Symmetries

Some of the most beautiful symmetries can be found on the wings of butterflies. Butterflies developed such flamboyant colors and figures for many reasons. One of which is to create a camouflage to protect themselves from predators. Some other species of butterflies developed such brilliant colors to be used in courtship, so that the same species can recognize each other. Finally, certain poisonous butterflies developed colorful patterns to warn prevent predators from eating them by accident.

Not all mandalas are artificially constructed for artistic purposes; they can be constructed naturally as well. The two flowers above have each formed a mandala-image. The petals and stigmas have effectively separated the structure into several sections, and thus they have naturally constructed a mandala of its own.

Snowflakes are symmetrical because they reflect the internal order of the water molecules as they arrange themselves in the solid state (the process of crystallization). Water molecules in the solid state, such as in ice and snow, form weak bonds (called hydrogen bonds) to one another. These ordered arrangements result in the basic symmetrical, hexagonal shape of the snowflake. In reality, there are many different types of snowflakes (as in the clich¿ that ‘no two snowflakes are alike’); this differentiation occurs because each snowflake is a separate crystal that is subject to the specific atmospheric conditions, notably temperature and humidity, under which it is formed.

Chinese opera performers mainly have two methods of makeup: masks and facial painting. The frequent on-stage change of masks, without the audience noticing, is a special technique, known as changing faces.  The funny thing is that, each of the faces should be symmetric and different from each other to show different characteristics.

In Chinese philosophy, the concept of yin-yang , which is often called “yin and yang”, is used to describe how seemingly opposite or contrary forces are interconnected and interdependent in the natural world; and, how they give rise to each other as they interrelate to one another. Many natural dualities (such as male and female, light and dark, high and low, hot and cold, water and fire, life and death, and so on) are thought of as physical manifestations of the yin-yang concept. The concept lies at the origins of many branches of classical Chinese science and philosophy, as well as being a primary guideline of traditional Chinese medicine,  and a central principle of different forms of Chinese martial art and exercise.

This is a character representing ” marriage” and only can be seen in a wedding in China. It is symmetric because it shows the connective relation between the bride and bridegroom. The half of the character can be recognized as one of the spouse.

This is a character representing ” marriage” and only can be seen in a wedding in China. It is symmetric because it shows the connective relation between the bride and bridegroom. The half of the character can be recognized as one of the spouse.

It has a D2 symmetry

Next time you’re in Baruch’s Vertical Campus, I encourage you to drop by the Mathematics Department Office on the 6th floor. Newly installed on the wall, in a utilitarian position behind the photocopier, is the base of a Sand Mandala which was made at Baruch last September by visiting Tibetan monks.

The base is not the mandala itself — that was swept away at the end of the day. The base is an embodiment of the mathematics underlying the mandala’s construction. It’s what’s left of the mandala after you’ve swept away the sand, the color, the decoration, the ritual — and the mandala itself. It’s nothing, and yet it’s something — it’s symmetry.

Rockefeller Center Painting

MM Pattern

Demetrios Benetos