The concept of symmetry is pretty familiar to all of us. We can recognize it in nature, and in arts and architecture as well . The word  "Symmetry" is derived from the Greek word "symmetrus"  meaning regular , well proportioned or harmonious. Hermann Weyl , a 20th century German philosopher and author of the classic book "Symmetry"  defines it as a transformation of an object which leaves it unchanged...i.e it can be divided into two of more identical parts. The branch of mathematics that deal with symmetry is called "Group theory" and finds wide application for investigating patterns in art and nature. Symmetry is found in crystals , animals , plants,  music , biological forms ...etc and plays an important role in chemistry , physics , crystallography, architecture ..especially decorative arts where the study of patterns helps in classification of artifacts.

In this and a couple of following posts we will look at symmetry from a recreational angle and we will mostly concentrate on geometric symmetry  by studying forms of leaves , bees' honeycomb , snowflakes , pottery , rangolis , mandalas , textiles, architecture , ornamentation .....etc . We will look at the classification of commonly seen objects around us into groups like dihedral / bilateral  symmetry , rotational symmetry , dilation symmetry ...etc .  

The most frequently observed symmetry are those related to reflection and rotation.

Fig -1  : Dihedral / Mirror symmetry

Let us consider reflection or mirror symmetry ( mathematically called dihedral/ bi-lateral  symmetry)  first where each half of the pattern is a mirror reflection of the other. These halves which are themselves asymmetrical are called enantiomorphs. Some of the  most common examples of mirror symmetry are the human face and the alphabet A ( symmetrical about the vertical axis) and the letter E ( symmetrical about the horizontal axis ). These belong to the symmetry order one ( as they have only one mirror line) and is denoted by the symbol D1. A higher number of lines indicates a higher order of symmetry and is denoted by Dn where "n" denotes the number of mirror lines. Some examples are shown in Fig 1 .

What other examples do we have of dihedral symmetry? We will discuss two more....music and words.If you think of music stretching out in time , it's mirror image is obtained by simply playing it backward! Music can be turned upside down by changing the high notes to a low note and vica versa. The Austrian composer Mozart is believed to have composed a piece of music in two part which displayed both types of reflection! The second part was just the first part inverted and played backward!

Palindromic words which read the same in both direction are said to have bi-lateral symmetry. Examples are ....malayalam , radar , rotor ,repaper, madam ...etc . The word "COOKBOOK" exhibits horizontal mirror symmetry whereas the word  "MAXIMUM" displays vertical mirror symmetry when the letters are written one below the other . The word "mud" when reflected in a mirror reads as "bum" and the word "SWIMS" reads the same when you turn it upside down. See how many such words you can discover.

Fig 2  : Cyclic / Rotational symmetry

Next we will look into Rotational symmetry ( mathematically called Cyclic symmetry) where the object remains unchanged  after rotation by an angle of "360 deg / n" where "n" is an interger. A circle is invariant under rotation through any angle and is said to have full rotational symmetry or C∞. An equilateral triangle has a rotational symmetry of C3 as well as dihedral symmetry of D3 . . If an object is invariant under many different types of transformations then the object is said to possess a high degree of symmetry.  
As a fun exercise can you classify all the alphabets (in capital form ) into their groups? What is the symmetry group of a parallelogram? ...a rhombus ? ...an 8  X 8 chessboard? ...the crossword pattern in today's newspaper ?.. the flowers in your garden? 

Flowers , starfish , snowflakes display rotational symmetry whereas insects and animals display dihedral symmetry

Some polyhedras ( 3D geometric models)  display both rotational and dihedral symmetry where some displays only one type.

Phaeodari

Remember the Platonic and Archimedean  solids we encountered in our earlier posts?  Some of these occur frequently in nature as complex crystalizations of minerals and appear as the skeletal remains of several species of amoebic sea creatures in the Radiolarian phylum. These creatures were studied and beautifully illustrated, more than a century ago ,  in 1904 by the Victorian-era biologist Ernst Haeckel in his Kunstformen der Nature.
I have reproduced one of his drawings alongside ( from Wikipedia )  .. Can you recognise the icosahedra form?

You can see some of  his  illustration in all its beauty and detail ( and in zoomable form) at the link given below. These photos give a lot of ideas and inspirations for creating beautiful repetitive patterns/ tessellations for your artistic expression .  http://algorithmic-worlds.net/Haeckel/haeckel.php



Symmetry is one idea by which man through ages has tried to comprehend and create order , beauty and perfection.

If you can lay your hands on the book entitled " Visual Symmetry" by Magdolna Hargittaione and István Hargittai  you will be enthralled by the sheer splendor of the imagery in the book. This is one of the finest books I have some across on symmetry and one can see it is a labour of love .I have reproduced a couple of pages from this book to whet your appetite for more , so that you make efforts to find and read this book .


 The other book I would recommend is " Symmetry"  by Hermann Weyl ..... an all time classic and a must read for both art and science oriented people.

In my next posts we will look at dilation symmetry, transalational symmetry in one direction ( frieze / border  patterns) , and in two directions ( wall paper/ tiling  patterns).  Until then start looking at objects and patterns in and around you ( starting from your face in the mirror ) and enjoy the beauty of
symmetry .
Note:
The next post on Frieze patterns has been posted on 4th Dec 11 at http://xploreandxpress.blogspot.com/2011/12/fun-with-mathematicssymmetry-in-art-and.html


 Note: Some of the pictures above are taken from the internet and used  only for education and inspiration . If I have violated  any copyright issues , please send me comments and I will remove the associated pictures.