SOMA News 27 Nov 2001 E-Mail.

# A Primer on Symmetry

I previously wrote a SOMA Newsletter about Symmetry ( 1999.09.09 SOMA and Symmetry ), and I am currently writing another about the Mathematics of Symmetry. But Thorleif correctly pointed out to me that I had never really explained what I meant by the word "Symmetry". This Newsletter is a long overdue attempt to correct that oversight.

Formally, Symmetry is a mathematical concept used to describe shapes that do not seem to change when they are rotated, or when they reflected by a mirror, or both at the same time. This is basically what I mean when I use the word, but here I will not enforce the rigor that a mathematician might. Flowers and animals display symmetry in their external shapes, but perhaps not with the precise exactness the term is supposed to imply. Let me begin with a few everyday examples of symmetry.

## Symmetries from A to Z

The capital letters of the alphabet are good examples. The letters A and C display what is called "Bilateral Symmetry". That just means that one side is the mirror image of the other side. The "Mirror Plane" is vertical for the letter A, and horizontal for the letter C, as represented by the dashed lines in the picture. These letters and others like them can be said to have a "2-fold Mirror Symmetry": they essentially look just like their mirror images, although a bit of rotating may be necessary to exactly superimpose the two shapes upon each other if the proper mirror plane is not chosen. The letter Z is an example of "2-fold Rotational Symmetry"; it looks the same after being rotated by 180° around its center. But it does not have mirror symmetry. Shapes like this are called "chiral", which means that they can not be superimposed on their mirror images. Like shoes or gloves, chiral shapes come in pairs, each of which is "different" from its mirror partner.

The letter X, by contrast, displays both rotational and mirror symmetries. Since it can be manipulated by rotations or reflections so that any of its four arms can appear in, say, the upper left corner, it is called an example of "4-fold symmetry". The letter F, by contrast, does not have either rotational or mirror symmetry. It could be said to be "asymmetric", or to have no symmetry, but I prefer to say that it has "1-fold Symmetry". It is, of course, chiral as well.

## Flowers for Algebra

The symmetries of flowers help me guess to which Family of plants they might belong. Pea and Bean flowers usually have 2-fold mirror symmetry like the letters A and C, as do Iris flowers. Members of the Mustard Family generally have flowers with 4-fold rotational symmetry. But they also have mirror symmetry, so they could more properly be said to have 8-fold symmetry. The "Rules of Symmetry" say that if a shape has any mirror symmetry, its total symmetry count is twice the count of its rotational symmetries. Many Families of flowers also exhibit rotational symmetries in multiples of three or five. The most amazing mathematical property found in flowers is to me the appearance of the Fibonacci numbers in the patterns of seeds in sunflowers. Compared to that surprise, I suppose complicated symmetries aren't unusual.

Flowers do not seem to exhibit chiral symmetries very often - for some innate reason, they seem to prefer to include mirror reflection planes in their designs. I assume they are exploiting some genetic programming shortcut. Twining vines come to mind as the best examples of strong handedness in the plant world. My favorite natural example of chirality is the snail's shell. If you examine a large number of snails, you will find that nearly all of any species of snail have the same handedness, though you will probably be able to find a rare "other-handed" specimen. The animal kingdom does not seem to be as diverse in symmetries as the plant kingdom. One beautiful exception is the echinoderm family, which includes starfishes and sea urchins, whose members often exhibit 5-fold radial symmetry. However, the truly spectacular symmetries to be found in plants and animals seem to be relegated to the microscopic world, in pollens and planktons and such.

## Showy Snow

Here are some beautiful hexagonally symmetrical snow crystals, grown by some Caltech scientists. They are studying the detailed physics of how snow crystals form, and as part of their research are creating snow crystals in the laboratory. They are even beginning to engineer "designer" snow crystals with patterns of their own choosing. Snow crystals typically have 6-fold rotational symmetry, but since they also have mirror symmetry, they really have 12-fold total symmetry. Click here to visit a truly wonderous website, "www.SnowCrystals.net"

Symmetry is also often found in human art, in cultures throughout the ages and around the world. Symmetry is often a major part of the designs seen in quilts and mosaics, as is the mathematically related field of "tiling a surface". I think that the ability to create and recognize symmetrical patterns must be hard-wired deep inside the human brain, since the pleasures of symmetry seem to be so universal. Maybe this ability came about to help us recognize and remember the plants and animals around us. Perhaps it is part of whatever allows us to envision abstract mathematics. I wonder if apes and dolphins, or perhaps dogs and cats, can recognize symmetries.

Most of the previous examples of symmetry can be considered to be two-dimensional cases. Almost all animals with backbones, and many without them, are three-dimensional examples of 2-fold bilateral mirror symmetry. More complex examples are the Platonic Solids of geometry, which are defined as "polyhedra with regular polygons as faces". That is, they are all of the three dimensional shapes that can be built using only identical triangles, squares, or pentagons. The Platonic Solids are good models of how to count the number of symmetries in a shape. For instance, the cube can have any one of its six faces on top, and then any of four sides to the front. Six times four, times two for mirror symmetry, gives a total of 48-fold symmetry. Or an octahedron can have any one of its eight triangles on top, times three sides, times two: 48-fold again. One of my favorite geometric shapes is what we in the States call a "soccer ball", and which everybody else more appropriately calls a "football". It has 120-fold icosohedral symmetry: any of the 12 pentagons on top, times any of five rotated to the front, times two. Or, any of the 20 hexagons on top, times 3 rotations, times two. The most symmetrical shape of all is the sphere - it has an infinite number of rotational symmetries. Times two, of course, for mirror reflection.

## Prize Winning Symmetry

This year can be considered a particularly special one for Symmetry, because of the 2001 Nobel Prize in Chemistry. Many molecules come in two mirror image forms, called "enantiomers", one of which can be called "right-handed" and the other "left-handed". Although they have exactly the same chemical formula, they have symmetrically different shapes, and can behave completely differently in living organisms. For example, one enantiomer of the drug thalidomide caused terrible birth defects in the 1960s, while the other mirror form was a valuable medicine. The Nobel winners, two from the United States and one from Japan, devised special catalysts to ensure that only the correct form of a desired chemical compound is produced, a process called "asymmetric synthesis".

There are many other examples of symmetry that come to mind, such as the arrangements of atoms in crystals, and the way fruit trees are often planted in precise rows that cause interesting optical effects as you drive past them. And, of course, what these two examples have in common. And a property called "supersymmetry" is an important part of the "Grand Unification Theories" of physicists as they try to turn their cosmological speculations into a "Theory Of Everything". It is possible that symmetries in 11 dimensions may help them explain how the Universe began, and what will happen to it in the end, and all of the messy little details of all the stuff in between. In this article, I haven't really tried to "define" the concept of Symmetry in any rigorous mathematical or linguistic sense, but I have tried instead to impart the "flavor" of Symmetry, and perhaps a bit of why the subject continues to fascinate me so much.

Submitted by Bob Allen, WBØFVK <wb0fvk@arrl.net>.

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