Forms of Mathematical Symmetry

Mathematical symmetry, according to the dictionary, is "a geometrical or other regularity that is possessed by a mathematical object and is characterized by the operations that leave the object invariant." The word itself is derived from the Latin symmetria, which simply refers to an "agreement in dimension, proportion and arrangement."

But what does this mean, exactly?

Human beings (and other animals) are certainly familiar with a great many examples of symmetry in everyday life. This concept absolutely abounds in nature: The human body certainly posses a certain degree of superficial symmetry - cut it down the middle and the two halves will basically be identical (but opposite).

Likewise, there is a symmetry of many internal organs (two kidneys, two lungs, etc.) and of other natural phenomena, such as plants and even geological formations. Symmetry is everywhere.

What does this have to do with mathematics?

Quantifying of Symmetry

Mathematicians have found many wonderful uses for the idea of symmetry, beginning as far back as the early Greek and Alexandrian geometers such as Euclid and Pythagoras, who were able to make ample practical use of the symmetry of certain shapes and formations in developing their clever proofs.

Today, the mathematical idea of symmetry has grown to the point where one can point so several very distinct "forms" of symmetry, each of which hold their own value within the world of mathematics.

First, there is what is called reflectional symmetry. This might also be referred to as "mirror-image" symmetry, as it merely means that a certain object (or function, in mathematics) will be identical if viewed in respect to its mirror image. As an example of this, one might look to the function f(x)=x2, which any student of mathematics will easily recognize as a simple parabola which, when graphed, possesses reflectional symmetry along the x-axis of a Cartesian coordinate system.

Rotational symmetry, on the other hand, means that a mathematical expression is equal to another mathematical expression after having been "rotated" around a certain point by a certain degree. One example of rotational symmetry might be found in a function which is known as an "odd" function, which refers to a function with 180-degree rotational symmetry, such as: f(x)=x (which translates to a line passing through the point (0,0) at a forty-five degree angle. When one half of this line is rotated around the origin by 180 degrees, it will equal the other half. This is rotational symmetry.

Translational Symmetry is somewhat less well-known than these first two, and refers to two expressions which remain identical after undergoing any number of "translations." For instance, if one begins with two symmetrical expressions, x and y, these could be considered to be translationally symmetrical if they remained in the same relation to one another after each undergoing a certain translation (for instance, if one was to multiply both expressions by 2, or take them to a certain power).

While these are three of the most common forms of symmetry, they are in no way alone in the mathematical world. By combining or expanding these, one can also find definitions of helical symmetry (a combination of rotational and translational symmetry, such as that found in springs or drill bits), fractal symmetry (which is a symmetry of magnification, where a pattern remains the same on a large scale and at an arbitrarily small scale), or any of a number of other, more obscure examples.

Mathematical Uses of Symmetry

Despite serving only as an interesting mathematical curiosity, the notion of symmetry has proven positively invaluable in many areas of mathematics. Any calculus class will require at least a basic knowledge of symmetrical functions (at the very least, such a knowledge will aid the student in their understanding), and any engineer, architect, artist, etc. will certainly be able to testify to the importance of these concepts.

Where the idea of symmetry truly seems to shine, however, is in the applied sciences. To list all possible applications of mathematical symmetry at use in physics, biology and chemistry would take the course of entire books, to be sure.

Just take a look at the structure of DNA to get an excellent example of helical symmetry in the natural world (a symmetry which made it possible for the structure of DNA to be discovered in the first place), or look into the subject of particle physics to find that certain subatomic particles are defined, in part, by their symmetry (or lack thereof) with each other. In fact, symmetry continues to play a large role in the search by physicists to find a "theory of everything," one of the leading candidates of which is the aptly-titled, "Supersymmetry," which posits that at the very moment of the creation of the universe, all forces and particles possessed perfect symmetry with one another.

Clearly, this is a massive topic which mathematicians and scientists have only just begun to understand, and which will prove endlessly exciting as human understanding continues the inevitable march onward.

Vitruvian Man - An example of symmetry

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