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 a*symmetry*, dis* symmetry* , anti*symmetry*. Symmetry and the lack of symmetry characterise the phenomena in our natural and artificial environment, as well as our ideas about the world.

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

In its *everyday use *** symmetry** is associated with its most frequent manifestations, like

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.

Concepts:

Introduction

Geometrical symmetry

Fuzzy symmetries

Our ability to geometrize

Harmony and proportion

A unifying concept

Two major symmetry classes

Organization

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

Hands

Heterochiral

Homochiral

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

Sculptures

Logos

Two and three fold rotational symmetry

Five and many fold rotational symmetry

Flowers

Rotational motifs in buildings

Folk art

Creating rotational patterns

Reflection & rotation

Combining symmetries

Rotational and mirror symmetry in flowers

Primitive organisms

In overhead lighting

Logos

In sculpture and display

In architecture

Radical symmetry

Paper cutting

Fact to consider

Snowflakes

Hexagonal symmetry

Diversity in shape

Uniform growth

Looking back

6000 photos

Artificial snowflakes

Buildings from above

Polygons

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

Duality

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

Balloons

The origin of shapes

Walnuts

Molecules

Antisymmetry

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

Papercutting

Inducing the feeling of motion

Helix & spiral

Helices

Spirals

In natural phenomena

In art

Shells

Life forms

Towers

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

Proportions

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

Decorations

Artistic patterns

Dancing with symmetry

Facades

Diamonds & glass

Crystals

Minerals

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

Packing

Quasi crystals

Serendipity