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Symmetry

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Symmetry describes a state of balance or proportion. In daily life, we use it to mean something looks harmonious. In mathematics, the term has a much more precise definition. An object is symmetric if it remains invariant under certain transformations. Invariance means the object does not change its overall appearance after the transformation occurs. These transformations can include moving, flipping, or rotating an object. The absence of symmetry is known as asymmetry.

Asymmetric (PSF)
Asymmetric (PSF)

Geometric symmetry involves specific types of movements that preserve an object's shape. Reflectional symmetry, also called mirror symmetry, occurs when a line or plane divides an object into two identical mirror images. Rotational symmetry happens when an object is turned around a fixed point without changing its appearance. For example, a triskelion shows three-fold rotational symmetry.

The armoured triskelion on the flag of the Isle of Man
The armoured triskelion on the flag of the Isle of Man

Translational symmetry occurs when every point of an object moves the same distance in the same direction. Helical symmetry combines both translation and rotation along a screw axis. Scale symmetry means an object does not change its shape when it is expanded or contracted. Fractals are a famous example of this, as their smaller parts look like their larger parts.

BigPlatoBig
BigPlatoBig

Mathematical symmetry extends far beyond simple shapes into abstract logic and algebra. In logic, a relation is symmetric if the connection works both ways. For instance, the relation "is the same age as" is symmetric because if Paul is the same age as Mary, Mary is also the same age as Paul. In broader mathematics, an object is symmetric if a specific operation preserves a certain property. The collection of these operations is called a group. This concept appears in many fields, such as symmetric matrices in linear algebra or symmetric probability distributions in statistics.

In the field of physics, symmetry is a fundamental tool for understanding the universe. Physicists define symmetry as invariance under coordinate transformations. Many scientists believe that the laws of nature originate from these symmetries. The Nobel laureate PW Anderson suggested in 1972 that physics is essentially the study of symmetry. This connection is explained by Noether's theorem, which links continuous mathematical symmetries to conserved quantities like energy or momentum. Furthermore, Wigner's classification shows that the symmetries of physical laws determine the properties of particles in nature.

Biology shows diverse uses of symmetry in body shapes and structures. Many animals, including humans, possess bilateral symmetry. This means they are symmetric across a sagittal plane that divides the body into left and right halves. This structure is useful for animals that move in a specific direction. However, internal organs are often arranged asymmetrically. Other organisms, like sea anemones, exhibit radial or rotational symmetry. This allows them to respond to threats from any direction. Echinoderms, such as starfish and sea urchins, often display fivefold symmetry.

Chance and a Half, Posing
Chance and a Half, Posing

Human perception is highly tuned to recognize certain types of symmetry. In 1897, Ernst Mach observed that humans are especially sensitive to vertical reflection symmetry, such as in the human face. Modern neuroimaging studies have confirmed this special sensitivity. When people perceive symmetry, activity is found in the extrastriate regions of the occipital cortex, including areas like V4 and the lateral occipital complex. This process is incredibly fast. Humans can detect reflectional symmetry in as little as 100 to 150 milliseconds. This efficiency helps the brain group objects and organize visual information.

Symmetry also plays a vital role in architecture and the arts. The Taj Mahal is a famous example of architectural balance. It displays bilateral symmetry when viewed from the side and fourfold symmetry when viewed from above.

Taj Mahal, Agra views from around (85)
Taj Mahal, Agra views from around (85)

Islamic architecture often uses complex patterns involving rotation and translation. You can see symmetry in the intricate designs of the Alhambra or in Persian rugs.

Farsh1
Farsh1

Even the process of making pottery on a wheel utilizes symmetry.

Makingpottery
Makingpottery

From the layout of floor plans to the design of individual tiles, symmetry creates a sense of order and beauty in our built environment.

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🖼️ Images & Media (10)
File:Asymmetric (PSF).svg
Asymmetric (PSF).svg
File:Sphere symmetry group o.svg
Sphere symmetry group o.svg
File:BigPlatoBig.png
BigPlatoBig.png
File:The armoured triskelion on the flag of the Isle of Man.svg
The armoured triskelion on the flag of...
File:Chance and a Half, Posing.jpg
Chance and a Half, Posing.jpg
File:Taj Mahal, Agra views from around (85).JPG
Taj Mahal, Agra views from around (85).JPG
File:Makingpottery.jpg
Makingpottery.jpg
File:Farsh1.jpg
Farsh1.jpg
File:kitchen kaleid.svg
kitchen kaleid.svg
File:Major and minor triads, circles, dozenal.png
Major and minor triads, circles, dozenal.png
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