What Is Symmetry?

We may learn the etymology of the word from The Words of Mathematics by S. Schwartzman:

symmetric (adjective), symmetry (noun): the first element from the Greek sun- "together with," from the Indo-European root ksun "with." The second element is from Greek metron "a measure." The Indo-European root is probably me- "to measure." Suppose two points are symmetric with respect to a line; if you measure the distance between one of the points and the line of symmetry, then "together with" that measurement you have simultaneously also measured the distance between the other point and the line of symmetry; the two distances are equal.

As a matter of fact, the symmetry in line is just one appearance of the term symmetry in geometry. In mathematics, in general, the word "symmetry" may also be encountered in non-geometric contexts.

In plane geometry, we have symmetries in a line and in a point, the latter is often called a half-turn. In 3D, similarly, we study symmetries in a plane, a line, and a point. Accordingly, two figures are symmetric (in a plane, a line, or a point) if points of one uniquely correspond to the points of the other by means of a symmetry. To obtain a symmetric figure one performs a symmetry transform that maps points into their symmetric counterparts. A figure may be its own image under a symmetry, in which case it is said to be symmetric or possess a plane of symmetry, or an axis of symmetry, or a center of symmetry. When symmetry is associated with a transformation it is often referred to as a reflection.

A circle has a center of symmetry whilst every line through the center serves as an axis of symmetry. A square is symmetric in its center, the two diagonals, and the two lines joining the midpoints of the opposite sides. Both shapes have additional symmetries. A circle maps onto itself with any rotation around its center. A square maps onto itself by any rotation through a multiple of $90^{\circ}.$ The isometries of a figure form a group known as the symmetry group of the figure. The classification of symmetry groups is a matter of intense study. The symmetry group of a regular n-gon is called dihedral - $D_n$ - group. Infinite strips of a repeated motif (known as border designs or frieze patterns) may have $7$ different symmetries. Wall paper patterns may have $17$ different types of symmetry. (The proof of the latter fact was given by George Polya in 1924.)

An apparent lack of symmetry in the definition of the symmetry in a circle suggests that, in mathematics, the concepts of symmetry may transcend our intuitive notions.

In algebra, symmetric group is a group of permutation and a relation may or may not be symmetric. A polynomial of several variables also may or may not be symmetric. For example, polynomial $P(x,y)=x^{3}+xy+y^{3}$ is symmetric because it does not change under a permutation of its variables: $P(x,y)=P(y,x).$

The argument by symmetry resembles that by similarity. In a triangle, there are three pairs of medians. Should anything be true for any one pair of medians that is shown true without invoking any specific properties of sides or angles of the triangle, then it could be argued by symmetry that the same is true for any other pair of medians. One of the proofs of Morley's theorem supplies another example of an argument by symmetry.

[E. Barbeau, pp. 21-22] refers to an example by [D. Wells]. Wells observes that from a non-symmetric equality $a=b+c$ one can deduce (via double squaring and rearrangement of terms) a symmetric equality

$a^{4} + b^{4} + c^{4} = 2b^{2}c^{2} + 2c^{2}a^{2} + 2a^{2}b^{2}.$

and asks where does the symmetry come from. He argues that the double squaring eliminates the effect of minus signs in the expressions, $a - b - c,$ $b - c - a,$ or $c - a - b,$ and leads to the same result as it would for the expression $a + b + c.$ This implies that all four expressions are factors of

$P(a, b, c) = a^{4} + b^{4} + c^{4} - 2b^{2}c^{2} - 2c^{2}a^{2} - 2a^{2}b^{2}.$

E. Barbeau observes that factoring $P(a, b, c)$

$P(a, b, c) = (a + b + c)(a - b - c)(b - c - a)(c - a - b)$

makes the nature of the symmetry evident and notes that on the right, by Heron's formula, we have 16 times the area of a triangle with sides a, b, c which may be expected to be a symmetric form in the three variables.

It is also possible to observe that, since

$P(a, b, c) = P(\pm a, \pm b, \pm c),$

the fact that $P(a, b, c)$ has one of the factors, say, $(a - b - c),$ makes it indeed obvious that it has other three factors as well. So if

$P(b + c, b, c) = 0$

checks out we, by symmetry, have grounds to claim that

$P(a, b, c) = k(a + b + c)(a - b - c)(b - c - a)(c - a - b),$

for some k. That $k = 1$ can be seen by comparing any two equal terms on both sides, say $a^4.$

Analogous situations arise in geometry. For example, in the proof of the Isoperimetric Theorem for quadrilaterals, given a kite $ABCD$ with $AB=BC$ and $AD=CD,$ the ellipse with foci at $B$ and $D$ passing through $A$ also, by symmetry, passes through $C.$

References

E. J. Barbeau, Mathematical Fallacies, Flaws, and Flimflam, MAA, 2000
E. J. Borowski & J. M. Borwein, The Harper Collins Dictionary of Mathematics, Harper Perennial, 1991
J. H. Conway, H. Burgiel, C. Goodman-Strauss, The Symmetries of Things, A K Peters, 2008
H. S. M. Coxeter, Introduction to Geometry, John Wiley & Sons, 1961
J. Daintith, R. D. Nelson (eds), The Penguin Dictionary of Mathematics, Penguin Books, 1989
S. Schwartzman, The Words of Mathematics, MAA, 1994
P. Zeitz, The Art and Craft of Problem Solving, John Wiley & Sons, 1999
D. Wells, You are a Mathematician, John Wiley & Sons, 1995, p. 88

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