Bi-Symmetric Polyhedral Cages with Maximally Connected Faces and Small Holes

1. Introduction

A few years ago, J. Heddle created an artificial protein cage made out of 24 so-called rings themselves made out of 11 copies of the same protein called TRAP [1]. The nano-cage appears to be a regular structure made out of 24 hendecagons, with some small holes, which is mathematically impossible. It was shown that the faces of the corresponding structure are not regular but that the deformation of their edge lengths and angles is less than half a percent [2] making them look regular for the naked eye. These small deformations can easily be absorbed by the proteins and the termini where the faces are linked together. A small nano-cage made out of the same ring but counting only 12 of them was made in the same lab[3]. This was also identified as a nearly regular structure but one where the deformations are approximately 2.5%.

These discoveries led us to define polyhedral cages [4] (p-cages for short) as assemblies of polygons with holes, requiring that each face shares edges with at least 3 other faces and also imposing that of any two adjacent edges of a face only one of them can be shared. While the faces must be planar convex polygons, the holes can have any shape. In what follows, we only consider convex p-cages, as defined in [4,5].

P-cages are said to be regular if all the faces are regular and near-miss if they are slightly deformed. Homogeneous p-cages are made out of polygons with the same number of edges while bi-homogeneous p-cages are made out of 2 types of polygons. In [5] we defined symmetric p-cages as p-cages for which any two faces can be mapped into each other via a rotation which is an automorphism of the p-cage. Similarly, bi-symmetric p-cages are bi-homogeneous p-cages for which any 2 faces belonging to the same family can be mapped into each other via a rotation automorphism of the p-cage [6].

When we consider near-miss p-cages we must decide how large a deformation we are willing to consider. In [4,5] we have identified all regular and near-miss symmetric p-cages made out of polygons with up to 20 edges and with deformations not exceeding 10%. In [6] we have identified the bi-symmetric p-cages where each face of a given type are only sharing edges with faces of the other type, again restricting ourselves to 10% deformation and polygons with up to 20 edges.

The aim of this paper is to identify bi-symmetric p-cages where the faces have a maximum number of neighbours. As it is not possible for all the faces to have 6 neighbours [7], we consider p-cages for which the faces of the first type have 5 neighbours each while the faces of the second type have 6.

So far, a number of artificial protein cages have been experimentally generated [8,9,10,11,12,13]. The main aim is to develop new drug delivery methods [14,15,16,17,18]. The idea is to encapsulate the drug inside the cage while specific receptors are linked to the holes outside the cage to bind with the cells that are targeted such as cancer cells [14]. Once absorbed by cell, the protein-cage can release the drug in the cytoplasm [19]. This will lead to more efficient drug delivery as a smaller amount of the drug must be provided with the bonus of reduced side effects as only the targeted cells will be affected.

While the virus capsids found in nature are essentially based on the geometry of platonic solids [20] some of the cages created experimentally exhibit somewhat different structures [2,3,21]. It is then natural to ask the question as to what geometries are possible for such cages [22,23,24].

The aim of this paper is to find further potential geometries for nano-protein cages, identifying all bi-symmetric ones with maximal connectivity between the faces as well exhibiting small holes.

2. Methodology

As described in [4] to construct a p-cage one must first consider the planar graph obtained by joining the centres of the neighbouring faces of the p-cage with straight lines. We call the resulting object the hole polyhedron graph because its faces describe how many faces surround each hole. The nodes of the hole polyhedron graph correspond to the faces of the p-cage while the edges describe how the faces are joined together.

Our aim in this paper is to construct p-cages made out of 2 families of faces such that each face of a given family can be mapped by an automorphism rotation of the p-cage to any other face of the same family.

In what follows we define as the valency of a face the number of neighbour that it has. As we are interested by p-cages with the maximum number of neighbours, we impose that the faces belonging to the first family has valency 5 while the other faces valency 6. As we are interested only in p-cages with small holes we also impose that the faces of the hole polyhedron graphs are triangles, squares or any mix of each.

There are only 9 such graphs [7] as listed in Table 1:

The naming convention of the graphs is described in [7] but the end of these names is particularly useful as in Vn_m, n and m correspond respectively to the numbers of pentavalent and hexavalent nodes and hence also correspond to the numbers of p-cage faces with 5 and 6 neighbours respectively.

In what follows, we denote by $P_1$ and $P_2$ the numbers of edges of the faces belonging to the families 1 and 2 respectively. When a polygon with P edges has n neighbours its has $P-n$ edges which must be distributed between the n edges adjacent to neighbour faces. If we use the labels a, b,c, d, e, to label the numbers of edges given to each hole by the first family of face and A, B,C, D, E, F for the second family, we have

$$\begin{array}{c}\hfill \mathtt{a}+\mathtt{b}+\mathtt{c}+\mathtt{d}+\mathtt{e}+\mathtt{5}=P_1,\mathtt{A}+\mathtt{B}+\mathtt{C}+\mathtt{D}+\mathtt{E}+\mathtt{F}+\mathtt{6}=P_2\end{array}$$

(1)

for the faces belonging respectively to the first and the second family. We must then find out how distribute them on each of the identified hole polyhedron graphs in such a way that the equivalence between the faces is preserved. We now consider each of these 9 graphs one by one:

For HAP6 the corners of the valency 6 nodes must all be identical, but the corners around the valency 5 nodes can be arbitrary (Figure 1a).

For TTP6 the corners of the valency 6 nodes must be alternating A-B-A-B-A-B but then the corners of the valency 5 nodes can be arbitrary (Figure 1b).

For TOP6 the corners of the valency 5 nodes are arbitrary and mapped as in Figure 2a. The corners of the valency 6 nodes must then alternate between A and B (Figure 2a).

For PD the equivalence between two adjacent valency 6 nodes is achieved either via a 5-fold rotation around the pyramid or via a $2\pi$ rotation around the centre of the link joining them. In either case we see that the corners around the valency 5 nodes must all be equal. Then the corners of the valency 6 nodes must be alternatingly A and B (Figure 2b).

For IP5 we see that by symmetry, the corners of the triangles joining the valency 6 nodes must all be the same so the corners of the valency 5 nodes are also all identical. Then for the valency 6 nodes we have the sequence A-B-C-A-B-C (Figure 3a).

For SDP50 the corners around the valency 5 nodes must all be the same but then the corners around the valency 6 nodes are arbitrary as shown in the figure (Figure 3b).

The p-cages TTM3 and TOM3 do not lead to any p-cages with deformations below 10% and the best TCM3 p-cages having deformations exceeding 8% are not good candidates for protein cages as the faces are too irregular for proteins rings to adjust to the deformations. For this reason the construction of these 3 families of cages is described in the supplementary material.

To construct the p-cages we follow a method similar to the one described in [5,6]: we derive the most general parametrisation for the different faces which we then use to determine numerically the faces which are least irregular.

First we notice that each of the identified hole polyhedron graphs are built from an Archimedean solid, the dual of an Archimedean solid or an antiprism. Given a face of any of the 2 families, all the other faces of the same family will be obtained by applying the symmetry of the regular solid underlying the hole polyhedron graph. We label each face with its family type j, set to 1 for the pentavalent faces and 2 for the hexavalent faces, as well as an index number i. Each face will belong to a plane plane going through the vector ${\mathit{V}}_{j,i}$ and spanned by the orthonormal basis vector ${\mathit{v}}_{j,i,1}$ and ${\mathit{v}}_{j,i,2}$ . We also chose ${\mathit{V}}_{j,i}$ such that it is orthogonal to the considered plane. The planes and faces with index $i=1$ are called the reference plane and face respectively. The remaining faces of the p-cage can then be obtained by applying a rotation to the reference faces. We can then determine the lines of intersection between adjacent faces knowing that the shared edges will lie on these lines. The vertices of the reference faces can then be parametrised in the reference plane restricting the vertices belonging to two faces to lie on the corresponding intersection lines. The vertices belonging to a hole will belong to the reference plane without any further restriction.

We then have to identify the face configuration for which the faces are as regular as possible.

First we denote ${\mathit{n}}_i$, $i=0,P_j-1$, the vertices of a face, ordered anticlockwise when seen from outside the p-cage and ${\mathit{m}}_{f_j}$ the normal to the reference face of type j pointing out of the p-cage. We then define the edge vector ${\mathit{s}}_i={\mathit{n}}_i-{\mathit{n}}_{(i+1)mod{P}_j}$ so that the edge lengths are given by $d_i=\left|{\mathit{s}}_i\right|$ and compute the angle ${\alpha }_i$ between adjacent edges, by evaluating ${\mathit{s}}_i\times {\mathit{s}}_{i+1}$ and

$$\begin{array}{ccc}\hfill \mathrm{if}({\mathit{s}}_i\times {\mathit{s}}_{i+1})·{\mathit{m}}_{f_j}\ge 0& :& {\alpha }_i=\pi -\mathrm{acos}\left({\displaystyle \frac{({\mathit{s}}_i·{\mathit{s}}_{i+1})}{|{\mathit{s}}_i\left|\right|{\mathit{s}}_{i+1}|}}\right),\hfill \\ \hfill \mathrm{if}({\mathit{s}}_i\times {\mathit{s}}_{i+1})·{\mathit{m}}_{f_j}<0& :& {\alpha }_i=\pi +\mathrm{acos}\left({\displaystyle \frac{({\mathit{s}}_i·{\mathit{s}}_{i+1})}{|{\mathit{s}}_i\left|\right|{\mathit{s}}_{i+1}|}}\right).\hfill \end{array}$$

(2)

Notice that ${\alpha }_i$ in (2) corresponds to the angle inside the face which is larger than $\pi$ if the face is not convex. For a regular P-polygon, $\alpha =\pi (1-{\displaystyle \frac{2}{P}})$.

After computing the $d_i$ and ${\alpha }_i$ we can minimise the deviation from the regularity energy

$$\begin{array}{c}\hfill E={\displaystyle \frac{1}{P_1+P_2}}\left(E_1+E_2+c_c{E}_{\mathrm{Fconv}}+c_{pc}{E}_{\mathrm{Pconv}}\right)\end{array}$$

(3)

where

$$\begin{array}{c}\hfill E_j=\sum _{i=0}^{P_j-1}\left[c_l{\left({\displaystyle \frac{d_i-L}{L}}\right)}^2+c_a{\left({\displaystyle \frac{{\alpha }_i-\pi (1-\frac{2}{P_j})}{\pi (1-\frac{2}{P_j})}}\right)}^2\right],j=1,2\end{array}$$

(4)

with the 3 weight factors $c_l$, $c_a$ and $c_c$. $E_{\mathrm{Fconv}}$, given explicitly by

$$\begin{array}{c}\hfill E_{\mathrm{Fconv}}=\sum _{j=1,2}{\displaystyle \frac{1}{P_j}}\sum _i\left[H\left(-({\mathit{s}}_i\times {\mathit{s}}_{i+1})·{\mathit{m}}_{f_j}\right)\right]\end{array}$$

(5)

where $H\left(x\right)$ is the Heaviside function. (5), is 0 unless the polygon defined by the vertices is concave.

To compute $E_{\mathrm{Pconv}}$ we find ${\mathit{C}}_i$, the position of the centre of face i and if we consider 2 adjacent faces ${\mathit{C}}_i$ and ${\mathit{C}}_j$ with normal vectors ${\mathit{m}}_i$ and ${\mathit{m}}_j$ respectively. The p-cage is convex by if for all pairs of adjacent faces, the distance between the centres of the 2 faces is smaller than the distance between ${\mathit{C}}_i+m_i$ and ${\mathit{C}}_j+m_j$. This is used to define $E_{\mathrm{Pconv}}$ as

$$\begin{array}{c}\hfill E_{\mathrm{Pconv}}=\sum _{j=1,2}\left[{\displaystyle \frac{1}{P_j}}\sum _{i\in I_j}\left[H\left(|{\mathit{C}}_i-{\mathit{C}}_j{|}^2-{|{\mathit{C}}_i+{\mathit{m}}_i-{\mathit{C}}_j-{\mathit{m}}_j|}^2\right)\right]\right].\end{array}$$

(6)

where $I_j$ is the set of faces, of any type, adjacent to the reference face j. Notice that $E_{\mathrm{Pconv}}$ is 0 unless the p-cage is concave.

These last two terms are used in the simulated annealing optimisation procedure to enforce convexity of the faces and the p-cage by taking large values for $c_c$ and $c_{pc}$. In (3) we divide the sum by $P_1+P_2$ to make the parametrisation of the optimising algorithm easier.

3. Notation

In what follows we denote with $R_{\mathit{w}}\left(\theta \right)$ a rotation of angle $\theta$ around the vector $\mathit{w}$. We also denote $R_x\left(\theta \right)$, $R_y\left(\theta \right)$ and $R_z\left(\theta \right)$ the rotation of an angle $\theta$ around the respective axis x, y and z.

The plane of face $j,i$ can be parametrised using the point $V_{j,i}$ contained in the plane and the plane basis vectors $v_{j,i,1}$ and $v_{j,i,2}$ as

$$\begin{array}{c}\hfill {\mathcal{P}}_{j,i}(t_1,t_2)={\mathit{V}}_{j,i}+t_1{\mathit{v}}_{j,i,1}+t_2{\mathit{v}}_{j,i.2},\end{array}$$

(7)

Given two such planes with parametrisation ${\mathcal{P}}_{j_1,i}$ and ${\mathcal{P}}_{j_2,k}$ we must find the intersecting line which determines where the edge shared by the two adjacent faces is. First we define the normal vectors to each planes, ${\mathit{p}}_{j_1,i}$ and ${\mathit{p}}_{j_1,}k$, as well as the vector ${\mathit{u}}_{j_1,i;j_{2}k}$ parallel to the plane intersection:

$$\begin{array}{c}\hfill {\mathit{p}}_{j_1,i}={\mathit{v}}_{i,1}\times {\mathit{v}}_{i,2},{\mathit{p}}_{j_2,k}={\mathit{v}}_{j,1}\times {\mathit{v}}_{j,2},{\mathit{u}}_{j_1,i;j_2,k}={\mathit{p}}_{j_1,i}\times {\mathit{p}}_{j_2,k}.\end{array}$$

(8)

Any point $U_{j1,i1,j2,i2}$ on that line is in the range of the parametrisations of both planes:

$$\begin{array}{c}\hfill {\mathit{U}}_{j_1,i;j_2,k}={\mathit{V}}_{j_1,i}+t_1{\mathit{v}}_{j_1,i,1}+t_2{\mathit{v}}_{j_1,i,2}={\mathit{V}}_{j_2,k}+s_1{\mathit{v}}_{j_2,k,1}+s_2{\mathit{v}}_{j_2,k,2}.\end{array}$$

(9)

If that point is perpendicular to $u_{j1,i1,j2,i2}$ then a relation holds among $t_1$, $t_2$ and $s_1$, $s_2$, obtained by multiplying both sides of (9) by ${\mathit{u}}_{j_1,i;j_2,k}$ which, as detailed in [25], when inserted back into (9) gives

$$\begin{array}{ccc}\hfill {\mathit{U}}_{j_1,i;j_2,k}& =& {\mathit{V}}_{j_1,i}+{\displaystyle \frac{({\mathit{p}}_{j_2,k}·({\mathit{V}}_{j_2,k}-{\mathit{V}}_{j_1,i}))({\mathit{u}}_{j_1,i;j_2,k}·{\mathit{v}}_{j_1,i,2})+({\mathit{u}}_{j_1,i;j_2,k}·{\mathit{V}}_{j_1,i})({\mathit{p}}_{j_2,k}·{\mathit{v}}_{j_1,i,2})}{({\mathit{p}}_{j_2,k}·{\mathit{v}}_{j_1,i,1})({\mathit{u}}_{j_1,i;j_2,k}·{\mathit{v}}_{j_1,i,2})-({\mathit{u}}_{j_1,i;j_2,k}·{\mathit{v}}_{j_1,i,1})({\mathit{p}}_{j_2,k}·{\mathit{v}}_{j_1,i,2})}}\times \hfill \\ & & \left({\mathit{v}}_{j_1,i,1}-{\displaystyle \frac{({\mathit{u}}_{j_1,i;j_2,k}·{\mathit{v}}_{j_1,i,1})}{({\mathit{u}}_{j_1,i;j_2,k}·{\mathit{v}}_{j_1,i,2})}}{\mathit{v}}_{j_1,i,2}\right)-{\displaystyle \frac{({\mathit{u}}_{j_1,i;j_2,k}·{\mathit{V}}_{j_1,i})}{({\mathit{u}}_{j_1,i;j_2,k}·{\mathit{v}}_{j_1,i,2})}}{\mathit{v}}_{j_1,i,2}.\hfill \end{array}$$

(10)

To obtain the intersection point of 2 coplanar lines, ${\mathit{U}}_{j_1,i}+\lambda {\mathit{u}}_{j_1,i}$ and ${\mathit{U}}_{j_2,k}+\mu {\mathit{u}}_{j_2,k}$ where we assume ${\mathit{u}}_{j_1,i}$ and ${\mathit{u}}_{j_2,k}$ to be normalised to 1, we need to find the value of $\lambda$ so that

$$\begin{array}{c}\hfill {\mathit{U}}_{j_1,i}+\lambda {\mathit{u}}_{j_1,i}={\mathit{U}}_{j_2,k}+\mu {\mathit{u}}_{j_2,k}\end{array}$$

(11)

Multiplying (11) by respectively ${\mathit{u}}_{j_1,i}$ and ${\mathit{u}}_{j_2,k}$ we obtain

$$\begin{array}{cc}\hfill \lambda & =({\mathit{U}}_{j_2,k}-{\mathit{U}}_{j_1,i}){\mathit{u}}_{j_1,i}+\mu ({\mathit{u}}_{j_2,k}·{\mathit{u}}_{j_1,i})\hfill \\ \hfill \mu & =({\mathit{U}}_{j_1,i}-{\mathit{U}}_{j_2,k}){\mathit{u}}_{j_2,k}+\lambda ({\mathit{u}}_{j_1,i}·{\mathit{u}}_{j_2,k}).\hfill \end{array}$$

(12)

Substituting the second equation in the first one we obtain for the point of intersection

$$\begin{array}{c}\hfill P_{j_1,i;j_2,k}=U_{j_1,i}+u_{j_1,i}\left({\displaystyle \frac{(({\mathit{U}}_{j_2,k}-{\mathit{U}}_{j_1,i})·({\mathit{u}}_{j_1,i}-{\mathit{u}}_{j_2,k}({\mathit{u}}_{j_1,i}·{\mathit{u}}_{j_2,k})))}{1-{({\mathit{u}}_{j_1,i}·{\mathit{u}}_{j_2,k})}^2}}\right).\end{array}$$

(13)

In what follows, ${\mathit{P}}_{j_1,i;j_2,k}$ will denote the intersection point between the following 3 faces: the reference face of type 1, the face i of type $j_i$ and the face k, of type $j_2$. This corresponds to the intersection point between the two lines ${\mathit{U}}_{1,1;j_1,i}+\lambda {\mathit{u}}_{1,1;p{j}_1,i}$ and ${\mathit{U}}_{1,1;j_2,k}+\lambda {\mathit{u}}_{1,1;j_2,k}$ obtained using (13). Similarly ${\mathit{Q}}_{j_1,i;j_2,k}$ will denote the intersection point between the reference face of type 2, the face i of type $j_i$ and the face k of type $j_2$.

To denote face i of type j we use the notation $F_{j,i}$ and each face with index $i>1$ is related to the references face of the correct type via a rotation

$$\begin{array}{c}\hfill F_{1,i}={\mathcal{R}}_{1,i}{F}_{1,1},F_{2,i}={\mathcal{R}}_{2,i}{F}_{2,1}.\end{array}$$

(14)

The spanning vectors ${\mathit{V}}_{j,i}$, ${\mathit{v}}_{j,i,1}$ and ${\mathit{v}}_{j,i,1}$ will then be obtained via the same rotation:

$$\begin{array}{c}\hfill \mathit{V}j,i={\mathcal{R}}_{j,i}{\mathit{V}}_{j,1},\mathit{v}j,i,1={\mathcal{R}}_{j,i}{\mathit{v}}_{j,1,1},\mathit{v}j,i,1={\mathcal{R}}_{j,i}{\mathit{v}}_{j,1,1}.\end{array}$$

(15)

4. Parametrisation

4.1. HAP6

The symmetry of the HAP6 p-cage is that of the hexagonal anti-prism. For the reference face vectors we have choose the following:

$$\begin{array}{cccc}\hfill {\mathit{V}}_{1,1}& =S_1{\left(0,cos\left({\theta }_2\right),-sin\left({\theta }_2\right)\right)}^t,\hfill & \hfill {\mathit{V}}_{2,1}& =S_2{(0,0,1)}^t\hfill \end{array}$$

(16)

$$\begin{array}{cccc}\hfill {\mathit{v}}_{1,1,1}& =(1,0,0),\hfill & \hfill {\mathit{v}}_{1,1,2}& =(0,sin\left({\theta }_2\right),cos\left({\theta }_2\right))\hfill \end{array}$$

(17)

$$\begin{array}{cccc}\hfill {\mathit{v}}_{2,1,1}& =(1,0,0),\hfill & \hfill {\mathit{v}}_{2,1,2}& =(0,1,0).\hfill \end{array}$$

(18)

and the rotations between the faces are

$$\begin{array}{cc}\hfill {\mathcal{R}}_{1,i}& =R_z\left({\displaystyle \frac{i\pi }{3}}\right),i=2\dots 6{\mathcal{R}}_{1,i}=R_z\left({\displaystyle \frac{(1+2i)\pi }{6}}+\sigma \right)R_y\left(\pi \right),1=7\dots 12\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill {\mathcal{R}}_{2,2}& =R_z\left({\displaystyle \frac{\pi +\sigma }{6}})\right)R_y\left(\pi \right)\hfill \end{array}$$

(20)

where $\sigma$ is an angle which allows the top and bottom halves of the p-cages to be shifted sideways with respect to each others.

Figure 4. Parametrisation of the hexagonal antiprism p-cage.

Figure 4. Parametrisation of the hexagonal antiprism p-cage.

The nodes of the reference face of type 1 shared with the adjacent faces are

$$\begin{array}{cccc}\hfill {\mathit{n}}_1& =P_{1,2;1,7}+k_1{\mathit{u}}_{1,1;1,2},\hfill & \hfill {\mathit{n}}_2& =P_{1,2;1,7}+k_2{\mathit{u}}_{1,1;1,2},\hfill \end{array}$$

(21)

$$\begin{array}{cccc}\hfill {\mathit{n}}_3& =P_{1,2;2,1}+k_3{\mathit{u}}_{2,1;1,1},\hfill & \hfill {\mathit{n}}_4& =P_{1,2;2,1}+k_4{\mathit{u}}_{2,1;1,1},\hfill \end{array}$$

(22)

$$\begin{array}{cccc}\hfill {\mathit{n}}_5& =R_z\left({\displaystyle \frac{-\pi }{3}}\right){\mathit{n}}_2\hfill & \hfill {\mathit{n}}_6& =R_z\left({\displaystyle \frac{-\pi }{3}}\right){\mathit{n}}_1\hfill \end{array}$$

(23)

$$\begin{array}{cccc}\hfill {\mathit{n}}_7& =P_{1,6;1,12}+k_7{\mathit{u}}_{1,1;1,12},\hfill & \hfill {\mathit{n}}_8& =R_z\left({\displaystyle \frac{-\pi }{6}}+\sigma \right)R_y\left(\pi \right){\mathit{n}}_7,\hfill \end{array}$$

(24)

$$\begin{array}{cccc}\hfill {\mathit{n}}_{10}& =P_{1,7;1,12}+k_{1}0{\mathit{u}}_{1,1;1,7},\hfill & \hfill {\mathit{n}}_9& =R_z\left({\displaystyle \frac{\pi }{6}}+\sigma \right)R_y\left(\pi \right){\mathit{n}}_{10}.\hfill \end{array}$$

(25)

for some parameters $k_i,i=1\dots 8$. The nodes of the reference face of type 2 are then given by

$$\begin{array}{c}\hfill {\mathit{m}}_{1+2i}=R_z\left({\displaystyle \frac{i\pi }{3}}\right){\mathit{n}}_4,{\mathit{m}}_{2+2i}=R_z\left({\displaystyle \frac{i\pi }{3}}\right){\mathit{n}}_{3}i=0\dots 5.\end{array}$$

(26)

The optimization parameters are $\theta$, $\sigma$, $S_1$, $S_2$, $k_1$, $k_2$, $k_3$, $k_4$, $k_7$, $k_{10}$ as well as the coordinates of the non-shared vertices within the plane of the faces for both reference faces.

For the optimisation to work well it helps to start from a reasonable initial configuration. We notice that the 10 shared edges between the reference face and its 5 neighbours form a pentagon and we can easily determine the coordinates of its vertices. As an initial configuration we can locate the shared vertices so that the edges of the pentagon are split in 3 equal parts. The unshared vertices can then be be placed equally spaced between these vertices. This is done explicitly in the code supplied on Zenodo. For the other parameters, we choose as initial values $\theta ={20}^{\circ }$, $\varphi =\sigma =0$ and $S_1=S_2=20$.

4.2. TTP6

The underlying symmetry of the TTP6 p-cage is that of the truncated tetrahedron. We have oriented the tetrahedron as depicted in Figure 5.

The coordinates of the vertices of the centred tetrahedron are

$$\begin{array}{c}\hfill \mathit{A}=(0,0,1),\mathit{B}=(0,-{\displaystyle \frac{\sqrt{8}}{3}},-{\displaystyle \frac{1}{3}}),\mathit{C}=(\sqrt{{\displaystyle \frac{2}{3}}},{\displaystyle \frac{\sqrt{2}}{3}},-{\displaystyle \frac{1}{3}}),\mathit{D}=(-\sqrt{{\displaystyle \frac{2}{3}}},{\displaystyle \frac{\sqrt{2}}{3}},-{\displaystyle \frac{1}{3}}),\end{array}$$

(27)

while the centres of the reference faces are given by

$$\begin{array}{cccc}\hfill {\mathit{F}}_{2,1}={\displaystyle \frac{\mathit{A}+\mathit{B}+\mathit{C}}{3}}& =(\sqrt{{\displaystyle \frac{2}{27}}},-{\displaystyle \frac{\sqrt{2}}{9}},{\displaystyle \frac{1}{9}}),\hfill & \hfill {\mathit{F}}_{1,1}& =\mathit{A}+{\displaystyle \frac{\mathit{B}-\mathit{A}}{3}}=\left(0,-{\displaystyle \frac{\sqrt{8}}{9}},{\displaystyle \frac{5}{9}}\right),\hfill \end{array}$$

(28)

and the rotations linking the faces are

$$\begin{array}{ccccc}\hfill {\mathcal{R}}_{1,2}& =R_z\left({\displaystyle \frac{2\pi }{3}}\right),\hfill & \hfill {\mathcal{R}}_{1,3}& =R_z\left({\displaystyle \frac{4\pi }{3}}\right),\hfill & \hfill \\ \hfill {\mathcal{R}}_{1,1+j+3i}& =R_{F_{2,1}}\left({\displaystyle \frac{i2\pi }{3}}\right)R_z\left({\displaystyle \frac{j2\pi }{3}}\right),\hfill & \hfill {\mathcal{R}}_{1,10+j}& =R_{F_{2,2}}\left({\displaystyle \frac{2\pi }{3}}\right)R_z\left({\displaystyle \frac{j2\pi }{3}}\right),\hfill & \hfill j=0,1,2,i=1,2\\ \hfill {\mathcal{R}}_{2,2}& =R_z\left({\displaystyle \frac{4\pi }{3}}\right),\hfill & \hfill {\mathcal{R}}_{2,3}& =R_z\left({\displaystyle \frac{2\pi }{3}}\right),\hfill & \hfill {\mathcal{R}}_{2,4}=R_{\mathit{B}}\left({\displaystyle \frac{4\pi }{3}}\right).\end{array}$$

(29)

We choese the following vectors to parametrise the reference planes:

$$\begin{array}{cccc}\hfill {\mathit{V}}_{1,1}& =S_1{R}_z\left(\varphi \right)\left(0,-cos\left(\theta \right),sin\left(\theta \right)\right),\hfill & \hfill {\mathit{V}}_{2,1}& =S_2{\mathit{F}}_{2,1},\hfill \\ \hfill {\mathit{v}}_{1,1,1}& =R_z\left(\varphi \right)(1,0,0),\hfill & \hfill {\mathit{v}}_{2,1,1}& ={\displaystyle \frac{\mathit{C}-\mathit{B}}{|\mathit{C}-\mathit{B}|}}=\left({\displaystyle \frac{1}{2}},{\displaystyle \frac{\sqrt{3}}{2}},0\right),\hfill \\ \hfill {\mathit{v}}_{1.1,2}& =R_z\left(\varphi \right)\left(0,sin\left(\theta \right),cos\left(\theta \right)\right),\hfill & \hfill {\mathit{v}}_{2,1,2}& ={\displaystyle \frac{{\mathit{F}}_{2,1}-\mathit{A}}{|{\mathit{F}}_{2,1}-\mathit{A}|}}=\left({\displaystyle \frac{1}{2\sqrt{3}}},-{\displaystyle \frac{1}{6}},-{\displaystyle \frac{\sqrt{8}}{3}}\right),\hfill \end{array}$$

(30)

where $S_1$ and $S_2$ are scaling parameters. The shared vertices of the reference faces are then given by

$$\begin{array}{cccccc}\hfill {\mathit{n}}_1& =P_{2,1;1,2}+k_1{\mathit{u}}_{1,1;1,2},\hfill & \hfill {\mathit{n}}_2& =P_{2,1;1,2}+k_2{\mathit{u}}_{1,1;1,2},\hfill & \hfill {\mathit{n}}_3& =R_z\left({\displaystyle \frac{-2\pi }{3}}\right){\mathit{n}}_2,\hfill \\ \hfill {\mathit{n}}_4& =R_z\left({\displaystyle \frac{-2\pi }{3}}\right){\mathit{n}}_1,\hfill & \hfill {\mathit{n}}_5& =P_{1,3;2,2}+k_5{\mathit{u}}_{1,1;2,2},\hfill & \hfill {\mathit{n}}_6& =P_{1,3;1,2,2}+k_6{\mathit{u}}_{1,1;2,2},\hfill \\ \hfill {\mathit{n}}_7& =P_{2,2;1,5}+k_7{\mathit{u}}_{1,1;1,5},\hfill & \hfill {\mathit{n}}_8& =R_{{\mathit{F}}_1}\left({\displaystyle \frac{2\pi }{3}}\right)R_z\left({\displaystyle \frac{2\pi }{3}}\right){\mathit{n}}_7,\hfill & \hfill {\mathit{n}}_9& =P_{1,5;2,1}+k_9{\mathit{u}}_{1,1;2,1},\hfill \\ \hfill {\mathit{n}}_{10}& =P_{1,5;2,1}+k_{10}{\mathit{u}}_{1,1;2,1},\hfill & \hfill {\mathit{m}}_1& ={\mathit{n}}_{10},{\mathit{m}}_2={\mathit{n}}_9,\hfill & \hfill {\mathit{m}}_3& =R_{{\mathit{F}}_{2,1}}\left({\displaystyle \frac{2\pi }{3}}\right)R_z\left({\displaystyle \frac{2\pi }{3}}\right){\mathit{n}}_6,\hfill \\ \hfill {\mathit{m}}_4& =R_{{\mathit{F}}_{2,1}}\left({\displaystyle \frac{2\pi }{3}}\right)R_z\left({\displaystyle \frac{2\pi }{3}}\right){\mathit{n}}_5,\hfill & \hfill {\mathit{m}}_i& =R_{{\mathit{F}}_{2,1}}\left({\displaystyle \frac{2\pi }{3}}\right){\mathit{m}}_{i-4}i=5\dots 12.\hfill \end{array}$$

(31)

The optimization parameters are $\theta$, $\varphi$, $S_1$, $S_2$, $k_1$, $k_2$, $k_5$, $k_6$, $k_7$, $k_9$, $k_{10}$ as well as the planar coordinates of the non-shared vertices for both reference faces. As initial parameter values we have used $\theta =\mathrm{acos}\left(\sqrt{8/33}\right)$, $\varphi =0$ and $S_1=3.75$, $S_2=9.375$. The other parameters are determined as described in the HAP6 section.

4.3. TOP6

The underlying symmetry of the TOP6 p-cage is that of the truncated octahedron. We orient the octahedron as depicted in Figure 6.

For the vectors normal to the reference faces we use

$$\begin{array}{cccc}\hfill {\mathit{F}}_{1,1}& =\left(0,-cos\left(\theta \right),sin\left(\theta \right)\right),\hfill & \hfill {\mathit{F}}_{2,1}& =\left({\displaystyle \frac{1}{\sqrt{3}}},-{\displaystyle \frac{1}{\sqrt{3}}},{\displaystyle \frac{1}{\sqrt{3}}}\right)\hfill \end{array}$$

(32)

and the rotations linking the faces to the reference faces are

$$\begin{array}{cccc}\hfill {\mathcal{R}}_{1,2}& =R_z\left({\displaystyle \frac{\pi }{2}}\right),\hfill & \hfill {\mathcal{R}}_{1,3}& =R_z\left(\pi \right),{\mathcal{R}}_{1,4}=R_z\left({\displaystyle \frac{3\pi }{2}}\right),\hfill \\ \hfill {\mathcal{R}}_{2,2}& =R_z\left({\displaystyle \frac{\pi }{2}}\right),\hfill & \hfill {\mathcal{R}}_{2,3}& =R_z\left(\pi \right),{\mathcal{R}}_{2,4}=R_z\left({\displaystyle \frac{3\pi }{2}}\right),\hfill \\ \hfill {\mathcal{R}}_{1,4+i}& =R_x\left({\displaystyle \frac{\pi }{2}}\right)R_z\left({\displaystyle \frac{(i-1)\pi }{2}}\right),\hfill & \hfill {\mathcal{R}}_{1,8+i}& =R_x\left(\pi \right)R_z\left({\displaystyle \frac{(i-1)\pi }{2}}\right),i=1\dots 4\hfill \\ \hfill {\mathcal{R}}_{1,12+i}& =R_x\left({\displaystyle \frac{3\pi }{2}}\right)R_z\left({\displaystyle \frac{(i-1)\pi }{2}}\right),\hfill & \hfill {\mathcal{R}}_{1,16+i}& =R_y\left({\displaystyle \frac{\pi }{2}}\right)R_z\left({\displaystyle \frac{(i-1)\pi }{2}}\right),i=1\dots 4\hfill \\ \hfill {\mathcal{R}}_{1,20+i}& =R_y\left({\displaystyle \frac{3\pi }{2}}\right)R_z\left({\displaystyle \frac{(i-1)\pi }{2}}\right),\hfill & \hfill {\mathcal{R}}_{2,4+i}& =R_x\left(\pi \right)R_z\left({\displaystyle \frac{(i-1)\pi }{2}}\right),i=1\dots 4.\hfill \end{array}$$

(33)

We choose the following vectors to parametrise the reference plane:

$$\begin{array}{cc}\hfill {\mathit{V}}_{1,1}& =S_1{R}_z\left(\varphi \right){\mathit{F}}_{1,1},{\mathit{v}}_{1,1,1}=R_z\left(\varphi \right)(1,0,0),\hfill \\ \hfill {\mathit{v}}_{1,1,2}& =R_z\left(\varphi \right)({\mathit{v}}_{1,1,1}\times {\mathit{V}}_{1,1})=R_z\left(\varphi \right)\left(0,sin\left(\theta \right),cos\left(\theta \right)\right),\hfill \\ \hfill {\mathit{V}}_{2,1}& =S_2{\mathit{F}}_{2,1},{\mathit{v}}_{2,1,1}=\left({\displaystyle \frac{1}{\sqrt{2}}},{\displaystyle \frac{1}{\sqrt{2}}},0\right),{\mathit{v}}_{2,1,2}={\mathit{v}}_{1,1,1}\times {\mathit{V}}_{2,1}=\left(-{\displaystyle \frac{1}{\sqrt{6}}},{\displaystyle \frac{1}{\sqrt{6}}},\sqrt{{\displaystyle \frac{2}{3}}}\right),\hfill \end{array}$$

(34)

where $S_1$ and $S_2$ are scaling parameters.

The shared vertices of the reference faces are then given by

$$\begin{array}{cccccc}\hfill {\mathit{n}}_1& =P_{2,1;1,2}+k_1{\mathit{u}}_{1,1;1,2},\hfill & \hfill {\mathit{n}}_2& =P_{2,1;1,2}+k_2{\mathit{u}}_{1,1;1,2},\hfill & \hfill {\mathit{n}}_3& =R_z\left({\displaystyle \frac{-\pi }{2}}\right){\mathit{n}}_2,\hfill \\ \hfill {\mathit{n}}_4& =R_z\left({\displaystyle \frac{-\pi }{2}}\right){\mathit{n}}_1,\hfill & \hfill {\mathit{n}}_5& =P_{1,4;2,4}+k_5{\mathit{u}}_{1,1;2,4},\hfill & \hfill {\mathit{n}}_6& =P_{1,4;1,2,4}+k_6{\mathit{u}}_{1,1;2,4},\hfill \\ \hfill {\mathit{n}}_7& =P_{2,4;1,7}+k_7{\mathit{u}}_{1,1;1,7},\hfill & \hfill {\mathit{n}}_8& =R_x\left({\displaystyle \frac{\pi }{2}}\right)R_z\left(\pi \right){\mathit{n}}_7,\hfill & \hfill {\mathit{n}}_9& =P_{1,7;2,1}+k_9{\mathit{u}}_{1,1;2,1},\hfill \\ \hfill {\mathit{n}}_{10}& =P_{1,7;2,1}+k_{10}{\mathit{u}}_{1,1;2,1},\hfill & \hfill {\mathit{m}}_1& ={\mathit{n}}_{10},{\mathit{m}}_2={\mathit{n}}_9,\hfill & \hfill {\mathit{m}}_3& =R_{{\mathit{F}}_{2,1}}\left({\displaystyle \frac{2\pi }{3}}\right)R_z\left({\displaystyle \frac{\pi }{2}}\right){\mathit{n}}_6,\hfill \\ \hfill {\mathit{m}}_4& =R_{{\mathit{F}}_{2,1}}\left({\displaystyle \frac{2\pi }{3}}\right)R_z\left({\displaystyle \frac{\pi }{2}}\right){\mathit{n}}_5,\hfill & \hfill {\mathit{m}}_i& =R_{{\mathit{F}}_{2,1}}\left({\displaystyle \frac{2\pi }{3}}\right){\mathit{m}}_{i-4},\hfill & \hfill i& =5\dots 12.\hfill \end{array}$$

(35)

The optimization parameters are $\theta$, $\varphi$, $S_1$, $S_2$, $k_1$, $k_2$, $k_5$, $k_6$, $k_7$, $k_9$, $k_{10}$ as well as the planar coordinates of the non-shared vertices for both reference faces. As initial parameter values we choose $\theta ={65}^{\circ }$, $\varphi =0$ and $S_1=S_2=4.7$. The other parameters are determined as described in the HAP6 section.

4.4. PD

The underlying symmetry of the PD p-cage is that of the dual of the truncated icosahedron. The pentagons sit on the vertices of the icosahedron while the hexagon areplaced at the centres of the triangular faces of the icosahedron.

The coordinates of the icosahedron are given by the coordinates of 3 perpendicular golden ratio rectangles

$$\begin{array}{c}\hfill (0,\pm 1,\pm \phi ),(\pm \phi ,0,\pm 1,)(\pm 1,\pm \phi ,0),\end{array}$$

(36)

where $\phi =(1+\sqrt{5})/2$ is the golden ratio. The type 1 faces are placed on the vertices of the icosahedron while the type 2 faces are placed at the centre of the icosahedron faces

$$\begin{array}{c}\hfill {\mathit{F}}_{1,1}=(0,-1,\phi ),{\mathit{F}}_{2,1}=({\mathit{F}}_{1,1}+{\mathit{F}}_{1,2}+{\mathit{F}}_{1,6})/3=({\displaystyle \frac{\phi }{3}},0,{\displaystyle \frac{1+2\phi }{3}}).\end{array}$$

(37)

Defining

$$\mathit{g}={\mathit{F}}_{11}\times {\mathit{e}}_x=(0,{\varphi }_g,1),$$

(38)

the rotations linking the faces to the reference faces are

$$\begin{array}{cc}\hfill {\mathcal{R}}_{1,2}& =R_z\left(\pi \right),{\mathcal{R}}_{1,2+i}=R_{{\mathit{F}}_{1,1}}\left(i{\displaystyle \frac{2\pi }{5}}\right)R_z\left(\pi \right),i=1\dots 4\hfill \\ \hfill {\mathcal{R}}_{1,6+i}& =R_{\mathit{g}}\left(\pi \right){\mathcal{R}}_{1,i},i=1\dots 6{\mathcal{R}}_{2,1+i}=R_{{\mathit{F}}_{1,1}}\left(i{\displaystyle \frac{2\pi }{5}}\right),i=1\dots 4\hfill \\ \hfill {\mathcal{R}}_{2,5+i}& =R_{\mathit{g}}\left(\pi \right),i=1\dots 5\hfill \\ \hfill {\mathcal{R}}_{2,11}& =R_{{\mathit{F}}_{2,1}}\left({\displaystyle \frac{4\pi }{5}}\right),{\mathcal{R}}_{2,12+1}=R_{{\mathit{F}}_{2,1}}\left(i{\displaystyle \frac{2\pi }{5}}\right)R_{{\mathit{F}}_{2,1}}\left({\displaystyle \frac{4\pi }{5}}\right),i=1\dots 4,\hfill \\ \hfill {\mathcal{R}}_{2,12}& =R_{{\mathit{F}}_{2,1}}\left({\displaystyle \frac{6\pi }{5}}\right),{\mathcal{R}}_{2,16+1}=R_{{\mathit{F}}_{2,1}}\left(i{\displaystyle \frac{2\pi }{5}}\right)R_{{\mathit{F}}_{2,1}}\left({\displaystyle \frac{6\pi }{5}}\right),i=1\dots 4.\hfill \end{array}$$

(39)

We choose the following vectors to parametrise the reference plane:

$$\begin{array}{cc}\hfill {\mathit{V}}_{1,1}& =S_1{\mathit{F}}_{1,1},{\mathit{v}}_{1,1,1}=(1,0,0),{\mathit{v}}_{1.1,2}={\mathit{v}}_{1,1,1}\times {\mathit{V}}_{1,1}={\displaystyle \frac{\sqrt{2}}{\sqrt{\sqrt{5}+5}}}\left(0,\phi ,1\right),\hfill \\ \hfill {\mathit{V}}_{2,1}& =S_2{\mathit{F}}_{2,1},{\mathit{v}}_{2,1,1}=\left(0,1,0\right),{\mathit{v}}_{2,1,2}={\mathit{v}}_{1,1,1}\times {\mathit{V}}_{2,1}={\displaystyle \frac{\sqrt{6}}{3\sqrt{3\sqrt{5}+7}}}\left(-\sqrt{5}-2,0,\phi \right),\hfill \end{array}$$

(40)

where $S_1$ and $S_2$ are scaling parameters.

The shared vertices of the reference faces are given by

$$\begin{array}{cccc}\hfill {\mathit{n}}_1& =P_{2,5;2,1}+k_1{\mathit{u}}_{1,1;2,1},\hfill & \hfill {\mathit{n}}_2& =P_{2,5;2,1}+k_2{\mathit{u}}_{1,1;2,1},\hfill \\ \hfill {\mathit{n}}_{1+2i}& =R_{{\mathit{F}}_{1,1}}\left({\displaystyle \frac{i2\pi }{5}}\right){\mathit{n}}_1,\hfill & \hfill {\mathit{n}}_{2+2i}& =R_{{\mathit{F}}_{1,1}}\left({\displaystyle \frac{i2\pi }{5}}\right){\mathit{n}}_2,i=1\dots 4\hfill \\ \hfill {\mathit{m}}_1& ={\mathit{n}}_2,\hfill & \hfill {\mathit{m}}_2& ={\mathit{n}}_1,\hfill \\ \hfill {\mathit{m}}_{1+2i}& =R_{{\mathit{F}}_{2,1}}\left({\displaystyle \frac{i\pi }{3}}\right){\mathit{m}}_1,\hfill & \hfill {\mathit{m}}_{2+2i}& =R_{{\mathit{F}}_{2,1}}\left({\displaystyle \frac{i\pi }{3}}\right){\mathit{m}}_2,i=1\dots 5.\hfill \end{array}$$

(41)

The optimization parameters are $S_1$, $S_2$, $k_1$, $k_2$, $k_3$, $k_4$ as well as the planar coordinates of the non-shared vertices for both reference faces. As initial parameters values we choose $S_1=14.14$, $S_2=18$ while the other parameters are determined as described in the HAP6 section.

4.5. IP5

The underlying symmetry of the IP5 p-cage is that of the dual of the icosidodecahedron.

Figure 8. Parametrisation of the icosidodecahedron with a pyramid on the pentagonal faces p-cage. a) Icosidodecahedron vectors. b) Mapping of vertices.

Figure 8. Parametrisation of the icosidodecahedron with a pyramid on the pentagonal faces p-cage. a) Icosidodecahedron vectors. b) Mapping of vertices.

The type 1 faces are placed at the centres of the pentagons of the icosidodecahedron. The vectors pointing to these centres correspond to the 12 vertices of the icosahedron. The type 2 hexagonal faces are placed on the vertices of the icosidodecahedron which correspond to the midpoint of icosidodecahedron edges:

$$\begin{array}{cccccc}\hfill {\mathit{F}}_{1,1}& =(0,-1,\phi ),\hfill & \hfill {\mathit{F}}_{1,2}& =R_z\left(\pi \right){\mathit{F}}_{1,1}=(0,1,\phi ),\hfill & \hfill {\mathit{F}}_{2,1}& ={\displaystyle \frac{1}{2}}({\mathit{F}}_{1,1}+{\mathit{F}}_{1,2})=(0,0,\phi ).\hfill \end{array}$$

(42)

The rotations linking the faces to the reference faces are

$$\begin{array}{cc}\hfill {\mathcal{R}}_{1,2}& =R_z\left(\pi \right),{\mathcal{R}}_{1,2+i}=R_{{\mathit{F}}_{1,1}}\left(i{\displaystyle \frac{2\pi }{5}}\right)R_z\left(\pi \right),i=1\dots 4\hfill \\ \hfill {\mathcal{R}}_{1,6+i}& =R_{\mathit{g}}\left(\pi \right){\mathcal{R}}_{1,i}i=1\dots 6,{\mathcal{R}}_{2,1+i}=R_{{\mathit{F}}_{1,1}}\left(i{\displaystyle \frac{i2\pi }{5}}\right),i=1\dots 4\hfill \\ \hfill {\mathcal{R}}_{2,6}& =R_{{\mathit{G}}_1}\left({\displaystyle \frac{4\pi }{3}}\right),{\mathcal{R}}_{2,6+i}=R_{{\mathit{F}}_{1,1}}\left(i{\displaystyle \frac{2\pi }{5}}\right){\mathcal{R}}_{2,6}i=1\dots 4\hfill \\ \hfill {\mathcal{R}}_{2,10+i}& =R_{\mathit{g}}\left(\pi \right){\mathcal{R}}_{2,i},i=1\dots 10\hfill \\ \hfill {\mathcal{R}}_{2,21}& =R_{{\mathit{F}}_{1,6}}\left({\displaystyle \frac{6\pi }{5}}\right){\mathcal{R}}_{2,6},{\mathcal{R}}_{2,21+2i}=R_{{\mathit{F}}_{1,1}}\left(i{\displaystyle \frac{2\pi }{5}}\right){\mathcal{R}}_{2,21},\hfill \\ \hfill {\mathcal{R}}_{2,22}& =R_{{\mathit{F}}_{1,6}}\left({\displaystyle \frac{8\pi }{5}}\right){\mathcal{R}}_{2,6},{\mathcal{R}}_{2,22+2i}=R_{{\mathit{F}}_{1,1}}\left(i{\displaystyle \frac{2\pi }{5}}\right){\mathcal{R}}_{2,22},\hfill \end{array}$$

(43)

where $\mathit{g}={\mathit{F}}_{11}\times {\mathit{e}}_y$ and

$$\begin{array}{cccccc}\hfill {\mathit{F}}_{1,2}& =R_z\left(\pi \right){\mathit{F}}_{1,1}=(0,1,\phi ),\hfill & \hfill {\mathit{F}}_{1,6}& =(\phi ,0,1,)\hfill & \hfill {\mathit{G}}_1& ={\displaystyle \frac{1}{3}}({\mathit{F}}_{1,1}+{\mathit{F}}_{1,2}+{\mathit{F}}_{1,6}).\hfill \end{array}$$

(44)

We choose the following vectors to parametrise the reference plane:

$$\begin{array}{cc}\hfill {\mathit{V}}_{1,1}& =S_1{\mathit{F}}_{1,1},{\mathit{v}}_{1,1,1}=(1,0,0),{\mathit{v}}_{1.1,2}={\mathit{v}}_{1,1,1}\times {\mathit{V}}_{1,1}={\displaystyle \frac{\sqrt{2}}{\sqrt{\sqrt{5}+5}}}\left(0,\phi ,1\right),\hfill \\ \hfill {\mathit{V}}_{2,1}& =S_2{\mathit{F}}_{2,1},{\mathit{v}}_{2,1,1}=\left(0,1,0\right),{\mathit{v}}_{2,1,2}=\left(-1,0,0\right).\hfill \end{array}$$

(45)

where $S_1$ and $S_2$ are scaling parameters.

The shared vertices of the reference faces are given by

$$\begin{array}{cccccc}\hfill {\mathit{m}}_1& =Q_{1,2;2,7}+k_1{\mathit{u}}_{2,1;2,7},\hfill & \hfill {\mathit{m}}_2& =Q_{1,2;2,7}+k_2{\mathit{u}}_{2,1;2,7},\hfill & \hfill {\mathit{m}}_3& =R_{{\mathit{G}}_2}\left({\displaystyle \frac{4\pi }{3}}\right){\mathit{m}}_2,\hfill \\ \hfill {\mathit{m}}_4& =R_{{\mathit{G}}_2}\left({\displaystyle \frac{4\pi }{3}}\right){\mathit{m}}_1,\hfill & \hfill {\mathit{m}}_5& =Q_{2,2;1,1}+k_5{\mathit{u}}_{2,1;1,1},\hfill & \hfill {\mathit{m}}_6& =Q_{2,2;1,1}+k_6{\mathit{u}}_{2,1;1,1},\hfill \\ \hfill {\mathit{m}}_{6+i}& =R_{{\mathit{F}}_{2,1}}\left(\pi \right){\mathit{m}}_i,\hfill & \hfill i& =1\dots 6,\hfill \\ \hfill {\mathit{n}}_1& ={\mathit{m}}_6,\hfill & \hfill {\mathit{n}}_{1+2i}& =R_{{\mathit{F}}_{1,1}}\left(i{\displaystyle \frac{2\pi }{5}}\right){\mathit{n}}_1,\hfill & \hfill i& =1\dots 4.\hfill \\ \hfill {\mathit{n}}_2& ={\mathit{m}}_5,\hfill & \hfill {\mathit{n}}_{2+2i}& =R_{{\mathit{F}}_{1,1}}\left(i{\displaystyle \frac{2\pi }{5}}\right){\mathit{n}}_2,\hfill & \hfill i& =1\dots 4.\hfill \end{array}$$

(46)

The optimization parameters are $S_1$, $S_2$, $k_1$, $k_2$, $k_5$, $k_6$ as well as the planar coordinates of the non-shared vertices for both reference faces. As initial parameters values we choose $S_1=3$, $S_2=3.5$ while the other parameters are determined as described in the HAP6 section.

4.6. SDP5

The underlying symmetry of that of the SDP5 p-cage is that of the snub dodecahedron.

Figure 9. Parametrisation of the cage built on a snub dodecahedron with pyramids on the pentagons p-cage. a) Dodecahedron node numbering. b) Face labelling. The green dots correspond to the vectors ${\mathit{A}}_{v,i}$ c) Mapping of vertices.

Figure 9. Parametrisation of the cage built on a snub dodecahedron with pyramids on the pentagons p-cage. a) Dodecahedron node numbering. b) Face labelling. The green dots correspond to the vectors ${\mathit{A}}_{v,i}$ c) Mapping of vertices.

The normal vectors to the reference planes are

$$\begin{array}{c}\hfill {\mathit{F}}_{1,1}=\left(0,-1,\phi \right),{\mathit{F}}_{2,1}=\left(0,0,1\right),\end{array}$$

(47)

Defining ${\mathit{G}}_{i,j,k}=({\mathit{F}}_{1,i}+{\mathit{F}}_{1,j}+{\mathit{F}}_{1,k})/3$, the rotation axe vectors are

$$\begin{array}{cccc}\hfill {\mathit{A}}_{v,1}& ={\mathit{G}}_{1,2,3}=\left({\displaystyle \frac{\phi }{3}},0,{\displaystyle \frac{2\phi +1}{3}}\right),\hfill & \hfill {\mathit{A}}_{v,3}& ={\mathit{G}}_{1,3,5}=\left({\displaystyle \frac{{\phi }^2}{3}},-{\displaystyle \frac{{\phi }^2}{3}},{\displaystyle \frac{{\phi }^2}{3}}\right),\hfill \\ \hfill {\mathit{A}}_{v,4}& ={\mathit{G}}_{1,4,7}=\left(-{\displaystyle \frac{{\phi }^2}{3}},-{\displaystyle \frac{{\phi }^2}{3}},{\displaystyle \frac{{\phi }^2}{3}},\right),\hfill & \hfill {\mathit{A}}_{v,5}& ={\mathit{G}}_{2,3,6}=\left({\displaystyle \frac{{\phi }^2}{3}},{\displaystyle \frac{{\phi }^2}{3}},{\displaystyle \frac{{\phi }^2}{3}},\right),\hfill \\ \hfill {\mathit{A}}_{v,8}& ={\mathit{G}}_{2,6,8}=\left(0,{\displaystyle \frac{2\phi +1}{3}},{\displaystyle \frac{\phi }{3}}\right),\hfill & \hfill {\mathit{A}}_{v,9}& ={\mathit{G}}_{3,5,9}=\left({\displaystyle \frac{2\phi +1}{3}},-{\displaystyle \frac{\phi }{3}},0\right),\hfill \\ \hfill {\mathit{A}}_{v,14}& ={\mathit{G}}_{7,10,11}=\left(-{\displaystyle \frac{{\phi }^2}{3}},-{\displaystyle \frac{{\phi }^2}{3}},-{\displaystyle \frac{{\phi }^2}{3}}\right),\hfill & \hfill {\mathit{A}}_{v,15}& ={\mathit{G}}_{6,9,12}=\left({\displaystyle \frac{{\phi }^2}{3}},{\displaystyle \frac{{\phi }^2}{3}},-{\displaystyle \frac{{\phi }^2}{3}}\right).\hfill \end{array}$$

(48)

The rotations linking the faces to the reference faces are given by

$$\begin{array}{cccccc}\hfill {\mathcal{R}}_{1,2}& =R_{{\mathit{A}}_{v,1}}\left({\displaystyle \frac{4\pi }{3}}\right),\hfill & \hfill {\mathcal{R}}_{1,3}& =R_{{\mathit{A}}_{v,1}}\left({\displaystyle \frac{2\pi }{3}}\right),\hfill & \hfill {\mathcal{R}}_{1,4}& =R_{{\mathit{A}}_{v,4}}\left({\displaystyle \frac{2\pi }{3}}\right),\hfill \\ \hfill {\mathcal{R}}_{1,5}& =R_{{\mathit{A}}_{v,3}}\left({\displaystyle \frac{2\pi }{3}}\right),\hfill & \hfill {\mathcal{R}}_{1,6}& =R_{{\mathit{A}}_{v,5}}\left({\displaystyle \frac{4\pi }{3}}\right){\mathcal{R}}_{1,2},\hfill & \hfill {\mathcal{R}}_{1,7}& =R_{{\mathit{A}}_{v,4}}\left({\displaystyle \frac{4\pi }{3}}\right),\hfill \\ \hfill {\mathcal{R}}_{1,8}& =R_{{\mathit{A}}_{v,8}}\left({\displaystyle \frac{4\pi }{3}}\right){\mathcal{R}}_{1,2},\hfill & \hfill {\mathcal{R}}_{1,9}& =R_{{\mathit{A}}_{v,9}}\left({\displaystyle \frac{4\pi }{3}}\right){\mathcal{R}}_{1,3},\hfill & \hfill {\mathcal{R}}_{1,10}& =R_{{\mathit{A}}_{v,14}}\left({\displaystyle \frac{2\pi }{3}}\right){\mathcal{R}}_{1,7},\hfill \\ \hfill {\mathcal{R}}_{1,11}& =R_{{\mathit{A}}_{v,14}}\left({\displaystyle \frac{4\pi }{3}}\right){\mathcal{R}}_{1,7},\hfill & \hfill {\mathcal{R}}_{1,12}& =R_{{\mathit{A}}_{v,15}}\left({\displaystyle \frac{4\pi }{3}}\right){\mathcal{R}}_{1,6},\hfill \\ \hfill {\mathcal{R}}_{2,1+5j+i}& ={\mathcal{R}}_{1,j+1}{R}_{{\mathit{F}}_{1,1}}\left(i{\displaystyle \frac{2\pi }{3}}\right),\hfill & \hfill i& =0\dots 4,j=0\dots 11.\hfill \end{array}$$

(49)

We choose the following vectors to parametrise the reference plane:

$$\begin{array}{cc}\hfill {\mathit{V}}_{1,1}& =S_1{\mathit{F}}_{1,1}=S_1\left(0,-1,\phi \right),{\mathit{v}}_{1,1,1}={\widehat{\mathit{e}}}_x,{\mathit{v}}_{1,1,2}=\left(0,\phi {\displaystyle \frac{\sqrt{2}}{\sqrt{\sqrt{5}+5}}},{\displaystyle \frac{\sqrt{2}}{\sqrt{\sqrt{5}+5}}}\right),\hfill \\ \hfill {\mathit{V}}_{2,1}& =S_2{R}_z\left(\varphi \right)R_x\left(\theta \right){\widehat{\mathit{e}}}_z,{\mathit{v}}_{2,1,1}=R_z\left(\varphi \right)R_x\left(\theta \right){\widehat{\mathit{e}}}_x,{\mathit{v}}_{2,1,2}=R_z\left(\varphi \right)R_x\left(\theta \right){\widehat{\mathit{e}}}_y.\hfill \end{array}$$

(50)

With these we have

$$\begin{array}{cccccc}\hfill {\mathit{m}}_1& =Q_{2,18;2,2}+k_1{\mathit{u}}_{2,1;2,2},\hfill & \hfill {\mathit{m}}_2& =Q_{2,18;2,2}+k_2{\mathit{u}}_{2,1;2,2},\hfill & \hfill {\mathit{m}}_3& =Q_{2,2;1,1}+k_3{\mathit{u}}_{2,1;1,1},\hfill \\ \hfill {\mathit{m}}_4& =Q_{2,2;1,1}+k_4{\mathit{u}}_{2,1;1,1},\hfill & \hfill {\mathit{m}}_5& =R_{{\mathit{A}}_{1,1}}\left({\displaystyle \frac{8\pi }{5}}\right){\mathit{m}}_2,\hfill & \hfill {\mathit{m}}_6& =R_{{\mathit{A}}_{1,1}}\left({\displaystyle \frac{8\pi }{5}}\right){\mathit{m}}_1,\hfill \\ \hfill {\mathit{m}}_7& =Q_{2,5;2,10}+k_7{\mathit{u}}_{2,1;2,10},\hfill & \hfill {\mathit{m}}_8& =Q_{2,5;2,10}+k_8{\mathit{u}}_{2,1;2,10},\hfill & \hfill {\mathit{m}}_9& =Q_{2,10;2,9}+k_9{\mathit{u}}_{2,1;2,9},\hfill \\ \hfill {\mathit{m}}_{10}& =Q_{2,10;2,9}+k_{10}{\mathit{u}}_{2,1;2,9},\hfill & \hfill {\mathit{m}}_{11}& =R_{{\mathit{F}}_{1,1}}\left({\displaystyle \frac{4\pi }{5}}\right){\mathit{m}}_{10},\hfill & \hfill {\mathit{m}}_{12}& =R_{{\mathit{F}}_{1,1}}\left({\displaystyle \frac{4\pi }{5}}\right){\mathit{m}}_9,\hfill \\ \hfill {\mathit{n}}_1& ={\mathit{m}}_4,\hfill & \hfill {\mathit{n}}_{1+2i}& =R_{{\mathit{F}}_{1,1}}\left(i{\displaystyle \frac{2\pi }{5}}\right){\mathit{n}}_1,\hfill & \hfill i& =1\dots 4.\hfill \\ \hfill {\mathit{n}}_2& ={\mathit{m}}_3,\hfill & \hfill {\mathit{n}}_{2+2i}& =R_{{\mathit{F}}_{1,1}}\left(i{\displaystyle \frac{2\pi }{5}}\right){\mathit{n}}_2,\hfill & \hfill i& =1\dots 4.\hfill \end{array}$$

(51)

The optimization parameters are $\Theta$, $\varphi$, $S_1$, $S_2$, $k_1$, $k_2$, $k_3$, $k_4$, $k_7$, $k_8$,$k_9$, $k_{10}$ as well as the planar coordinates of the non-shared vertices for both reference faces. As initial parameter values we use $\theta =5^{\circ }$, $\varphi =-{25}^{\circ }$, $S_1=4$, $S_2=7.5$ while the other parameters are determined as described in the HAP6 section.

5. Results

We used a computer program to optimise (3) for each configuration described above with the restrictions that each face contributes up to 3 of its edges to each hole. That restriction was chosen to avoid p-cages with very large holes. A python program was used to generate all the combinations of the labels a to e and A to F satisfying (1) where each label may take the values 1, 2 or 3 which led to over half a million configurations to optimize. We used a simulated annealing method to minimize (3) for 200 values of $c_l$ and $c_a$, with the constraint $c_l+c_a=2$, taking $c_c=c_{pc}=500$ and the selected the p-cage with the smalest value of max(${\Delta }_l,{\Delta }_a$).

The p-cages are named according to the graphs they are build from, the size of the face and the number of edges contributing to the holes as follows : GRAPH_P$P_1$_P$P_2$_a_b_c_d_e-A_B_C_D_E_F where the italic symbols are replaced by their value. To avoid very long names, we also use a simplification when n successive hole-edge labels have the same values, $\alpha$, where we replace the sequence $\alpha \_\alpha \_\dots \alpha$ by nx$\alpha$. So for example, IP5_P10_P12_5x1-6x1 is equivalent to IP5_P10_P12_1_1_1_1_1-1_1_1_1_1_1 or SDP5_P15_P21_5x2-2x2_2x3_2_3 is equivalent to SDP5_P15_P21_2_2_2_2_2-2_2_3_3_2_3.

Our first result was that none of these configurations led to regular p-cages. The best HAP6 p-cage has a deformation just under under 5%. The two type 2 faces, at the top and bottom of the p-cage, are surrounded by two rings of 5 type 1 faces joined together (Figure 10a).

The least deformed p-cages are of the type PD: PD_P20_P24_5x3-6x3 has deformations ${\Delta }_l=0.0084$, ${\Delta }_a=0.0084$ and PD_P10_P12_5x1-6x1 which is similar in structure but with smaller polygons has deformations just exceeding 1%: ${\Delta }_l=0.0109$, ${\Delta }_a=0.0109$ (Figure 10b–d).

The IP5 p-cages are made out of respectively 12 and 36 faces of type 1 and 2. The p-cages IP5_P10_P12_5x1-6x1 is nearly regular with deformations ${\Delta }_l=0.0161$, ${\Delta }_a=0.0161$. (Figure 10e).

The p-cage IP5_P10_P14_5x1-2_2x1_2_2x1 is interesting in that its hole-edge labels are not identical and still has relatively small deformations ${\Delta }_l=0.0296$, ${\Delta }_a=0.0295$ (Figure 10f). The TOP6 p-cages are made out of respectively 24 and 8 faces of type 1 and 2. The p-cage TOP6_P11_P12_2_4x1-6x1 is particularly interesting in that it is made out of the polygons, hendecagons and dodecagons, from which the nano-cages used, separately, in [2,3] are made. Its deformations are relatively small too ${\Delta }_l=0.0096$, ${\Delta }_a=0.0093$ (Figure 10g). The p-cage TOP6_P16_P18_3_4x2-6x2 is another example of a p-cage with differing hole-edge labels and with still relatively small deformations: ${\Delta }_l=0.0234$, ${\Delta }_a=0.0234$ (Figure 10h).

The TTP6 p-cages are made out of respectively 12 and 4 faces of type 1 and 2. The p-cages TTP6_P20_P24_5x3-6x3, TTP6_P15_P18_5x2-6x2 and TTP6_P10_P12_5x1-6x1 all have a similar structure with respective deformations ${\Delta }_l=0.0229$, ${\Delta }_a=0.0229$, ${\Delta }_l=0.0261$, ${\Delta }_a=0.0261$ and ${\Delta }_l=0.0297$, ${\Delta }_a=0.0297$ (Figure 11a–c).

The SDP5 p-cages are made out of respectively 12 and 60 faces of type 1 and 2. These are the largest bi-symmetric p-cages with maximal connectivity between faces. The p-cages SDP5_P20_P24_5x3-6x3, SDP5_P15_P18_5x2-6x2 and SDP5_P10_P12_5x1-6x1 have a similar structure with deformations respectively ${\Delta }_l=0.0212$, ${\Delta }_a=0.0211$, ${\Delta }_l=0.0261$, ${\Delta }_a=0.0261$ and ${\Delta }_l=0.0326$, ${\Delta }_a=0.0326$ (Figure 11d,f,g). The p-cage SDP5_P15_P21_5x2-2x2_2x3_2_3 is another example of a p-cage with non identical hole-edge numbers and relatively small deformations ${\Delta }_l=0.0214$, ${\Delta }_a=0.0189$ (Figure 11e).

The TCM4 p-cages are made out of respectively 24 faces of each type. The configuration TCM3 lead to 68 p-cages with deformation below 10% but the least deformed p-cage, TCM4_P20_P24_5x3-6x3, has a deformation of over 8%: ${\Delta }_l=0.0805$, ${\Delta }_a=0.0690$ (Figure 11h ). While aesthetically pleasing, it does not constitute a good candidate for a protein cage.

6. Conclusion

In this paper we have constructed near-miss p-cages made out of two types of faces such that the faces of one family have 5 neighbours while the others have 6. Moreover each face of a given family can be mapped to any other face of the same family via a rotation automorphism of the p-cage.

We have shown that the p-cages can be constructed by placing the faces on the vertices of the so called hole-polyhedron, the edges of which describes how the faces are attached to each others. The bi-symmetry of the p-cage and the constraint of each face having 5 or 6 neighbours restricts the hole-polyhedron graphs to be one of 9 graphs identified in [7]. From these 9 hole-polyhedron graphs 2 leads to p-cages with deformation exceeding 10% (TTM3 and TOM3 p-cages) while the TCM3 p-cages have quite large deformations exceeding 8%. The other 6 types of p-cages have much smaller deformations, a number of them being approximately $1\%$.

If we restrict ourselves to small P, so to a small number of hole-edges, the following p-cages are good candidates for artificial protein cages: PD_P10_P12_5x1-6x1 (deformation $1.09\%$), TOP6_P11_P12_2_4x1-6x1 (deformation $0.096\%$), IP5_P10_P12_5x1-6x1 (deformation $1.61\%$), TTP6_P10_P12_5x1-6x1 (deformation $3\%$), IP5_P10_P14_5x2-2_2x1_2_2x1 (deformation $3\%$) or SDP5_P10_P24_5x1-6x1 (deformation $3.26\%$). So each type of p-cage, except possibly HAP6, are potential geometries for good p-cages. Notice that SDP5_P10_P24_5x1-6x1 is the largest of such p-cages as it is made out of 72 faces. The TTP6 p-cages are the smallest with only 16 faces, while the TOP6 and PD p-cages have 32 faces while the IP5 p-cages have 42.