Applications of Symmetries

What is symmetry?

SYMMETRY is a phenomenon, a class of properties and a concept present in (1) all scientific disciplines, and a variety of arts.

Symmetry bridges different disciplines, sciences and arts and even different cultures.

The term symmetry is of ancient Greek origin. Its meaning is in close association with the related terms of asymmetry, dis symmetry , antisymmetry. Symmetry and the lack of symmetry characterise the phenomena in our natural and artificial environment, as well as our ideas about the world.

Traditional meaning of symmetry

The meaning of this term went through a fabulous transformation during its use for dozens of centuries. The proper translation of the Greek term symmetria – (from the prefix sym [common] and the noun metros [measure]) – is 'common measure'. The Greeks interpreted this word, as the harmony of the different parts of an object, the good proportions between its constituent parts. Later this meaning was transferred to e.g., the rhythm of poems, of music, the cosmos ('well-ordered system of the universe as contrast of chaos'). Therefore the Latin and the modern European languages used its translations like harmony, proportion until the Renaissance. In wider sense, balance, equilibrium belonged also to this family of synonyms.  Some way symmetry was always related to beauty, truth and good. (These relative-meanings determined its application in the arts, the sciences, and the ethics, respectively.) Symmetry was not only related to such positive values, it became even a symbol of seeking for perfectness.

Common meaning of symmetry

In its everyday use symmetry is associated with its most frequent manifestations, like reflection or, in other words, mirror-symmetry, rotation (rotational symmetry), and repetition (translational symmetry). A few further geometrical appearances of symmetry belong also to this class of interpretations, like glide reflection , similitude , affine projection, perspective, topological symmetry . All they are associated with the observation, that one performed a certain geometric operation (a transformation) on an object; and during that transformation one (or more) geometric properti(es) of that geometric object did not change (were conserved). That/those property/ies proved to be invariant under the given transformation. They are called 'symmetry' in everyday life.

Generalised, contemporary meaning of symmetry

In generalised meaning one can speak about symmetry if
under any (not certainly geometric) transformation (operation),
at least one (not certainly geometric) property
of the (not certainly geometric) object
is left invariant (intact).


Thus we made a generalisation in 3 respects: to
any transformation,
any object, and its
any property.


This generalised meaning of symmetry made possible to apply symmetry to materialised objects in the physical and the organic nature, to products of our mind, etc. Over geometric (morphological) symmetries, we can discuss functional symmetries and asymmetries (e.g., in the human brain), gauge symmetries (of physical phenomena); properties, like colour, tone, shadiness, weight, etc. (of artistic objects).

Asymmetry: The lack of symmetry

Dissymmetry: The observed object is symmetric in its main features, but this symmetry is slightly distorted (e.g., an arabesque ornament)

Antisymmetry: The observed object is symmetric in one of its properties, but one of its other properties changes to its opposite (e.g., a chess-board)

The concept of symmetry is ubiquous and concepts such as those listed below are themselvesstructured symmetrically. Groups I to IX cover on point group symmetries such as bilateral symmetry, chirality, rotational symmetry, and the symmetries of polyhedra. Groups XI to XV cover space group symmetries such as translational symmetry, helices and spirals, planar patterns, and the symmetry of crystals. Group X, explores antisymmetry, the symmetry of opposites in which geometrical symmetry is combined with color changes or other property reversals.

     Geometrical symmetry
     Fuzzy symmetries
     Our ability to geometrize
     Harmony and proportion
     A unifying concept
     Two major symmetry classes
Bilateral symmetry
     Mirror symmetry
     In plants
     The human body
     The human face
     Double heads
     In architecture
     In religion and music
Shape & movement
     Forward motion: bilateral symmetry
     In land, sea, and air vehicles
     Vertical motion: cylindrical symmetry
     Spherical symmetry
     More cylindrical symmetries
Right hand, left hand
     Molecular Chirality
     Chirality and life
     Creating Chiral shapes by dissection
     Recipe for La coupe du roi
     Right brain, left brain
     Right and left in the universe?
Pinwheels & windmills
     Rotational symmetry
     Make your own pinwheel
     Rotating blades
     Two and three fold rotational symmetry
     Five and many fold rotational symmetry
     Rotational motifs in buildings
     Folk art
     Creating rotational patterns
Reflection & rotation
     Combining symmetries
     Rotational and mirror symmetry in flowers
     Primitive organisms
     In overhead lighting
     In sculpture and display
     In architecture
     Radical symmetry
     Paper cutting
     Fact to consider
     Hexagonal symmetry
     Diversity in shape
     Uniform growth
     Looking back
     6000 photos
     Artificial snowflakes
Buildings from above
     Symmetries of regular polygons
     Famous shapes
     Two to six fold symmetries
     Eight and many fold symmetries
     Round shaped buildings
Cubes & other polyhedra
     The cube and its symmetries
     The five regular polyhedra
     Symmetries of the regular polyhedra
     Models of the regular polyhedra
     Regular polyhedra in nature
     The regular polyhedra in a planetary model
     Artistic dodechahedra and icosahedra
     Star polyhedra
     Archimedean polyhedra
     The Buckyball molecule
     Prisms and antiprisms
     The sphere
     Polyhedra in sculptures
Balloons, walnuts & molecules
     The origin of shapes
     Antimirror symmetry
     Antirotational symmetry
     Antisymmetry in the universe
     More subtle examples of antisymmetry
     Antisymmetry in geography
     Antisymmetry as a literary device
Repeating everything
     Translational symmetry
     Repetitive symmetry
     The seven classes of band patterns
     Hungarian needlework
     Greek ornaments
     Roman border designs
     Egyptian border designs
     Mexican patterns
     Native American designs
     Arabic patterns
     Persian border designs
     Japanese border designs
     Chinese lattice designs
     Building decorations
     Inducing the feeling of motion
Helix & spiral
     In natural phenomena
     In art
     Life forms
     The Fibonacci numbers
     In leaves and plants
     In decorative design
     The golden ratio
     The golden section
     The golden rectangle
     The logarithmic spiral
     Logarithmic spiral and golden section
Bees & engineering
     Patterns from circles
     Patterns from hexagons
     Hexagonal designs, human made and natural
     Covering the surface with regular polygons
     Patterns from pentagons
Rhythm on the wall
     Planar patterns
     Creating planar patterns
     Filling the surface completely
     Decorative patterns
     Seventeen symmetry classes for planar patterns
     Native American designs
     Artistic patterns
     Dancing with symmetry
Diamonds & glass
     External symmetry of crystals and the magic number 32
     Stereographic projections
     Preparations of stereographic projections
     Internal structure of crystals
     The magic number 230
     Diamond and graphite
     Diamond and glass
     Quasi crystals