The Klein 4-Group

What is the common factor linking book-flips, solitaire, twelve-tone music and the solution of quartic equations?   Answer: {K_4}.

Symmetries of a Book — or a Brick

The four symmetric configurations of a book under 3D rotations.}

The four symmetric configurations of a book under 3D rotations.}

Take a book, place it on the table and draw a rectangle around it. How many ways can the book fit into the rectangle? Clearly, once any single corner of the book is put at the top left corner of the rectangle, there is no further lee-way; the positions of the remaining three corners are determined. Thus, there are four ways the book can fit into the rectangle. They are shown — for a randomly chosen book — in the figure above.

The four orientations of the book can be described in terms of simple rotations, starting from the upright configuration:

  1. Place the book upright with front cover up (that is, do not move it).
  2. Rotate through 180º about X-axis (horizontal line through centre).
  3. Rotate through 180º about Y-axis (vertical line through centre).
  4. Rotate through 180º about Z-axis (line through centre perpendicular to book).

We use the four symbols {\{I, X, Y, Z\}} for these operations. We denote the combination or composition of two operations by {*}. It is clear that any operation performed twice brings us back to the original position:

\displaystyle X * X = Y * Y = Z * Z = I

Moreover, combining any two of {\{X, Y, Z\}} gives the third, for example, {X*Y = Z}. This means that {X*Y*Z = I}. Drawing up a full table of combinations of two operations, we get the following table (the Cayley table):

Klein-4-CayleyTableThis is a simple example of a group, a set of elements together with a rule for combining them. A group operation must satisfy four conditions: closure, associativity, identity and inverse. The book symmetries are a realization of the Klein 4-group, {K_4}.

The Klein 4-Group

The group {K_4} was introduced by Felix Klein in his study of the roots of polynomial equations, solution of cubics and quartics and the unsolvability of the quintic equation. The orientations of a book, or symmetries of a rectangle, are just one way to describe the group. A more formal representation is to express the rotation operations {\{ X, Y, Z\}} as 3-by-3 matrices. The identity matrix \mathsf{I} represents the identity transformation I and the rotations through 180° about the axes are

\displaystyle \mathsf{X} = \begin{bmatrix} +1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{bmatrix} \quad \mathsf{Y} = \begin{bmatrix} -1 & 0 & 0 \\ 0 & +1 & 0 \\ 0 & 0 & -1 \end{bmatrix} \quad \mathsf{Z} = \begin{bmatrix} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & +1 \end{bmatrix}

It is a simple matter to show that these matrices satisfy the same multiplication table as shown above. Thus {\{\mathsf{I},\mathsf{X},\mathsf{Y},\mathsf{Z}\}} is a representation of the group {K_4} in the special orthogonal group {\mathrm{SO}(3)} of rotations in \mathbb{R}^3.

The book — or at least its infinitely thin idealization — can be thought of as a 2-dimensional object. Suppose we wish to remain in the plane; then only the operations {I} and {Z} can be performed. But we can replace the other two rotations by reflections in the {X} and {Y} axes. Then we represent the four operations by four 2-by-2 matrices in the orthogonal group {\mathrm{O}(2)} of isometries (distance preserving transformations) in \mathbb{R}^2. The matrices are the identity matrix \mathsf{I}_2 and the three matrices

\displaystyle \mathsf{X}_2 = \begin{pmatrix} +1 & 0 \\ 0 & -1 \end{pmatrix} \quad \mathsf{Y}_2 = \begin{pmatrix} -1 & 0 \\ 0 & +1 \end{pmatrix} \quad \mathsf{Z}_2 = \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}
Again, it is simple to show that the matrices {\{\mathsf{I}_2,\mathsf{X}_2,\mathsf{Y}_2,\mathsf{Z}_2\}} have the same Cayley table as {K_4}. Thus, they are a representation of the group {K_4} in terms of matrices in 2-dimensional space. The configurations are shown below.

The four symmetric configurations of a book under 2D reflections and rotations.

The four symmetric configurations of a book under 2D reflections and rotations.

Other Applications

The Klein 4-group {K_4} is also useful for musicians working on twelve-tone composition. In the twelve-tone technique — aka dodecaphony — the composer starts with a tone row containing all the notes of the chromatic scale. This tone row can then be transformed using reflection (left-right flip), inversion (up-down flip) or a combination of these (rotation through 180º). These transformations are completely equivalent to the symmetries of a rectangle, embodied in the group {K_4}.

A prime a tone row. Transformations are the retrograde, inversion and retrograde-inversion.

A prime a tone row. Transformations are the retrograde, inversion and retrograde-inversion.

Another application of {K_4} is to the solution of the board-game called solitaire. Bialostocki (1998) showed, on the basis of simple group theory using the Klein 4-group, that there are precisely five possible finishing positions with just one peg remaining, or just two positions if rotations are ignored.

Initial position (left) and the five possible final positions (right).

Initial position (left) and the five possible final positions (right).

Finally, we mention that the Klein 4-group occurs in Galois’ Theorem, which gives conditions under which a polynomial equation is soluble in radicals: {K_4} is a component in the chain of subgroups {S_4 \supset A_4 \supset K_4 \supset I}, giving a basis for the solution of the quartic equation.

Sources

♦  Bialostocki, Arie, 1998: An application of elementary group theory to central solitaire. College Math. Jour., 29, 208–112.


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