Symmetry

An isometry or rigid motion in \({\mathbb R}^n\) is a distance-preserving function \(f\) from \({\mathbb R}^n\) to \({\mathbb R}^n\). This means that \(f\) must satisfy \[\begin{aligned}\end{aligned}\] for all \({\mathbf x}, {\mathbf y} \in {\mathbb R}^n\). It is not difficult to show that \(f\) must be a one-to-one map. By , any element in \(O(n)\) is an isometry on \({\mathbb R}^n\); however, \(O(n)\) does not include all possible isometries on \({\mathbb R}^n\). Translation by a vector \({\mathbf x}\), \(T_{\mathbf y}({\mathbf x}) = {\mathbf x} + {\mathbf y}\) is also an isometry (); however, \(T\) cannot be in \(O(n)\) since it is not a linear map.

We are mostly interested in isometries in \({\mathbb R}^2\). In fact, the only isometries in \({\mathbb R}^2\) are rotations and reflections about the origin, translations, and combinations of the two. For example, a glide reflection is a translation followed by a reflection (). In \({\mathbb R}^n\) all isometries are given in the same manner. The proof is very easy to generalize.

For any arbitrary isometry, \(f\), \(T_{\mathbf x} f\) will fix the origin for some vector \({\mathbf x}\) in \({\mathbb R}^2\); hence, \(T_{\mathbf x} f({\mathbf y}) = A {\mathbf y}\) for some matrix \(A \in O(2)\). Consequently, \(f({\mathbf y}) = A {\mathbf y} + {\mathbf x}\). Given the isometries \[\begin{aligned}f({\mathbf y}) & = A {\mathbf y} + {\mathbf x}_1 \\ g({\mathbf y}) & = B {\mathbf y} + {\mathbf x}_2\end{aligned}\], their composition is \[\begin{aligned}\end{aligned}\]. This last computation allows us to identify the group of isometries on \({\mathbb R}^2\) with \(E(2)\).

A symmetry group in \({\mathbb R}^n\) is a subgroup of the group of isometries on \({\mathbb R}^n\) that fixes a set of points \(X \subset {\mathbb R}^n\). It is important to realize that the symmetry group of \(X\) depends both on \({\mathbb R}^n\) and on \(X\). For example, the symmetry group of the origin in \({\mathbb R}^1\) is \({\mathbb Z}_2\), but the symmetry group of the origin in \({\mathbb R}^2\) is \(O(2)\).

Suppose that we wish to study wallpaper patterns in the plane or crystals in three dimensions. Wallpaper patterns are simply repeating patterns in the plane (). The analogs of wallpaper patterns in \({\mathbb R}^3\) are crystals, which we can think of as repeating patterns of molecules in three dimensions (). The mathematical equivalent of a wallpaper or crystal pattern is called a lattice.

Let us examine wallpaper patterns in the plane a little more closely. Suppose that \({\mathbf x}\) and \({\mathbf y}\) are linearly independent vectors in \({\mathbb R}^2\); that is, one vector cannot be a scalar multiple of the other. A lattice of \({\mathbf x}\) and \({\mathbf y}\) is the set of all linear combinations \(m {\mathbf x} + n {\mathbf y}\), where \(m\) and \(n\) are integers. The vectors \({\mathbf x}\) and \({\mathbf y}\) are said to be a basis for the lattice.

Notice that a lattice can have several bases. For example, the vectors \((1,1)^\transpose\) and \((2,0)^\transpose\) have the same lattice as the vectors \((-1, 1)^\transpose\) and \((-1, -1)^\transpose\) (). However, any lattice is completely determined by a basis. Given two bases for the same lattice, say \(\{ {\mathbf x}_1, {\mathbf x}_2 \}\) and \(\{ {\mathbf y}_1, {\mathbf y}_2 \}\), we can write \[\begin{aligned}{\mathbf y}_1 & = \alpha_1 {\mathbf x}_1 + \alpha_2 {\mathbf x}_2 \\ {\mathbf y}_2 & = \beta_1 {\mathbf x}_1 + \beta_2 {\mathbf x}_2\end{aligned}\], where \(\alpha_1\), \(\alpha_2\), \(\beta_1\), and \(\beta_2\) are integers. The matrix corresponding to this transformation is \[\begin{aligned}\end{aligned}\]. If we wish to give \({\mathbf x}_1\) and \({\mathbf x}_2\) in terms of \({\mathbf y}_1\) and \({\mathbf y}_2\), we need only calculate \(U^{-1}\); that is, \[\begin{aligned}\end{aligned}\]. Since \(U\) has integer entries, \(U^{-1}\) must also have integer entries; hence the determinants of both \(U\) and \(U^{-1}\) must be integers. Because \(U U^{-1} = I\), \[\begin{aligned}\end{aligned}\] consequently, \(\det(U) = \pm 1\). A matrix with determinant \(\pm 1\) and integer entries is called unimodular. For example, the matrix \[\begin{aligned}\end{aligned}\] is unimodular. It should be clear that there is a minimum length for vectors in a lattice.

Condensed — the full section is in Judson, Abstract Algebra: Theory and Applications.

Symmetry groups have intrigued mathematicians for a long time. Leonardo da Vinci was probably the first person to know all of the point groups. At the International Congress of Mathematicians in 1900, David Hilbert gave a now-famous address outlining 23 problems to guide mathematics in the twentieth century. Hilbert's eighteenth problem asked whether or not crystallographic groups in \(n\) dimensions were always finite. In 1910, L. Bieberbach proved that crystallographic groups are finite in every dimension. Finding out how many of these groups there are in each dimension is another matter. In \({\mathbb R}^3\) there are \(230\) different space groups; in \({\mathbb R}^4\) there are \(4894\); in \({\mathbb R}^5\) there are \(222097\). No one has been able to compute the number of space groups for \({\mathbb R}^6\) and beyond (oeis.org/A006227). It is interesting to note that the crystallographic groups were found mathematically for \({\mathbb R}^3\) before the \(230\) different types of crystals were actually discovered in nature.