# Symmetry

state; balance of object

Symmetry (from Greek συμμετρία symmetria "agreement in dimensions, due proportion, arrangement") has two meanings. The first is a vague sense of harmonious and beautiful proportion and balance. The second is an exact mathematical "patterned self-similarity" that can be demonstrated with the rules of a formal system, such as geometry or physics. Although these two meanings of "symmetry" can sometimes be told apart, they are related, so they are here discussed together. Mathematical 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 and as an aspect of abstract objects, theoretic models, language, music and even knowledge itself.

Arranged alphabetically by author or source:
A · B · C · D · E · F · G · H · I · J · K · L · M · N · O · P · Q · R · S · T · U · V · W · X · Y · Z · See also · External links

## A

• The earth is a cylindrical column, surrounded by air; it floats upright in the centre of the universe without support or anything to stand on, yet it does not fall because, being in the centre, it has no preferred direction towards which to lean; if it did, this would disturb the symmetry and balance of the whole. The spherical heavens enclose the atmosphere 'like the bark of a tree', and there are several layers of this enclosure to accommodate the various stellar objects.

## B

• Guided only by their feeling for symmetry, simplicity, and generality, and an indefinable sense of the fitness of things, creative mathematicians now, as in the past, are inspired by the art of mathematics rather than by any prospect of ultimate usefulness.

## D

• Men achieve tranquillity through moderation in pleasure and through the symmetry of life. Want and superfluity are apt to upset them and to cause great perturbations in the soul. The souls that are rent by violent conflicts are neither stable nor tranquil. ...one ought not to desire other men's blessings, and one ought not to envy those who have more, but rather, comparing his life with that of those who fare worse, and laying to heart their sufferings, deem himself blest of fortune in that he lives and fares so much better than they. Holding fast to this saying you will pass your life in greater tranquillity and will avert not a few of the plagues of life—envy and jealousy and bitterness of mind.
• What is this world? A complex whole, subject to endless revolutions. All these revolutions show a continual tendency to destruction; a swift succession of beings who follow one another, press forward, and vanish; a fleeting symmetry; the order of a moment. I reproached you just now with estimating the perfection of things by your own capacity; and I might accuse you here of measuring its duration by the length of your own days. You judge of the continuous existence of the world, as an ephemeral insect might judge of yours. The world is eternal for you, as you are eternal to the being that lives but for one instant. Yet the insect is the more reasonable of the two. For what a prodigious succession of ephemeral generations attests your eternity! What an immeasurable tradition! Yet shall we all pass away, without the possibility of assigning either the real extension that we filled in space, or the precise time that we shall have endured. Time, matter, space—all, it may be, are no more than a point.
• Denis Diderot, Lettre sur les aveugles [Letter on the Blind] (1749)

## G

• Physicists describe the two properties of physical laws—that they do not depend on when or where you use them—as symmetries of nature. By this usage physicists mean that nature treats every moment in time and every location in space identically—symmetrically—by ensuring that the same fundamental laws are in operation. Much in the same manner that they affect art and music, such symmetries are deeply satisfying; they highlight an order and coherence in the workings of nature. The elegance of rich, complex, and diverse phenomena emerging from a simple set of universal laws is at least part of what physicists mean when they invoke the term "beautiful."
• Brian Greene, The Elegant Universe (1999) Ch. 7 The "Super" in Superstrings.
• One overarching lesson... during the past hundred years is that the known laws of physics are associated with principles of symmetry. Special relativity is based on the symmetry embodied in the principle of relativity—the symmetry between all constant-velocity vantage points. The gravitational force, as embodied in the general theory of relativity, is based on the equivalence principle—the extension of the principle of relativity to embrace all vantage points regardless of the complexity of their states of motion. And the strong, weak, and electromagnetic forces are based on the... gauge symmetry principles. ...String theory takes us down another notch on the scale of explanatory depth because all of these symmetry principles, as well as another—supersymmetry—emerge from its structure.
• Brian Greene, The Elegant Universe (1999, 2003) Ch. 15 "Prospects."

## J

• To let the reader sometimes complete the symmetry between words and to do no more than suggest it. In this painting of our life given to us by our memories, everything is moving and depends on our point of view.

## L

• The introduction of symmetries belongs here too, silhouettes in inkblots, etc. Likewise the gradation we establish in the order of creatures: all this is not in the things but in us. In general we cannot remember too often that when we observe nature, and especially the ordering of nature, it is always ourselves alone we are observing.
• Nature has been kind to us by being governed by universal laws, rather than by mere parochial bylaws. A hydrogen atom on Earth... or even in a galaxy that is ten billion light years away, behaves in precisely the same manner. And this is true in any direction we look and at any time. ...such properties ...are called symmetries and they reflect immunity to changes in location, orientation, or the time... If not for these (and other) symmetries, any hope of deciphering nature's grand design would have been lost...

## M

• For every symmetry there comes a constraint. ...If physics is to look the same when the origin of time is shifted... [o]nly those processes that conserve energy are allowed. ...If physical law is to be immune to the arbitrary displacement of our spatial axes, then nature requires the conservation of linear momentum. ...If the laws are to be unaffected by the arbitrary rotation of a coordinate system, then angular momentum must be conserved. ...If the laws are to be the same for all inertial observers, then the space-time interval must be invariant. ...[A]nother constraint ...so beautiful as to make one jaw drop in wonder ...symmetry creates force. ...[T]he symmetry of identical particles forces matter ...to be enrolled as either fermion or boson ...Bosons, typified by the photon, carry the the fundamental forces that cause fermions to attract and repel. Fermions, led by electrons and quarks, become constituents of ordinary matter. ...Gravity. Electromagnetism. The strong force. The weak force. Each fundamental interaction is called into being by the requirements of a particular local symmetry.
• Michael Munowitz, Knowing: The Nature of Physical Law (2005)

## O

• Some people focus more on sonics. Some people focus more on story. I focus on both sonics and story, but music sometimes, just music itself, can turn into more of a maths problem. I guess everything in life is a math problem, but it can be more about an empirical route to getting the symmetry that you want, and this vibe, sonically.

## P

• Beauty is our weapon against nature; by it we make objects, giving them limit, symmetry, proportion. Beauty halts and freezes the melting flux of nature.

## R

• Graphic design, which evokes the symmetria of Vituvius, the dynamic symmetry of Hambidge, the asymmetry of Mondrian; which is a good gestalt, generated by intuition or by computer, by invention or by a system of coordinates, is not good design if it does not communicate.

## W

• We have simply arrived too late in the history of the universe to see this primordial simplicity easily … But although the symmetries are hidden from us, we can sense that they are latent in nature, governing everything about us. That's the most exciting idea I know: that nature is much simpler than it looks. Nothing makes me more hopeful that our generation of human beings may actually hold the key to the universe in our hands—that perhaps in our lifetimes we may be able to tell why all of what we see in this immense universe of galaxies and particles is logically inevitable.

## Y

• Nature seems to take advantage of the simple mathematical representations of the symmetry laws. When one pauses to consider the elegance and the beautiful perfection of the mathematical reasoning involved and contrast it with the complex and far-reaching physical consequences, a deep sense of respect for the power of the symmetry laws never fails to develop.
• By the late Nineties, we had become a more visual nation. Big-money taste moved to global standards - new architecture, design and show-off contemporary art. The Sloane domestic aesthetic - symmetry, class symbolism and brown furniture - became as unfashionable as it had been hot in the early Eighties.