\section{Introduction}
\label{sec:introduction}
One of the important problems for discrete geometry in general, 
is to find a continuum associated with a given discrete object 
and compare their geometries. 
For a triangulation of a continuous surface, 
the continuum is the continuous surface itself (for example, see \cite{Hildebrandt-Polthier-Wardetzky}). 
A typical question is 
how geometric data of the triangulation converges to 
the corresponding geometric data 
of the continuous surface when meshes get finer.
What do we do with discrete objects with no obvious underlying continuum?
To address the issue, in the present paper, we study a discrete surface,
whose notion is defined in \cite{Kotani-Naito-Omori} as a trivalent graph in $\R^3$.
We introduce a method to subdivide a given discrete surface $M$, 
discuss convergence of the sequence $\{ M_{i} \}$ of the iteratively subdivided discrete surfaces,
and find a continuous object as its limit  when there is no obvious underlying surface for $M$.
\par
Let us state more precisely. 
Let $X=(V, E, F)$ be a trivalent topological surface graph, 
where $V$ denotes the set of vertices, 
and $E$ the set of edges.
We often identify a graph $X$ with the set $V$ of its vertices. 
Although $X$ is a one-dimensional object, 
it is convenient to consider a circuit, 
a closed simple curve without self-intersections, 
as a ``face'' of $X$.
Since we assume that $X$ is a surface graph, 
the notion of faces is well-defined, 
and $F$ denotes the set of faces.
An $n$-gonal face is 
$f=\{ v_0, \ldots, v_{n-1} \}$ with the ordered vertices $v_i \in V$ in the circuit of the length $n$. 
Let us denote $F$ the set of faces in $X$. 
Two faces are said to be neighbored when they share a common edge.
For later use, we also introduce the notion of ``leaf''.
The set  of a face $f$ and its neighboring faces is called a leaf with a core face $f$ and is denoted by $L(f)$.
\par
Given a discrete surface $\Phi \colon X \to M = \Phi(X) \subset \R^3$,
where $X$ is a trivalent graph and  $\Phi$ is a piecewise linear map, 
and let $\MV$, $\ME$, $\MF$ be the image set of $V$, $E$, $F$.
Note that through the paper we write $X$ for a topological graph 
and $M$ for a graph (discrete surface) realized in $\R^3$. 
\par
Let $\{X_{i} \}$ be a sequence of Goldberg-Coxeter construction (GC-construction, for detail see Section \ref{sec:3}) iteratively constructed from $X_{0} = X$. 
For a given $M_{i} = \Phi_{i}(X_{i})$, 
its subdivision $M_{i+1}$ is construct iteratively by the following two steps:
\begin{enumerate}
\item
  Solving the Dirichlet energy minimizing equation for $X_{i+1}$ 
  with the boundary condition $\Phi_{i}(X_i)$, 
\item
  Replace $\Phi_{i}(X_{i})$ by the barycenter of its nearest neighbors, and rename it as $M_{i}$.
\end{enumerate}
More precisely, we do the process leafwise.
We call $\{ M_i \}$ \emph{a sequence of subdivisions of a discrete surface} $M$
and prove $\{ M_i \}$ forms a Cauchy sequence in the Hausdorff topology (Theorem \ref{cauchy})
and show the energy monotonicity formula (Theorem \ref{monotonicity}).
Note that the above subdivision method is a modification of what introduced in \cite{Tao}, 
and we call this procedure the Goldberg-Coxeter subdivision (GC-subdivision).
\par
The limit of this Cauchy sequence 
$\mathcal {M}_{\infty} = \overline{\cup M_i}$ 
is divided into three kinds of sets:
\begin{equation*}
  \mathcal {M}_{\infty} = \MM_{\MF} \cup \MM_{\MV}  \cup \MM_{\MS}.
\end{equation*} 
The first two come from accumulating points of leafwise convergence and the third one appears from global accumulation.
Given a leaf with its center $\v{f}^{(i)}$, which is an $n$-gon in $M_{i}$, 
its GC-subdivision is an $n$-gon $\v{f}^{(i+1)}$ in $M_{i+1}$ and its neighboring $n$ hexagons (see Figure~\ref{fig-b}).
The first one is the set $\MM_{\MF}$ of accumulating points associated with each face in $M_i$.
We prove in Lemma \ref{lem-d}, 
for a fixed face $\v{f}^{(i)}$ in $M_i$,  
$\{ \v{f}^{(i)}_{k}  \in M_{i+k} \} $ with $\v{f}^{(i)}_0 = \v{f}^{(i)}$ form a converged sequence
and all vertices of $\{ \v{f}^{(i)}_k \}$ 
converges to the barycenter $\v{f}^{(i)}_{\infty}$ 
of the original face $\v{f}^{(i)}$ and also of all $\v{f}^{(i)}_k$. 
We call it an accumulating point associated with the face and put 
\begin{equation*}
  \MM_\MF \coloneqq \bigcup_i  \{ \v{f}^{(i)}_{\infty} \mid \text{ the barycenter of all faces } \v{f}^{(i)} \in M_{i} \}.
\end{equation*}
\par
The second one is the set of all vertices, replaced as in the above step,  i.e., 
\begin{math}
  \mathcal{M}_{\MV} = \cup_{i} M_i.
\end{math}
\par
The regularity of the limit set is not trivial at all, 
although we have the energy monotonicity formula (Theorem \ref{monotonicity}).
It seems a balancing condition plays an important role.
For example, when we take a \ce{C60}, a polygonal graph on the sphere, which does not satisfy the balancing condition, 
we obtain a pathological shape as the limit of its subdivisions (Section \ref{sec:numerical}).
\par
We also prove the convergence to a point in $\MM_{\MF}$ is of $C^1$ class 
in the sense that the corresponding normal vectors 
converges to a unique unit vector independent of the choice of converging sequence in $\MM_{\MF}$.
\par
The third one is the set $\MM_{\MS}$ of the rest of the accumulating points.
We know little about $\MM_{\MS}$ in general, 
however, we prove an un-branched discrete surface do not have such $\MM_{\MS}$. 

\begin{thm}[Theorem \ref{cauchy} and Theorem \ref{thm-f}]
  A sequence $\{M_i\}$ of iteratively subdivided discrete surface constructed 
  forms a discrete surface $M$ form a Cauchy sequence in Hausdorff topology. 
  The limit set $\MM_{\infty}$ consists of $\MM_{\MV}$, $\MM_{\MF}$ and $\MM_{\MS}$.
  When $M$ is un-branched, $\MM_{\MS} $ is empty.
\end{thm}

The first statement of the above theorem was proved by the last author with a little different subdividing method in \cite{Tao}.
In the present paper, we propose a modified method.
The proof needs some modification but almost the same. We explain why we think the modified one is better.
\begin{thm}[Theorem \ref{monotonicity}]
  The total Dirichlet energy $E_D(M_i)$ is bounded when it is subjected to a finite domain at the initial stage $M_0$.
  Moreover it monotonically decreases if $M_0$ contains no $n$-gonal faces with $n > 6$ 
  and contains an $n$-gonal face with $n < 6$.
\end{thm}

\par
A trivalent graph is said to be ``branched'' when an edge is shared with more than two faces.
We note that the condition ``un-branched'' is necessary for a graph to be considered as a ``surface'' is shown in Section \ref{section:k4}. 
The $K_{4}$-lattice is the triply periodic trivalent graph in $\R^3$ discovered by T.~Sunada \cite{Sunada}, 
which is the one of the two structures that satisfy the strong-isotropic property.  
The $K_{4}$-lattice is branched. Actually each edge is shared by 10 faces. 
The numerical computation shows each leaf of the $K_{4}$-lattice converges to a smooth leaf.
Two leaves, however, which have all common neighboring faces 
but one does not converge to the same leaf.
{"config": "arxiv", "file": "1806.03531/sec_introduction.tex"}