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Before diving into group theory, let’s understand affine transformations and Euclidean transformations for representing symmetry operations. The latter half of this page explains Seitz notation for compact representation.

Affine Transformation

A three-dimensional coordinate \(X=\begin{pmatrix}x\\y\\z\end{pmatrix}\) can be transformed by a 3×3 matrix \(R=\begin{pmatrix}
R_{11} & R_{12} & R_{13} \\
R_{21} & R_{22} & R_{23} \\
R_{31} & R_{32} & R_{33} \\
\end{pmatrix}\), then translated by \(t=\begin{pmatrix}t_x\\t_y\\t_z\end{pmatrix}\) to produce \(X’=\begin{pmatrix}x’\\y’\\z’\end{pmatrix}\). This operation can be expressed using matrix multiplication and addition as follows:
$$X’ = R X + t$$An affine transformation matrix can express this operation with a single matrix multiplication. For three dimensions, we add one dummy dimension to create a 4×4 matrix as follows:
$$A = \begin{pmatrix}
R_{11} & R_{12} & R_{13} & t_x\\
R_{21} & R_{22} & R_{23} & t_y\\
R_{31} & R_{32} & R_{33} & t_z\\
0 & 0 & 0 & 1
\end{pmatrix}= \begin{pmatrix}
\begin{matrix} \\ \large{R} \\ \\ \end{matrix} & \begin{matrix} \\ \large{t} \\ \\ \end{matrix} \\
\begin{matrix}0&0&0\end{matrix} & 1
\end{pmatrix}
$$The final row (row 4) of affine transformation matrix \(A\) is always \(0,0,0,1\). The 3×3 part denoted by \(R\) performs rotation, deformation, and scaling operations, while the 3×1 part denoted by \(t\) performs translation. We add one dummy dimension to the coordinate \((x, y, z)\) to be transformed, with the final component set to \(1\), and perform the following transformation:
$$A \begin{pmatrix}x\\y\\z\\1\end{pmatrix} =
\begin{pmatrix}
R_{11}x+R_{12}y+R_{13}z+t_x\\
R_{21}x+R_{22}y+R_{23}z+t_y\\
R_{31}x+R_{32}y+R_{33}z+t_z\\
1\end{pmatrix} =
\begin{pmatrix} R \begin{pmatrix}x\\y\\z\end{pmatrix} + t \\ 1 \end{pmatrix}
$$The first three components of the transformed vector correspond to the transformed coordinates, and the final component will always be \(1\). Although affine transformations have the slight disadvantage of increasing dimensionality and computational cost, they have the major advantage of allowing rotation, deformation, scaling, and translation to all be computed with a single matrix multiplication.

For example,
$$A = \begin{pmatrix}
\cos(\pi/3) & -\sin(\pi/3) & 0 & 1/4\\
\sin(\pi/3) & \cos(\pi/3) & 0 & 1/2\\
0 & 0 & 1 & 0\\
0 & 0 & 0 & 1
\end{pmatrix}
$$represents an affine transformation that rotates around the Z-axis by \(\pi/3\) and then translates by 1/4 in the X-axis direction and 1/2 in the Y-axis direction. Convenient, isn’t it?

Affine transformations are used very broadly across many fields, not limited to crystallography. They are particularly essential in computer graphics.

Euclidean Transformation (Symmetry Operations)

In crystallography, “symmetry operations” are a type of affine transformation. However, the operation must not change the volume or deformation. We certainly don’t want atoms to change size! The requirement that volume and deformation do not change is equivalent to the requirement that the Euclidean distance between any two points mapped by the affine transformation is invariant. The condition for an affine transformation \(A\) to preserve Euclidean distance is that the submatrix \(R\) (excluding the final row and column of the \(A\) matrix) must be an orthogonal matrix, that is:$$
R R^{\mathrm{T}} = R^{\mathrm{T}} R =I
$$where \(R^{\mathrm{T}}\) denotes the transpose of \(R\) and \(I\) is the identity matrix. This condition ensures that the affine transformation \(A\) preserves distance, angles, and volume. However, orientation may change. The determinant \(\det R\) of an orthogonal matrix \(R\) is always \(\pm1\); if \(\det R=1\), a right-handed system remains right-handed, and if \(\det R=-1\), it is converted to a left-handed system. The meaning of determinants and orthogonal matrices is explained in detail on a separate page.

An affine transformation that preserves the distance between two points, as described above, is called a Euclidean transformation. It corresponds to the matrix representation of symmetry operations in crystallography.

Classification of Symmetry Operations

Three-dimensional Euclidean transformations (i.e., symmetry operations) can be classified as follows based on their degrees of freedom, orientation preservation, and translational nature.

Degrees of freedom is the number of parameters needed to uniquely determine a transformation. For example, the identity transformation is simply a 4×4 identity matrix, and no matrix components can be modified—zero degrees of freedom. For rotation, two parameters specify the direction of the rotation axis, two specify its position in space, and one specifies the rotation amount, for a total of five degrees of freedom. For glide reflection, two parameters specify the normal direction of the plane, one specifies the distance from the origin to the plane, and two more specify the direction and amount of translation, for a total of five degrees of freedom.

The top five operations in the table do not involve translation. If applied repeatedly, they eventually return to the original location. Combining these operations creates “point groups.” The bottom three operations involve translation. Mixing and combining all of these creates “space groups.” Although the number of combinations might seem infinite, there are only 230 three-dimensional space groups. Careless combinations fail to satisfy the requirements for being a group.

Seitz Notation

While the 4×4 matrix representation of affine transformations makes the details of calculations easy to understand, it consumes considerable space when writing. For compact representation of symmetry operations, a notation called Seitz notation (or Seitz symbol) is convenient. In this notation, a symmetry operation is expressed as follows:

$$ \{R|t\}$$where \(R\) corresponds to the linear operation and \(t\) corresponds to the translation operation. There are several conventions for this notation, but here we explain the convention adopted in ITA (Table 1.4.2.1-3). For \(R\), a subscript indicates the direction of the axis and a superscript indicates the sign of the rotation direction for rotation or rotoinversion operations. However, for the identity (\(1\)), inversion (\(\bar{1}\)), 2-fold rotation (\(2\)), and reflection operations (\(m\)), the rotation direction is not needed. For example, the symbol \(\bar{3}^+_{111}\) means a 3-fold rotoinversion operation (120-degree rotation followed by inversion) in the \([111]\) axis direction in the \(+\) direction (right-hand screw direction). In matrix form, it is:$$
\bar{3}^+_{111}=\begin{pmatrix} 0 & 0 & -1 \\ -1 & 0 & 0 \\ 0 & -1 & 0 \\ \end{pmatrix}
$$The translation \(t\) is expressed by comma-separated translation components as \(\frac{1}{2}, \frac{1}{2},0\). However, if all translation components are zero, it is simply written as \(0\). For example, the Seitz symbol \(\{ \bar{3}^+_{111} | \frac{1}{2}, \frac{1}{2},0 \}\) expressed as a 4×4 affine matrix is:$$
\textstyle \{ \bar{3}^+_{111} | \frac{1}{2}, \frac{1}{2},0 \} = \begin{pmatrix} 0 & 0 & -1 & \frac{1}{2} \\ -1 & 0 & 0 & \frac{1}{2} \\ 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 1 \end{pmatrix}$$

Note that Seitz notation is used in the space with unit cell vectors as the basis (i.e., fractional coordinate space). Naturally, the direction of axes and amounts of translation are expressed as multiples of the unit cell vectors (\(a, b, c\)). Please be mindful of this.

Below, we summarize in tables all possible notations for \(R\) in Seitz symbols allowed for three-dimensional crystals.

Triclinic, Monoclinic, Orthorhombic, Tetragonal, and Cubic Crystal Systems

Trigonal (Hexagonal Setting) and Hexagonal Crystal Systems

Trigonal (Rhombohedral Setting)

Representation of Symmetry Operations Using Matrices and Seitz Notation

All symmetry operations can be expressed as affine (Euclidean) transformations and Seitz notation. The following are concrete examples.

Reflection

Rotation

In general, the matrix representation of a rotation operation with rotation axis passing through the origin and parallel to unit vector \((x,y,z)\) with rotation angle \(\theta\) is as follows:
$$\begin{pmatrix}
x^2(1-\cos\theta) +\cos\theta & x y(1-\cos\theta) – z\sin\theta & z x (1-\cos\theta) +y \sin\theta &0 \\
x y (1-\cos\theta) +z \sin\theta& y^2 (1-\cos\theta) +\cos\theta& y z (1-\cos\theta) -x \sin\theta &0 \\
z x (1-\cos\theta) -y \sin\theta& y z (1-\cos\theta) + x \sin\theta& z^2 (1-\cos\theta) +\cos\theta &0 \\
0&0&0&1
\end{pmatrix}$$

Rotoinversion

Translation

Simple translation operation translating by \(u, v, w\) along the X, Y, Z axes
$$\begin{pmatrix}
1&0&0&u\\
0&1&0&v\\
0&0&1&w\\
0&0&0&1
\end{pmatrix}
=\{1|u,v,w\}$$

Screw

Glide

Footnotes

1. If we use the Cartesian coordinate system instead of the hexagonal setting unit cell vectors as the basis, the matrix representation becomes: $\begin{pmatrix}
   \frac{1}{2}&\frac{\sqrt{3}}{2}&0&0\\
   -\frac{\sqrt{3}}{2}&\frac{1}{2}&0&0\\
   0&0&-1&0\\
   0&0&0&1
   \end{pmatrix}$ ↩︎
2. If we use the Cartesian coordinate system instead of the hexagonal setting unit cell vectors as the basis, the matrix representation becomes: $\begin{pmatrix}
   \frac{1}{2}&-\frac{\sqrt{3}}{2}&0&0\\
   \frac{\sqrt{3}}{2}&\frac{1}{2}&0&0\\
   0&0&1&\frac{1}{3}\\
   0&0&0&1
   \end{pmatrix}$ ↩︎