Symmetry

[2][3][a] In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations, such as translation, reflection, rotation, or scaling.

[17] 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.

"[18] See Noether's theorem (which, in greatly simplified form, states that for every continuous mathematical symmetry, there is a corresponding conserved quantity such as energy or momentum; a conserved current, in Noether's original language);[19] and also, Wigner's classification, which says that the symmetries of the laws of physics determine the properties of the particles found in nature.

Bilateral animals, including humans, are more or less symmetric with respect to the sagittal plane which divides the body into left and right halves.

[22] Plants and sessile (attached) animals such as sea anemones often have radial or rotational symmetry, which suits them because food or threats may arrive from any direction.

The control of the symmetry of molecules produced in modern chemical synthesis contributes to the ability of scientists to offer therapeutic interventions with minimal side effects.

A rigorous understanding of symmetry explains fundamental observations in quantum chemistry, and in the applied areas of spectroscopy and crystallography.

Ernst Mach made this observation in his book "The analysis of sensations" (1897),[27] and this implies that perception of symmetry is not a general response to all types of regularities.

[29] Studies of human perception and psychophysics have shown that detection of symmetry is fast, efficient and robust to perturbations.

Sasaki et al.[31] used functional magnetic resonance imaging (fMRI) to compare responses for patterns with symmetrical or random dots.

Pottery created using a wheel acquires full rotational symmetry in its cross-section, while allowing substantial freedom of shape in the vertical direction.

Upon this inherently symmetrical starting point, potters from ancient times onwards have added patterns that modify the rotational symmetry to achieve visual objectives.

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.

Not surprisingly, rectangular rugs have typically the symmetries of a rectangle—that is, motifs that are reflected across both the horizontal and vertical axes (see Klein four-group § Geometry).

Escher and the many applications of tessellation in art and craft forms such as wallpaper, ceramic tilework such as in Islamic geometric decoration, batik, ikat, carpet-making, and many kinds of textile and embroidery patterns.

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.

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 tonal centers.

[49][50] The first extended composition consistently based on symmetrical pitch relations was probably Alban Berg's Quartet, Op.

[54] Symmetry can be found in various forms in literature, a simple example being the palindrome where a brief text reads the same forwards or backwards.