Contents 1 In mathematics 1.1 In geometry 1.2 In logic 1.3 Other areas of mathematics 2 In science and nature 2.1 In physics 2.2 In biology 2.3 In chemistry 3 In social interactions 4 In the arts 4.1 In architecture 4.2 In pottery and metal vessels 4.3 In quilts 4.4 In carpets and rugs 4.5 In music 4.5.1 Musical form 4.5.2 Pitch structures 4.5.3 Equivalency 4.6 In other arts and crafts 4.7 In aesthetics 4.8 In literature 5 See also 6 Notes 7 References 8 Further reading 9 External links In mathematics[edit]
In geometry[edit]
Main article:
The triskelion has 3-fold rotational symmetry. A geometric shape or object is symmetric if it can be divided into two or more identical pieces that are arranged in an organized fashion.[5] This means that an object is symmetric if there is a transformation that moves individual pieces of the object but doesn't change the overall shape. The type of symmetry is determined by the way the pieces are organized, or by the type of transformation: An object has reflectional symmetry (line or mirror symmetry) if there
is a line going through it which divides it into two pieces which are
mirror images of each other.[6]
An object has rotational symmetry if the object can be rotated about a
fixed point without changing the overall shape.[7]
An object has translational symmetry if it can be translated without
changing its overall shape.[8]
An object has helical symmetry if it can be simultaneously translated
and rotated in three-dimensional space along a line known as a screw
axis.[9]
An object has scale symmetry if it does not change shape when it is
expanded or contracted.[10]
In logic[edit]
A dyadic relation R is symmetric if and only if, whenever it's true
that Rab, it's true that Rba.[12] 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 ⊽).
Other areas of mathematics[edit]
Main article:
Many animals are approximately mirror-symmetric, though internal organs are often arranged asymmetrically. In biology[edit]
Further information: symmetry in biology and facial symmetry
In biology, the notion of symmetry is mostly used explicitly to
describe body shapes. Bilateral animals, including humans, are more or
less symmetric with respect to the sagittal plane which divides the
body into left and right halves.[19] Animals that move in one
direction necessarily have upper and lower sides, head and tail ends,
and therefore a left and a right. The head becomes specialized with a
mouth and sense organs, and the body becomes bilaterally symmetric for
the purpose of movement, with symmetrical pairs of muscles and
skeletal elements, though internal organs often remain asymmetric.[20]
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. Fivefold symmetry is found in
the echinoderms, the group that includes starfish, sea urchins, and
sea lilies.[21]
In biology, the notion of symmetry is also used as in physics, that is
to say to describe the properties of the objects studied, including
their interactions. A remarkable property of biological evolution is
the changes of symmetry corresponding to the appearance of new parts
and dynamics.[22][23]
In chemistry[edit]
Main article: molecular symmetry
The ceiling of Lotfollah mosque, Isfahan,
Further information:
Seen from the side, the
Clay pots thrown on a pottery wheel acquire rotational symmetry. Since the earliest uses of pottery wheels to help shape clay vessels, pottery has had a strong relationship to symmetry. 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. 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.[32] In quilts[edit] 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.[33] In carpets and rugs[edit] Persian rug with rectangular symmetry 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
Major and minor triads on the white piano keys are symmetrical to the D. (compare article) (file)
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
Celtic knotwork Symmetries appear in the design of objects of all kinds. Examples
include beadwork, furniture, sand paintings, knotwork, masks, and
musical instruments. Symmetries are central to the art of M.C. Escher
and the many applications of tessellation in art and craft forms such
as wallpaper, ceramic tilework, batik, ikat, carpet-making, and many
kinds of textile and embroidery patterns.[39]
In aesthetics[edit]
Main article:
Automorphism
Burnside's lemma
Chirality
Even and odd functions
Notes[edit] ^ For example,
References[edit] ^ "symmetry". Online Etymology Dictionary.
^ Zee, A. (2007). Fearful Symmetry. Princeton, N.J.: Princeton
University Press. ISBN 978-0-691-13482-6.
^
Further reading[edit] The Equation That Couldn't Be Solved: How Mathematical Genius
Discovered the
External links[edit] Look up symmetry in Wiktionary, the free dictionary. Wikimedia Commons has media related to Symmetry. Dutch:
v t e
Concepts Algorithm Catenary Fractal Golden ratio Plastic number Hyperboloid structure Minimal surface Paraboloid Perspective Camera lucida Camera obscura Projective geometry Proportion Architecture Human Symmetry
Tessellation
Forms Algorithmic art
Anamorphic art
Computer art
4D art
Girih Jali Muqarnas Zellige Knotting Architecture Geodesic dome Islamic Mughal Pyramid Vastu shastra Music Origami Textiles String art Sculpture Tiling Artworks List of works designed with the golden ratio Continuum Octacube Pi Pi in the Sky Buildings Hagia Sophia
Pantheon
Parthenon
Artists Renaissance Paolo Uccello Piero della Francesca Albrecht Dürer Leonardo da Vinci Vitruvian Man Parmigianino Self-portrait in a Convex Mirror 19th–20th Century William Blake The Ancient of Days Newton Jean Metzinger Danseuse au café L'Oiseau bleu Man Ray René Magritte La condition humaine Salvador Dalí Crucifixion The Swallow's Tail Giorgio de Chirico M. C. Escher Circle Limit III Print Gallery Relativity Reptiles Waterfall Contemporary Martin and Erik Demaine Scott Draves Jan Dibbets John Ernest Helaman Ferguson Peter Forakis Bathsheba Grossman George W. Hart Desmond Paul Henry John A. Hiigli Anthony Hill Charles Jencks Garden of Cosmic Speculation Robert Longhurst István Orosz Hinke Osinga Hamid Naderi Yeganeh A Bird in Flight Boat Tony Robbin Oliver Sin Hiroshi Sugimoto Daina Taimina Roman Verostko Theorists Ancient Polykleitos Canon Vitruvius De architectura Renaissance Luca Pacioli De divina proportione Piero della Francesca De prospectiva pingendi Leon Battista Alberti De pictura De re aedificatoria Sebastiano Serlio Regole generali d'architettura Andrea Palladio I quattro libri dell'architettura Albrecht Dürer Vier Bücher von Menschlicher Proportion Romantic Frederik Macody Lund "Ad Quadratum" Jay Hambidge "The Greek Vase" Samuel Colman "Nature's Harmonic Unity" Modern Owen Jones The Grammar of Ornament Ernest Hanbury Hankin The Drawing of Geometric Patterns in Saracenic Art G. H. Hardy A Mathematician's Apology George David Birkhoff Aesthetic Measure Douglas Hofstadter Gödel, Escher, Bach Nikos Salingaros The 'Life' of a Carpet Publications Journal of
Organizations Ars Mathematica
The Bridges Organization
European Society for
Related topics Droste effect Mathematical beauty Patterns in nature Sacred geometry Category v t e Patterns in nature Patterns Crack Dune Foam Meander Phyllotaxis Soap bubble Symmetry in crystals Quasicrystals in flowers in biology Tessellation Vortex street Wave Widmanstätten pattern Causes Pattern formation Biology Natural selection Camouflage Mimicry Sexual selection Mathematics Chaos theory Fractal Logarithmic spiral Physics Crystal Fluid dynamics Plateau's laws Self-organization People Plato Pythagoras Empedocles Leonardo Fibonacci Liber Abaci Adolf Zeising Ernst Haeckel Joseph Plateau Wilson Bentley D'Arcy Wentworth Thompson On Growth and Form Alan Turing The Chemical Basis of Morphogenesis Aristid Lindenmayer Benoît Mandelbrot How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension Related Pattern recognition Emergence M |