# Classical and Quantum Gases on a Semiregular Mesh
## 2 Lattice-Gas Models on a Spherical Mesh  
As anticipated in the Introduction, we hereafter explore the possibility of unusual orderings in a system of particles occupying the nodes of a spherical mesh, chosen to be sufficiently regular so that some polyhedral (Platonic) arrangement can occur.  
We focus our attention on the pentakis dodecahedron [23] (PD, see Figure 1), a Catalan solid with 32 vertices obtained by augmenting the dodecahedron with 12 right pyramids on its pentagonal faces, in such a way that the resulting polyhedron is dual to the truncated icosahedron (clearly, the pyramidal apices form the vertices of an icosahedron). Even though the PD is not inscribable, implying that no spherical mesh can be drawn from its vertices, it is straightforward to obtain a biscribed solid by a small distortion of the PD that preserves its connectivity properties and its full icosahedral symmetry. A biscribed solid is any convex polyhedron that has concentric circumscribed and inscribed spheres, where the sphere center is also the centroid of the vertices. The five Platonic solids are biscribed solids, but none of the Archimedean or Catalan solids are. As for the PD, it suffices to adjust the height of the pentagonal pyramids only slightly to force all the vertices to be on the same sphere. The outcome of this construction is the biscribed form of the PD. It is this variant of the PD that is considered hereafter.  
The PD has 60 faces (isosceles triangles) and 90 edges (60 short and 30 long). We call PD mesh the skeleton of the PD, i.e., the mesh formed by its edges. The PD mesh is the finite analog of a lattice; the nodes of the mesh (i.e., the PD vertices) are its \"sites\". While five edges depart from an icosahedral vertex/site, the number of edges departing from a dodecahedral vertex/site is six (in this sense, icosahedral and dodecahedral sites are \"inequivalent\"). Setting the circumscribed radius equal to 1, the short-edge length (i.e., the shortest distance in the mesh) is \(\ell_{1}=\sqrt{30\big{(}15-\sqrt{15(5+2\sqrt{5})}\big{)}}\,/15\simeq 0.640 85\ldots\), whereas the long-edge length (the second shortest distance in the mesh) is \(\ell_{2}=\big{(}\sqrt{15}-\sqrt{3}\big{)}/3\simeq 0.71364\ldots\)The phase behavior of a lattice-gas model on the PD mesh is better studied in the grand-canonical ensemble. Upon increasing the chemical potential \(\mu\) at fixed \(T\), the mesh becomes increasingly populated, with the possibility of \"transitions\" between qualitatively distinct arrangements. Denoting \(c_{i}=0,1\) the occupation number of site \(i\), we call first (second) neighbors any two sites/particles that are linked by a short (long) edge. According to our nomenclature, an icosahedral site has five first-neighbor sites and no second-neighbor site, whereas a dodecahedral site has three first neighbors and three second neighbors (see Figure 1). With these specifications, the grand Hamiltonian of our model reads:  
\[H=V_{1}\sum_{\langle i,j\rangle}c_{i}c_{j}+V_{2}\sum_{\langle\langle k,l\rangle \rangle}c_{k}c_{l}-\mu\sum_{i}c_{i}\,, \tag{1}\]  
where \(V_{1}>0\) (\(V_{2}\)) is the coupling between two first (second) neighbor particles. This model can mimic a system of particles adsorbed on a \"substrate\" sculpted like a PD mesh (think of, e.g., the interstices between atoms in a C\({}_{60}\) molecule), and interacting via a spherically-symmetric potential with a hard core followed, at larger distances, by a soft short-range repulsion. By suitably tuning the ratio \(\gamma=V_{2}/V_{1}\) between the couplings, we anticipate the existence of a competition between icosahedral order and dodecahedral order at low temperature, with the further possibility of arrangements with intermediate order.  
In the following, we use \(V_{1}\) as the unit of energy; in turn, this defines a reduced temperature, \(T^{*}=k_{B}T/V_{1}\) (\(k_{B}\) being the Boltzmann constant), and a reduced chemical potential, \(\mu^{*}=\mu/V_{1}\).