Symmetry
Natural forms such as snowflakes, sea shells, water droplets, mineral crystals, and faces exhibit "symmetry." Often, symmetry correlates with beauty: a visual stimulus translated directly to sensations of pleasantness without words or conscious thought. In this post we'll explore what mathematicians mean by symmetry and see how symmetry inspires work at Differential Geometry. Our aim is not maximum generality, but bridging familiar qualitative experiences of "symmetry" with mathematical structure.
To start, we'll distinguish "objects" and "transformations." Loosely, objects are shapes, such as circles, disks (filled circles), spheres, line segments, squares (filled or not), and other polygons. In the same qualitative sense, transformations are operations performed on objects, such as rotation, translation, reflection, uniform or non-uniform stretching, and folding. Linguistically, we are thinking of objects as nouns, and transformations as verbs.
Every object has at least one symmetry: "Do nothing," or "leave it alone." Mathematicians call this the identity transformation. This is a fundamental property, which we record for later: Doing nothing is a symmetry.
As we're using the terms "object" and "transformation," they are mathematical abstractions. Part of their interest, however, is an approximate correspondence with real snowflakes, sea shells, water droplets, crystals, and faces.
Rotation
Before trying to define symmetry, let's look at what happens to a square, an octagon, and a circle under rotation about their common center. Before reading further, try to put into words what you observe.
The initial orientations of a square, octagon, and circle are shown in light gray. When these three objects are rotated, the square and octagon do not always "line up with" with their initial orientations. The circle, however, does.
The blue flashes visually emphasize the symmetry. When one of the rotated shapes coincides with its initial orientation, it is drawn in a heavy blue line rather than a thin black line. After one-quarter turn, the square "lines up with itself" and flashes blue. After one-eighth of a turn, the octagon lines up with itself and flashes blue. The circle continually lines up with itself, and is always drawn with a heavy blue line.
Because the square lines up with itself after one-quarter turn, it also lines up with itself after two-quarters turns, or one-half turn; and after three-quarters turns; and four-quarters turns; and so on. This turns out to be a fundamental property of symmetry: A symmetry followed by a symmetry yields a symmetry. In math-speak, A composition of symmetries is a symmetry.
Similarly, the octagon lines up with itself after one-eighth turn, and therefore also lines up with itself after two-eighths turns, or one-quarter turn; and after three-eighths turns; and four-eighths turns, or one-half turn; and so on.
In all this talk of "turns," we implicitly mean counterclockwise turns as shown in the animation loop. We can also, however, imagine "running the film backward" or rotating clockwise. Rotating a square clockwise one-quarter turn also lines up the square with itself, and similarly for two or more clockwise quarter turns.
We may usefully view a clockwise quarter turn as a counterclockwise negative one-quarter turn. In this way, each whole number \(k\) (positive, negative, or zero) corresponds to the transformation "counterclockwise rotation by \(k\) quarter turns." If \(k\) is positive we obtain "actual" counterclockwise rotations. If \(k\) is negative, we obtain clockwise rotations. If \(k\) is zero, we recover the identity transformation.
Suppose we rotate the square counterclockwise by one-quarter turn, then clockwise by one-quarter turn. Not only does the square line up with itself (a composition of symmetries is a symmetry), but each point of the square returns to its initial position: The composition of these symmetries is the identity transformation. A third fundamental property of symmetry captures this general fact: Undoing a symmetry is a symmetry. A bit more carefully, Every symmetry has an inverse transformation, and the inverse of a symmetry is a symmetry.
In summary, the square is symmetric under rotations by a whole number of quarter turns. The octagon is symmetric under rotations by a whole number of eighths of a turn. Every rotation symmetry of the octagon is also a rotation symmetry of the square, essentially because two eighths is one quarter. In this sense, the octagon is "more symmetric" than the square.
The circle is symmetric under every rotation about its center. Loosely, the circle is "perfectly symmetric" under rotation: It does not flash blue, but is constantly lighted.
Incidentally, the symmetry of the square has an invisible manifestation: The animation loop does not show one full turn, but only one quarter turn! This permitted the file to be only one-quarter the size it would otherwise have been.
Groups
Mathematicians favor abstraction to the extent of viewing verbs (transformations) as nouns (things). To a mathematician, the three "fundamental properties" of symmetry listed above motivate the mathematical definition of an algebraic structure known as a group. Our aim here, again, is not maximum generality. We merely record enough of the general definition to capture qualitatively what we mean by symmetry.
Suppose we have an object \(S\) (for shape) and a collection \(G\) (for group) of transformations of \(S\) satisfying three conditions:
- The identity transformation is in \(G\).
- The composition of two arbitrary transformations in \(G\) is in \(G\).
- For every transformation in \(G\), the inverse transformation exists and is in \(G\).
Then we say \(G\), together with composition of transformations, constitutes a symmetry group of \(S\).
Legalistically, the set of rotations by a whole number of quarter turns is a symmetry group of the regular octagon, or of the circle. In practice, we usually assume further that "the symmetry group" \(G\) of \(S\) contains all transformations satisfying the three conditions above. Thus, "the rotation group" of the octagon comprises rotations by a whole number of eighths of a turn, while "the rotation group" of the circle comprises all rotations about the center, regardless of rotation angle.
Rotations are not the only symmetries, but for technical reasons they play a disproportionate role at Differential Geometry. In passing, we mention a few other classes of symmetry.
Reflection
In geometry, "reflection" refers to an operation analogous to a mirror. In the plane, a mirror is a line; in space, a mirror is a plane. In either case, there is a unique direction perpendicular to the mirror. Reflection "reverses" the direction perpendicular to the mirror and acts on the mirror itself as the identity transformation.
Every reflection is its own inverse: Reversing the direction of a line then reversing again effects the identity transformation. This "involutive property" gives reflections a discrete character, in contrast to the continuous character of rotations. Loosely, reflection is all or nothing; we cannot "reflect partway."
The square, octagon, and circle all admit reflection symmetries. You may enjoy investigating them yourself, either by making careful drawings of polygons with labeled vertices, or by physically cutting out paper shapes and manipulating them in space. (Spoiler: A square has four axes of reflection symmetry, two passing through pairs of opposite corners, and two passing through opposite midpoints of sides. A similar analysis holds for an octagon, or a circle.)
Incidentally, no set of reflection symmetries is a group. (!) See if you can determine which of the three properties do not hold. Further, while arbitrary plane rotations about the origin may be performed in whatever order we like without changing the composition, the order in which we perform reflections matters. (!!) In order to study reflections, some kind of algebraic notation is advisable.
Although reflections alone do not constitute a group, multiple mirrors "generate" pleasing symmetries by successive composition (reflections of reflections, and so forth). Kaleidoscopes are based on pairs of mirrors set at carefully chosen angles.
Reflections are closely connected to a "paradox" that, apparently, has caused heated public exchanges by philosophers: When we look at our reflection in a mirror, our left and right sides are exchanged, but not our head and feet. Why is this? In the book The Ambidextrous Universe, Martin Gardner discusses this "paradox" and many other fascinating aspects of reflection symmetry clearly and in engaging detail.
Translation
In geometry, translation means spatial shifting. Think of an infinite line with equally-spaced spots separated by one unit. Translating this line-with-spots rightward by one unit is a symmetry. Translating rightward by an arbitrary positive integer \(k\) is therefore also a symmetry. Leftward translation may be viewed as rightward translation by a negative amount. Translation by \(0\) is the identity transformation.
Like discrete rotations (quarter turn, eighth turn, etc.) for polygons above, integers correspond to symmetries of our line-with-dots. Unlike the situation for rotations, there is no "full turn," no positive integer \(k\) for which translation by \(k\) is the identity. Objects with translation symmetry are necessarily unbounded.
A roll of wallpaper has translation symmetry along the roll if we imagine the roll to continue infinitely far in both directions. Discrete groups of Euclidean plane motions involving translation along only one direction are usually called frieze groups, after the linear decorative patterns.
A tiled floor typically manifests translation symmetry along two non-parallel directions if we mentally continue the pattern outward forever to cover the entire plane. The same is true of wallpaper patterns if we imagine laying infinitely many strips of infinitely long wallpaper side by side. Discrete groups of Euclidean plane motions involving translations along two non-parallel directions are usually called wallpaper groups.
Möbius Transformations
Rotations, reflections, and translations are all symmetries of Euclidean geometry, with its concepts of distance and angle. Other, less-familiar mathematical structures also have symmetries. Among the richest easily-visualized examples are transformations of a sphere (soap bubble) that preserve angles: Möbius transformations. These include rotations of a sphere, but also include dilations, uniform scaling about the origin viewed as transformations of the sphere under stereographic projection, and symmetries obtained by composing rotations and dilations.
What Is Geometry?
Snowflakes are nearly unchanged (or invariant) under rotations by one-sixth of a turn, or by reflections in certain planes. Water droplets are nearly invariant under rotations about a vertical axis, or if they are spherical, about an arbitrary axis. Sea shells are nearly invariant under transformations effected by simultaneous rotation and scaling.
The concept of symmetry arose from natural forms that are invariant under certain spatial transformations, and led to the concept of a group. In 1872, the mathematician Felix Klein published the Erlangen Program, which loosely takes the opposite view: An abstract group acting on a set of objects is the primitive concept, and a "geometric property" is anything invariant under the group.
Abstract groups are usually defined using algebra: vectors (ordered lists of real or complex numbers), matrices (rectangular arrays of real or complex numbers), and abstract objects constructed from these. Although building physical objects in a Euclidean space of four or more dimensions is impossible, algebra handles the necessary concepts and calculations with ease, treating a list of seven real numbers as a geometric location in seven-dimensional space. Algebra, guided by geometric intuition in three-space, allows us to "project" objects in high-dimensional spaces to objects in three-space, analogously to casting a shadow. Remarkably, the results generally conform with our spatial intuition.
This perspective of starting with a "group action" is close in spirit to how I often design images and jewelry. A large majority of pendant and earring designs at Differential Geometry originate from abstract rotation symmetries of four- and six-dimensional space. The "shadows" in three-space of high-dimensional forms swept by continuous symmetries generally do not have continuous symmetries, but do often have pleasing discrete symmetries, analogous to flowers and snowflakes. But although physical forms such as Harmonia, Luminoso, Intimo, and others are familiar-looking, perhaps visually compelling, they would have been difficult or impossible to find and describe without using abstract symmetry, reaching into geometrically-inaccessible spaces, and retrieving objects found there.