\documentclass{mc} %%%%% journal info -- DO NOT CHANGE %%%%%%%%% \setcounter{page}{429} \renewcommand\thisnumber{2} \renewcommand\thisyear {2013} \renewcommand\thismonth{xxx} \renewcommand\thisvolume{18} \renewcommand\datereceived{April 3, 2013} \renewcommand\dateaccepted{June 23, 2013} \def\CC{\mathbb C} \def\RR{\mathbb R} \def\QQ{\mathbb Q} \def\ZZ{\mathbb Z} \def\NN{\mathbb N} \usepackage[displaymath,mathlines,pagewise]{lineno} \begin{document} %\begin{linenumbers} \markboth{J.\,Kovi\v{c}}{Centrally symmetric convex polyhedra with regular polygonal faces} \title{Centrally symmetric convex polyhedra with regular polygonal faces} \author[J.\,Kovi\v{c}]{Jurij Kovi\v{c}\affil{1}\comma\corrauth} \address{\affilnum{1}\ Institute of Mathematics, Physics and Mechanics, University of Ljubljana, Jadranska 19, SL-1\,000 Ljubljana, Slovenia} \email{{\tt jurij.kovic@siol.net} (J.\,Kovi\v{c})} \begin{abstract} First we prove that the class $C_{I}$ of centrally symmetric convex polyhedra with regular polygonal faces consists of 4 of the 5 Platonic, 9 of the 13 Archimedean, 13 of the 92 Johnson solids and two infinite families of $2n$-prisms and $(2n+1)$-antiprisms. Then we show how the presented maps of their halves (obtained by identification of all pairs of antipodal points) in the projective plane can be used for obtaining their flag graphs and symmetry-type graphs. Finally, we study some linear dependence relations between polyhedra of the class $C_{I}$. \end{abstract} \keywords{map, Platonic solid, Archimedean solid, Johnson solid, flag graph, convex polyhedron, projective plane} \ams{37F20, 57M10} \maketitle \section{Introduction} Any centrally symmetric convex (hence: spherical) polyhedron $\mathcal C$ admits identification of pairs of antipodal points $x$ and $x^{*}$; thus the \emph{map} (i.e. embedding of a graph in a compact surface) of its \emph{half} $\mathcal C/2 = \mathcal C/_{x \equiv x^{*}}$ has the Euler characteristic $E = v - e + f = 1$ (where $v$, $e$ and $f$ are the numbers of the vertices, edges and faces of the map, respectively) and can be drawn in a \emph{projective plane} (represented as a disc with identified antipodal points \cite{PisaServa}). Thus the flag graph of $\mathcal C/2$ can be easily constructed in the projective plane, too, while the flag graph of $\mathcal C$ is exactly a 2-sheet cover space \cite{Fulton, Hatcher} over $\mathcal C/2$. It is well known that the class $C$ of convex polyhedra with regular polygonal faces consists of 5 \emph{Platonic solids}, 13 \emph{Archimedean solids} \cite{PisZit}, the class of 92 non-uniform (i.e. having at least two orbits of vertices) \emph{Johnson solids} \cite{Johnson} and two infinite families of prisms and antiprisms. Among these solids we will find a subset $C_{I} \subset C$ of centrally symmetric solids and present the maps of their halves $\mathcal C/2$ (obtained by identification of all pairs of antipodal points) in the projective plane modelled as a disk with identified antipodal points. From these maps we can deduce the corresponding \emph{flag graphs} and \emph{symmetry-type graphs} which can be used for the classification of \emph{maps}, tilings and polyhedra, too \cite{Lins, Kovic, Massey, Orba}. The maps of the halves of four Platonic solids (hemi-cube, hemi-octahedron, hemi-dodecahedron, hemi-icosahedron) and the maps of regular and semi-regular spherical polyhedra can also be found in Wikipedia (Regular polyhedron, Spherical polyhedron). For convex uniform polyhedra we use the usual notation ($p.q.r.\dots$), describing the cyclical sequence of regular $n$-gonal faces ($n = 3, 4, 5, \dots$) around any vertex. {\sc Structure:} First some general propositions about centrally symmetric polyhedra are given (Section 2), and then the solids $\mathcal C \in C_{I}$ are identified (Section 3). After that the maps of their halves in the projective plane are presented and we show how to construct the corresponding flag graph and symmetry-type graph (Section~4). From these maps the number of faces 3,4,5,6,8,10 for each $\mathcal C/2$ can be easily found, too. The corresponding vectors $n = (n_{10},n_{8},n_{6},n_{5},n_{4},n_{3})$ can be used to solve the following problem (Section 5): Is it possible to take $a$ copies of a polyhedron $A \in C_{I}$ and $b$ copies of polyhedron $B \in C_{I}$ and by dissecting their boundary into faces construct a polyhedron $C \in C_{I}$ so that no faces are left unused? \section{Centrally symmetric solids} The sets of vertices, edges and faces of a polyhedron $\mathcal C$ are denoted by $V(\mathcal C)$, $E(\mathcal C)$, $F(\mathcal C)$, respectively. The \textit{central point} (or the centre) of a polyhedron $\mathcal C \in C_{I}$ is defined as the point $O$ fixed by the central inversion $c$ preserving $\mathcal C$. The \emph{antipodal elements} (vertices, edges, faces) of a vertex $v \in V(\mathcal C)$, an edge $e \in E(\mathcal C)$ and a face $f \in F(\mathcal C)$ are denoted by $c(v) = v^{*}$, $c(e) = e^{*}$, $c(f) = f^{*}$, respectively. Here are some necessary (although not sufficient) conditions for $\mathcal C$ belonging to the class $C_{I}$: \begin{proposition}\label{nec} Let $\mathcal C \in C_{I}$. Then: \begin{itemize} \item[$($i$)$] Any pair of antipodal edges $e,e^{*}\!\!\in\!E(\mathcal C)$ is parallel; likewise, any pair of antipodal faces $f,f^{*} \in F(\mathcal C)$ is parallel, too. \item[$($ii$)$] The numbers $\#v$, $\#e$, $\#f$ of vertices, edges and faces of $C$ must be even. \item[$($iii$)$] The numbers of each class of faces with the same number $(3,4,5,\dots)$ of edges must be even. \end{itemize} \end{proposition} \begin{proof} (i): Let $O$ be the central point of $\mathcal C \in C_{I}$. For any vertex $v$ let $\vec OV = \vec v$ be the vector with the starting point $O$ and the ending point in $v$. Let $e(u,v) \in E(\mathcal C)$. Then $\vec e^{*} = \vec u^{*} - \vec v^{*} = -\vec u - (- \vec v) = -\overrightarrow{(u - v)} = -\vec e$ for any $u,v \in V(\mathcal C)$, hence the vectors $\vec e^{*}$ and $\vec e$ are parallel. Consequently, all the edges of faces $f$ and $f^{*}$ are parallel, hence $f$ and $f^{*}$ must be parallel, too. (ii): Obviously $v \neq v^{*}$, $e \neq e^{*}$,$f \neq f^{*}$ for each $v \in V(\mathcal C)$, $e \in E(\mathcal C)$, $f \in F(\mathcal C)$. (iii): Faces $f$ and $f^{*}$ have the same number of edges. \end{proof} \begin{corollary} Tetrahedron (3.3.3) and truncated tetrahedron (6.6.3) are not in the class $C_{I}$. \end{corollary} \begin{proof} None of the four faces of (3.3.3) has a parallel face. The same holds for the four hexagonal faces of (6.6.3). Hence, by Proposition~\ref{nec}(i), these two solids cannot be in $C_{I}$. \end{proof} We shall say that a polyhedron has a rotation $R$ if it is symmetric by rotation $R$. Similarly, it is symmetric by a reflection, we shall say it has a reflection. \begin{proposition}\label{not} $\,$ \begin{itemize} \item[$($i$)$] If $\mathcal C \in C$ has two orthogonal reflection planes $\Pi$ and $\Omega$, but it is not preserved by the reflection over a plane orthogonal both to $\Pi$ and $\Omega$, then $\mathcal C \notin C_{I}$. \item[$($ii$)$] If $\mathcal C \in C$ has a rotation $R$ for the angle $\pi$, but it has not a reflection plane orthogonal to the axis of $R$, then $\mathcal C \notin C_{I}$. \item[$($iii$)$] If $\mathcal C \in C$ has a a reflection plane $\Pi$ but it has not a rotation $R$ for the angle $\pi$ with an axis orthogonal to $\Pi$, then $\mathcal C \notin C_{I}$. \end{itemize} \end{proposition} \begin{proof} (i): Let the central point $O$ of $\mathcal C$ be the origin of the Cartesian coordinate system with axes $(x)$ and $(y)$ in the plane $\Pi$ and $(y)$ and $(z)$ in $\Omega$. Then the reflections $Z_{\Pi}$ and $Z_{\Omega}$ transform a vertex $v$ with coordinates $v(x,y,z)$ into $v_{\Pi} = v(-x,y,z)$ and $v_{\Omega} = v(x,-y,z)$, respectively. If there is also the central inversion $c$, then $c(v) = v(-x,-y,-z)$. Hence there should also be the reflection $v(x,y,z) \rightarrow v(x,y,-z)$. (ii) and (iii) are proved similarly as (i), using the fact that the rotation $R$ sends the point $(x,y,z)$ into the point $(-x,-y,z)$. \end{proof} \begin{proposition}\label{suf} $\,$ \begin{itemize} \item[$($i$)$] If a polyhedron $\mathcal C \in C$ is symmetrical by the following two operations: \begin{itemize} \item[a$)$] reflection $Z$ over a plane $\Pi$; \item[b$)$] rotation $R_{\pi}$ for the angle $\pi$ around an axis $a$, orthogonal to $\Pi$; \end{itemize} then $\mathcal C \in C_{I}$. \item[$($ii$)$] If a polyhedron $\mathcal C \in C$ is preserved by the reflections over three mutually orthogonal planes, then $\mathcal C \in C_{I}$. \end{itemize} \end{proposition} \begin{proof} (i): The composition of reflection $Z$ and rotation $R_{\pi}$ sends any point $(x,y,z)$ into its antipodal point $(-x,-y,-z)$: $ZR_{\pi} = R_{\pi}Z = c$. (ii): The composition $Z_{1}Z_{2}$ of two reflections $Z_{1}$ and $Z_{2}$ over two orthogonal planes produces a rotation for the angle $\pi$ around the axis $a$, which is orthogonal to the third plane, and we can use (i). \end{proof} \begin{corollary}\label{Platon} The cube (4.4.4), the octahedron (3.3.3.3), the dodecahedron (5.5.5) and the icosahedron (3.3.3.3.3) are in the class $C_{I}$. \end{corollary} \begin{proof} For (4.4.4) and (3.3.3.3) this is true by Proposition~\ref{suf}(i), while for (5.5.5) and (3.3.3.3.3) this is true by Proposition~\ref{suf}(ii). \end{proof} All Platonic and Archimedean solids can be obtained from a tetrahedron using the operations \emph{medial} $Me(\mathcal C)$, \emph{truncation} $Tr(\mathcal C)$, \emph{dual} $Du(\mathcal C)$ and \emph{snub} $Sn(\mathcal C)$ \cite{PisZit}. \begin{proposition}\label{MeTrDu} $\,$ \begin{itemize} \item[$($i$)$] If the solid $\mathcal C$ belongs to the class $C_{I}$, the same holds for its truncation $Tr(\mathcal C)$, $Me(\mathcal C)$ and dual $Du(\mathcal C)$. \item[$($ii$)$] However, there are solids such that $\mathcal C \notin C_{I}$ and $Me(C)\in C_{I}$. \item[$($iii$)$] Likewise, there are solids $\mathcal C \in C_{I}$ such that $Me(C)\notin C_{I}$ or $Sn(\mathcal C) \notin C_{I}$. \end{itemize} \end{proposition} \begin{proof} (i): From the definitions of operations $Tr$, $Me$ and $Du$ it follows that they do not have any impact on the central symmetry of the solid. (ii): We already know that the tetrahedron (3.3.3) is not in $C_{I}$, while its medial-the octahedron (3.3.3.3)-is. (iii): The cube (4.4.4) and the dodecahedron (5.5.5) are in $C_{I}$, while the snub cube (4.3.3.3.3) and the snub dodecahedron (5.3.3.3.3) are not, as we can conclude by Proposition~\ref{not}(ii). \end{proof} \begin{proposition}\label{4anti} The 4-antiprism is not centrally symmetric. \end{proposition} \begin{proof} Suppose the 4-antiprism is centrally symmetric. Then the central inversion $c$ sends the vertices 1,2,3,4 of one square face into vertices $c(1) = 1^{*}$, $c(2) = 2^{*}$, $c(3) = 3^{*}$, $c(4) = 4^{*}$ of the other square face (Figure~\ref{why} left). For any face $f$ the face $c(f) = f^{*}$ has no common point with the face $f$, hence $v \neq v^{*}$ for any vertex $v$. Since $v$ and $v^{*}$ do not belong to the same face, they cannot be adjacent vertices. Therefore $A = 3^{*}$ or $A = 4^{*}$. Likewise $B = 4^{*}$ or $B = 1^{*}$. Likewise $C = 1^{*}$ or $C = 2^{*}$ and $D = 2^{*}$ or $D = 3^{*}$. As soon as we choose one of the possibilities for $A$, then $B$, $C$ and $D$ are determined by the above relations. \begin{figure}[h!] \centering \includegraphics[scale=0.7]{4antiprism.eps} \caption{\it The 4-antiprism and why it is not centrally symmetric} \label{why} \end{figure} In the first case (Figure~\ref{why} in the middle), the antipod of triangle $\Delta(124^{*})$ cannot be the triangle $\Delta(1^{*}2^{*}4)$, since the antipod of the edge $14^{*}$ is not the edge $1^{*}4$. In the second case (Figure~\ref{why} right), the antipod of the triangle $\Delta(123^{*})$ is not the triangle $\Delta(1^{*}2^{*}3)$, since the antipod of the edge $23^{*}$ is not the edge $2^{*}3$. \end{proof} \begin{proposition}\label{ant} The $n$-antiprism belongs to the class $C_{I}$ if and only if $n$ is odd. \end{proposition} \begin{proof} If the antiprism $(N.3.3)$, $N \geq 3$ has the central symmetry $c$, then $c$ sends the vertices of the upper $n$-gon $1,2,3,4,\dots,n$ into vertices $1^{*},2^{*},3^{*},4^{*},\dots,N^{*}$ of the upper $N$-gon (Figure~\ref{antiprismproof}). The antipod of the triangle $\Delta(1,2,X^{*})$ must be the triangle $\Delta(1^{*},2^{*},X)$. The vertex $X$ belongs to the upper $N$-gon. Along the upper $N$-gon we have $X-1$ (grey) triangles $\Delta(1,2,X^{*})$, $\Delta(2,3,(X+1)^{*})$,\dots, $\Delta(X-1,X,1^{*})$. The same number of (white) triangles is between vertices $X^{*}$ and $1$ along the lower $N$-gon: $\Delta(X^{*}, (X+1)^{*},2)$, $\Delta((X+1)^{*},(X+2)^{*},3)$,\dots, $\Delta(1^{*},2^{*},X)$. Therefore it is $X - 1 \equiv 2 - X$ (mod $N$), hence $2X \equiv 3$ (mod $N$). But this is possible only if $N$ is an odd number, since the remainder of $2X$ modulo $2n$ is always an even number. Therefore such labeling of the triangles as shown in Figure~\ref{antiprismproof} is possible only if $N = 2n+1$, and it is not possible if $N = 2n$. \end{proof} \begin{figure}[h!] \centering \includegraphics[scale=0.8]{antiprismproof.eps} \caption{\it Triangles of a centrally symmetric antiprism} \label{antiprismproof} \end{figure} \section{Determination of the class $C_{I}$} \begin{theorem} The class $C_{I}$ consists of the following solids: \begin{itemize} \item[$($i$)$] four of the five Platonic solids: Cube (4.4.4), Octahedron (3.3.3.3), Dodecahedron (5.5.5) and Icosahedron (3.3.3.3.3); \item[$($ii$)$] nine of the 13 Archimedean solids: Truncated Cube (8.8.3), Truncated Dodecahedron (10.10.3), Truncated Octahedron (4.6.6), Truncated Icosahedron (5.6.6), Truncated Cuboctahedron (8.4.6), Cuboctahedron (4.3.4.3), Icosidodecahedron (5.3.5.3), Rhombicuboctahedron (4.4.3.4), Rhombicosidodecahe-\linebreak dron (5.4.3.4); \item[$($iii$)$] the infinite families of $2n$-prisms and $(2n+1)$-antiprisms; \item[$($iv$)$] 13 Johnson solids: J15, J28, J31, J36, J39, J43, J55, J59, J67, J69, J73, J80, J91. \end{itemize} All these solids satisfy the condition of Proposition~\ref{suf}(i) (this will help us to draw the maps of their halves in Section~\ref{maps}). \end{theorem} \begin{proof} (i): These solids are in the class $C_{I}$ by Corolary~\ref{Platon}. (ii): All these solids are duals, medials or truncations of solids being in $C_{I}$, hence by Proposition~\ref{MeTrDu}(i) they are in $C_{I}$, too. (iii): The $2n$-prisms are in $C_{I}$ by Proposition~\ref{suf}(i). The $(2n+1)$-prisms have odd number of faces 4. The result on antiprisms is given in Proposition~\ref{ant}. \begin{figure}[h!] \centering \includegraphics[scale=0.7]{J15J91a.eps} \caption{\it Two centrally symmetric Johnson solids:J15 and J91} \label{J15J91a} \end{figure} (iv): Either using computer programs for polyhedra (like Great Stella) or with the help of 3D-models of Johnson solids it is easy to see that all these 13 solids satisfy one or both of the conditions of Proposition~\ref{suf} (see Table 1). The arguments why the other 79 Johnson solids are not in the class $C_{I}$ are given in Table 2. \end{proof} \begin{table}[h!] \centering \begin{tabular}{|c||c|c||} \hline % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ... Johnson solid $\mathcal C$ & Jxx & $\mathcal C \in C_{I}$ \\ \hline \hline elongated square dipyramid & J15 & by Proposition~\ref{suf}(ii) \\ \hline square orthobicupola & J28 & by Proposition~\ref{suf}(ii)\\ \hline pentagonal gyrobicupola & J31 & by Proposition~\ref{suf}(i) \\ \hline elongated triangular gyrobicupola & J36 & by Proposition~\ref{suf}(i) \\ \hline elongated pentagonal gyrobicupola & J39 & by Proposition~\ref{suf}(i)\\ \hline elongated pentagonal gyrobirotunda & J43 & by Proposition~\ref{suf}(i)\\ \hline parabiaugmented hexagonal prism & J55 & by Proposition~\ref{suf}(ii)\\ \hline parabiaugmented dodecahedron & J59 & by Proposition~\ref{suf}(i)\\ \hline biaugmented truncated cube & J67 & by Proposition~\ref{suf}(ii)\\ \hline parabiaugmented truncated dodecahedron & J69 & by Proposition~\ref{suf}(i)\\ \hline parabigyrate rhombicosidodecahedron & J73 & by Proposition~\ref{suf}(i)\\ \hline paradiminished rhombicosidodecahedron & J80 & by Proposition~\ref{suf}(i)\\ \hline bilunabirotunda & J91 & by Proposition~\ref{suf}(ii)\\ \hline \end{tabular} \caption{\it The 13 Johnson solids belonging to the class $C_{I}$} \end{table} And here are the Johnson solids not in the class $C_{I}$: \begin{table}[h!] \centering \begin{tabular}{|c||c|c||} \hline Johnsons solid $\mathcal C$ & Jxx & eliminated since \\ \hline \hline square pyramid & J1 & only one face $4$\\ \hline pentagonal pyramid & J2 & only one face $5$\\ \hline triangular cupola & J3 & only one face $6$ \\ \hline square cupola & J4 & only one face $8$\\ \hline pentagonal cupola & J5 & only one face $10$\\ \hline pentagonal rotunda & J6 & only one face $10$ \\ \hline elongated triangular pyramid & J7 & 3 faces with 4 edges \\ \hline elongated square pyramid & J8 & 5 faces with 4 edges \\ \hline elongated pentagonal pyramid & J9 & only one face 5\\ \hline gyroelongated square pyramid & J10 & only one face 4 \\ \hline gyroelongated pentagonal pyramid & J11 & only one face 5 \\ \hline triangular dipyramid & J12 & $v = 5$ odd number \\ \hline pentagonal dipyramid & J13 & $v = 7$ odd number \\ \hline elongated triangular dipyramid & J14 & $f = 9$ odd number \\ \hline elongated pentagonal dipyramid & J16 & 5 faces 4 \\ \hline gyroelongated square dipyramid & J17 & it contains a 4-antiprism\\ \hline elongated triangular cupola & J18 & only one face 6\\ \hline elongated square cupola & J19 & only one face 8\\ \hline elongated pentagonal cupola & J20 & only one face 10\\ \hline elongated pentagonal rotunda & J21 & only one face 10 \\ \hline gyroelongated triangular cupola & J22 & only one face 6 \\ \hline gyroelongated square cupola & J23 & only one face 8 \\ \hline \end{tabular} \end{table} \begin{table}[h!] \centering \begin{tabular}{|c||c|c||} \hline gyroelongated pentagonal cupola & J24 & only one face 10 \\ \hline gyroelongated pentagonal rotunda & J25 & only one face 10 \\ \hline gyrobifastigium & J26 & by Proposition~\ref{not}(ii) \\ \hline triangular orthobicupola & J27 & by Proposition~\ref{not}(ii)\\ \hline square gyrobicupola & J29 & by Proposition~\ref{not}(iii)\\ \hline pentagonal orthobicupola & J30 & by Proposition~\ref{not}(iii)\\ \hline pentagonal gyrobicupola & J32 & 7 faces with 5 edges\\ \hline pentagonal gyrocupolarotunda & J33 & 7 faces with 5 edges \\ \hline pentagonal orthobirotunda & J34 & by Proposition~\ref{not}(ii)\\ \hline elongated triangular orthobicupola & J35 & by Proposition~\ref{not}(ii)\\ \hline elongated square gyrobicupola & J37 & by Proposition~\ref{not}(ii)\\ \hline elongated pentagonal orthobicupola & J38 & by Proposition~\ref{not}(ii)\\ \hline elongated pentagonal orthocupolarotunda & J40 & 7 faces with 5 edges\\ \hline elongated pentagonal gyrocupolarotunda & J41 & 7 faces with 5 edges \\ \hline elongated pentagonal orthobirotunda & J42 & by Proposition~\ref{not}(ii)\\ \hline gyroelongated triangular bicupola & J44 & by Proposition~\ref{not}(ii) \\ \hline gyroelongated square bicupola & J45 & by Proposition~\ref{not}(ii)\\ \hline gyroelongated pentagonal bicupola & J46 & by Proposition~\ref{not}(ii) \\ \hline gyroelongated pentagonal cupolarotunda & J47 & 7 faces 5\\ \hline gyroelongated pentagonal birotunda & J48 & by Proposition~\ref{not}(ii) \\ \hline augmented triangular prism & J49 & $v = 7$ odd number \\ \hline biaugmented triangular prism & J50 & only one face 4 \\ \hline triaugmented triangular prism & J51 & $v = 9$ odd number \\ \hline augmented pentagonal prism & J52 & $v = 11$ odd number \\ \hline biaugmented pentagonal prism & J53 & 3 faces 4\\ \hline augmented hexagonal prism & J54 & $f = 11$ odd number \\ \hline parabiaugmented hexagonal prism & J56 & by Proposition~\ref{not}(ii)\\ \hline triaugmented hexagonal prism & J57 & 3 faces 4\\ \hline augmented dodecahedron & J58 & $v = 21$ odd number \\ \hline metabiaugmented dodecahedron & J60 & by Proposition~\ref{not}(ii)\\ \hline triaugmented dodecahedron & J61 & $v = 23$ odd number \\ \hline metadiminished dodecahedron & J62 & by Proposition~\ref{not}(ii)\\ \hline tridiminished icosahedron & J63 & $v = 9$ odd number \\ \hline augmented tridiminished icosahedron & J64 & 3 faces 5\\ \hline augmented truncated tetrahedron & J65 & 3 faces 6\\ \hline augmented truncated cube & J66 & 5 faces 8\\ \hline augmented truncated dodecahedron & J68 & $v = 65$ odd number \\ \hline metabiaugmented truncated dodecahedron & J70 & by Proposition~\ref{not}(i)\\ \hline triaugmented truncated dodecahedron & J71 & $v = 75$ odd number \\ \hline gyrate rhombicosidodecahedron & J72 & by Proposition~\ref{not}(iii)\\ \hline metabigyrate rhombicosidodecahedron & J74 & by Proposition~\ref{not}(i)\\ \hline trigyrate rhombicosidodecahedron & J75 & by Proposition~\ref{not}(iii)\\ \hline diminished rhombicosidodecahedron & J76 & by Proposition~\ref{not}(iii)\\ \hline \end{tabular} \end{table} \begin{table}[h!] \centering \begin{tabular}{|c||c|c|c||} \hline paragyrate diminished rhombicosidodecahedron & J77 & by Proposition~\ref{not}(iii)\\ \hline metagyrate diminished rhombicosidodecahedron & J78 & by Proposition~\ref{not}(iii)\\ \hline bigyrate diminished rhombicosidodecahedron & J79 & by Proposition~\ref{not}(iii)\\ \hline metadiminished rhombicosidodekahedron & J81 & by Proposition~\ref{not}(i)\\ \hline gyrate bidiminished rhombicosidodekahedron & J82 & by Proposition~\ref{not}(iii)\\ \hline tridiminished rhombicosidodekahedron & J83 & $v = 45$ odd number \\ \hline snub disphenoid & J84 & by Proposition~\ref{not}(i)\\ \hline snub square antiprism & J85 & by Proposition~\ref{not}(ii)\\ \hline sphenocorona & J86 & by Proposition~\ref{not}(i)\\ \hline augmented sphenocorona & J87 & by Proposition~\ref{not}(iii)\\ \hline augmented sphenocorona & J88 & by Proposition~\ref{not}(i)\\ \hline hebesphenomegacorona & J89 & 3 faces 4 \\ \hline disphenocingulum & J90 & by Proposition~\ref{not}(i)\\ \hline triangular hebesphenorotunda & J92 & only one face 6\\ \hline \end{tabular} \caption{\it The 79 Johnson solids not being in the class $C_{I}$} \end{table} \section{Maps of $\mathcal C/2$, $\mathcal C \in C_{I}$ in the projective plane}\label{maps} \begin{figure}[h!] \centering \includegraphics[scale=0.55]{Platproj2.eps} \caption{\it The halves of Platonic solids in the projective plane} \label{Platproj} \end{figure} \begin{figure}[h!] \centering \includegraphics[scale=0.55]{Aproj3.eps} \caption{\it The halves of Archimedean solids in the projective plane} \label{PAproj} \end{figure} \begin{figure}[h!] \centering \includegraphics[scale=0.55]{ten1.eps} \label{maps10a} \end{figure} \begin{figure}[h!] \centering \includegraphics[scale=0.55]{ten2b.eps} \caption{\it The halves of Johnson solids in the projective plane} \label{maps10} \end{figure} The flag graphs of halves of solids $\mathcal C \in C_{I}$ can now be deduced from Figures~\ref{Platproj}, ~\ref{PAproj} and ~\ref{maps10}. For J31 this is done in Figure ~\ref{flaggraphJ15}. Now it is easy to obtain the symmetry-type graphs of any $\mathcal C \in C_{I}$. For J15 this is done in Figure~\ref{symmetrygraphJ15}. \begin{figure}[h!] \centering \includegraphics[scale=0.55]{prizmainantiprizma2.eps} \caption{\it The halves of 6-prism and 5-antiprism in the projective plane} \label{pa} \end{figure} \begin{figure}[h!] \centering \includegraphics[scale=0.5]{flaggraphJ15.eps} \caption{\it Flags and flag graph of (J15)/2} \label{flaggraphJ15} \end{figure} \begin{figure}[t] \centering \includegraphics[scale=0.55]{symgraph.eps} \caption{\it Representative flags of orbits and symmetry-type graph of J15} \label{symmetrygraphJ15} \end{figure} \section{Spectral analysis of faces of $\mathcal C/2$ for $\mathcal C \in C_{I}$}\label{spectra} \begin{definition} For any polyhedron $\mathcal P$ with regular faces having at most $n$ vertices let the vector $\textbf{s}(\mathcal C) = (f_{n},\dots, f_{6},f_{5},f_{4},f_{3})$ denote its »spectral vector, counting the numbers $f_{i}$ of its faces with $i$ vertices. In the corresponding »spectral codes« $S(\mathcal P)$ (see the right column of Table 3) only the nonzero numbers $f_{i}$ are given. \end{definition} \vspace*{-.3cm} \begin{table}[h!] \centering \begin{tabular}{|c|c|c|c|c|c|c|c|} \hline % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ... $\mathcal C$ & $n_{10}$ & $n_{8}$ & $n_{6}$ &$n_{5}$ & $n_{4}$ & $n_{3}$ & $S(\mathcal C/2)$\\ \hline \hline $(4.4.4)$ & & & & & 3 & & $4_{3}$\\ \hline $(5.5.5)$ & & & &6 & & &$5_{6}$\\ \hline $(3.3.3.3)$ & & & & & & 4 &$3_{4}$\\ \hline $(3.3.3.3.3)$ & & & & & & 10 &$3_{10}$\\ \hline \hline $(4.6.6)$ & & & 4 & & 3 & &$6_{4}4_{3}$\\ \hline $(5.6.6)$ & & & 10 & 6 & & &$6_{10}5_{6}$\\ \hline $(8.8.3)$ & & 3 & & & & 4 &$8_{3}3_{4}$\\ \hline $(10.10.3)$ & 6 & & & & & 10 &$10_{6}3_{10}$\\ \hline $(8.4.6)$ & & 3 & 4 & & 6 & &$8_{3}6_{4}4_{6}$\\ \hline $(3.4.4.4)$ & & & & & 9 & 4 &$4_{9}3_{4}$\\ \hline $(4.3.4.3)$ & & & & & 3 & 4 &$4_{3}3_{4}$\\ \hline $(5.3.5.3)$ & & & & 6 & & 10 &$5_{6}3_{10}$\\ \hline $(5.4.3.4)$ & & & & 6 & 15 & 10 &$5_{6}4_{15}3_{10}$\\ \hline \hline J15 & & & & & 2 & 4 &$4_{2}3_{4}$\\ \hline J28 & & & & & 5 & 4 &$4_{5}3_{4}$\\ \hline J31 & & & & 1 & 5 & 5 &$5_{1}4_{5}3_{5}$\\ \hline J36 & & & & & 6 & 4 &$4_{6}3_{4}$\\ \hline J39 & & & & 1 & 10 & 5 &$5_{1}4_{10}3_{5}$\\ \hline J43 & & & & 6 & 5 & 10 &$5_{6}4_{5}3_{10}$\\ \hline J55 & & & & & 4 & 4 &$4_{4}3_{4}$\\ \hline J59 & & & & 5 & & 5 &$5_{5}3_{5}$\\ \hline J67 & & 2 & & & 5 & 8 &$8_{2}4_{5}3_{8}$\\ \hline J69 & 5 & & & & 5 & 15 &$10_{5}4_{5}3_{15}$\\ \hline J73& & & & 6 & 15 & 10 &$5_{6}4_{15}3_{10}$\\ \hline J80 & 1 & & & 5 & 10 & 5 &$10_{1}5_{5}4_{10}3_{5}$\\ \hline J91 & & & & 2 & 1 & 4 &$5_{2}4_{1}3_{4}$\\ \hline \end{tabular} \caption{\it Spectral vectors of $\mathcal C/2$ for $\mathcal C \in C_{I}$} \end{table} \subsection{Linear dependence relations between polyhedra} \label{linear} The concept of the spectral vector paves the way to the study of linear dependence relations between any polyhedra. \begin{definition}\label{linear} Polyhedra $P_{1}, \dots, P_{m}$ are linearly dependent, if their corresponding spectral vectors $\textbf{s}(P_{i})$ are linearly dependent. \end{definition} In other words: there are $a_{1},\dots,a_{m} \in \mathbb Z$ not all equal to zero such that \[ a_{1}\textbf{s}(P_{1}) + \dots + a_{m}\textbf{s}(P_{m}) = \textbf{0}. \] Using spectral vectors we can also define such concepts as ``collinearity'' and ``coplanarity'' of polyhedra. \begin{definition}\label{coplanar} Let $A, B, C$ be any polyhedra. If it is possible to take $a$ copies of $A$ and $b$ copies of $B$ and by dissecting their boundary into faces construct $c$ copies of a polyhedron $C$ so that no faces are left unused, we say that the solids $A$, $B$, $C$ are \emph{coplanar} and we write this symbolically as $aA + bB = cC$. Similarly, we write $aA = bB$ and say that $A$ and $B$ are collinear, if the relation $a\textbf{s}(A) = b\textbf{s}(B)$ holds for their corresponding spectral vectors. \end{definition} Using the information gathered in Table 3 we can now easily solve questions about linear dependence relations polyhedra from $C_{I}$ (since for the corresponding spectral vectors we obviously have the relation $\textbf{s}(\mathcal C) = 2 \textbf{s}(\mathcal C/2)$. \begin{example} Are the polyhedra J55, J59 and J73 coplanar? To answer this we have to solve the vector equation $a \textbf{s}($J$55/2) + b \textbf{s}($J$59/2) = c \textbf{s}($J$73/2)$, or, equivalently, $a(4_{4} + 3_{4}) + b(5_{5} + 3_{5}) = c(5_{6} + 4_{10} + 3_{10})$. From this we obtain the following system of three linear equations: $5b = 6c$, $4a = 10c$, $4a + 5b = 10c$. Thus $b = 6c/5$, $a = 5c/2$ and $4(5c/2) + 5(6c/5) = 10c$, hence $10c + 6c = 10c$ and $c = 0$. Thus J55, J59 and J73 are not coplanar. Some examples of coplanar solids from $C_{I}$ are: \begin{itemize} \item[] J31, J59 and J59, since $5_{1}4_{5}3_{5} + 5_{5}3_{5} = 5_64_{5}3_{10}$, hence J31 + J59 = J43; \item[] J31, (4.4.4) and J39, since $3\cdot 5_{1}4_{5}3_{5} + 5\cdot 4_{3} = 3\cdot 5_{1}4_{10}3_{5}$, hence $3\cdot $J$31 + \linebreak 5\cdot(4.4.4) = 3\cdot $J$39$; \item[] J15, (3.4.4.4) and J39, since $4\cdot 4_{2}3_{4} + 3\cdot 4_{9}3_{4} = 7\cdot 4_{5}3_{4}$, hence $4\cdot $J$15 + \linebreak 3\cdot(3.4.4.4) = 7\cdot $J$28$. \end{itemize} Other ``linear polyhedral equations'', as $aA + bB = cC + dD$, may be treated in a similar way, too. \end{example} \begin{thebibliography}{99} \bibitem{Fulton} {\sc W.\,Fulton}, \textit{Algebraic topology}, Springer-Verlag, New York, Berlin, Heidelberg, 1995. \bibitem{Hatcher} {\sc A.\,Hatcher}, \textit{Algebraic topology}, Cambridge University press, Cambridge, 2002. \bibitem{Johnson} {\sc N.\,W.\,Johnson}, \emph{Convex polyhedra with regular faces}, Canad. J. Math. \textbf{18}(1966), 169--200. \bibitem{Kovic} {\sc J.\,Kovi\v{c}}, \emph{Symmetry-type graphs of Platonic and Archimedean solids}, Math. Commun. \textbf{16}(2011), 491--507. \bibitem{Lins} {\sc S.\,Lins}, \emph{Graph-encoded maps}, J. Combin. Theory Ser. B \textbf{32}(1982), 171--181. \bibitem{Massey} {\sc W.\,Massey}, \textit{Algebraic topology: An introduction}, Springer-Verlag, New York, 1967. \bibitem{Orba} {\sc A.\,Orbani\'{c}}, \textit{Edge-transitive maps}, PhD thesis, University of Ljubljana, Ljubljana, 2006. \bibitem{PisaServa} {\sc T.\,Pisanski, B.\,Servatius}, \textit{Configurations, incidence structures from a graphical viewpoint}, Birkh\"{a}user, Boston, 2012. \bibitem{PisZit} {\sc T.\,Pisanski, A.\,\v{Z}itnik}, \emph{Representations of graphs and maps}, Preprint series \textbf{42}(2004), 924. \end{thebibliography} %\end{linenumbers} \end{document}