## 3 ICI Problem in Underground Mining VLC Environments and ADR-Based Solutions
### ADR with the Hemi-Dodecahedron Structure  
To present a more efficient ICI mitigation alternative in underground mining VLC systems, a new ADR scheme that follows a regular hemi-dodecahedron structure is proposed in this work. This proposal fulfills all the general benefits of ADRs. Its geometric shape was chosen because it takes the best part of the ADRs of flat side faces, and of the ADRs of spherical surfaces, without overloading of PDs the \(R_{x}\). With these features, we can generate greater granularity together with a compact design and easy insertion in a mining environment, specifically in the helmets of mining workers. The physical structure of the proposed ADR is composed of six pentagonal faces, as we see in Figure 6b. Assuming that the ADR is always vertical, one of the faces will be pointing up, with a PD located in the center of its pentagonal surface, which we will call \(PD_{0}\). The lateral faces, also of pentagonal surface, will be five in total and each one will have a PD on its surface, which we will call \(PD_{i}\) with \(1\leq i\leq 5\). As mentioned, all PDs, both those of the lateral faces and those of the upper face, will be located in the center of the pentagonal surfaces, all of them having the same FoV and optical and electrical performances. The location of \(PD_{0}\) will be called \(P_{0}\). The \(PD_{i}\)s will be located uniformly and equidistant from each other, in a circle whose center coincides in projection with \(P_{0}\), and has radius \(r_{h}\). The side and top view of the proposed ADR is shown in Figure 6a,b, respectively, with regular pentagonal surfaces of side \(L\), central angle \(\omega\), and apothem \(ap\). By Geometrically, and since \(\omega\) is the angle formed by two lines that join the pentagon center with two consecutive vertices, whose general expression is \(\omega=2\pi/N\), where \(N\) is the sides number of the polygon, it is possible to express \(ap\) in terms of \(\omega\) and \(L\) as follows,  
\[ap=\frac{L}{2 tan(\omega/2)}. \tag{20}\]  
To maintain the same hybrid position and orientation coordinate system, and PDs normal vectors defined in Section 3.2, is necessary to define and delimit the elevation and azimuth angles that PDs can take. Thus, we will first define a normal vector \(\mathbf{n}_{PD}^{0}\) of \(PD_{0}\) located at \(P_{0}=(x_{PD_{0}},y_{PD_{0}},z_{PD_{0}})\), given as  
\[\mathbf{n}_{PD}^{0}=\left(x_{PD_{0}},y_{PD_{0}},z_{PD_{0}},\alpha_{PD_{0}},\beta_{ PD_{0}}\right), \tag{21}\]  
and a normal vector \(\mathbf{n}_{PD}^{i}\) of \(PD_{i}\) located at \(P_{i}=(x_{PD_{i}},y_{PD_{i}},z_{PD_{i}})\), given as  
\[\mathbf{n}_{PD}^{i}=\left(x_{PD_{i}},y_{PD_{i}},z_{PD_{i}},\alpha_{PD_{i}},\beta_{ PD_{i}}\right). \tag{22}\]  
Being \(PD_{0}\), in the upper face, its inclination angle will be null, whereas the \(PD_{i}\)s in the lateral faces will have an inclination angle \(\alpha_{PD_{i}}\in[0,\pi]\). We assume that due to the symmetry of regular polyhedra, all \(PD_{i}\) will have the same \(\alpha_{PD_{i}}\), which will be the same inclination angle, \(\alpha_{p}\) as the pentagons of the lateral faces. Therefore, when the hemi-dodecahedron ADR is vertically oriented, \(\alpha_{PD_{i}}\) will be defined as  
\[\alpha_{PD_{i}}= \begin{cases}0^{\circ},&\text{if }i=0\\ \alpha_{p},&\text{if }i=1,\dots,5\end{cases} . \tag{23}\]According to the symmetric and equidistant distribution that each \(PD_{i}\) will have in the circumference of radius \(r_{p}\), we will define the azimuth angle as \(\beta_{PD_{i}}\in[0,2\pi]\). As there are five lateral PDs, one on each side of the hemi-dodecahedron, we will determine that one of the PDs will have a fixed \(\beta_{1}\), in \(\beta_{PD_{1}}\)=\(0^{\circ}\). Due to this assumption, we can leave the general expression of \(\beta_{PD_{i}}\) defined as  
\[\beta_{PD_{i}}=\frac{2\pi(i-1)}{5}\qquad for\ \ 1\leq i\leq 5. \tag{24}\]  
Thus, the PDs will be specifically oriented in 0, \(2\pi/5\), \(4\pi/5\), \(6\pi/5\), and \(8\pi/5\).