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THEOREM 10.2 The only possible symmetries for a frieze pattern are:
• horizontal translations along the midline,
• rotations of 180° around points on the midline,
• reflections in vertical lines perpendicular to the midline,
• a reflection in the horizontal midline,
• glide reflections using the midline.
Proof The midline must be fixed by any symmetry of the frieze. Now consider what various isometries would do to the midline. Translations that are not parallel to this midline change the position of the midline. Rotations that are not 180° or 0° change the inclination of the midline. Rotations of 180° that are not centered on the midline produce a line parallel to but different from the midline. Reflections in a line that is not parallel or perpendicular to the midline change the inclination of the line. Reflections in a horizontal line other than the midline change the position of the midline. The same is true for glide reflections in other horizontal lines. Thus we have eliminated all possibilities except those listed in the theorem. □
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By definition, any frieze will have an infinite number of translational symmetries along its midline. We can use the other possible symmetries to classify and count the frieze groups.
THEOREM 10.3 There exist exactly seven symmetry groups for friezes.
Proof Since all friezes have translations, let us consider the other types of symmetries and the various combinations of symmetries. We will see that many of these combinations cannot occur.
We will use the following abbreviations:
• H is the reflection in the horizontal midline.
• V is a reflection in a vertical line.
• R is a rotation of 180° about a center on the midline.
• G is a glide reflection using the midline.
The following table lists all possible combinations:
\begin{array}{c|cccc|c}
\hline&\text{H}&\text{V}&\text{R}&\text{G}&\text{Result}\\\hline
\text{1.}&\text{Yes}&\text{Yes}&\text{Yes}&\text{Yes}&\text{X pattern}\\
\text{2.}&\text{Yes}&\text{Yes}&\text{Yes}&\text{No}&\text{not possible}\\
\text{3.}&\text{Yes}&\text{Yes}&\text{No}&\text{Yes}&\text{not possible}\\
\text{4.}&\text{Yes}&\text{Yes}&\text{No}&\text{No}&\text{not possible}\\
\text{5.}&\text{Yes}&\text{No}&\text{Yes}&\text{Yes}&\text{not possible}\\
\text{6.}&\text{Yes}&\text{No}&\text{Yes}&\text{No}&\text{not possible}\\
\text{7.}&\text{Yes}&\text{No}&\text{No}&\text{Yes}&\text{E pattern}\\
\text{8.}&\text{Yes}&\text{No}&\text{No}&\text{No}&\text{not possible}\\
\text{9.}&\text{No}&\text{Yes}&\text{Yes}&\text{Yes}&\text{∨∧ pattern}\\
\text{10.}&\text{No}&\text{Yes}&\text{Yes}&\text{No}&\text{not possible}\\
\text{11.}&\text{No}&\text{Yes}&\text{No}&\text{Yes}&\text{not possible}\\
\text{12.}&\text{No}&\text{Yes}&\text{No}&\text{No}&\text{W pattern}\\
\text{13.}&\text{No}&\text{No}&\text{Yes}&\text{Yes}&\text{not possible}\\
\text{14.}&\text{No}&\text{No}&\text{Yes}&\text{No}&\text{Z pattern}\\
\text{15.}&\text{No}&\text{No}&\text{No}&\text{Yes}&\text{┌└ pattern}\\
\text{16.}&\text{No}&\text{No}&\text{No}&\text{No}&\text{P pattern}\\\hline
\end{array}In this table we have shown seven frieze patterns, each having a different combination of symmetries. Why are there no others? Think about some of the possible compositions.
If a frieze has the horizontal reflection $H$ and we compose that with a translation along the midline, the result is $G$, a glide reflection. This eliminates options 2, 4, 6, and 8.
If a frieze has $H$ and has vertical reflections, the compositions $H◦V$ are 180° rotations (because the composition of two such reflections is direct and is not the identity). This eliminates option 3 (and 4 again).
The composition $G◦R$ is opposite so is either $H$ or $V$. However, the upper half of the pattern will end up on top again, so the composition must equal $V$. This eliminates option 13 (and 5 again).
Composing $V◦R$ equals either $H$ or $G$, depending on whether or not the mirror for $V$ passes through the center for $R$. Either result eliminates option 10.
The composition $G◦V$ is direct, so equals either $R$ or a translation. However, the upper half of the pattern ends up on the bottom this time, so the composition must equal $R$. This eliminates option 11 (and 3 again).
Therefore we are left with only options 1, 7, 9, 12, 14, 15, and 16, and we have shown friezes to match each of these possibilities. |
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