Symmetry and Symmetry Operations/Elements
Symmetry is the property whereby an object’s state is completely unchanged (invariant) before and after applying a certain transformation. Such a transformation is called a symmetry operation. In the park example on the previous page, applying a translation (parallel movement) left the park’s state completely unchanged. In other words, the park possesses a symmetry element called a translational element. The concepts may feel confusing at this point; the following explains symmetry operations and symmetry elements in detail.
Classification and Properties of Symmetry Operations
A symmetry operation is an operation that leaves an object in space unchanged. Detailed mathematical treatment will be deferred to Chapter 2; here we give an overview of the various symmetry operations. Symmetry operations fall into two broad categories: operations without translation and operations with translation.
Operations Without Translation
Operations without translation always return to the original position after being repeated a finite number of times. In other words, there must always exist at least one fixed point in space. Combining such operations yields what is called a point group, which is explained in detail in another chapter.
| Identity | Do nothing (leave unchanged) |
| Inversion | Take a point in space as the origin and reverse the signs of all x, y, z coordinates |
| Reflection | Reflect through a plane in space, as in a mirror |
| Rotation | Rotate about a line in space |
| Rotoinversion | Rotation followed immediately by inversion through a point on the rotation axis |
Identity Operation
The operation of “doing nothing.” This may seem pointless, but including it is what makes the set of operations satisfy the mathematical structure of a group.
Inversion Operation
This operation takes a point in space as the origin and inverts the x, y, z coordinates of everything around it to their negatives. A right hand is converted to a left hand. Repeating the operation twice returns everything to its original position.
Reflection Operation
This operation reflects everything around a plane in space as if in a mirror. A right hand is converted to a left hand. Repeating twice returns to the original position.
Rotation Operation
This operation rotates objects about a line (the rotation axis) in space. While the rotation angle can in general take any value, for crystals (figures with translational symmetry) it is restricted to \(\frac{\pi}{3}\), \(\frac{\pi}{2}\), \(\frac{2\pi}{3}\), \(\pi\), \(\frac{4\pi}{3}\), \(\frac{3\pi}{2}\), \(\frac{5\pi}{3}\), \(2\pi\) (i.e., \(60°, 90°, 120°, 180°, 240°, 270°, 300°, 360°\)). A right hand remains a right hand. If the rotation angle is \(\theta\), repeating the operation \(n = 2\pi/\theta\) times returns to the original position; \(n\) is called the order of the rotation. When \(n = 1\) the operation does nothing, so it is identified with the identity operation.
Rotoinversion Operation
Rotoinversion is a rotation followed immediately by an inversion through a point on the rotation axis — remember it as “rotate then invert.” It may be the hardest operation for beginners to visualise. Why treat it as a single operation rather than two separate ones? Naming and classifying this combined operation as one simplifies many subsequent discussions. A right hand is converted to a left hand. As with rotation, for crystals the angle is restricted to the same set of values. The order is defined as \(n = 2\pi/\theta\). If \(n\) is even, repeating \(n\) times returns to the original; if \(n\) is odd, \(2n\) repetitions are needed. Order \(n=1\) is simply the inversion operation; order \(n=2\) is equivalent to reflection through the plane perpendicular to the axis.
Operations With Translation
Operations with translation move away from the original position each time they are repeated; no fixed point exists in space. Every crystal necessarily has such operations, and any figure that has such operations is a crystal. The set of operations both with and without translation constitutes the space group.
| Lattice translation | Simple parallel movement |
| Screw | Rotation followed by translation along the rotation axis |
| Glide | Reflection followed by translation parallel to the mirror plane |
Lattice Translation
Moving in a certain direction by a certain distance, without changing orientation. A right hand remains a right hand.
Screw
Rotation followed by translation along the rotation axis — think of a spiral staircase. A right hand remains a right hand. Although it looks like two operations combined, it is classified as one, just like rotoinversion.
Glide
Reflection through a plane followed by translation parallel to that plane. A right hand is converted to a left hand. Two sequential operations, but counted as one independent operation.
Summary
The operations introduced so far are actions of rotating or moving an object. When the object is unchanged by such an action, that action is a symmetry operation. All symmetry operations are transformations that preserve volume and shape — called Euclidean (or isometric) transformations (see “2.1. Matrix Representation of Symmetry Operations and Seitz Notation” for details).
There are five types of symmetry operations without translation: identity, inversion, reflection, rotation, and rotoinversion. Note that the identity operation is equivalent to a 360° rotation, and inversion and reflection are special cases of rotoinversion; bearing this in mind deepens understanding of the notation described next. There are three types with translation: lattice translation, screw, and glide.
The symmetry operations described in this section are all possible symmetry operations in three-dimensional space. There are surprisingly few, but that is just the way it is.
Symmetry Elements
Every object in the world is invariant under the identity operation, and a regular tetrahedron is invariant under 120° rotation, 240° rotation, reflection, and so on. However, a symmetry operation is just an operation (the act of rotating or moving something) — it is not a property or element that an object possesses. What should we call the property of an object that allows a given symmetry operation? This is the concept of a symmetry element.
A symmetry element is a virtual point, line, or plane that exists at the location where a symmetry operation is performed and that embodies the nature of that operation (reflection, rotation, etc.). In the diagrams of symmetry operations they are shown as green points, lines, or planes. They are of course imaginary concepts, not physical objects. Why introduce this idea? A simple example explains.
Suppose we have a figure that is invariant under a \(90°\) rotation. If a \(90°\) rotation is allowed, then \(180°\), \(270°\), and \(360°\) rotations must also be allowed — all four operations are simultaneously present. Describing this figure by saying “invariant under rotations of \(90°\), \(180°\), \(270°\), and \(360°\)” is cumbersome. Enter the symmetry element: we say the figure possesses the symmetry element “\(4\)” (represented by a virtual green line). The element “\(4\)” encompasses all four operations (\(90°\), \(180°\), \(270°\), \(360°\) rotations), and the figure is invariant under any of them. With symmetry-element notation, the symmetry of this figure can be described in a single word: “it has \(4\).”
A “symmetry operation” is an action that leaves an object unchanged; a “symmetry element” is a property an object possesses. The two are easily confused, but keeping them distinct is good practice.
Note two exceptions to the rule that “a symmetry element is a point, line, or plane at the location of the operation.” First, the symmetry element corresponding to the identity operation has no fixed geometric form (it exists everywhere in space, so to speak). Second, the translational element is a symmetry element that every crystal must have; it is not located at a specific position but is defined purely by direction and magnitude — it is nothing other than the concept of the unit cell.
Notation for Symmetry Elements
The following presents the internationally standard notation for symmetry elements, as established by the International Union of Crystallography. Only symmetry elements that appear in crystals (objects with translational symmetry) are included. Symmetry elements are written with numbers and letters; the rule is that numbers are not italicised, whereas letters are italicised. For symmetry operations, it is conventional to use the Seitz notation described later, which is based on the element notation below.
Symmetry Elements Without Translation
| Name | Symbol | Description |
|---|---|---|
| Identity | \(1\) | Equivalent to 360° rotation |
| Inversion | \(\bar{1}\ (=i)\) | Equivalent to 360° rotoinversion; also written \(i\) |
| Reflection | \(m\ (=\bar{2})\) | Equivalent to 180° rotoinversion |
| Rotation | \(2\), \(3\), \(4\), \(6\) | Denoted by the order (360°/rotation angle) |
| Rotoinversion | \(\bar{1}\ (=i)\), \(\bar{2}\ (=m)\), \(\bar{3}\), \(\bar{4}\), \(\bar{6}\) | Denoted by the order (360°/rotation angle) |
Rotation and rotoinversion elements are expressed not by the rotation angle itself but by the value “360° ÷ rotation angle,” which is called the order of the rotation or rotoinversion.
Rotation elements are written simply as the order. For example, \(4\) is read “four,” “4-fold rotation1,” or “four.” Rotoinversion elements are written with a bar over the order symbol. For example, \(\bar{4}\) is read “bar four,” “4-fold rotoinversion,” etc.
Symmetry Elements With Translation
| Name | Symbol | Description |
|---|---|---|
| Lattice translation | \(P\) \(A, B, C\) \(I\) \(F\) \(R\) |
Primitive lattice Base-centred lattice Body-centred lattice Face-centred lattice Rhombohedral lattice |
| Screw | \(2_1\) \(3_1\), \(3_2\) \(4_1\), \(4_2\), \(4_3\) \(6_1\), \(6_2\), \(6_3\), \(6_4\), \(6_5\) |
First number: rotation order; second number: translation fraction (translation = axis period × second number / first number) |
| Glide | \(a\), \(b\), \(c\) \(n\) \(e\) \(d\) |
Axial glide planes Diagonal glide plane Double glide plane Diamond glide plane |
Symmetry elements with translation exist only in crystals. Not just ordinary translation elements, but also screw and glide elements can only exist within crystals.
Translation elements are written as capital letters: \(P, A, B, C, I, F, R\) — seven symbols representing the types of unit cell. Their meanings are explained on the next page, “1.3. Lattice Parameters, Crystal Systems, and Bravais Lattices.”
Screw elements are written with two numbers: the first (normal size) gives the rotation order; the second (subscript) gives the translation fraction. The screw axis direction always coincides with a translation vector of the crystal (a vector connecting two lattice points)2. The translation per operation is defined as (subscript / order) × the minimum period along the screw axis. For example, a crystal with symmetry element \(4_3\) is invariant under “rotate 90° and translate 3/4 of the axis period along the axis.” Individual screw elements are described in detail in “3.1. Symmetry Elements in Detail.”
Glide elements are written as lowercase letters. There are many symbols, and they are explained in “3.1. Symmetry Elements in Detail.”
Footnotes
- In Japanese, the expression “n-fold rotation” (n回回転) has the character 回 (meaning “times” and “rotation”) appearing twice, which feels awkward. “n-fold rotation” is the natural English equivalent and avoids this issue. ↩︎
- Placing a screw axis in any other direction would generate new symmetry elements, and redefining the unit cell to accommodate them would ultimately force the screw axis to coincide with a translation vector anyway. ↩︎