## 2 Orbit Trap Method  
In [17], [PERSON] et al. introduced a method for generating artistic images with wallpaper symmetries. The method was based on the mappings for which the chaotic attractor possesses a wallpaper symmetry [15] and the orbit trap method [16; 19]. Moreover, they proposed a method for generating colour maps using an exponential function. In this section, we briefly present the method of [PERSON] et al. In general, in the method for each point in the considered area, we iterate a function starting from the point and check whether the transformed point falls into one of the orbit traps (orbit trap is a bounded area in the plane). When the point falls into one of the orbit traps, the point is considered to be trapped, and we end the iteration process. Next, a colour is assigned to the starting point based on the distance from the centre of the orbit trap to the trapped point. If the point is not trapped in the given number of iterations, then it gets a background colour. To obtain images with wallpaper symmetry, we need to use a proper function that is iterated and some orbit traps. [PERSON] et al. based their method on the functions introduced in [15]. Depending on the symmetry type, we need to use different functions. For the symmetry type p2, pm, pg, cm, pmm, cmm, pmg, pgg, p4, p4m, and p4g, we use function \(f:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}\) of the following form [15]:  
\[f\bigg{(} \begin{bmatrix}x\\ y\end{bmatrix} \bigg{)}\ =\ [1,\cos x,\cos 2x,\sin x,\sin 2x] \begin{bmatrix}P_{00}& \ldots&P_{04}\\ \vdots&\vdots&\vdots\\ P_{04}&\ldots&P_{44}\end{bmatrix} \cdot \begin{bmatrix}1\\ \cos y\\ \cos 2y\\ \sin y\\ \sin 2y\end{bmatrix} \mod \begin{bmatrix}2\pi\\ 2\pi\end{bmatrix} , \tag{1}\]  
where \(P_{00},P_{01},\ldots,P_{44}\in\mathbb{R}^{2}\). In [15], [PERSON] et al. derived conditions that the points \(P_{00},P_{01},\ldots,P_{44}\) must satisfy to obtain each of the symmetry type. For completeness, we gather the conditions in Appendix A. To define functions for the p3, p3m1, p3m, p6, and p6m symmetries, let us firstly define the vectors [15]:  
\[v_{0}= \begin{bmatrix}1\\ -\frac{1}{\sqrt{3}}\end{bmatrix} ,\ v_{1}\ =\ \begin{bmatrix}0\\ -\frac{2}{\sqrt{3}}\end{bmatrix} ,\ v_{2}\ =\ v_{0}+v_{1}, \tag{2}\]  
and the following mappings:\[R_{3}\bigg{(} \begin{bmatrix}x\\ y\end{bmatrix} \bigg{)} = \begin{bmatrix}-\frac{1}{2}&-\frac{\sqrt{3}}{2}\\ \frac{\sqrt{3}}{2}&-\frac{1}{2}\end{bmatrix}\begin{bmatrix}x\\ y\end{bmatrix} , \tag{3}\] \[R_{6}\bigg{(} \begin{bmatrix}x\\ y\end{bmatrix} \bigg{)} = \begin{bmatrix}\frac{1}{2}&-\frac{\sqrt{3}}{2}\\ \frac{\sqrt{3}}{2}&\frac{1}{2}\end{bmatrix}\begin{bmatrix}x\\ y\end{bmatrix} . \tag{4}\]  
Let us note that \(R_{3}\) is a counter-clockwise rotation by \(120^{\circ}\), whereas \(R_{6}\) is a counter-clockwise rotation by \(60^{\circ}\). Moreover, for \(k\in\{3,6\}\), let us define [15]  
\[g_{k}(p,v,G_{v}) = \sum_{i=\ 0}^{k-1}R_{k}^{i}(G_{v})\cos(R_{k}^{i}(v)\cdot p), \tag{5}\] \[h_{k}(p,v,H_{v}) = \sum_{i=\ 0}^{k-1}R_{k}^{i}(H_{v})\sin(R_{k}^{i}(v)\cdot p), \tag{6}\]  
where \(p,v,G_{v},H_{v}\in\mathbb{R}^{2}\), and \(\cdot\) is the dot product. For each of the \(v_{0},v_{1},v_{2}\) vectors, let us arbitrarily select \(G_{v_{0}},H_{v_{0}},G_{v_{1}},H_{v_{1}},G_{v_{2}},H_{v_{2}}\in\mathbb{R}^{2}\), respectively. Now, for each of the p3, p3m1, p31m, p6, p6m symmetries, we define the \(f:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}\) function in the following way [15]:  
* p3 symmetry (\(k\ =\ 3\)) and p6 symmetry (\(k\ =\ 6\)) \[f(p) = \sum_{j\ =\ 0}^{2}\Big{(}g_{k}(p,v_{j},G_{v_{j}})+h_{k}(p,v_{j},H_{v_{j} })\Big{)}\mod\quad \begin{bmatrix}2\pi\\ 2\pi\end{bmatrix} ,\] (7)
* p3m1 and p31m symmetries (\(k\ =\ 3\)), and p6m symmetry (\(k\ =\ 6\)) \[f(p) =\sum_{j\ =\ 0}^{2}\Big{(}g_{k}(p,v_{j},G_{v_{j}})+g_{k}(p, \sigma(v_{j}),\sigma(G_{v_{j}}))\] (8) \[+h_{k}(p,v_{j},H_{v_{j}})+h_{k}(p,\sigma(v_{j}),\sigma(H_{v_{j}})) \Big{)}\mod\quad \begin{bmatrix}2\pi\\ 2\pi\end{bmatrix} ,\] where \[\sigma\bigg{(} \begin{bmatrix}x\\ y\end{bmatrix} \bigg{)}=