Symmetry in Nature
Symmetry surrounds you. Look down at your body. Look at the shapes on the screen. Look at
the buildings on your street. Look at your cat or dog.Symmetryis variously defined as
"proportion," "perfect, or harmonious proportions," and "a structure that allows an object to be
divided into parts of an equal shape and size." When you think of symmetry, you probably
think of some combination of all these definitions. That's because symmetry, whether in
biology, architecture, art, or geometry reflects all of those definitions.
The two main types of symmetry arereflectiveandrotational. Reflective, or line, symmetry
means that one half of an image is the mirror image of the other half (think of a butterfly's
wings). Rotational symmetry means that the object or image can be turned around a center
point and match itself some number of times (as in a five-pointed star).
In biology, there are three classifications of symmetry found in living organisms.Point
symmetry(a kind of reflective symmetry) means that any straight cut through the center point
divides the organism into mirroring halves. Some floating animals with radiating parts, and
some microscopic protozoa fit into this category. Animals with this layout are all very
small.Radial symmetry(a kind of rotational symmetry) means that a cone or disk shape is
symmetrical around a central axis. Starfish, sea anemones, jellyfish, and some flowers have
radial symmetry. Lastly,planeorbilateral symmetry(also reflective symmetry) means that a
body can be divided by a central (sagittal) plane into two equal halves that form mirror images
of each other. Human beings, insects, and mammals all show bilateral symmetry.
Man is naturally attracted to symmetry. Very often we consider a face beautiful when the
features are symmetrically arranged. We are drawn to even proportions. In this, we are not
alone. Many animals choose mates on the basis of symmetry, or a lack of asymmetrical
features. Biologists believe the absence of asymmetry is an indicator of fitness (good genes),
since only a healthy organism can maintain a symmetrical plan throughout its development in
the face of environmental stresses, such as illness or lack of food. A symmetrical animal is
usually a healthy animal. The same goes for humans.
Symmetrical forms can be found in the inanimate world as well. The planets, with slight
variation due to chance, exhibit radial symmetry. Snowflakes also provide an example of
radial symmetry. All snowflakes show a hexagonal symmetry around an axis that runs
perpendicular to their face. Every one sixth of a revolution around this axis produces a design
identical to the original. The fact that all snowflakes have this sort of symmetry is due to the
way water molecules arrange themselves when ice forms. It's a reminder that symmetry is
part of the structure of the world around us.
Things to Think About...
1. What advantage does a bilaterally symmetrical structure have for humans? For horses?
2. Most animals are not symmetrical with respect to a cross-sectional plane (i.e., one that
parallels your waist). Why?
3. Why are planets spherical? What would happen if they were lopsided? Try spinning a top
with an object stuck to its side. What happens?
4. If a stone falls into a pond, in which direction do the resulting waves travel? Why?
:: PRINTER FRIENDLY
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Symmetry: A ‘Key to Nature’s Secrets’
OCTOBER 27, 2011
Steven Weinberg
E-MAIL PRINT SHARE
Mike King
The five regular polyhedra. Steven Weinberg writes that ‘they satisfy the symmetry requirement that every face, every
edge, and every corner should be precisely the same as every other face, edge, or corner.... Plato argued in Timaeus that
these were the shapes of the bodies making up the elements: earth consists of little cubes, while fire, air, and water are
made of polyhedra with four, eight, and twenty identical faces, respectively. The fifth regular polyhedron, with twelve
identical faces, was supposed by Plato to symbolize the cosmos.’
When I first started doing research in the late 1950s, physics seemed to me to be in a dismal state.
There had been a great success a decade earlier in quantum electrodynamics, the theory of
electrons and light and their interactions. Physicists then had learned how to calculate things like
the strength of the electron’s magnetic field with a precision unprecedented in all of science. But
now we were confronted with newly discovered esoteric particles—muons and dozens of types of
mesons and baryons—most existing nowhere in nature except in cosmic rays. And we had to deal
with mysterious forces: strong nuclear forces that hold partiicles together inside atomic nuclei, and
weak nuclear forces that can change the nature of these particles. We did not have a theory that
would describe these particles and forces, and when we took a stab at a possible theory, we found
that either we could not calculate its consequences, or when we could, we would come up with
nonsensical results, like infinite energies or infinite probabilities. Nature, like an enemy, seemed
intent on concealing from us its master plan.
At the same time, we did have a valuable key to nature’s secrets. The laws of nature evidently
obeyed certain principles of symmetry, whose consequences we could work out and compare with
observation, even without a detailed theory of particles and forces. There were symmetries that
dictated that certain distinct processes all go at the same rate, and that also dictated the existence
of families of distinct particles that all have the same mass. Once we observed such equalities of
rates or of masses, we could infer the existence of a symmetry, and this we thought would give us
a clearer idea of the further observations that should be made, and of the sort of underlying
theories that might or might not be possible. It was like having a spy in the enemy’s high
command.1
1.
I had better pause to say something about what physicists mean by principles of symmetry. In
conversations with friends who are not physicists or mathematicians, I find that they often take
symmetry to mean the identity of the two sides of something symmetrical, like the human face or
a butterfly. That is indeed a kind of symmetry, but it is only one simple example of a huge variety
of possible symmetries.
The Oxford English Dictionary tells us that symmetry is “the quality of being made up of exactly
similar parts.” A cube gives a good example. Every face, every edge, and every corner is just the
same as every other face, edge, or corner. This is why cubes make good dice: if a cubical die is
honestly made, when it is cast it has an equal chance of landing on any of its six faces.
The cube is one example of a small group of regular polyhedra—solid bodies with flat planes for
faces, which satisfy the symmetry requirement that every face, every edge, and every corner
should be precisely the same as every other face, edge, or corner. Thus the regular polyhedron
called a triangular pyramid has four faces, each an equilateral triangle of the same size; six edges,
at each of which two faces meet at the same angle; and four corners, at each of which three faces
come together at the same angles. (See illustration on this page.)
These regular polyhedra fascinated Plato. He learned (probably from the mathematician
Theaetetus) that regular polyhedra come in only five possible shapes, and he argued
in Timaeus that these were the shapes of the bodies making up the elements: earth consists of little
cubes, while fire, air, and water are made of polyhedra with four, eight, and twenty identical faces,
respectively. The fifth regular polyhedron, with twelve identical faces, was supposed by Plato to
symbolize the cosmos. Plato offered no evidence for all this—he wrote in Timaeusmore as a poet
than as a scientist, and the symmetries of these five bodies representing the elements evidently had
a powerful hold on his poetic imagination.
The regular polyhedra in fact have nothing to do with the atoms that make up the material world,
but they provide useful examples of a way of looking at symmetries, a way that is particularly
congenial to physicists. A symmetry is a principle of invariance. That is, it tells us that something
does not change its appearance when we make certain changes in our point of view—for instance,
by rotating it or moving it. In addition to describing a cube by saying that it has six identical
square faces, we can also say that its appearance does not change if we rotate it in certain ways—
for instance by 90° around any direction parallel to the cube’s edges.
The set of all such transformations of point of view that will leave a particular object looking the
same is called that object’s invariance group. This may seem like a fancy way of talking about
things like cubes, but often in physics we make guesses about invariance groups, and test them
experimentally, even when we know nothing else about the thing that is supposed to have the
conjectured symmetry. There is a large and elegant branch of mathematics known as group theory,
which catalogs and explores all possible invariance groups, and is described for general readers in
two recently published books: Symmetry: A Journey into the Patterns of Nature by Marcus du
Sautoy and Why Beauty Is Truth: A History of Symmetry by Ian Stewart.
2.
The symmetries that offered the way out of the problems of elementary particle physics in the
1950s were not the symmetries of objects, not even objects as important as atoms, but the
symmetries of laws. A law of nature can be said to respect a certain symmetry if that law remains
the same when we change the point of view from which we observe natural phenomena in certain
definite ways. The particular set of ways that we can change our point of view without changing
the law defines that symmetry.
Laws of nature, in the modern sense of mathematical equations that tell us precisely what will
happen in various circumstances, first appeared as the laws of motion and gravitation that Newton
developed as a basis for understanding Kepler’s description of the solar system. From the
beginning, Newton’s laws incorporated symmetry: the laws that we observe to govern motion and
gravitation do not change their form if we reset our clocks, or if we change the point from which
distances are measured, or if we rotate our entire laboratory so it faces in a different direction.2
There is another less obvious symmetry, known today as Galilean invariance, that had been
anticipated in the fourteenth century by Jean Buridan and Nicole Oresme: the laws of nature that
we discover do not change their form if we observe nature within a moving laboratory, traveling at
constant velocity. The fact that the earth is speeding around the sun, for instance, does not affect
the laws of motion of material objects that we observe on the earth’s surface.3
Newton and his successors took these principles of invariance pretty much for granted, as an
implicit basis for their theories, so it was quite a wrench when these principles themselves became
a subject of serious physical investigation. The crux of Einstein’s 1905 Special Theory of
Relativity was a modification of Galilean invariance. This was motivated in part by the persistent
failure of physicists to find any effect of the earth’s motion on the measured speed of light,
analogous to the effect of a boat’s motion on the observed speed of water waves.
It is still true in Special Relativity that making observations from a moving laboratory does not
change the form of the observed laws of nature, but the effect of this motion on measured
distances and times is different in Special Relativity from what Newton had thought. Motion
causes lengths to shrink and clocks to slow down in such a way that the speed of light remains a
constant, whatever the speed of the observer. This new symmetry, known as Lorentz
invariance,4required profound departures from Newtonian physics, including the convertibility of
energy and mass.
The advent and success of Special Relativity alerted physicists in the twentieth century to the
importance of symmetry principles. But by themselves, the symmetries of space and time that are
incorporated in the Special Theory of Relativity could not take us very far. One can imagine a
great variety of theories of particles and forces that would be consistent with these space-time
symmetries. Fortunately it was already clear in the 1950s that the laws of nature, whatever they
are, also respect symmetries of other kinds, having nothing directly to do with space and time.
There are four forces that allow particles to interact with one another: the familiar gravity and
electromagnetism, and the less well-known weak nuclear force (which is responsible for certain
types of radioactive decay) and strong nuclear force (which binds protons and neutrons in the
nucleus of an atom). (I am writing of a time, during the 1950s, before the formulation of the
modern Standard Model, in which the three known forces other than gravity are now united in a
single theory.) It had been known since the 1930s that the unknown laws that govern the strong
nuclear force respect a symmetry between protons and neutrons, the two particles that make up
atomic nuclei.
Even though the equations governing the strong forces were not known, the observations of
nuclear properties had revealed that whatever these equations are, they must not change if
everywhere in these equations we replace the symbol representing protons with that representing
neutrons, and vice versa. Not only that, but the equations are also unchanged if we replace the
symbols representing protons and neutrons with algebraic combinations of these symbols that
represent superpositions of protons and neutrons, superpositions that might for instance have a 40
percent chance of being a proton and a 60 percent chance of being a neutron. It is like replacing a
photo of Alice or of Bob with a picture in which photos of both Alice and Bob are superimposed.
One consequence of this symmetry is that the nuclear force between two protons is not only equal
to the force between two neutrons—it is also related to the force between a proton and a neutron.
Then as more and more types of particles were discovered, it was found in the 1960s that this
proton–neutron symmetry was part of a larger symmetry group: not only are the proton and
neutron related by this symmetry to each other, they are also related to six other subatomic
particles, known as hyperons. The symmetry among these eight particles came to be called “the
eightfold way.” All the particles that feel the strong nuclear force fall into similar symmetrical
families, with eight, ten, or more members.
Mike King
A spinning nucleus ejects an electron while decaying, as does its reflection in a mirror. The electron is ejected in the
direction of the nuclear spin (represented by the vertical arrow) in the real world, but opposite to the direction of spin in
the mirror, violating mirror symmetry. Steven Weinberg writes, ‘In 1957 experiments showed convincingly that, while the
electromagnetic and strong nuclear forces do obey mirror symmetry, the weak nuclear force does not. Experiments
showed, for example, that it was possible to distinguish a cobalt nucleus in the process of decaying—as a result of the
weak nuclear force—from its mirror image, spinning in the opposite direction.’ Adapted from an illustration in A.
Zee, Fearful Symmetry: The Search for Beauty in Modern Physics (Princeton University Press, 2007).
But there was something puzzling about these internal symmetries: unlike the symmetries of space
and time, these new symmetries were clearly neither universal nor exact. Electromagnetic
phenomena did not respect these symmetries: protons and some hyperons are electrically charged;
neutrons and other hyperons are not. Also, the masses of protons and neutrons differ by about 0.14
percent, and their masses differ from those of the lightest hyperon by 19 percent. If symmetry
principles are an expression of the simplicity of nature at the deepest level, what are we to make of
a symmetry that applies to only some forces, and even there is only approximate?
An even more puzzling discovery about symmetry was made in 1956–1957. The principle of
mirror symmetry states that physical laws do not change if we observe nature in a mirror, which
reverses distances perpendicular to the mirror (that is, something far behind your head looks in the
mirror as if it is far behind your image, and hence far in front of you). This is not a rotation—there
is no way of rotating your point of view that has the effect of reversing directions in and out of a
mirror, but not sideways or vertically. It had generally been taken for granted that mirror
symmetry, like the other symmetries of space and time, was exact and universal, but in 1957
experiments showed convincingly that, while the electromagnetic and strong nuclear forces do
obey mirror symmetry, the weak nuclear force does not. Experiments showed, for example, that it
was possible to distinguish a cobalt nucleus in the process of decaying—as a result of the weak
nuclear force—from its mirror image, spinning in the opposite direction. (See illustration on this
page.)
So we had a double mystery: What causes the observed violations of the eightfold way symmetry
and of mirror symmetry? Theorists offered several possible answers, but as we will see, this was
the wrong question.
The 1960s and 1970s witnessed a great expansion of our conception of the sort of symmetry that
might be possible in physics. The approximate proton–neutron symmetry was originally
understood to be rigid, in the sense that the equations governing the strong nuclear forces were
supposed to be unchanged only if we changed protons and neutrons into mixtures of each other in
the same way everywhere in space and time (physicists somewhat confusingly use the adjective
“global” for what I am here calling rigid symmetries).
But what if the equations obeyed a more demanding symmetry, one that was local, in the sense
that the equations would also be unchanged if we changed neutrons and protons into different
mixtures of each other at different times and locations? In order to allow the different local
mixtures to interact with one another without changing the equations, such a local symmetry
would require some way for protons and neutrons to exert force on each other. Much as photons
(the massless particles of light) are required to carry the electromagnetic force, a new massless
particle, the gluon, would be needed to carry the force between protons and neutrons. It was hoped
that this sort of theory of symmetrical forces might somehow explain the strong nuclear force that
holds neutrons and protons together in atomic nuclei.
Conceptions of symmetry also expanded in a different direction. Theorists began in the 1960s to
consider the possibility of symmetries that are “broken.” That is, the underlying equations of
physics might respect symmetries that are nevertheless not apparent in the actual physical states
observed. The physical states that are possible in nature are represented by solutions of the
equations of physics. When we have a broken symmetry, the solutions of the equations do not
respect the symmetries of the equations themselves.5
The elliptical orbits of planets in the solar system provide a good example. The equations
governing the gravitational field of the sun, and the motions of bodies in that field, respect
rotational symmetry—there is nothing in these equations that distinguishes one direction in space
from another. A circular planetary orbit of the sort imagined by Plato would also respect this
symmetry, but the elliptical orbits actually encountered in the solar system do not: the long axis of
an ellipse points in a definite direction in space.
At first it was widely thought that broken symmetry might have something to do with the small
known violations of symmetries like mirror symmetry or the eightfold way. This was a false lead.
A broken symmetry is nothing like an approximate symmetry, and is useless for putting particles
into families like those of the eightfold way.
But broken symmetries do have consequences that can be checked empirically. Because of the
spherical symmetry of the equations governing the sun’s gravitational field, the long axis of an
elliptical planetary orbit can point in any direction in space. This makes these orbits acutely
sensitive to any small perturbation that violates the symmetry, like the gravitational field of other
planets. For instance, these perturbations cause the long axis of Mercury’s orbit to swing around
360° every 2,254 centuries.
In the 1960s theorists realized that the strong nuclear forces have a broken symmetry, known as
chiral symmetry. Chiral symmetry is like the proton–neutron symmetry mentioned above, except
that the symmetry transformations can be different for particles spinning clockwise or
counterclockwise. The breaking of this symmetry requires the existence of the subatomic particles
called pi mesons. The pi meson is in a sense the analog of the slow change in orientation of an
elliptical planetary orbit; just as small perturbations can make large changes in an orbit’s
orientation, pi mesons can be created in collisions of neutrons and protons with relatively low
energy.
The path out of the dismal state of particle physics in the 1950s turned out to lead through local
and broken symmetries. First, electromagnetic and weak nuclear forces were found to be governed
by a broken local symmetry. (The experiments now underway at Fermilab in Illinois and the new
accelerator at CERN in Switzerland have as their first aim to pin down just what it is that breaks
this symmetry.) Then the strong nuclear forces were found to be described by a different local
symmetry. The resulting theory of strong, weak, and electromagnetic forces is what is now known
as the Standard Model, and does a good job of accounting for virtually all phenomena observed in
our laboratories.
3.
It would take far more space than I have here to go into details about these symmetries and the
Standard Model, or about other proposed symmetries that go beyond those of the Standard Model.
Instead I want to take up one aspect of symmetry that as far as I know has not yet been described
for general readers. When the Standard Model was put in its present form in the early 1970s,
theorists to their delight encountered something quite unexpected. It turned out that the Standard
Model obeys certain symmetries that are accidental, in the sense that, though they are not the exact
local symmetries on which the Standard Model is based, they are automatic consequences of the
Standard Model. These accidental symmetries accounted for a good deal of what had seemed so
mysterious in earlier years, and raised interesting new possibilities.
The origin of accidental symmetries lies in the fact that acceptable theories of elementary particles
tend to be of a particularly simple type. The reason has to do with avoidance of the nonsensical
infinities I mentioned at the outset. In theories that are sufficiently simple these infinities can be
canceled by a mathematical process called “renormalization.” In this process, certain physical
constants, like masses and charges, are carefully redefined so that the infinite terms are canceled
out, without affecting the results of the theory. In these simple theories, known as
“renormalizable” theories, only a small number of particles can interact at any given location and
time, and then the energy of interaction can depend in only a simple way on how the particles are
moving and spinning.
For a long time many of us thought that to avoid intractable infinities, these renormalizable
theories were the only ones physically possible. This posed a serious problem, because Einstein’s
successful theory of gravitation, the General Theory of Relativity, is not a renormalizable theory;
the fundamental symmetry of the theory, known as general covariance (which says that the
equations have the same form whatever coordinates we use to describe events in space and time),
does not allow any sufficiently simple interactions. In the 1970s it became clear that there are
circumstances in which nonrenormalizable theories are allowed without incurring nonsensical
infinities, but that the relatively complicated interactions that make these theories
nonrenormalizable are expected, under normal circumstances, to be so weak that physicists can
usually ignore them and still get reliable approximate results.
This is a good thing. It means that to a good approximation there are only a few kinds of
renormalizable theories that we need to consider as possible descriptions of nature.
Now, it just so happens that under the constraints imposed by Lorentz invariance and the exact
local symmetries of the Standard Model, the most general renormalizable theory of strong and
electromagnetic forces simply can’t be complicated enough to violate mirror symmetry.6 Thus, the
mirror symmetry of the electromagnetic and strong nuclear forces is an accident, having nothing
to do with any symmetry built into nature at a fundamental level. The weak nuclear forces do not
respect mirror symmetry because there was never any reason why they should. Instead of asking
what breaks mirror symmetry, we should have been asking why there should be any mirror
symmetry at all. And now we know. It is accidental.
The proton–neutron symmetry is explained in a similar way. The Standard Model does not
actually refer to protons and neutrons, but to the particles of which they are composed, known as
quarks and gluons.7 The proton consists of two quarks of a type called “up” and one of a type
called “down”; the neutron consists of two down quarks and an up quark. It just so happens that in
the most general renormalizable theory of quarks and gluons satisfying the symmetries of the
Standard Model, the only things that can violate the proton–neutron symmetry are the masses of
the quarks. The up and down quark masses are not at all equal—the down quark is nearly twice as
heavy as the up quark—because there is no reason why they should be equal. But these masses are
both very small—most of the masses of the protons and neutrons come from the strong nuclear
force, not from the quark masses. To the extent that quark masses can be neglected, then, we have
an accidental approximate symmetry between protons and neutrons. Chiral symmetry and the
eightfold way arise in the same accidental way.
So mirror symmetry and the proton–neutron symmetry and its generalizations are not fundamental
at all, but just accidents, approximate consequences of deeper principles. To the extent that these
symmetries were our spies in the high command of nature, we were exaggerating their importance,
as also often happens with real spies.
The recognition of accidental symmetry not only resolved the old puzzle about approximate
symmetries; it also opened up exciting new possibilities. It turned out that there are certain
symmetries that could not be violated in any theory that has the same particles and the same exact
local symmetries as the Standard Model and that is simple enough to be renormalizable.8 If really
valid, these symmetries, known as lepton and baryon conservation,9 would dictate that neutrinos
(particles that feel only the weak and gravitational forces) have no mass, and that protons and
many atomic nuclei are absolutely stable. Now, on experimental grounds these symmetries had
been known long before the advent of the Standard Model, and had generally been thought to be
exactly valid. But if they are actually accidental symmetries of the Standard Model, like the
accidental proton–neutron symmetry of the strong forces, then they too might be only
approximate. As I mentioned earlier, we now understand that interactions that make the theory
nonrenormalizable are not impossible, though they are likely to be extremely weak. Once one
admits such more complicated nonrenormalizable interactions, the neutrino no longer has to be
strictly massless, and the proton no longer has to be absolutely stable.
There are in fact possible nonrenormalizable interactions that would give the neutrino a tiny mass,
of the order of one hundred millionth of the electron mass, and give protons a finite average
lifetime, though one so long that typical protons in matter today will last much longer than the
universe already has. Experiments in recent years have revealed that neutrinos do indeed have
such masses. Experiments are under way to detect the tiny fraction of protons that decay in a year
or so, and I would bet that these decays will eventually be observed. If protons do decay, the
universe will eventually contain only lighter particles like neutrinos and photons. Matter as we
know it will be gone.
I said that I would be concerned here with the symmetries of laws, not of objects, but there is one
thing that is so important that I need to say a bit about it. It is the universe. As far as we can see,
when averaged over sufficiently large scales containing many galaxies, the universe seems to have
no preferred position, and no preferred directions—it is symmetrical. But this too may be an
accident.
There is an attractive theory, called chaotic inflation, according to which the universe began
without any special spatial symmetries, in a completely chaotic state. Here and there by accident
the fields pervading the universe were more or less uniform, and according to the gravitational
field equations it is these patches of space that then underwent an exponentially rapid expansion,
known as inflation, leading to something like our present universe, with all nonuniformities in
these patches smoothed out by the expansion. In different patches of space the symmetries of the
laws of nature would be broken in different ways. Much of the universe is still chaotic, and it is
only in the patches that inflated sufficiently (and in which symmetries were broken in the right
ways) that life could arise, so any beings who study the universe will find themselves in such
patches.
This is all quite speculative. There is observational evidence for an exponential early expansion,
which has left its traces in the microwave radiation filling the universe, but as yet no evidence for
an earlier period of chaos. If it turns out that chaotic inflation is correct, then much of what we
observe in nature will be due to the accident of our particular location, an accident that can never
be explained, except by the fact that it is only in such locations that anyone could live.
1.
1
This article is based in part on a talk given at a conference devoted to symmetry at the Technical
University of Budapest in August 2009. ↩
2.
2
For reasons that are difficult to explain without mathematics, these symmetries imply important
conservation laws: the conservation of energy, momentum, and angular momentum (or spin). Some
other symmetries imply the conservation of other quantities, such as electric charge. ↩
3.
3
Strictly speaking, Galilean invariance applies only approximately to the motion of the earth, since
the earth is not moving in a straight line at constant speed. It is true that the earth's motion in its
orbit does not affect the laws we observe, but this is because gravity balances the effects of the
centrifugal force caused by the earth's curved motion. This too is dictated by a symmetry, but the
symmetry here is Einstein's principle of general covariance, the basis of the general theory of
relativity. ↩
4.
4
Lorentz had tried to explain the constancy of the observed speed of light by studying the effect of
motion on particles of matter. Einstein was instead explaining the same observation by a change in
one of nature's fundamental symmetries. ↩
5.
5
Consider the equation x 3 equals x. This equation has a symmetry under the transformation that
replaces x with– x; if we replace x with– x, we get the same equation. The equation has a
solution x = 0, which respects the symmetry;–0 = 0. But it also has a solution in which x = 1. This
does not respect the symmetry;–1 is not equal to 1. This is a broken symmetry. Of course, this
equation is not much like the equations of physics. ↩
6.
6
Honesty compels me to admit that here I am gliding over some technical complications. ↩
7.
7
These particles are not observed experimentally, not because they are too heavy to be produced
(gluons are massless, and some quarks are quite light), but because the strong nuclear forces bind
them together in composite states like protons and neutrons. ↩
8.
8
Again, I admit to passing over some technical complications. ↩
9.
9
Lepton number is defined as the number of electrons and similar heavier charged particles plus the
number of neutrinos, minus the number of their antiparticles. (This conservation law requires the
neutrino to be massless because neutrinos and antineutrinos, respectively, spin only
counterclockwise and clockwise around their directions of motion. If neutrinos have any mass then
they travel at less than the speed of light, so it is possible to reverse their apparent direction of
motion by travelling faster past them, hence converting the spin from counterclockwise to
clockwise, and neutrinos to antineutrinos, which changes the lepton number.) Baryon number is
proportional to the number of quarks minus the number of antiquarks. ↩
http://www.nybooks.com/articles/archives/2011/oct/27/symmetry-key-naturessecrets/?pagination=false
Symmetry
From Wikipedia, the free encyclopedia
For other uses, see Symmetry (disambiguation).
Sphere symmetrical group o.
Leonardo da Vinci's Vitruvian Man (ca. 1487) is often used as a representation of symmetry
in the human body and, by extension, the natural universe.
Symmetric arcades of a portico in the Great Mosque of Kairouanalso called the Mosque of
Uqba, inTunisia.
Symmetry (from Greek συμμετρεῖν symmetría "measure together") generally conveys two
primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing
proportionality and balance;[1][2] such that it reflects beauty or perfection. The second
meaning is a precise and well-defined concept of balance or "patterned self-similarity" that
can be demonstrated or proved according to the rules of a formal system: by geometry,
through physics or otherwise.
Although the meanings are distinguishable in some contexts, both meanings of "symmetry"
are related and discussed in parallel.[2][3]
The precise notions of symmetry have various measures and operational definitions. For
example, symmetry may be observed

with respect to the passage of time;

as a spatial relationship;

through geometric transformations such as scaling, reflection, and rotation;

through other kinds of functional transformations;[4] and

as an aspect of abstract objects, theoretic models, language, music and
even knowledge itself.[5][6]
This article describes these notions of symmetry from four perspectives. The first is that of
symmetry in geometry, which is the most familiar type of symmetry for many people. The
second perspective is the more general meaning of symmetry in mathematics as a whole.
The third perspective describes symmetry as it relates to science and technology. In this
context, symmetries underlie some of the most profound results found in modern physics,
including aspects ofspace and time. Finally, a fourth perspective discusses symmetry in
the humanities, covering its rich and varied use in history, architecture, art, and religion.
The opposite of symmetry is asymmetry.
Contents
[hide]


1 In geometry
o
1.1 Reflection symmetry
o
1.2 Point reflection and other involutive isometries
o
1.3 Rotational symmetry
o
1.4 Translational symmetry
o
1.5 Glide reflection symmetry
o
1.6 Rotoreflection symmetry
o
1.7 Helical symmetry
o
1.8 Non-isometric symmetries
o
1.9 Scale symmetry and fractals
2 In mathematics
o
2.1 Mathematical model for symmetry


o
2.2 Symmetric functions
o
2.3 Symmetry in logic
3 In science
o
3.1 Symmetry in physics
o
3.2 Symmetry in physical objects

3.2.1 Classical objects

3.2.2 Quantum objects

3.2.3 Consequences of quantum symmetry
o
3.3 Generalizations of symmetry
o
3.4 Symmetry in biology
o
3.5 Symmetry in chemistry
4 In history, religion, and culture
o
4.1 Symmetry in social interactions
o
4.2 Symmetry in architecture
o
4.3 Symmetry in pottery and metal vessels
o
4.4 Symmetry in quilts
o
4.5 Symmetry in carpets and rugs
o
4.6 Symmetry in music

4.6.1 Musical form

4.6.2 Pitch structures

4.6.3 Equivalency
o
4.7 Symmetry in other arts and crafts
o
4.8 Symmetry in aesthetics

5 See also

6 References

7 External links
[edit]In geometry
The most familiar type of symmetry for many people is geometrical symmetry. Formally, this
means symmetry under a sub-group of the Euclidean group ofisometries in two or three
dimensional Euclidean space. These isometries consist of reflections, rotations, translations
and combinations of these basic operations.[7]
[edit]Reflection symmetry
Main article: reflection symmetry
A butterfly with bilateral symmetry
Reflection symmetry, mirror symmetry, mirror-image symmetry, or bilateral symmetry is
symmetry with respect to reflection.
In 1D, there is a point of symmetry. In 2D there is an axis of symmetry, in 3D a plane of
symmetry. An object or figure which is indistinguishable from its transformed image is called
mirror symmetric (see mirror image).
The axis of symmetry of a two-dimensional figure is a line such that, if a perpendicular is
constructed, any two points lying on the perpendicular at equal distances from the axis of
symmetry are identical. Another way to think about it is that if the shape were to be folded
in half over the axis, the two halves would be identical: the two halves are each other's
mirror image. Thus a square has four axes of symmetry, because there are four different
ways to fold it and have the edges all match. A circle has infinitely many axes of symmetry,
for the same reason.
If the letter T is reflected along a vertical axis, it appears the same. This is sometimes called
vertical symmetry. One can better use an unambiguous formulation; e.g., "T has a vertical
symmetry axis" or "T has left-right symmetry".
The triangles with this symmetry are isosceles, the quadrilaterals with this symmetry are
the kites and the isosceles trapezoids.
For each line or plane of reflection, the symmetry group is isomorphic with Cs (see point
groups in three dimensions), one of the three types of order two (involutions), hence
algebraically C2. The fundamental domain is a half-plane or half-space.
Bilateria (bilateral animals, including humans) are more or less symmetric with respect to
the sagittal plane.
[edit]Point reflection and other involutive isometries
Reflection symmetry can be generalized to other isometries of m-dimensional space which
are involutions, such as
(x1, … xm) ↦ (−x1, … −xk, xk+1, … xm)
in certain system of Cartesian coordinates. This reflects the space along a m−kdimensional affine subspace. If k = m, then such transformation is known as point reflection,
which on the plane(m = 2) is the same as the half-turn (180°) rotation; see below.
Such "reflection" keeps orientation if and only if k is even. This implies that for m = 3 (as well
for other odd m) a point reflection changes orientation of the space, like mirror-image
symmetry. That's why in physics the term P-symmetry is used for both point reflection and
mirror symmetry (P stands for parity).
[edit]Rotational symmetry
Main article: rotational symmetry
Rotational symmetry is symmetry with respect to some or all rotations in m-dimensional
Euclidean space. Rotations are direct isometries; i.e., isometries preserving orientation.
Therefore a symmetry group of rotational symmetry is a subgroup of E+(m).
Symmetry with respect to all rotations about all points implies translational symmetry with
respect to all translations, and the symmetry group is the whole E+(m). This does not apply
for objects because it makes space homogeneous, but it may apply for physical laws.
For symmetry with respect to rotations about a point we can take that point as origin. These
rotations form the special orthogonal group SO(m), the group ofm × m orthogonal
matrices with determinant 1. For m = 3 this is the rotation group SO(3).
In another meaning of the word, the rotation group of an object is the symmetry group
within E+(m), the group of direct isometries; in other words, the intersection of the full
symmetry group and the group of direct isometries. For chiral objects it is the same as the
full symmetry group.
Laws of physics are SO(3)-invariant if they do not distinguish different directions in space.
Because of Noether's theorem, rotational symmetry of a physical system is equivalent to the
angular momentum conservation law. See also rotational invariance.
[edit]Translational symmetry
Main article: Translational symmetry
Translational symmetry leaves an object invariant under a discrete or continuous group
of translations
.
[edit]Glide reflection symmetry
A glide reflection symmetry (in 3D: a glide plane symmetry) means that a reflection in a line
or plane combined with a translation along the line / in the plane, results in the same object.
It implies translational symmetry with twice the translation vector. The symmetry group is
isomorphic with Z.
[edit]Rotoreflection symmetry
In 3D, rotoreflection or improper rotation in the strict sense is rotation about an axis,
combined with reflection in a plane perpendicular to that axis. As symmetry groups with
regard to a roto-reflection we can distinguish:

the angle has no common divisor with 360°, the symmetry group is not discrete

2n-fold rotoreflection (angle of 180°/n) with symmetry group S2n of order 2n (not to
be confused with symmetric groups, for which the same notation is used; abstract
group C2n); a special case is n = 1, inversion, because it does not depend on the axis
and the plane, it is characterized by just the point of inversion.

Cnh (angle of 360°/n); for odd n this is generated by a single symmetry, and the
abstract group is C2n, for even n this is not a basic symmetry but a combination. See
also point groups in three dimensions.
[edit]Helical symmetry
A drill bit with helical symmetry.
See also: Screw axis
Helical symmetry is the kind of symmetry seen in such everyday objects
as springs, Slinky toys, drill bits, and augers. It can be thought of as rotational symmetry
along with translation along the axis of rotation, the screw axis. The concept of helical
symmetry can be visualized as the tracing in three-dimensional space that results from
rotating an object at an even angular speed while simultaneously moving at another even
speed along its axis of rotation (translation). At any one point in time, these two motions
combine to give a coiling angle that helps define the properties of the tracing. When the
tracing object rotates quickly and translates slowly, the coiling angle will be close to 0°.
Conversely, if the rotation is slow and the translation is speedy, the coiling angle will
approach 90°.
Three main classes of helical symmetry can be distinguished based on the interplay of the
angle of coiling and translation symmetries along the axis:
Infinite helical symmetry
If there are no distinguishing features along the length of a helix or helix-like object, the
object will have infinite symmetry much like that of a circle, but with the additional
requirement of translation along the long axis of the object to return it to its original
appearance. A helix-like object is one that has at every point the regular angle of coiling of a
helix, but which can also have a cross section of indefinitely high complexity, provided only
that precisely the same cross section exists (usually after a rotation) at every point along the
length of the object. Simple examples include evenly coiled springs, slinkies, drill bits,
and augers. Stated more precisely, an object has infinite helical symmetries if for any small
rotation of the object around its central axis there exists a point nearby (the translation
distance) on that axis at which the object will appear exactly as it did before. It is this infinite
helical symmetry that gives rise to the curious illusion of movement along the length of an
auger or screw bit that is being rotated. It also provides the mechanically useful ability of
such devices to move materials along their length, provided that they are combined with a
force such as gravity or friction that allows the materials to resist simply rotating along with
the drill or auger.
n-fold helical symmetry
If the requirement that every cross section of the helical object be identical is relaxed,
additional lesser helical symmetries become possible. For example, the cross section of the
helical object may change, but still repeats itself in a regular fashion along the axis of the
helical object. Consequently, objects of this type will exhibit a symmetry after a rotation by
some fixed angle θ and a translation by some fixed distance, but will not in general be
invariant for any rotation angle. If the angle (rotation) at which the symmetry occurs divides
evenly into a full circle (360°), the result is the helical equivalent of a regular polygon. This
case is called n-fold helical symmetry, where n = 360°; e.g., double helix. This concept can be
further generalized to include cases where
is a multiple of 360° – that is, the cycle does
eventually repeat, but only after more than one full rotation of the helical object.
Non-repeating helical symmetry
This is the case in which the angle of rotation θ required to observe the symmetry
is irrational. The angle of rotation never repeats exactly no matter how many times the helix
is rotated. Such symmetries are created by using a non-repeating point group in two
dimensions. DNA is an example of this type of non-repeating helical symmetry.[citation needed]
[edit]Non-isometric symmetries
A wider definition of geometric symmetry allows operations from a larger group than the
Euclidean group of isometries. Examples of larger geometric symmetry groups are:

The group of similarity transformations; i.e., affine transformations represented by
a matrix A that is a scalar times an orthogonal matrix. Thus homothety is added, selfsimilarity is considered a symmetry.

The group of affine transformations represented by a matrix A with determinant 1
or −1; i.e., the transformations which preserve area.
This adds, e.g., oblique reflection symmetry.

The group of all bijective affine transformations.

The group of Möbius transformations which preserve cross-ratios.
This adds, e.g., inversive reflections such as circle reflection on the plane.
In Felix Klein's Erlangen program, each possible group of symmetries defines a geometry in
which objects that are related by a member of the symmetry group are considered to be
equivalent. For example, the Euclidean group defines Euclidean geometry, whereas the
group of Möbius transformations defines projective geometry.
[edit]Scale symmetry and fractals
Scale symmetry refers to the idea that if an object is expanded or reduced in size, the new
object has the same properties as the original. Scale symmetry is notable for the fact that it
does notexist for most physical systems, a point that was first discerned by Galileo. Simple
examples of the lack of scale symmetry in the physical world include the difference in the
strength and size of the legs of elephants versus mice, and the observation that if a candle
made of soft wax was enlarged to the size of a tall tree, it would immediately collapse under
its own weight.
A more subtle form of scale symmetry is demonstrated by fractals. As conceived by Benoît
Mandelbrot, fractals are a mathematical concept in which the structure of a complex form
looks similar or even exactly the same no matter what degree of magnification is used to
examine it. A coast is an example of a naturally occurring fractal, since it retains roughly
comparable and similar-appearing complexity at every level from the view of a satellite to a
microscopic examination of how the water laps up against individual grains of sand. The
branching of trees, which enables children to use small twigs as stand-ins for full trees
in dioramas, is another example.
This similarity to naturally occurring phenomena provides fractals with an everyday
familiarity not typically seen with mathematically generated functions. As a consequence,
they can produce strikingly beautiful results such as the Mandelbrot set. Intriguingly, fractals
have also found a place in CG, or computer-generated movie effects, where their ability to
create very complex curves with fractal symmetries results in more realistic virtual worlds.
[edit]In mathematics
Main article: Symmetry in mathematics
In formal terms, we say that a mathematical object is symmetric with respect to a
given mathematical operation, if, when applied to the object, this operation preserves some
property of the object. The set of operations that preserve a given property of the object
form a group. Two objects are symmetric to each other with respect to a given group of
operations if one is obtained from the other by some of the operations (and vice versa).
[edit]Mathematical model for symmetry
The set of all symmetry operations considered, on all objects in a set X, can be modeled as
a group action g : G × X → X, where the image of g in G and x in X is written as g·x. If, for
some g, g·x= y then x and y are said to be symmetrical to each other. For each object x,
operations g for which g·x = x form a group, the symmetry group of the object, a subgroup
of G. If the symmetry group of x is the trivial group then x is said to be asymmetric,
otherwise symmetric.
A general example is that G is a group of bijections g: V → V acting on the set of
functions x: V → W by (gx)(v) = x[g−1(v)] (or a restricted set of such functions that is closed
under the group action). Thus a group of bijections of space induces a group action on
"objects" in it. The symmetry group of x consists of all g for which x(v) = x[g(v)] for all v. G is
the symmetry group of the space itself, and of any object that is uniform throughout space.
Some subgroups of G may not be the symmetry group of any object. For example, if the
group contains for every v and w in V a gsuch that g(v) = w, then only the symmetry groups
of constant functions x contain that group. However, the symmetry group of constant
functions is G itself.
In a modified version for vector fields, we have (gx)(v) = h(g, x[g−1(v)]) where h rotates any
vectors and pseudovectors in x, and inverts any vectors (but not pseudovectors) according to
rotation and inversion in g, see symmetry in physics. The symmetry group of x consists of
all g for which x(v) = h(g, x[g(v)]) for all v. In this case the symmetry group of a constant
function may be a proper subgroup of G: a constant vector has only rotational symmetry
with respect to rotation about an axis if that axis is in the direction of the vector, and only
inversion symmetry if it is zero.
For a common notion of symmetry in Euclidean space, G is the Euclidean group E(n), the
group of isometries, and V is the Euclidean space. The rotation group of an object is the
symmetry group if G is restricted to E+(n), the group of direct isometries. (For
generalizations, see the next subsection.) Objects can be modeled as functions x, of which a
value may represent a selection of properties such as color, density, chemical composition,
etc. Depending on the selection we consider just symmetries of sets of points (x is just
a Boolean function of position v), or, at the other extreme; e.g., symmetry of right and left
hand with all their structure.
For a given symmetry group, the properties of part of the object, fully define the whole
object. Considering points equivalent which, due to the symmetry, have the same
properties, the equivalence classes are the orbits of the group action on the space itself. We
need the value of x at one point in every orbit to define the full object. A set of such
representatives forms a fundamental domain. The smallest fundamental domain does not
have a symmetry; in this sense, one can say that symmetry relies upon asymmetry.
An object with a desired symmetry can be produced by choosing for every orbit a single
function value. Starting from a given object x we can, e.g.:

Take the values in a fundamental domain (i.e., add copies of the object).

Take for each orbit some kind of average or sum of the values of x at the points of
the orbit (ditto, where the copies may overlap).
If it is desired to have no more symmetry than that in the symmetry group, then the object
to be copied should be asymmetric.
As pointed out above, some groups of isometries are not the symmetry group of any object,
except in the modified model for vector fields. For example, this applies in 1D for the group
of all translations. The fundamental domain is only one point, so we can not make it
asymmetric, so any "pattern" invariant under translation is also invariant under reflection
(these are the uniform "patterns").
In the vector field version continuous translational symmetry does not imply reflectional
symmetry: the function value is constant, but if it contains nonzero vectors, there is no
reflectional symmetry. If there is also reflectional symmetry, the constant function value
contains no nonzero vectors, but it may contain nonzero pseudovectors. A corresponding 3D
example is an infinitecylinder with a current perpendicular to the axis; the magnetic
field (a pseudovector) is, in the direction of the cylinder, constant, but nonzero. For vectors
(in particular the current density) we have symmetry in every plane perpendicular to the
cylinder, as well as cylindrical symmetry. This cylindrical symmetry without mirror planes
through the axis is also only possible in the vector field version of the symmetry concept. A
similar example is a cylinder rotating about its axis, where magnetic field and current density
are replaced by angular momentum and velocity, respectively.
A symmetry group is said to act transitively on a repeated feature of an object if, for every
pair of occurrences of the feature there is a symmetry operation mapping the first to the
second. For example, in 1D, the symmetry group of {…, 1, 2, 5, 6, 9, 10, 13, 14, …} acts
transitively on all these points, while {…, 1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, …} does not act
transitively on all points. Equivalently, the first set is only one conjugacy class with respect to
isometries, while the second has two classes.
[edit]Symmetric functions
Main article: symmetric function
A symmetric function is a function which is unchanged by any permutation of its variables.
For example, x + y + z and xy + yz + xz are symmetric functions, whereas x2 – yz is not.
A function may be unchanged by a sub-group of all the permutations of its variables. For
example, ac + 3ab + bc is unchanged if a and b are exchanged; its symmetry group is
isomorphic to C2.
[edit]Symmetry in logic
A dyadic relation R is symmetric if and only if, whenever it's true that Rab, it's true that Rba.
Thus, "is the same age as" is symmetrical, for if Paul is the same age as Mary, then Mary is
the same age as Paul.
Symmetric binary logical connectives are and (∧, or &), or (∨, or |), biconditional (if and only
if) (↔), nand (not-and, or ⊼), xor (not-biconditional, or ⊻), and nor (not-or, or ⊽).
[edit]In science
[edit]Symmetry in physics
Main article: Symmetry in physics
Symmetry in physics has been generalized to mean invariance—that is, lack of change—
under any kind of transformation, for example arbitrary coordinate transformations. This
concept has become one of the most powerful tools of theoretical physics, as it has become
evident that practically all laws of nature originate in symmetries. In fact, this role inspired
the Nobel laureate PW Anderson to write in his widely read 1972 article More is
Different that "it is only slightly overstating the case to say that physics is the study of
symmetry." See Noether's theorem (which, in greatly simplified form, states that for every
continuous mathematical symmetry, there is a corresponding conserved quantity; a
conserved current, in Noether's original language); and also,Wigner's classification, which
says that the symmetries of the laws of physics determine the properties of the particles
found in nature.
[edit]Symmetry in physical objects
[edit]Classical objects
Although an everyday object may appear exactly the same after a symmetry operation such
as a rotation or an exchange of two identical parts has been performed on it, it is readily
apparent that such a symmetry is true only as an approximation for any ordinary physical
object.
For example, if one rotates a precisely machined aluminum equilateral triangle 120 degrees
around its center, a casual observer brought in before and after the rotation will be unable
to decide whether or not such a rotation took place. However, the reality is that each corner
of a triangle will always appear unique when examined with sufficient precision. An observer
armed with sufficiently detailed measuring equipment such as optical or electron
microscopes will not be fooled; he will immediately recognize that the object has been
rotated by looking for details such ascrystals or minor deformities.
Such simple thought experiments show that assertions of symmetry in everyday physical
objects are always a matter of approximate similarity rather than of precise mathematical
sameness. The most important consequence of this approximate nature of symmetries in
everyday physical objects is that such symmetries have minimal or no impacts on the physics
of such objects. Consequently, only the deeper symmetries of space and time play a major
role in classical physics; that is, the physics of large, everyday objects.
[edit]Quantum objects
Remarkably, there exists a realm of physics for which mathematical assertions of simple
symmetries in real objects cease to be approximations. That is the domain of quantum
physics, which for the most part is the physics of very small, very simple objects such
as electrons, protons, light, and atoms.
Unlike everyday objects, objects such as electrons have very limited numbers of
configurations, called states, in which they can exist. This means that when symmetry
operations such as exchanging the positions of components are applied to them, the
resulting new configurations often cannot be distinguished from the originals no matter how
diligent an observer is. Consequently, for sufficiently small and simple objects the generic
mathematical symmetry assertion F(x) = x ceases to be approximate, and instead becomes
an experimentally precise and accurate description of the situation in the real world.
[edit]Consequences of quantum symmetry
While it makes sense that symmetries could become exact when applied to very simple
objects, the immediate intuition is that such a detail should not affect the physics of such
objects in any significant way. This is in part because it is very difficult to view the concept of
exact similarity as physically meaningful. Our mental picture of such situations is invariably
the same one we use for large objects: We picture objects or configurations that are very,
very similar, but for which if we could "look closer" we would still be able to tell the
difference.
However, the assumption that exact symmetries in very small objects should not make any
difference in their physics was discovered in the early 1900s to be spectacularly incorrect.
The situation was succinctly summarized by Richard Feynman in the direct transcripts of
his Feynman Lectures on Physics, Volume III, Section 3.4, Identical particles. (Unfortunately,
the quote was edited out of the printed version of the same lecture.)
… if there is a physical situation in which it is impossible to tell which way it happened,
it always interferes; it never fails.
The word "interferes" in this context is a quick way of saying that such objects fall under the
rules of quantum mechanics, in which they behave more like waves that interfere than like
everyday large objects.
In short, when an object becomes so simple that a symmetry assertion of the form F(x) =
x becomes an exact statement of experimentally verifiable sameness, x ceases to follow the
rules ofclassical physics and must instead be modeled using the more complex, and often far
less intuitive, rules of quantum physics.
This transition also provides an important insight into why the mathematics of symmetry are
so deeply intertwined with those of quantum mechanics. When physical systems make the
transition from symmetries that are approximate to ones that are exact, the mathematical
expressions of those symmetries cease to be approximations and are transformed into
precise definitions of the underlying nature of the objects. From that point on, the
correlation of such objects to their mathematical descriptions becomes so close that it is
difficult to separate the two.
[edit]Generalizations of symmetry
If we have a given set of objects with some structure, then it is possible for a symmetry to
merely convert only one object into another, instead of acting upon all possible objects
simultaneously. This requires a generalization from the concept of symmetry group to that
of a groupoid. Indeed, A. Connes in his book "Non-commutative geometry" writes that
Heisenberg discovered quantum mechanics by considering the groupoid of transitions of the
hydrogen spectrum.
The notion of groupoid also leads to notions of multiple groupoids, namely sets with many
compatible groupoid structures, a structure which trivialises to abelian groups if one
restricts to groups. This leads to prospects of higher order symmetry which have been a little
explored, as follows.
The automorphisms of a set, or a set with some structure, form a group, which models a
homotopy 1-type. The automorphisms of a group G naturally form a crossed
module
, and crossed modules give an algebraic model of homotopy 2-types. At
the next stage, automorphisms of a crossed module fit into a structure known as a crossed
square, and this structure is known to give an algebraic model of homotopy 3-types. It is not
known how this procedure of generalising symmetry may be continued, although crossed ncubes have been defined and used in algebraic topology, and these structures are only
slowly being brought into theoretical physics.[8][9]
Physicists have come up with other directions of generalization, such
as supersymmetry and quantum groups, yet the different options are indistinguishable
during various circumstances.
[edit]Symmetry in biology
Further information: symmetry (biology) and facial symmetry
[edit]Symmetry in chemistry
Main article: molecular symmetry
Symmetry is important to chemistry because it explains observations
in spectroscopy, quantum chemistry and crystallography. It draws heavily on group theory.
[edit]In history, religion, and culture
In any human endeavor for which an impressive visual result is part of the desired objective,
symmetries play a profound role. The innate appeal of symmetry can be found in our
reactions to happening across highly symmetrical natural objects, such as precisely formed
crystals or beautifully spiraled seashells. Our first reaction in finding such an object often is
to wonder whether we have found an object created by a fellow human, followed quickly by
surprise that the symmetries that caught our attention are derived from nature itself. In
both reactions we give away our inclination to view symmetries both as beautiful and, in
some fashion, informative of the world around us.[citation needed]
[edit]Symmetry in social interactions
People observe the symmetrical nature, often including asymmetrical balance, of social
interactions in a variety of contexts. These include assessments of reciprocity, empathy,
apology, dialog, respect, justice, and revenge. Symmetrical interactions send the message
"we are all the same" while asymmetrical interactions send the message "I am special;
better than you." Peer relationships are based on symmetry, power relationships are based
on asymmetry.[10]
[edit]Symmetry in architecture
Ceiling of Lotfollah mosque, Isfahan, Iran. Has rotational symmetry of order 32 and 32 lines
of reflection.
Leaning Tower of Pisa
The Taj Mahal has bilateral symmetry.
Another human endeavor in which the visual result plays a vital part in the overall result
is architecture. Both in ancient and modern times, the ability of a large structure to impress
or even intimidate its viewers has often been a major part of its purpose, and the use of
symmetry is an inescapable aspect of how to accomplish such goals.
Just a few examples of ancient architectures that made powerful use of symmetry to
impress those around them included the Egyptian Pyramids, the Greek Parthenon, the first
and second Temple of Jerusalem, China'sForbidden City, Cambodia's Angkor Wat complex,
and the many temples and pyramids of ancient Pre-Columbian civilizations. More recent
historical examples of architectures emphasizing symmetries includeGothic
architecture cathedrals, and American President Thomas Jefferson's Monticello home.
The Taj Mahal is also an example of symmetry.[11]
An interesting example of a broken symmetry in architecture is the Leaning Tower of Pisa,
whose notoriety stems in no small part not for the intended symmetry of its design, but for
the violation of that symmetry from the lean that developed while it was still under
construction. Modern examples of architectures that make impressive or complex use of
various symmetries include Australia's Sydney Opera House and Houston, Texas's
simplerAstrodome.
Symmetry finds its ways into architecture at every scale, from the overall external views,
through the layout of the individual floor plans, and down to the design of individual
building elements such as intricately carved doors, stained glass windows, tile
mosaics, friezes, stairwells, stair rails, and balustrades. For sheer complexity and
sophistication in the exploitation of symmetry as an architectural element, Islamic buildings
such as the Taj Mahal often eclipse those of other cultures and ages, due in part to the
general prohibition of Islam against using images of people or animals.[12][13]
[edit]Symmetry in pottery and metal vessels
Persian vessel (4th millennium BC)
Since the earliest uses of pottery wheels to help shape clay vessels, pottery has had a strong
relationship to symmetry. As a minimum, pottery created using a wheel necessarily begins
with full rotational symmetry in its cross-section, while allowing substantial freedom of
shape in the vertical direction. Upon this inherently symmetrical starting point cultures from
ancient times have tended to add further patterns that tend to exploit or in many cases
reduce the original full rotational symmetry to a point where some specific visual objective is
achieved. For example, Persian pottery dating from the fourth millennium BC and earlier
used symmetric zigzags, squares, cross-hatchings, and repetitions of figures to produce more
complex and visually striking overall designs.
Cast metal vessels lacked the inherent rotational symmetry of wheel-made pottery, but
otherwise provided a similar opportunity to decorate their surfaces with patterns pleasing to
those who used them. The ancient Chinese, for example, used symmetrical patterns in their
bronze castings as early as the 17th century BC. Bronze vessels exhibited both a bilateral
main motif and a repetitive translated border design.[14][15][16]
[edit]Symmetry in quilts
Kitchen Kaleidoscope Block
As quilts are made from square blocks (usually 9, 16, or 25 pieces to a block) with each
smaller piece usually consisting of fabric triangles, the craft lends itself readily to the
application of symmetry.[17]
[edit]Symmetry in carpets and rugs
Persian rug.
A long tradition of the use of symmetry in carpet and rug patterns spans a variety of
cultures. American Navajo Indians used bold diagonals and rectangular motifs.
Many Oriental rugs have intricate reflected centers and borders that translate a pattern. Not
surprisingly, rectangular rugs typically use quadrilateral symmetry—that is, motifs that are
reflected across both the horizontal and vertical axes.[18][19]
[edit]Symmetry in music
Major and minor triads on the white piano keys are symmetrical to the D. (compare
article) (file)
Symmetry is not restricted to the visual arts. Its role in the history of music touches many
aspects of the creation and perception of music.
[edit]Musical form
Symmetry has been used as a formal constraint by many composers, such as the arch (swell)
form (ABCBA) used by Steve Reich, Béla Bartók, and James Tenney. In classical music, Bach
used the symmetry concepts of permutation and invariance.[20]
[edit]Pitch structures
Symmetry is also an important consideration in the formation of scales and chords,
traditional or tonal music being made up of non-symmetrical groups of pitches, such as
the diatonic scale or the major chord. Symmetrical scales or chords, such as the whole tone
scale, augmented chord, or diminished seventh chord (diminished-diminished seventh), are
said to lack direction or a sense of forward motion, are ambiguous as to the key or tonal
center, and have a less specific diatonic functionality. However, composers such as Alban
Berg, Béla Bartók, and George Perle have used axes of symmetry and/or interval cycles in an
analogous way to keys or non-tonal tonalcenters.
Perle (1992) explains "C–E, D–F#, [and] Eb–G, are different instances of the same interval …
the other kind of identity. … has to do with axes of symmetry. C–E belongs to a family of
symmetrically related dyads as follows:"
D D# E F F# G G#
D C# C B A# A G#
Thus in addition to being part of the interval-4 family, C–E is also a part of the sum-4 family
(with C equal to 0).
2 3 4 5
6
7 8
+ 2 1 0 11 10 9 8
4 4 4 4
4
4 4
Interval cycles are symmetrical and thus non-diatonic. However, a seven pitch segment of C5
(the cycle of fifths, which are enharmonic with the cycle of fourths) will produce the diatonic
major scale. Cyclic tonal progressions in the works of Romantic composers such as Gustav
Mahler and Richard Wagner form a link with the cyclic pitch successions in the atonal music
of Modernists such as Bartók, Alexander Scriabin, Edgard Varèse, and the Vienna school. At
the same time, these progressions signal the end of tonality.
The first extended composition consistently based on symmetrical pitch relations was
probably Alban Berg's Quartet, Op. 3 (1910). (Perle, 1990)
[edit]Equivalency
Tone rows or pitch class sets which are invariant under retrograde are horizontally
symmetrical, under inversion vertically. See also Asymmetric rhythm.
[edit]Symmetry in other arts and crafts
Celtic knotwork
The concept of symmetry is applied to the design of objects of all shapes and sizes. Other
examples include beadwork, furniture, sand paintings,knotwork, masks, musical
instruments, and many other endeavors.
[edit]Symmetry in aesthetics
Main article: Symmetry (physical attractiveness)
The relationship of symmetry to aesthetics is complex. Certain simple symmetries, and in
particular bilateral symmetry, seem to be deeply ingrained in the inherent perception by
humans of the likely health or fitness of other living creatures, as can be seen by the simple
experiment of distorting one side of the image of an attractive face and asking viewers to
rate the attractiveness of the resulting image. Consequently, such symmetries that mimic
biology tend to have an innate appeal that in turn drives a powerful tendency to create
artifacts with similar symmetry. One only needs to imagine the difficulty in trying to market
a highly asymmetrical car or truck to general automotive buyers to understand the power of
biologically inspired symmetries such as bilateral symmetry.
Another more subtle appeal of symmetry is that of simplicity, which in turn has an
implication of safety, security, and familiarity.[citation needed] A highly symmetrical room, for
example, is unavoidably also a room in which anything out of place or potentially
threatening can be identified easily and quickly.[citation needed] For example, people who have
grown up in houses full of exact right angles and precisely identical artifacts can find their
first experience in staying in a room with no exact right angles and no exactly identical
artifacts to be highly disquieting.[citation needed]Symmetry thus can be a source of comfort not
only as an indicator of biological health, but also of a safe and well-understood living
environment.
Opposed to this is the tendency for excessive symmetry to be perceived as boring or
uninteresting. Humans in particular have a powerful desire to exploit new opportunities or
explore new possibilities, and an excessive degree of symmetry can convey a lack of such
opportunities.[citation needed] Most people display a preference for figures that have a certain
degree of simplicity and symmetry, but enough complexity to make them interesting.[21]
Yet another possibility is that when symmetries become too complex or too challenging, the
human mind has a tendency to "tune them out" and perceive them in yet another fashion:
as noisethat conveys no useful information.[citation needed]
Finally, perceptions and appreciation of symmetries are also dependent on cultural
background. The far greater use of complex geometric symmetries in many Islamic cultures,
for example, makes it more likely that people from such cultures will appreciate such art
forms (or, conversely, to rebel against them).[citation needed]
As in many human endeavors, the result of the confluence of many such factors is that
effective use of symmetry in art and architecture is complex, intuitive, and highly dependent
on the skills of the individuals who must weave and combine such factors within their own
creative work. Along with texture, color, proportion, and other factors, symmetry is a
powerful ingredient in any such synthesis; one only need to examine the Taj Mahal to
powerful role that symmetry plays in determining the aesthetic appeal of an object.
Modernist architecture rejects symmetry, stating only a bad architect relies on
symmetry;[citation needed] instead of symmetrical layout of blocks, masses and structures,
Modernist architecture relies on wings and balance of masses. This notion of getting rid of
symmetry was first encountered in International style. Some people find asymmetrical
layouts of buildings and structures revolutionizing; other find them restless, boring and
unnatural.
A few examples of the more explicit use of symmetries in art can be found in the remarkable
art of M.C. Escher, the creative design of the mathematical concept of a wallpaper group,
and the many applications (both mathematical and real world) of tiling.
[edit]See also
Symmetry in statistics

Skewness, asymmetry of a statistical distribution
Symmetry in games and puzzles

Symmetric games

Sudoku
Symmetry in literature

Palindrome
Moral symmetry

Empathy and Sympathy

Golden Rule

Reciprocity

Reflective equilibrium

Tit for tat
Other

Asymmetric rhythm

Asymmetry

Burnside's lemma

Chirality

M.C. Escher

Even and odd functions

Fixed points of isometry groups in Euclidean space – center of symmetry

Gödel, Escher, Bach

Ignacio Matte Blanco

Semimetric, which is sometimes translated as symmetric in Russian texts.

Spacetime symmetries

Spontaneous symmetry breaking

Symmetric relation

Symmetries of polyiamonds

Symmetries of polyominoes

Symmetry (biology)

Symmetry group

Time symmetry

Wallpaper group
[edit]References
1. ^ Penrose, Roger (2007). Fearful Symmetry. City: Princeton. ISBN 978-0-69113482-6.
2. ^ a b For example, Aristotle ascribed spherical shape to the heavenly bodies,
attributing this formally defined geometric measure of symmetry to the
natural order and perfection of the cosmos.
3. ^ Weyl 1982
4. ^ For example, operations such as moving across a regularly patterned tile
floor or rotating an eight-sided vase, or complex transformations of
equations or in the way music is played.
5. ^ See, e.g., Mainzer, Klaus (2005). Symmetry And Complexity: The Spirit and
Beauty of Nonlinear Science. World Scientific. ISBN 981-256-192-7.
6. ^ Symmetric objects can be material, such as a person, crystal, quilt, floor
tiles, or molecule, or it can be an abstract structure such as a mathematical
equation or a series of tones (music).
7. ^ `Higher dimensional group theory'
8. ^ n-category cafe – discussion of n-groups
9. ^ `Higher dimensional group theory'
10. ^ Emotional Competency Entry describing Symmetry
11. ^ Gregory Neil Derry (2002), What Science Is and How It Works, Princeton
University Press, p. 269
12. ^ Williams: Symmetry in Architecture
13. ^ Aslaksen: Mathematics in Art and Architecture
14. ^ Chinavoc: The Art of Chinese Bronzes
15. ^ Grant: Iranian Pottery in the Oriental Institute
16. ^ The Metropolitan Museum of Art – Islamic Art
17. ^ Quate: Exploring Geometry Through Quilts
18. ^ Mallet: Tribal Oriental Rugs
19. ^ Dilucchio: Navajo Rugs
20. ^ see ("Fugue No. 21," pdf or Shockwave)
21. ^ Arnheim, Rudolf (1969). Visual Thinking. University of California Press.
http://en.wikipedia.org/wiki/Symmetry
…………….
Why is the relationship between conservation laws and symmetries important? One reason
is that it allows for other conservation laws to be formulated. For example, for conduction
electrons in solids all locations in the solid are not equivalent. For one, some locations are
closer to nuclei than others. Therefore linear momentum of the electrons is not conserved.
(The total linear momentum of the complete solid is conserved in the absence of external
forces. In other words, if the solid is in otherwise empty space, it conserves its total linear
momentum. But that does not really help for describing the motion of the conduction
electrons.) However, if the solid is crystalline, its atomic structure is periodic. Periodicity is a
symmetry too. If you shift a system of conduction electrons in the interior of the crystal over
a whole number of periods, it makes no difference. That leads to a conserved quantity called
“crystal momentum,” {A.19}. It is important for optical applications of semiconductors.
Even in empty space there are additional symmetries that lead to important conservation
laws. The most important example of all is that it does not make a difference at what time
you start an experiment with a system of particles in empty space. The results will be the
same. That symmetry with respect to time shift gives rise to the law of conservation of
energy, maybe the most important conservation law in physics………………………..
http://www.eng.fsu.edu/~dommelen/quantum/style_a/physsym.html
n this century, I believe that Physics will enter into an another remarkable area of basic
understanding which we can not discover yet. No, I am not saying about the experimental
facts of CERN or Fermilab. I am saying about a completely new aspect of Physics which may
be related with some symmetry of nature. Really, symmetry is a very powerful tool for
physicists of understand the complicated natural phenomena more simple way.
Now, what is symmetry? we are very much concern about symmetry but it is
impossible to explain symmetry in a single sentence. Why? because symmetry is not a very
simple concept, some times it not very clear to us. Symmetry is itself many kinds, such as
rotation, translation, mirror etc. But there are also many types of symmetries which are very
much abstract and we can not realize them pictorially. Only tool to realize them is to study
the Group theory which is a very powerful concept of Mathematics. Once Sir Arthur
Eddington said that we require a super mathematics in which we don't know about the
system of our interest but by the virtue of that super mathematics a super mathematician
can extract the important properties of the system. Group Theory is such theory. Actually, I
am quite wrong because Group theory is not such a theory but its representation is very
likely to Eddington's imagination.
Actually Group theory is the study of symmetry in most smartest way. But how is
symmetry related with nature? obviously this is the big question. We know if any symmetry
is related to any matrix then the number of fundamental elements of the matrix are
reduced, i.e. we can study the very same matrix in simple way if we take into account its
symmetry. In very similar way we can apply the concept of symmetry in Physics. Suppose ,
we are studying a system which is spherically symmetric, then we can easily reduce the
variation of the system into only one variable instead of three.
In Physics, symmetry also plays another important role. It can tell about some kind of
conservation principle of nature. Every symmetry of nature are related to a conservation
principle of nature. For example, we are now studying a system which has a translational
symmetry, i.e. the system is invariant in every respect after a translation. We are considering
a translation along the X- axis . The total energy or the Hamiltonian of the system is,
the only dynamical quantity of the equation is momentum or P. Now the system is invariant
under the translation along X-axis, that means the Hamiltonian (H) of the system is invariant
under this translation. So, the only dynamical quantity i.e. linear momentum of the equation
being fixed, i.e. linear momentum of the system is conserved. If we see that system is
invariant under rotation then in same logic the angular momentum will be conserved.
Therefore , in every process in the nature there will be some symmetries and by the virtue of
these symmetries there will be some conserved quantity( like mass, charge, spin, baryon
number etc.).
Besides this, we can use the concept of symmetry to extract the complicated
phenomena from a complicated system( like nucleus, quarks, neutrino etc.). That is
symmetry is clue that our nature posses many kinds of mystery but those interrelated and
that is why nature is so beautiful.
RELATED ARTICLES ON SCIENCE 2.0

Quantum field theory in curved space time, quantum gravity....They are one and the
same.
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Time In Bed With Space
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Analyzing Actions: EM (2/3?)
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http://www.science20.com/not_exactly_believe_single_equation_can_explain_all_mysterie
s/blog/symmetry_and_principle_nature-85839
Quantum mechanics flummoxes physicists again
A fresh take on a classic experiment makes no progress in unifying quantum mechanics
and relativity.
Jon Cartwright
A 3-slit experiment has confirmed a basic rule of
quantum mechanics but failed to help physicists to reconcile the theory with
relativity.Science/ AAAS
If you ever want to get your head around the riddle that is quantum mechanics, look no
further than the double-slit experiment. This shows, with perfect simplicity, how just
watching a wave or a particle can change its behaviour. The idea is so unpalatable to
physicists that they have spent decades trying to find new ways to test it. The latest such
attempt, by physicists in Europe and Canada, used a three-slit version — but quantum
mechanics won out again.
In the standard double-slit experiment, a wide screen is shielded from an electron gun by a
wall containing two separated slits. If the electron gun is fired with one slit closed, a mound
of electrons forms on the screen beyond the open slit, trailing off to the left and right — the
sort of behaviour expected for particles. If the gun is fired when both slits are open,
however, electrons stack along the screen in comb-like divisions. This illustrates the
electrons interfering with each other — the hallmark of wave behaviour.
Such a crossover in behaviour — known as wave–particle duality — is perhaps not too hard
to swallow. But quantum mechanics gets weirder. Slow down the gun so that just one
electron at a time reaches the screen, and the interference pattern remains. Does each
electron pass through both slits at once and interfere with itself? The obvious way to answer
this question is to watch the slits as the gun fires, but as soon as you do this the interference
pattern disappears.
It's as if the electrons know when they're being watched and decide to behave as particles
again. According to Nobel laureate Richard Feynman, the phenomenon "has in it the heart of
quantum mechanics. In reality, it contains the only mystery".
Mind the gaps
The new three-slit version of the experiment, performed by Gregor Weihs at the University
of Innsbruck in Austria and his colleagues, sought to uncover gaps in our understanding of
quantum mechanics through which modern physics might make some headway. Perhaps the
greatest problem in modern physics is how to reconcile quantum mechanics, which allows
for seemingly instantaneous communication, with Einstein's theories of special and general
relativity, which imply that nothing should travel faster than light.
Weihs's group thought that a route to reconciliation could lie in Born's rule, a central tenet
of quantum mechanics that says interference should exist only between two paths, such as
the two paths of the double-slit experiment. If there were any three-way interference in the
three-slit version, Born's rule would break down and an area of quantum mechanics in which
relativity might take hold would be exposed.
To perform their experiment, Weihs and colleagues aimed a source of single photons (which,
like electrons, exhibit wave–particle duality) at a mask containing various open and closed
combinations of three slits. The authors fired photons repeatedly through the mask, while
building a probability distribution of photons arriving on a detector beyond it. From the
probabilities of each combination, they could calculate a crucial interference term, which
would highlight any three-path interference.
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As Weihs's group had secretly feared, the three-path interference term came to more or less
zero1. Co-author Ray Laflamme of the University of Waterloo in Ontario, Canada, "always
hoped for three-path interference", says Weihs. "But then he's more of a theoretician. If
there was three-path interference, there would be a Nobel prize waiting."
It is true that the experiment has yielded little for theorists to work with, but it's not all bad
news, as Markus Aspelmeyer at the University of Vienna points out. "The fact that one does
not observe deviations from quantum theory also has profound implications," he says. "It
suggests that the present theory is a good description of our physical world and that we
have to work harder to understand its fundamental message."
Weihs is now considering a more rigorous test of Born's rule with an interferometer, a highly
accurate device that employs a layout of mirrors and beam splitters in place of physical slits.
Still, Weihs and his colleagues probably feel they have worked hard enough already. Their
experiment involved the logging of billions of photons, a process that took over two years.
"It's becoming a little tedious, I must stress," says Weihs.

References
1. Sinha, U., Couteau, C., Jennewein, T., Laflamme, R. & Weihs, G.Science 329,
418-421 (2010). | Article | ChemPort |
Comments
If you find something abusive or inappropriate or which does not otherwise comply with
our Terms or Community Guidelines, please select the relevant'Report this comment' link.
Comments on this thread are vetted after posting.

#12218
The conclusion should be: yes all radiation is able to split and travel along all different routes
through the quantum vacuum lattice as long as these routes have the same length to be able
to interfere with itself.

o
Report this comment
o
2010-07-23 06:08:20 AM
o
Posted by: Leo Vuyk
#12233
A disappointing bit of reporting.
"Perhaps the greatest problem in modern physics is how to reconcile quantum mechanics,
which allows for seemingly instantaneous communication, with Einstein's theories of special
and general relativity, which imply that nothing should travel faster than light."
Anyone apparently unaware that there is a highly successful way of uniting quantum
mechanics and special relativity – this is where we get quantum field theories, the
foundation on which modern particle physics is built – surely has no business writing about
this particular topic.
Also, in the original article text, the problem of unifying quantum theory and gravitation is
no more than a very general and rather vague motivation. It's only mentioned in the
abstract, and there's no indication in the main text that a violation of Born's rule would have
told us anything definite about the connection between quantum mechanics and gravity (in
fact, gravity isn't even mentioned in the main text).
Surely, responsible reporting about this article should include pointing out that the link of
this experiment with the problem of quantum gravity is tenuous at best. Instead, we are told
that the experiment "failed to help physicists to reconcile the theory with relativity" - as if
that had been an even halfway realistic outcome at the time of reporting.
(Oh, and also: Born's rule? Much more than just a statement about two-way vs. three-way
interference. Those reading about Born's rule for the first time in this text are virtually
guaranteed to come away with the wrong impression.)

o
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o
2010-07-24 04:43:04 AM
o
Posted by: Markus Poessel
#12235
Although just an interested individual and not by any means a physicist, I have always
understood that wave collapse (a la double slit) may be due to interference from the
method of observation. When most articles on the double slit experiment and its possible
consequences are composed the method of observation is not always clear, certainly in
none professional pieces. Are observational methods active and without external particle
generation or are they passive in the true sense. I am told that this is discussed elsewhere
within the realms of scientific debate.

o
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o
2010-07-24 05:24:57 AM
o
Posted by: David Clarke
#12238
As I understand QM, there is no issue with the "single particle at any one time" experimental
result, for the simple reason that the distribution of end-points is determined by the wavefunction (the solution of Schrodinger's equation), which in turn is determined by the
geometry of the slits. The number of particles is only relevant when there is significant
particle-particle interaction, which is essentially zero for photons and assumed to be low
even for electrons, in this case.
o
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o
2010-07-24 07:31:06 AM
o

Posted by: David McCulloch
#12267
"Such a crossover in behaviour — known as wave–particle duality — is perhaps not too
hard to swallow."
If you understand that, step up and win your Nobel Prize! The wave-particle duality IS hard
to swallow and yet is essential to quantum "weirdess." That particles move as independent
objects yet appositely behave as is they were a disturbance in a medium is as
incommensurable to our minds as an ocean wave without the ocean.
"Slow down the gun so that just one electron at a time reaches the screen, and the
interference pattern remains. Does each electron pass through both slits at once and
interfere with itself? The obvious way to answer this question is to watch the slits as the gun
fires, but as soon as you do this the interference pattern disappears."
Self-contradictory statements here: Yes, the interference pattern DOES remain even as
electrons pass through "one at a time." (Again, how does "one at a time" ultimately relate to
a particle that is also wavelike? We don't know.) But this contradicts the next statement,
that the interference also disappears. More quantum "weirdness" or poor writing? No, the
double slit interference DOES remain even in the absence of statistical quantities of
particles, again, using that phrase so essential to pursuing physics well, we don't know why.

o
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o
2010-07-25 10:29:25 AM
o
Posted by: Denis Michael Reidy
#12269
Maybe a silly question, but if it's necessary for all paths to have an equal length, shouldn't
you do the three-slit experiment with two concentric cylinders? The source is at the center
of the two cylinders, the inner cylinder has the three slits and the outer cylinder is the
detector.
o
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o
2010-07-25 11:44:54 AM
o
Posted by: Brian DeCamp
http://www.nature.com/news/2010/100722/full/news.2010.371.html
A quantum take on certainty
Physicists show that in the iconic double-slit experiment, uncertainty can be eased.
Edwin Cartlidge
The double-slit experiment shows the dual waveparticle nature of photons.GIPHOTOSTOCK/SCIENCE PHOTO LIBRARY
An international group of physicists has found a way of measuring both the position and the
momentum of photons passing through the double-slit experiment, upending the idea that it
is impossible to measure both properties in the lab at the same time.
In the classic double-slit experiment, first done more than 200 years ago, light waves passing
through two parallel slits create a characteristic pattern of light and dark patches on a screen
positioned behind the slits. The patches correspond to the points on the screen where the
peaks and troughs of the waves diffracting out from the two slits combine with one another
either constructively or destructively.
In the early twentieth century, physicists showed that this interference pattern was evident
even when the intensity of the light was so low that photons pass through the apparatus
one at a time. In other words, individual photons seem to interfere with themselves, so light
exhibits both particle-like and wave-like properties.
However, placing detectors at the slits to determine which one a particle is passing through
destroys the interference pattern on the screen behind. This is a manifestation of Werner
Heisenberg's uncertainty principle, which states that it is not possible to precisely measure
both the position (which of the two slits has been traversed) and the momentum
(represented by the interference pattern) of a photon.
What quantum physicist Aephraim Steinberg of the University of Toronto in Canada and his
colleagues have now shown, however, is that it is possible to precisely measure photons'
position and obtain approximate information about their momentum1, in an approach
known as 'weak measurement'.
Steinberg's group sent photons one by one through a double slit by using a beam splitter
and two lengths of fibre-optic cable. Then they used an electronic detector to measure the
positions of photons at some distance away from the slits, and a calcite crystal in front of the
detector to change the polarization of the photon, and allow them to make a very rough
estimate of each photon's momentum from that change.
Average trajectory
By measuring the momentum of many photons, the researchers were able to work out the
average momentum of the photons at each position on the detector. They then repeated
the process at progressively greater distances from the slits, and so by "connecting the dots"
were able to trace out the average trajectories of the photons. They did this while still
recording an interference pattern at each detector position.
Intriguingly, the trajectories closely match those predicted by an unconventional
interpretation of quantum mechanics known as pilot-wave theory, in which each particle has
a well-defined trajectory that takes it through one slit while the associated wave passes
through both slits. The traditional interpretation of quantum mechanics, known as the
Copenhagen interpretation, dismisses the notion of trajectories, and maintains that it is
meaningless to ask what value a variable, such as momentum, has if that's not what is being
measured.
Steinberg stresses that his group's work does not challenge the uncertainty principle,
pointing out that the results could, in principle, be predicted with standard quantum
mechanics. But, he says, "it is not necessary to interpret the uncertainty principle as rigidly
as we are often taught to do", arguing that other interpretations of quantum mechanics,
such as the pilot-wave theory, might "help us to think in new ways".
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David Deutsch of the University of Oxford, UK, is not convinced that the experiment has told
us anything new about how the universe works. He says that although "it's quite cool to see
strange predictions verified", the results could have been obtained simply by "calculating
them using a computer and the equations of quantum mechanics".
"Experiments are only relevant in science when they are crucial tests between at least two
good explanatory theories," Deutsch says. "Here, there was only one, namely that the
equations of quantum mechanics really do describe reality."
But Steinberg thinks his work could have practical applications. He believes it could help to
improve logic gates for quantum computers, by allowing the gates to repeat an operation
deemed to have failed previously. "Under the normal interpretation of quantum mechanics
we can't pose the question of what happened at an earlier time," he says. "We need
something like weak measurement to even pose this question."

References
1.
Kocsis, S. et al. Science 332, 1170-1173 (2011). | Article | ChemPort |
http://www.nature.com/news/2011/110602/full/news.2011.344.html
Examining Capitalism Through Quantum Mechanics
Saturday, 28 July 2012 08:42By Michael Ortiz, Truthout | Op-Ed
http://truth-out.org/opinion/item/10446-examining-capitalism-through-quantummechanics
As human beings, we don’t just construct social realities and social systems, but
we literally help construct the physical universe of which we are a part. Therefore,
understanding the relationship between human beings and the quantum reality of
the universe becomes paramount if we seek to truly understand and transform
the social and structural systems of inequality that we have created for ourselves.
According to quantum mechanics, the subatomic level of reality exists in an
undifferentiated state of dynamic flux until a conscious observer measures it (or
looks at it), thus, giving that matter a particular form. In other words, an atom is
spread out all over the place as a wave of potential until a conscious observer
localizes it as an actual particle through that very act of observation.
The famous double-slit experiment actually captured this protean nature of the
quantum world. The double-slit experiment essentially launched particles through
a single slit, whereby each particle left a residual mark on the back wall where it
landed (creating a single band pattern). However, when particles were launched
through two slits, they left a residual interference pattern on the back wall (which
can only be created by waves that interfere with each other). Even when particles
were launched through the two slits one at a time, they still created an
interference pattern. (This occurrence is impossible according to classical quantum
physics.) So, in order to figure out how this interference pattern was occurring,
physicists placed a measuring device by the slits to observe the particles after they
were launched. Astonishingly, when the particles were launched with the
measuring device in place, they actually created a residual mark of a double band
pattern (which was expected in the first place). What physicists determined was
that, prior to being observed, each single particle actually existed as a wave of
potentials that simultaneously went through both slits at the same time; thus
interfering with itself and leaving a residual interference pattern. So in essence,
conscious observation then collapses the quantum wave function of particles and
thus localizes them at a fixed point.
Moreover, quantum superposition “holds that a physical system – such as an
electron – exists partly in all its particular, theoretically possible states (or,
configuration of its properties) simultaneously; but, when measured, it gives a
result corresponding to only one of the possible configurations (as described in
interpretation of quantum mechanics).”
The more we look at elementary particles, the more we realize that there is
actually no such thing as one electron or one photon on its own. A particle exists
only in relationship to the state that it finds itself in, with no generic or concrete
form. So, the more we examine “solid matter” in great detail, the less solid it
actually becomes.
(Photo: Earth with cogs and
wheels via Shutterstock)
Now, contradictory to contemporary quantum mechanics is the traditional
conception of solid matter as the “substance” of the universe. Why is this
important? Because “belief that the substance of the universe is matter (or
physical material) sets the precedent for people to accumulate as
many material possessions and riches as possible [especially under the system of
capitalism],” says UK author David Icke. Most of us in contemporary Western
culture have been socialized to view the world through a consumerist lens (among
a plethora of other social lenses) which implies that a solid, material realm
objectively exists. Furthermore, the system of capitalism creates the conditions
necessary for more and more people to actively participate in practices that
perpetuate the misconception that a solid, material world inexorably dictates our
perceptions and belief systems. Maximized material conquest and material gain
becomes the modus operandi of a capitalistic system.
Further illuminating the nature of capitalism, Chris Hedges states:
“The quest by a bankrupt elite in the final days of empire to accumulate greater
and greater wealth is modern society’s version of primitive fetishism … When the
most basic elements that sustain life are reduced to a cash product, life has no
intrinsic value. The extinguishing of ‘primitive’ societies, those that were defined
by animism and mysticism, those that celebrated ambiguity and mystery, those
that respected the centrality of the human imagination, removed the only
ideological counterweight to a self-devouring capitalist ideology.”
Here we see some of the characteristics of neoliberal capitalism which subscribe
to the notion that the world be defined in “material” terms. The ruling ideology of
capitalism has sought out to extinguish any alternative thought or knowledge that
understands the world in immaterial terms and replace it with the narrow
ideology of materialism, consumerism, commodification. The more people who
are complicit in capitalist ideology (among other forms of dominant ideologies),
the stronger the possibilities become to fetishize and develop the concept of “the
material.” all while the expropriation of vast forms of land, wealth, resources and
capital become normalized and accepted. Furthermore, once all “material”
resources have become accessed (or more importantly not accessed by the
majority of people), exploited and exhausted, then the majority of people become
even more subjected to the harsh and misleading conditions that capitalism
inflicts upon them.
So, as far as quantum mechanics is concerned, capitalism is based on the (false)
assumption that an absolute “material” world actually exists “out there.”
Traditional criticisms of capitalism typically focus on the exploitation of labor and
human bodies, as well as massive class inequalities and social injustice; however,
they leave out one crucial aspect in it all: that capitalist ideology and capitalist
operation mislead us about the nature of the universe (which includes the nature
of ourselves since we are part of the universe, as well). With that said, we can
actually use our knowledge of quantum mechanics to transform our perceptions
about the world around us, thus alleviating some of the conditions that capitalism
creates for us. Even Einstein alluded to the idea that we can utilize science to
“potentially change the world itself” by using “rational thinking and technology to
improve the conditions in which we live.” (1) As Peter Dreierstates:
“Einstein criticized capitalism’s ‘economic anarchy’ and the ‘oligarchy of private
capital, the enormous power of which cannot be effectively checked even by
democratically organized political society.’”
If Einstein could apply his knowledge of science and the quantum reality to social
injustice and systemic inequality, then there is no reason that we cannot do the
same here and now.
Given the fact that the underlying premise of capitalism acts in opposition to the
principles of quantum mechanics and, therefore, the nature of the universe itself
(as understood through quantum mechanics), then we should not be confounded
in the least when we experience the destructive consequences of a system that is
based on prodigious wealth and material accumulation. This systemic discord or
imbalance is bound to perpetuate the likes of environmental devastation and vast
human suffering. Furthermore, one of the unspoken consequences of capitalistic
operation is the alienation from one’s humanity and from nature.Not only are we
inundated by a social and economic matrix of domination every single day, but
that very matrix detaches us from the universe (or nature) in a sense. So, we
should not just look to eradicate the deleterious conditions of capitalism, but
rather, we should look to understand and work in accordance with the universe, so
that destructive systemic conditions do not even come into existence in the first
place.
Consequently, when we look at the world through the lens of quantum
mechanics, we see that the economic systems of capitalism, socialism and
communism actually have more in common with each other since they all are
based on material acquisition and distribution and on the assumption that our
world is a fundamentally material realm. However, we can use quantum
mechanics to create an entirely new way of viewing and operating inside of the
world, which would require a drastic philosophical and ideological change of epic
proportion. Epic change, perhaps, is a concept that we may need to start
entertaining.
Lastly, as if world hunger, poverty, class inequality, sickness and disease,
permanent war and ecological ruination weren’t enough to present a critical case
against capitalism, then consider the following. In relative terms to the rest of the
entire universe, quantum mechanics shows us just how narrow, constrictive and
destructive the system of capitalism actually is.
References:
(1) Dreier, Peter. 2012. “Albert Einstein: Radical Citizen and Scientist.” Truthout,
June 25.
This article is a Truthout original.
http://www.sustainabilitank.info/2012/07/capitalism-viewed-through-quantum-mechanicsresults-in-our-understanding-of-materialism-consumerism-commodification-as-a-very-onesided-view-of-the-universe-applying-quantum-mechanics-science-leads/
CROP CIRCLES AND QUANTUM MECHANICS
Many things have been said and written about the connection between crop circles, sound
and music. Although this connection seems to be obvious at first sight, it turns out not to
be plausible at all once we take a closer look at it!
If there is a connection, it is one with quantum mechanics, with super strings.
It's a connection with the nature of matter; not with music!
Indeed: a lot has been said and written about the presence of diatonic ratios in the crop
circles' geometry. But let us take a closer look at diatonic ratios: what are they really?
When we look at the different notes on for instance a piano, we see that they relate to each
other in a very specific way:
First octave 1
9/8
5/4
4/3
3/2
5/3
15/8
Second
octave
2
9/4
5/2
8/3
3
10/3
15/4
Third octave 4
9/2
5
16/3
6
20/3
15/2
Fourth
octave
8
9
10
32/3
12
40/3
15
Piano notes C
D
E
F
G
A
B
These ratios create the harmonics in music and are therefore called diatonic ratios.
They can also be shown geometrically, as follows:
These geometrical ratios are quite often found in the geometry of crop circles. In my
previous article I have shown that these ratios are a natural consequence of the type of
construction technique used to create the crop circles. They are a logical part of the crop
circles' geometry. As a result, the relationship "crop circles - music" seems to be obvious.
However there is another reason to assume that music - or at any rate sound - plays an
important role in the crop circle phenomenon.
In order to understand this we have to take a closer look at the nature of sound.
Sound consists of vibrations that move forward by means of air pressure differences.
Graphically it looks like this:
In other words, sound can be reproduced as a wave with peaks and lows. Sound that moves
through air shows peaks in places with the highest pressure, and lows in places with the
lowest pressure.
We can produce another tone that looks like this: When we produce both tones
simultaneously, we get this picture:
The waves in this graph can be added up. This is called interference: it is the way waves
correspond to each other, the way they can be added up. In the above example the result is
a horizontal line. The waves neutralise each other which results in no sound at all.
We can make this visible by taking melted paraffin and adding powder which will float
through the paraffin. We will then add sound. The sound will move forward through the
paraffin and the waves that arise this way will meet and interfere with each other. Together
they will form extreme peaks and extreme lows, and there are also places where they will
compensate each other. The latter happens when a peak meets a low: they extinguish each
other. This interference process creates a pattern with spots that vibrate fiercely and spots
that don't vibrate at all. An interference pattern arises. The powder will concentrate on the
spots with the least vibration, by which it makes the interference pattern clearly visible for
us!
The patterns that are created like this are often found in crop circles.
Does this mean there is a connection between music - or at least sound - and crop circles?
NO! This does not have to be a fact! It is even quite unlikely!
The above mentioned interference patterns arise with all sorts of waves, not just sound
waves. Water waves show a similar pattern, just like radio waves. The interference pattern
does not say anything about the type of wave involved. The only similarity between the
patterns in the paraffin and the crop circles are the interference patterns! There are no
indications whatsoever that the patterns in the crop are created by sound, or even that they
refer to sound. The only possible indication is the presence of interference or rather the
presence of interfering waves.
The chance that these are sound waves is minimal, since sound has a major drawback: it
always needs a medium through which it can move forward. A medium that can assume and
pass on the vibration of the sound source. Air is such a medium. If there is no medium sound
can not move forward. On the moon - with its vacuum - we can hear no sound. In other
words: sound is not a universal phenomenon!
This is not the case with Electro-magnetic waves, like for instance radio waves. These can
indeed move forward without the help of a medium. They are indeed universal.
The only similarity between sound waves and Electro-magnetic waves is the ability to
interfere. They can both cause interference patterns. Apart from that, these waves have
absolutely nothing in common. To put it differently: interference patterns say absolutely
nothing about the type of wave involved. They only say something about the interaction of
waves. The paraffin experiments visualise this interaction. They show us a graphic
reproduction of interference. A graphic reproduction with an unprecedented geometry!
They show us that interference can be represented geometrically and that it has a strongly
geometrical character. It shows the same geometry as we find in the crop formations.
But what about the diatonic ratios?
In order to understand their involvement we have to make a quantum leap, a leap into
quantum mechanics. Quantum mechanics reached a height with the development of the
Standard Model. All of the so far experimentally obtained data fitted perfectly in this model.
Every experiment confirmed the Standard Model. The only problem was that the Standard
Model was terribly ugly, complex and asymmetrical! And so, although everything seemed to
confirm the Standard Model, something also seemed to be wrong! Furthermore, the deeper
scientists went down to the depths of the subatomic world, the more they discovered
strange, exotic particles!
For these problems concerning the Standard Model, scientists found a solution by
developing the superstring theory.
When we make a string vibrate, we produce an audible tone. The pitch is determined by the
string's length. A string of a certain length will produce a tone that is specific to that length,
namely a tone with a frequency (=pitch) equal to the string's own vibration; a frequency in
which the string will start to resonate. This own frequency depends on the length of the
string. All of this enables us to play the violin!
Depending on the position of our fingers, we can play different notes. The note we hear is
determined by the part of the string between our fingers and the bridge. By changing the
distance between our fingers and the bridge, we play different notes.
Most essential here is the string, not the notes.
Analogue to this John Schwartz and Michael Green developed their superstring theory, with
infinitely thin strings that can resonate in different pitches. These vibrations correspond with
the various strings. Schwartz and Green believe that, if we magnify an elementary particle
under a microscope - which unfortunately we cannot do yet - we would no longer see
particles but vibrations: the vibrations of a superstring! The rest of the superstring theory
was further elaborated by Edward Witten.
We can now deduce that it is not the elementary particles which are fundamental, but the
harmonics created by the vibrating superstring. Matter therefore does not exist the way we
experience it. Since we cannot magnify subatomic particles with our current techniques, we
think of these particles as being elementary and fundamental. According to Schwartz, Green
and Witten however, this is only appearance. According to them, the string and its vibrations
are fundamental. Since a string is able to produce an infinite number of harmonics (just like
the many notes we can play on a violin) we will observe an infinite number of particles.
The superstring theory can deduce the character of particles from the resonating vibrations
of a string. It can also deduce Einstein's equations by stipulating the right conditions on
space-time. In fact, all theories before the superstring's - like Einstein's, Kaluza's and the
Standard Model - can be deduced from the superstring theory!
Conclusion
The geometry and forms that we find in crop circles are strongly related with the harmonics
and interference patterns created by vibrating strings.
However there is no reason whatsoever to assume on this basis a connection with music or
sound. If there is a connection at all, it is one of quantum mechanics. It's a reference to the
nature of matter. Matter that does not exist the way we experience it in our day-to-day life.
It's a reference to harmonics created by vibrating superstrings; the ones that give us the
sensation of matter!
Copyright: Bert Janssen, 1998.
http://cropcircleconnector.com/Dutch/CROPCIRCLESAN%20QUANTUMMECHANICS.html