non-physicists.

But the conclusion is that until the [price of gold] breaks down to a lower low, the current situation is viewed as a buying opportunity.

And a car racing enthusiast:

From your second scope shot is looks like the motor velocity suddenly breaks down to a lower velocity.

However, in both these cases, the authors are describing charts, and so "breaks down" may be describing a "sharp negative inflection" or a "very steep decline".

But it's harder to make a case against the following examples, starting with an academic mineralogy paper:

This phase [of a silica compound] breaks down to a lower hydrate near 200 degrees

and again in a discussion among video gamers:

Once you have a morph [video game character] out, hold down A, and you'll consume more energy to make the morph bigger. Very handy for when your morph breaks down to a lower level, and you don't have time to draw it again.

and again among, erm, survivalists:

Why, outside of all the compounds and elements in the universe, why is gunpowder immortal? Everything else breaks down to a lower energy state. Mountains crumble, houses fall, paint fades...

Each of these quotes perfectly captures the idea of "breaking down" as "reduction with transformation" to a state with lower power.