# Symmetry-Adapted Fourier Series for the Wallpaper Groups

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## Abstract

**:**

## 1. Introduction

## 2. Two-Dimensional Translational Symmetry

## 3. Full Wallpaper Group Symmetry

#### 3.1. Rotations

#### 3.2. Reflection Axes

#### 3.3. Glide Reflection Axes

## 4. Derivation of Fourier Coefficient Relations

#### 4.1. Rotation Axes

#### 4.2. Reflection Axes

#### 4.3. Glide Reflection Axes

#### 4.4. Combining Rotation Axes and (Glide) Reflection Axes

#### 4.5. Centering

#### 4.6. Structure Factors in Crystallography

## 5. Wallpaper Group Tables

## 6. Discussion and Conclusions

## Acknowledgments

## References

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**Figure 1.**$p6$ wallpaper group. (

**a**) Map of $({k}_{1},{k}_{2})$ points in domain D. Equivalent points, having equal Fourier coefficients ${c}_{{k}_{1},{k}_{2}}$, are assigned a same color and number. The points in the top left and bottom right white zones have equivalent points falling outside the $(-5\le {k}_{1}\le 5,-5\le {k}_{2}\le 5)$ range and are therefore not included in D. Note the “distorted” hexagonal symmetry; (

**b**) Domain ${D}_{6}$ containing one representative point of each set ${S}_{{k}_{1},{k}_{2}}$.

**Figure 2.**Real function $f(\overrightarrow{r}$) with $p6$ symmetry; the only non-zero Fourier coefficients are ${c}_{1,0}=-\frac{1}{2}$ and ${c}_{3,1}=\frac{1}{4}$. Basis vectors of the $p6$ unit cell as well as the asymmetric unit (bound by gray lines) are shown.

**Figure 3.**$p6mm$ wallpaper group. (

**a**) Map of $({k}_{1},{k}_{2})$ points in domain D; equivalent points are shown with a same color and number. Note the higher symmetry (less representative points) than for the $p6$ wallpaper group [see Figure 2(a)]; (

**b**) Domain ${D}_{6}\cup {D}_{12}$ containing representative points; (

**c**) Domain ${D}_{6}$ containing representative points with a cycle of 6; (

**d**) Domain ${D}_{12}$ containing representative points with a cycle of 12.

**Figure 4.**Real function $f(\overrightarrow{r}$) with $p6mm$ symmetry; the only non-zero Fourier coefficients are ${c}_{2,0}=\frac{1}{4}$ and ${c}_{2,1}=-1$. Basis vectors of the $p6mm$ unit cell as well as the asymmetric unit (bound by gray lines) are shown.

**Figure 5.**$pg$ wallpaper group. (

**a**) Map of $({k}_{1},{k}_{2})$ points in domain D. Equivalent points are shown with a same color and number; (

**b**) Domain ${D}_{1}^{0}\cup {D}_{2}$ containing representative points; (

**c**) Domain ${D}_{1}^{0}$ containing representative points with a cycle of 1, excluding $(0,0)$; (

**d**) Domain ${D}_{2}$ containing representative points with a cycle of 2. The equivalence relation [Equation (65)] involves a phase factor ${h}_{{k}_{1},{k}_{2}}={(-1)}^{{k}_{1}}$; points outside the representative domain ${D}_{1}^{0}\cup {D}_{2}$ for which ${h}_{{k}_{1},{k}_{2}}=1$ and $-1$ are marked by squares and discs, respectively. Vanishing Fourier coefficients are marked by the symbol “⌼”.

**Figure 6.**Real function $f(\overrightarrow{r}$) with $pg$ symmetry; the only non-zero independent Fourier coefficients are ${c}_{1,1}=\frac{1}{2}$ and ${c}_{2,1}=1+i$. Basis vectors of the $pg$ unit cell as well as the asymmetric unit (bound by gray lines) are shown.

**Figure 7.**$p3m1$ wallpaper group. (

**a**) Map of $({k}_{1},{k}_{2})$ points in domain D; equivalent points are shown with a same color and number; (

**b**) Domain ${D}_{3}\cup {D}_{6}$ containing representative points; (

**c**) Domain ${D}_{3}$ containing representative points with a cycle of 3; (

**d**) Domain ${D}_{6}$ containing representative points with a cycle of 6.

**Figure 8.**Real function $f(\overrightarrow{r}$) with $p3m1$ symmetry; the only non-zero Fourier coefficients are ${c}_{2,0}=\frac{1}{4}+i$ and ${c}_{2,1}=-1$ [implying ${c}_{2,2}=\frac{1}{4}-i$, see Equation (108)]. Basis vectors of the $p3m1$ unit cell as well as the asymmetric unit (bound by gray lines) are shown.

**Figure 9.**Incomplete realisations of the $p2gg$ wallpaper group: (

**a**) ${c}_{2,0}=1$; (

**b**) ${c}_{0,2}=1$; (

**c**) ${c}_{2,0}=2{c}_{0,2}=1$; (

**d**) ${c}_{1,1}=1$. The general asymmetric unit is shown with dashed lines, the actual asymmetric unit with full lines.

**Table 1.**Basis vectors $({\overrightarrow{a}}_{1},{\overrightarrow{a}}_{2})$ and reciprocal basis vectors $({\overrightarrow{b}}_{2},{\overrightarrow{b}}_{2})$ of the 5 planar Bravais lattices. For the non-centered lattices, ${\overrightarrow{a}}_{1}$ can always be chosen parallel to ${\overrightarrow{e}}_{x}$ without loss of generality. For the hexagonal lattice, the angle between the basis vectors is taken to be ${60}^{\circ}$ rather than ${120}^{\circ}$.

oblique | ${\overrightarrow{a}}_{1}=a(1,0)$ | ${\overrightarrow{b}}_{1}=\frac{2\pi}{a}(1,-\frac{{a}_{2x}}{{a}_{2y}})$ |

${\overrightarrow{a}}_{2}=({a}_{2x},{a}_{2y})$ | ${\overrightarrow{b}}_{2}=\frac{2\pi}{{a}_{2y}}(0,1)$ | |

rectangular | ${\overrightarrow{a}}_{1}=a(1,0)$ | ${\overrightarrow{b}}_{1}=\frac{2\pi}{a}(1,0)$ |

${\overrightarrow{a}}_{2}=b(0,1)$ | ${\overrightarrow{b}}_{2}=\frac{2\pi}{b}(0,1)$ | |

centered rectangular | ${\overrightarrow{a}}_{1}=(\frac{a}{2},-\frac{b}{2})$ | ${\overrightarrow{b}}_{1}=2\pi (\frac{1}{a},\frac{1}{b})$ |

${\overrightarrow{a}}_{2}=(\frac{a}{2},\frac{b}{2})$ | ${\overrightarrow{b}}_{2}=2\pi (\frac{1}{a},-\frac{1}{b})$ | |

square | ${\overrightarrow{a}}_{1}=a(1,0)$ | ${\overrightarrow{b}}_{1}=\frac{2\pi}{a}(1,0)$ |

${\overrightarrow{a}}_{2}=a(0,1)$ | ${\overrightarrow{b}}_{2}=\frac{2\pi}{a}(0,1)$ | |

hexagonal | ${\overrightarrow{a}}_{1}=a(1,0)$ | ${\overrightarrow{b}}_{1}=\frac{2\pi}{a}(1,-\frac{1}{\sqrt{3}})$ |

${\overrightarrow{a}}_{2}=a(\frac{1}{2},\frac{\sqrt{3}}{2})$ | ${\overrightarrow{b}}_{2}=\frac{2\pi}{a}(0,\frac{2}{\sqrt{3}})$ |

**Table 2.**Fourier coefficient properties for generating symmetry elements of the 16 non-trivial wallpaper groups.

symmetry operation | Bravais lattice | Fourier coefficients | wallpaper groups |
---|---|---|---|

⬮ | oblique | ${c}_{{k}_{1},{k}_{2}}={c}_{-{k}_{1},-{k}_{2}}$ | $p2$ |

rectangular | $p2mm$, $p2mg$, $p2gg$ | ||

centered rectangular | $c2mm$ | ||

▲ | hexagonal | ${c}_{{k}_{1},{k}_{2}}={c}_{-{k}_{1}+{k}_{2},-{k}_{1}}$ | $p3$, $p3m1$, $p31m$ |

$\phantom{{c}_{{k}_{1},{k}_{2}}}={c}_{-{k}_{2},{k}_{1}-{k}_{2}}$ | |||

◆ | square | ${c}_{{k}_{1},{k}_{2}}={c}_{-{k}_{2},{k}_{1}}$ | $p4$, $p4mm$, $p4gm$ |

$\phantom{{c}_{{k}_{1},{k}_{2}}}={c}_{-{k}_{1},-{k}_{2}}={c}_{{k}_{2},-{k}_{1}}$ | |||

⬢ | hexagonal | ${c}_{{k}_{1},{k}_{2}}={c}_{{k}_{2},-{k}_{1}+{k}_{2}}$ | $p6$, $p6mm$ |

$\phantom{{c}_{{k}_{1},{k}_{2}}}={c}_{-{k}_{1}+{k}_{2},-{k}_{1}}={c}_{-{k}_{1},-{k}_{2}}$ | |||

$\phantom{{c}_{{k}_{1},{k}_{2}}}={c}_{-{k}_{2},{k}_{1}-{k}_{2}}={c}_{{k}_{1}-{k}_{2},{k}_{1}}$ | |||

reflection axis #1 | rectangular | ${c}_{{k}_{1},{k}_{2}}={c}_{{k}_{1},-{k}_{2}}$ | $pm$, $p2mm$ |

square | $p4mm$ | ||

centered rectangular | ${c}_{{k}_{1},{k}_{2}}={c}_{{k}_{2},{k}_{1}}$ | $cm$, $c2mm$ | |

hexagonal | ${c}_{{k}_{1},{k}_{2}}={c}_{{k}_{1},{k}_{1}-{k}_{2}}$ | $p31m$, $p6mm$ | |

glide reflection axis #1 | rectangular | ${c}_{{k}_{1},{k}_{2}}={(-1)}^{{k}_{1}}{c}_{{k}_{1},-{k}_{2}}$ | $pg$ |

reflection axis #2 | rectangular | ${c}_{{k}_{1},{k}_{2}}={(-1)}^{{k}_{2}}{c}_{{k}_{1},-{k}_{2}}$ | $p2mg$ |

glide reflection axis #2 | rectangular | ${c}_{{k}_{1},{k}_{2}}={(-1)}^{{k}_{1}+{k}_{2}}{c}_{{k}_{1},-{k}_{2}}$ | $p2gg$ |

square | $p4gm$ | ||

reflection axis #3 | hexagonal | ${c}_{{k}_{1},{k}_{2}}={c}_{{k}_{2},{k}_{1}}$ | $p3m1$ |

**Table 3.**Fourier coefficient properties for generating symmetry elements of the centered rectangular wallpaper groups.

symmetry operation | Bravais lattice | Fourier coefficients | wallpaper groups |
---|---|---|---|

centering | centered rectangular | ${c}_{{q}_{1},{q}_{2}}={(-1)}^{{q}_{1}+{q}_{2}}{c}_{{q}_{1},{q}_{2}}$ | $cm$, $c2mm$ |

⬮ | centered rectangular | ${c}_{{q}_{1},{q}_{2}}={c}_{-{q}_{1},-{q}_{2}}$ | $c2mm$ |

reflection axis #1 | centered rectangular | ${c}_{{q}_{1},{q}_{2}}={c}_{{q}_{2},{q}_{1}}$ | $cm$, $c2mm$ |

$p1$ oblique | $\begin{array}{c}{\overrightarrow{a}}_{1}=a(1,0)\hfill \\ {\overrightarrow{a}}_{2}=({a}_{2x},{a}_{2y})\hfill \end{array}$ | $\begin{array}{c}{\overrightarrow{b}}_{1}=\frac{2\pi}{a}(1,-\frac{{a}_{2x}}{{a}_{2y}})\hfill \\ {\overrightarrow{b}}_{2}=\frac{2\pi}{{a}_{2y}}(0,1)\hfill \end{array}$ |

$D=\left\{(0,0)\right\}\cup {D}_{1}^{0}$ | ${D}_{\mathrm{min}}={D}_{1}^{0}$ | |

$\begin{array}{c}{c}_{0,1}={c}_{0,-1}^{*}=-2+4i\hfill \\ {c}_{1,0}={c}_{-1,0}^{*}=1-\frac{1}{2}i\hfill \end{array}$ | $({k}_{1},{k}_{2})\ne (0,0)$ | |

$c}_{{k}_{1},{k}_{2}$ | ||

$f(x,y)={c}_{0,0}+\sum _{({k}_{1},{k}_{2})\ne (0,0)}{c}_{{k}_{1},{k}_{2}}[cos\frac{2\pi \left[{k}_{1}{a}_{2y}x+(-{k}_{1}{a}_{2x}+{k}_{2}a)y\right]}{a{a}_{2y}}$ | ||

$\phantom{f(x,y)={c}_{0,0}+\sum _{({k}_{1},{k}_{2})\ne (0,0)}{c}_{{k}_{1},{k}_{2}}[}+isin\frac{2\pi \left[{k}_{1}{a}_{2y}x+(-{k}_{1}{a}_{2x}+{k}_{2}a)y\right]}{a{a}_{2y}}]$ | ||

$[\mathbb{R}]\phantom{\rule{4.pt}{0ex}}{c}_{-{k}_{1},-{k}_{2}}={c}_{{k}_{1},{k}_{2}}^{*}$ |

$p2$ oblique | $\begin{array}{c}{\overrightarrow{a}}_{1}=a(1,0)\hfill \\ {\overrightarrow{a}}_{2}=({a}_{2x},{a}_{2y})\hfill \end{array}$ | $\begin{array}{c}{\overrightarrow{b}}_{1}=\frac{2\pi}{a}(1,-\frac{{a}_{2x}}{{a}_{2y}})\hfill \\ {\overrightarrow{b}}_{2}=\frac{2\pi}{{a}_{2y}}(0,1)\hfill \end{array}$ |

$D=\left\{(0,0)\right\}\cup {D}_{2}$ | ${D}_{\mathrm{min}}={D}_{2}$ | |

$\begin{array}{c}{c}_{0,1}={c}_{0,1}^{*}=-\frac{1}{4}\hfill \\ {c}_{1,1}={c}_{1,1}^{*}=1\hfill \end{array}$ | $\begin{array}{c}({k}_{1}>0,0)\\ ({k}_{1},{k}_{2}>0)\end{array}$ | |

$c}_{{k}_{1},{k}_{2}}={c}_{-{k}_{1},-{k}_{2}$ | ||

$f(x,y)={c}_{0,0}+2\sum _{{k}_{1}>0}{c}_{{k}_{1},0}cos\frac{2\pi {k}_{1}({a}_{2y}x-{a}_{2x}y)}{a{a}_{2y}}$ | ||

$\phantom{f(x,y)={c}_{0,0}}+2\sum _{{k}_{1}}\sum _{{k}_{2}>0}{c}_{{k}_{1},{k}_{2}}cos\frac{2\pi \left[{k}_{1}{a}_{2y}x+(-{k}_{1}{a}_{2x}+{k}_{2}a)y\right]}{a{a}_{2y}}$ | ||

$[\mathbb{R}]\phantom{\rule{4.pt}{0ex}}{c}_{{k}_{1},{k}_{2}}={c}_{{k}_{1},{k}_{2}}^{*}$ | $c}_{{k}_{1},{k}_{2}}\in \mathbb{R$ |

$pm$ rectangular | $\begin{array}{c}{\overrightarrow{a}}_{1}=a(1,0)\hfill \\ {\overrightarrow{a}}_{2}=b(0,1)\hfill \end{array}$ | $\begin{array}{c}{\overrightarrow{b}}_{1}=\frac{2\pi}{a}(1,0)\hfill \\ {\overrightarrow{b}}_{2}=\frac{2\pi}{b}(0,1)\hfill \end{array}$ |

$D=\left\{(0,0)\right\}\cup {D}_{1}^{0}\cup {D}_{2}$ | ${D}_{\mathrm{min}}={D}_{1}^{0}\cup {D}_{2}$ | |

$\begin{array}{c}{c}_{1,0}={c}_{-1,0}^{*}=-\frac{1}{4}+\frac{1}{8}i\hfill \\ {c}_{1,1}={c}_{-1,1}^{*}=1-\frac{1}{4}i\hfill \end{array}$ | $\begin{array}{c}({k}_{1}\ne 0,0)\\ ({k}_{1},{k}_{2}>0)\end{array}$ | |

${D}_{1}^{0}$ | ${D}_{2}$ | |

$({k}_{1}\ne 0,0)$ | $({k}_{1},{k}_{2}>0)$ | |

$c}_{{k}_{1},{k}_{2}}={c}_{{k}_{1},-{k}_{2}$ | ||

$f(x,y)={c}_{0,0}+\sum _{{k}_{1}\ne 0}{c}_{{k}_{1},0}\left[cos\frac{2\pi {k}_{1}x}{a}+isin\frac{2\pi {k}_{1}x}{a}\right]$ | ||

$\phantom{f(x,y)={c}_{0,0}}+2\sum _{{k}_{1}}\sum _{{k}_{2}>0}{c}_{{k}_{1},{k}_{2}}\left[cos\frac{2\pi {k}_{1}x}{a}cos\frac{2\pi {k}_{2}y}{b}+isin\frac{2\pi {k}_{1}x}{a}cos\frac{2\pi {k}_{2}y}{b}\right]$ | ||

$[\mathbb{R}]\phantom{\rule{4.pt}{0ex}}{c}_{-{k}_{1},{k}_{2}}={c}_{{k}_{1},{k}_{2}}^{*}$ | $c}_{0,{k}_{2}}\in \mathbb{R$ |

$pg$ rectangular | $\begin{array}{c}{\overrightarrow{a}}_{1}=a(1,0)\hfill \\ {\overrightarrow{a}}_{2}=b(0,1)\hfill \end{array}$ | $\begin{array}{c}{\overrightarrow{b}}_{1}=\frac{2\pi}{a}(1,0)\hfill \\ {\overrightarrow{b}}_{2}=\frac{2\pi}{b}(0,1)\hfill \end{array}$ |

$D=\left\{(0,0)\right\}\cup {D}_{1}^{0}\cup {D}_{2}$ | ${D}_{\mathrm{min}}={D}_{1}^{0}\cup {D}_{2}$ | |

$\begin{array}{c}{c}_{1,1}=-{c}_{-1,1}^{*}=\frac{1}{2}\hfill \\ {c}_{2,1}={c}_{-2,1}^{*}=1+i\hfill \end{array}$ | $\begin{array}{c}({k}_{1}\phantom{\rule{4.pt}{0ex}}\mathrm{even}\phantom{\rule{4.pt}{0ex}}\ne 0,0)\\ ({k}_{1},{k}_{2}>0)\end{array}$ | |

${D}_{1}^{0}$ | ${D}_{2}$ | |

$({k}_{1}\phantom{\rule{4.pt}{0ex}}\mathrm{even}\phantom{\rule{4.pt}{0ex}}\ne 0,0)$ | $({k}_{1},{k}_{2}>0)$ | |

$c}_{{k}_{1},{k}_{2}}={(-1)}^{{k}_{1}}{c}_{{k}_{1},-{k}_{2}$ | ${c}_{2n+1,0}=0$ | |

$f(x,y)={c}_{0,0}+\sum _{\begin{array}{c}{k}_{1}\phantom{\rule{4.pt}{0ex}}\mathrm{even}\\ {k}_{1}\ne 0\end{array}}{c}_{{k}_{1},0}[cos\frac{2\pi {k}_{1}x}{a}+isin\frac{2\pi {k}_{1}x}{a}]$ | ||

$\phantom{f(x,y)={c}_{0,0}}+2\sum _{{k}_{2}>0}\{\sum _{{k}_{1}\phantom{\rule{4.pt}{0ex}}\mathrm{even}}{c}_{{k}_{1},{k}_{2}}\left[cos\frac{2\pi {k}_{1}x}{a}cos\frac{2\pi {k}_{2}y}{b}+isin\frac{2\pi {k}_{1}x}{a}cos\frac{2\pi {k}_{2}y}{b}\right]$ | ||

$\phantom{f(x,y)={c}_{0,0}+2\sum _{{k}_{2}>0}\{}+\sum _{{k}_{1}\phantom{\rule{4.pt}{0ex}}\mathrm{odd}}{c}_{{k}_{1},{k}_{2}}\left[-sin\frac{2\pi {k}_{1}x}{a}sin\frac{2\pi {k}_{2}y}{b}+icos\frac{2\pi {k}_{1}x}{a}sin\frac{2\pi {k}_{2}y}{b}\right]\}$ | ||

$[\mathbb{R}]\phantom{\rule{4.pt}{0ex}}{c}_{-{k}_{1},{k}_{2}}={(-1)}^{{k}_{1}}{c}_{{k}_{1},{k}_{2}}^{*}$ | $c}_{0,{k}_{2}}\in \mathbb{R$ |

$cm$ centered rectangular | $\begin{array}{c}{\overrightarrow{a}}_{1}=(\frac{a}{2},-\frac{b}{2})\hfill \\ {\overrightarrow{a}}_{2}=(\frac{a}{2},\frac{b}{2})\hfill \end{array}$ | $\begin{array}{c}{\overrightarrow{b}}_{1}=2\pi (\frac{1}{a},\frac{1}{b})\hfill \\ {\overrightarrow{b}}_{2}=2\pi (\frac{1}{a},-\frac{1}{b})\hfill \end{array}$ |

$D=\left\{(0,0)\right\}\cup {D}_{1}^{0}\cup {D}_{2}$ | ${D}_{\mathrm{min}}={D}_{1}^{0}\cup {D}_{2}$ | |

$\begin{array}{c}{c}_{0,1}={c}_{-1,0}^{*}=-\frac{1}{4}+\frac{1}{8}i\hfill \\ {c}_{1,1}={c}_{-1,-1}^{*}=\frac{1}{8}\hfill \end{array}$ |