The Unseen World of Symmetry: A Mathematical Riddle

Symmetry, a concept that resonates from the smallest subatomic particles to the grandest architectural designs, is more than just a visual phenomenon. It's a fundamental language of nature, a key to understanding the universe itself. But what lies beneath the surface of symmetrical objects? What invisible numbers and laws govern their existence?

The Language of Symmetry

Symmetry plays a crucial role in various scientific fields:

• Molecular Structure: The mathematics of symmetry helps us understand the possible structures of crystals.
• Microbiology: Symmetrical objects, like the swine flu virus, can use symmetry's efficiency to propagate rapidly.
• Biology: Symmetry communicates genetic information. Symmetrical features often signal good genes and a healthy upbringing, making an individual a more attractive mate.
• Particle Physics: The Large Hadron Collider at CERN relies on symmetry to predict the existence of fundamental particles, which appear to be facets of a symmetrical shape in a higher-dimensional space.

Galileo's Insight

Galileo Galilei eloquently captured the power of mathematics in understanding the world:

"The universe cannot be read until we have learned the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles, and other geometric figures, without which means it is humanly impossible to comprehend a single word."

Symmetry in Art and Architecture

Artists, like scientists, are fascinated by symmetry. However, their relationship with it is often more complex. Thomas Mann, in The Magic Mountain, describes a snowflake with "perfect precision" as "deathly, the very marrow of death."

Breaking Symmetry

Sometimes, the most captivating art involves setting up expectations of symmetry and then deliberately breaking them. A visit to the temples in Nikko, Japan, revealed a gateway with eight columns. Seven were perfectly symmetrical, but the eighth was intentionally turned upside down. This deliberate act, inspired by the Japanese essays in idleness from the 14th century, suggests that:

"In everything, uniformity is undesirable. Leaving something incomplete makes it interesting and gives one the feeling that there is room for growth."

Even when building the imperial palace, the Japanese always leave one place unfinished.

The Alhambra: A Palace of Symmetry

If forced to choose one building to live in for its celebration of symmetry, the Alhambra in Granada would be a prime candidate. The Moorish artists, forbidden from drawing things with souls, explored geometric art, embedding symmetry throughout the palace. Reflective symmetry in the water and intricate designs on the walls showcase the beauty and complexity of symmetry.

Galois and the Language of Motion

Évariste Galois, a revolutionary mathematician, developed a language to describe symmetry based on motion. Instead of viewing symmetry as static, Galois saw it as a series of transformations that leave an object looking the same.

Magic Trick Moves

Imagine a magic trick: you perform an action on an object, conceal it, and then reveal it, unchanged. These "magic trick moves" characterize the symmetry of an object. For example, rotating a tile in the Alhambra by 90 degrees and placing it back in its original position leaves the wall looking identical.

The Algebra of Symmetry

Galois realized that the interaction between symmetries is crucial. Combining one magic trick move with another results in a third magic trick move. This concept allows us to develop an algebra of symmetries, where letters represent rotations and reflections. The order in which these operations are performed can significantly impact the final result, distinguishing between objects with the same number of symmetries but different underlying structures.

Laws of Symmetry

Galois established laws governing how symmetries interact, creating "Sudoku tables" where no symmetry appears twice in any row or column. These rules allowed him to prove that only two objects exist with six symmetries, corresponding to the symmetries of a triangle and a six-pointed starfish.

The Power of Abstraction

Galois's language allows us to understand that seemingly different walls in the Alhambra possess the same underlying abstract symmetries. For example, walls with triangles and walls with six-pointed stars can share the same symmetry, known as 6-3-2. Similarly, a wall, a ceiling, and a floor can all represent the same symmetrical abstract object, 4-4-2.

Beyond the Visible

The power of Galois's mathematical language extends beyond the visible world. It allows us to create symmetrical objects in unseen dimensions, from two-dimensional to infinite-dimensional spaces. Mathematicians can use this language to explore and create symmetrical objects in high-dimensional spaces.

Conclusion

Symmetry is a profound and multifaceted concept that permeates the universe. From the beauty of snowflakes to the fundamental laws of physics, symmetry offers a glimpse into the underlying order of reality. By understanding the language of symmetry, we can unlock new insights into the unseen world and appreciate the elegance and interconnectedness of mathematics and the natural world.