text:
\section{Introduction}

\begin{figure}[htb]
\centering
 \includegraphics{puzzle-color}
\caption{Three polyominoes that fold along grid lines into a unit cube, from puzzles by Nikolai Beluhov \cite{Beluhov-2014}.}
 \label{puzzles}
\end{figure}

Given a piece of paper in the shape of a polyomino, i.e., a polygon in the plane formed by unit squares on the square lattice that are connected edge-to-edge, does it have a folded state
in the shape of a unit cube?  The standard rules of origami apply~\cite{GFALOP}; in particular, we allow each unit-square
face to be covered by multiple layers of paper.
Examples of this decision problem are given by the three puzzles by Nikolai Beluhov
\cite{Beluhov-2014} shown in \cref{puzzles}.
We encourage the reader to print out the puzzles and try folding them.

Prior work \cite{abdde-fpipc-18} studied this decision problem extensively,
introducing and analyzing several different models of folding.
Beluhov
\cite{Beluhov-2014} implicitly defined a \emph{grid model} with the puzzles
in \cref{puzzles}:
Fold only along grid lines of the polyomino;
allow only orthogonal fold angles%
\footnote{The fold angle of a fold measures the deviation from the flat (unfolded) state, i.e., $180^\circ$ minus the dihedral angle between the two incident faces.}
($\pm 90^\circ$ and $\pm 180^\circ$); and
forbid folding material strictly interior to the cube.
In this model, the prior work \cite{abdde-fpipc-18} characterizes which
\emph{tree-shaped} polyominoes (whose unit squares are connected edge-to-edge
to form a tree dual graph)
lying within a $3 \times n$ strip can fold into a unit cube,
and exhaustively characterizes which tree-shaped polyominoes of $\leq 14$
unit squares fold into a unit cube.

Notably, however, the polyominoes in \cref{puzzles} are not tree-shaped,
and their interior is not even simply connected:
The first puzzle has a hole, the second puzzle has two holes, and the
third puzzle has a degenerate (zero-area) hole or \emph{slit}.  Arguably, these holes are what
makes the puzzles fun and challenging.  Therefore, in this paper, we embark on
characterizing which polyominoes with hole(s) fold into a unit cube in the
grid model.
Although we do not obtain a complete characterization, we give many
interesting conditions under which a polyomino does or does not fold into a
unit cube.

The problem is sensitive to the choice of model.
The other main model that has been studied in past work
is the more flexible \emph{half-grid model},
which allows orthogonal and diagonal folds between half-integral points,
as in Figure~\ref{fig:45-diag}.
The prior work \cite{abdde-fpipc-18} shows that \textit{all} polyominoes of at
least ten unit squares can fold into a unit cube in the half-grid model,
leaving only a constant number of cases to explore,
which were tackled recently \cite{CubeFolding_CCCG2020}.
Therefore, we focus on the grid model, which matches the puzzles
of Beluhov \cite{Beluhov-2014}.

\begin{figure}[ht]
	\centering
	\includegraphics[scale=1.25]{lb9-half.pdf}
	\caption{One way to fold a cube in the half-grid model, adapted from~\cite[Fig.~5(b)]{abdde-fpipc-18}. In all our figures, solid/red lines denote mountain folds, dashed/blue lines denote valley folds, light/grey lines denote grid lines, and bold/black lines denote the polyomino boundary.}
	\label{fig:45-diag}
\end{figure}

If we generalize the target shape from a unit cube to polycube(s),
there are polyominoes that fold in the grid model into all polycubes
of at most a given surface area \cite{StripsGrid_CGTA}.
If we further forbid overlapping unit squares (polyhedron unfolding/nets
instead of origami), this fold-all-polycubes problem has been studied for
small polycubes \cite{AloBosCol-CCGA-10}, and there is extensive work on
finding polyominoes that fold into multiple (two or three or more)
different boxes
\cite{AbeDemDem-CCCG-11,Mitani-Uehara-2008,Shirakawa-Uehara-2013,Uehara-box-survey,Xu2015}.

\subsection*{Our Results}

\begin{compactenum}
\item
We show that any hole that is not one of five \emph{basic} shapes of holes (see Figure~\ref{fig:simple-holes1}) always guarantee that a polyomino containing the hole folds into a cube; see \cref{thm:singleHole} in \cref{subsec:fold-1hole}. Polyominoes with exactly one of the five basic holes only sometimes allow folding into a cube.
\item We identify combinations of two (of the five basic) holes that allow the polyomino to fold into a cube; see \cref{subsec:fold-2holes}.
\item We show that certain of the five basic holes or their combinations do not allow folding into a cube, that is, we show that subclasses of polyominoes with only specific basic hole(s) cannot be folded into a unit cube; see \cref{sec:not-fold}.
\item We present an algorithm that checks a necessary local condition for folding into a cube; see \cref{subsec:alg}.
\item Whether this condition also constitutes a sufficient condition remains an open question; see \cref{sec:concl}.
\item We conjecture that a slit of size 1 (see Figure~\ref{fig:simple-holes1}, second from left) never affects whether a polyomino can fold into a cube; see \cref{subsec:not-fold-slit}. However, we show that a slit of size 1 can be the deciding factor for foldability for larger polycubes.
\end{compactenum}

\section{Notation}\label{sec:not}
A \emph{polyomino} is a connected polygon $P$ in the plane formed by joining
together $|P|=n$ unit squares on the square lattice.
We refer to the vertices of the $n$ unit squares forming $P$
as the \emph{grid points} of $P$.
We view $P$ as an \emph{open} region (excluding its boundary) which includes
the $n$ open unit squares of the form $(x_i,x_i+1) \times (y_i,y_i+1)$ as well
as \textit{some} of the shared unit-length edges (and grid points)
among these $n$ unit squares.
Notably, we do not require $P$ to include the common edge between every
adjacent pair of squares; if such an edge is missing from $P$,
we call the edge a \emph{slit edge}.
But there must be at least $n-1$ unit-length edges in $P$ so that $P$ is
(interior-)connected.

A \emph{hole} of a polyomino $P$ is a bounded connected component of $P$'s
exterior, whose boundary is one of the connected components of $P$'s boundary
other than the outermost one.
We assume that $P$ has no holes that are just a single grid point,
because such holes do not affect foldability,
so we can fill them in (add them to~$P$).
We call a hole a \emph{slit} if it has zero area (and is more
than a single point), and thus consists entirely of one or more slit edges.
We call a hole \emph{basic} if it has one of the following shapes
(refer to see \cref{fig:simple-holes1}):
\begin{enumerate}
\item A unit square
\item A slit of size~1 (a single slit edge)
\item A slit of size~2 (L-shaped or straight)
\item A U-slit of size~3
\end{enumerate}

A \emph{unit cube \ensuremath{\mathcal{C}}\xspace} is a three-dimensional polyhedron with six unit-square faces and volume of $1$. We refer to the vertices of $\ensuremath{\mathcal{C}}\xspace$ as \emph{corners}.

In this paper, we study the problem of folding a given polyomino $P$ with holes to form $\ensuremath{\mathcal{C}}\xspace$, allowing
creases along edges of the lattice with fold angles of
$\pm 90^\circ$ or $\pm 180^\circ$.
In all our figures, solid/red lines denote mountain folds, dashed/blue lines denote valley folds, light/grey lines denote grid lines, bold/black lines denote the polyomino boundary, bold dotted/purple lines denote creases folded by $\pm 180^\circ$, and thin dotted/purple lines denoted creases folded by $\pm 90^\circ$ (the purple/dotted creases could be mountain or valley).

Some crease patterns give numbers on the unit squares to indicate which face
they fold onto in a ``real-world'' six-sided die, where opposite faces sum up to~$7$.

\begin{figure}[ht]
	\centering
	\includegraphics[page=3]{Cube1-color}
	\caption{The five basic holes: a unit square, a slit of size~1, a straight slit of size~2, an L-slit of size~2, and a U-slit of size~3.}
	\label{fig:simple-holes1}
\end{figure}

\section{Polyominoes That Do Fold}\label{sec:fold}
In this section, we present polyominoes that fold. We start with polyominoes that contain a hole guaranteeing foldability.

\subsection{Single-Hole Polyominoes}\label{subsec:fold-1hole}

In this section, we show that any hole different from a basic hole guarantees foldability.

\newcommand{basic}{basic}
\begin{theorem}\label{thm:singleHole}
If a polyomino $P$ contains a hole $h$ that is not basic, then $P$ folds into a cube.
\end{theorem}
\begin{proof}
By the definition of basic~hole, because $h$ is non-basic, it must be a superset of either two unit squares, a unit square and a unit slit, or a slit of size 3 that is not a U-slit.
In the case it contains two unit squares sharing a grid point, $h$ must be a superset of one of the holes in Figure~\ref{fig:cases}~(a)--(b) up to rotation.
If it contains a unit square and a unit slit sharing a grid point, then $h$ is a superset of Figure~\ref{fig:cases}~(e) up to reflection and rotation.
Else, $h$ must be a superset of the slits in Figure~\ref{fig:cases}~(c), (d), (f), (g) because those are all possible slits of size 3 that are not U-shaped up to reflection and rotation.
Then, we distinguish the cases where $h$ contains
\begin{compactitem}
\item  Two unit squares sharing an edge
\item  Two unit squares sharing a grid point
\item  A unit square and an incident slit
\item  A slit of length at least 3 (straight, zigzagged, L-shaped, or T-shaped)
\end{compactitem}

In a first step, we show that if $h$ contains one of the four above holes, we may assume that it contains exactly this hole. Let $h$ be a hole containing a hole $h'$ of the above type. By definition of a hole, $h$ needs to be enclosed by solid squares. Thus we can sequentially fold the squares of $P$ in columns to the left and right of~$h'$ on top of the columns directly left and right of~$h'$, respectively, as illustrated in \cref{fig:reduction}.
\begin{figure}[t]
	\centering
	\includegraphics[page=2]{Cube1-color}
	\caption{Folding strategy to reduce to seven cases.}
	\label{fig:reduction}
\end{figure}
Afterwards, we fold the squares of~$P$ in rows to the top and bottom of~$h'$ on top of the rows directly top and bottom of~$h'$, respectively.  We call the resulting polyomino~$P'$. Note that because~$h$ is a hole of $P$, all neighboring squares of~$h'$ exist in~$P'$. Thus we may assume that we are given one of the seven polyominoes depicted in \cref{fig:cases}, where striped  squares may or may not be present. Note that we can assume that no additional slits are present.

\begin{figure}[htbp]
	\centering
	\includegraphics[page=1]{Cube1-color}
	\caption{Any polyomino with a hole that is not basic~can be reduced to one of the seven illustrated cases; striped squares may or may not be present.}
	\label{fig:cases}
\end{figure}

Secondly, we present strategies to fold the polyomino into $\ensuremath{\mathcal{C}}\xspace$.
Note that the case if $h'$ consists of two squares sharing only a grid point, we can fold the top row on its neighboring row and obtain the case where $h'$ consist of a square and an incident slit. For an illustration of the folding strategies for the remaining six cases consider \cref{fig:foldings}.
\begin{figure}[htbp]
	\centering
	\includegraphics[page=4]{Cube1-color}
	\caption{Crease pattern of cube foldings. Mountain folds are shown in solid red, valley folds in dashed blue. Squares with the same number cover the same face of the cube.}
	\label{fig:foldings}
\end{figure}
\end{proof}

\begin{figure}[hp]
	\centering
	\includegraphics[page=1]{helpful-color}
	\caption{Four basic~holes may be helpful. Mountain folds are shown in solid red, valley folds in dashed blue.}
	\label{fig:helpful}
\end{figure}

\subsection*{Are basic~holes ever helpful?}
In fact, four of the five basic~holes sometimes allow foldability, as illustrated in \cref{fig:helpful}.
Note that the U-slit of size 3 reduces to the square hole by a $\pm 180^\circ$ mountain or valley fold.

 In \cref{thm:slit1}, we show that the slit of size 1 never helps to fold a rectangular polyomino.
 Moreover, we show in \cref{lem:slitNEW1} that the crease pattern around the slit behaves as if the slit was nonexistent, i.e., the only option to make use of the slit is to push part of the polyomino through the slit.
 In fact, we conjecture that the slit of size~1 never helps to fold a polyomino into~\ensuremath{\mathcal{C}}\xspace.

\subsection{Combinations of Two Basic Holes}\label{subsec:fold-2holes}

In this section, we consider combinations of two basic~holes that fold. For a polyomino with two holes, for which the lowest grid point $v$ of the upper hole has a higher y-coordinate than the upper grid point $w$ of the lower hole, we denote the number of unit-square rows between $w$ and $v$ as the \emph{number of rows between} the two holes. Analogously, we define the \emph{number of columns between} two holes.

\begin{theorem}
A polyomino with two vertical straight size-2 slits with at least two columns and an odd number of rows between them folds.
\end{theorem}
\begin{proof}
As in the previous section, we first fold all rows between the slits together to one row; this is possible because there is an odd number of rows between the slits. Then, all the surrounding rows and columns are folded towards the slits. Finally, we fold the columns between the slits to reduce their number to two or three. Depending on whether the number of columns between the slits was even or odd, this yields a shape as shown in \cref{fig:twoholes1}~(a) and~(b), respectively, where the striped squares may be (partially) present.
\begin{figure}[b!]
	\centering
	\includegraphics{two_holes-color.pdf}
	\caption{Combinations of two basic~holes that are foldable with and without (part of) the striped region. Mountain folds are shown in solid red, valley folds in dashed blue.
	}
	\label{fig:twoholes1}
\end{figure}
In all cases, the two shapes fold as indicated by the illustrated crease pattern. Note that in \cref{fig:twoholes1}~(b) the polyomino is of course connected, which implies that at least one square of the central column is part of the polyomino, i.e., a square with label 6 is used.
\end{proof}

If we have a polyomino with exactly two slits that have only one or no column between them, then the shape cannot be folded as can be verified by the algorithm of \cref{subsec:alg}.
In the following theorems we call a U-slit which has the open part at the bottom an A-slit. If the orientation of the U-slit is not relevant, then we call it a C-slit.

\begin{theorem}
A unit cube can be folded from any polyomino
with an A-slit and a unit-square hole/C-slit in the same column above it, having an even number of rows between them.
\end{theorem}

\begin{proof}
We can assume that the upper hole is a unit square, as the flaps generated by a C-slit can always be folded away.
Similar to before we fold away all surrounding rows and columns and reduce the number of rows between the A-slit and the unit-square hole to two. This yields the shape of \cref{fig:twoholes1}~(c), which can be folded as indicated by the crease pattern.
\end{proof}

Note that if the bottom slit is a U-slit, then the shape of \cref{fig:twoholes1}~(c) does not fold, again verified by the algorithm of \cref{subsec:alg}.

\begin{theorem}
A polyomino with an A-slit and a unit-square hole/C-slit below it and separated by an odd number of rows, folds, regardless in which columns they are.
\end{theorem}

\begin{proof}
As before, we assume that the lower hole is a unit square, fold away the surrounding rows and columns, and reduce the number of rows between the two slits/holes to one. Furthermore, we fold the columns between the slits/holes such that at most two columns remain between the two slits/holes. Consequently, we obtain one of the shapes shown in \cref{fig:twoholes1}~(d) to~(g). All of them fold, with or without the striped region. Note that the upper unit-square holes in \cref{fig:twoholes1}~(d) and~(e) can be replaced by an A-slit which can be folded away.
\end{proof}

Note that if the two holes are in the same or neighbored column(s) (\cref{fig:twoholes1}~(d) and~(e)), then independently of the orientation of the U-slits or whether they are unit-square holes, any combination folds, yielding Theorem~\ref{th:2us-holes}. In the other cases, the unit square incident to all three slit edges constitutes the only unit square that covers the face '1' in the unit cube.

\begin{theorem}\label{th:2us-holes}
 A polyomino with two unit-square holes which are in the same or in neighboured column(s) and have an odd number of rows between them folds.
\end{theorem}

\iffalse
\subsection{Polyominoes That Surprisingly Fold}\label{subsec:fold-surp}
{\tt Surprising stuff that folds (coauthor post Mar 27, 0:43 and Mar 26 3:15)(?)}
\textcolor{red}{Christiane}
In this section we list various polyominoes that surprisingly do fold, we leave the development of crease patterns to the reader.
\begin{itemize}
    \item A $3\times 6$ rectangle with a square hole and a C-shaped\todo{IK: did we call these U-shaped before?} hole (where the open part of the C-shape is directed towards the boundary)
    \item Any combination of a C-shaped and a unit-square hole, two C-shaped holes, a C-shaped and a U-shaped hole, a C-shaped and an A-shaped hole, or a C-shaped and an inverse C-shaped hole with even horizontal distance fold into a unit cube (as it reduces to a ring with a flap).
    \item A rectangle with a row of parallel vertical slits of length at least two folds if there exists any flap on its outer boundary, that is, an additional square attached to the boundary.
    \item
\end{itemize}
\fi

\section{Polyominoes That Do Not Fold}\label{sec:not-fold}
In this section, we identify basic~holes and combinations of basic~holes that do not allow the polyomino to fold.
First, we present some results that show how the paper is constrained around an interior grid point $v$. In particular, we consider situations when the induced polyomino of the four unit squares $A, B, C, D$ incident to $v$ is connected; for an example consider \cref{fig:180creases}.

\begin{lemma}\label{le:43}
Four unit squares incident to a polyomino grid point $v$ for which the induced polyomino is connected,  cannot cover more than three faces of $\ensuremath{\mathcal{C}}\xspace$.
\end{lemma}

\begin{proof}
 The grid point $v$ is incident to four unit squares in $P$. As grid points of $P$ are mapped to corners of $\ensuremath{\mathcal{C}}\xspace$ and all corners of $\ensuremath{\mathcal{C}}\xspace$ are incident to 3 faces, $v$ is incident to only 3 faces in $\ensuremath{\mathcal{C}}\xspace$.
\end{proof}

\begin{lemma}\label{le:4by4}
Four unit squares incident to a grid point $v$ not on the boundary of $P$ cannot cover more than two faces of $\ensuremath{\mathcal{C}}\xspace$. In particular, at least two collinear incident creases are folded by $\pm 180^\circ$.
\end{lemma}
\begin{proof}
Let $A$, $B$, $C$, and $D$ be the unit squares incident to $v$ in circular order; see the left of \cref{fig:180creases}.
By \cref{le:43}, $A$, $B$, $C$, and $D$ cover at most three faces of $\ensuremath{\mathcal{C}}\xspace$. Hence, at least two unit squares map to the same face of $\ensuremath{\mathcal{C}}\xspace$; these can be edge-adjacent or diagonal.

In the first case, assume without loss of generality that $A$ and $B$ map to the same face. Hence, the crease between them must be folded by $\pm 180^\circ$. Then $C$ and $D$ must also map to the same face of $\ensuremath{\mathcal{C}}\xspace$ to maintain the paper connected. Consequently, the crease between $C$ and $D$ is  folded by $\pm 180^\circ$.

In the latter case, let without loss of generality $A$ and $C$ map to the same face of $\ensuremath{\mathcal{C}}\xspace$. As they are both incident to $v$, only two options of folding those two unit squares on top of each other exist. Either the edge between $A$ and $B$ gets folded on top of the edge between $B$ and $C$, this leaves a diagonal fold on $B$, a contradiction, or the edge between $A$ and $D$ gets folded on top of the edge between $B$ and $C$, which results in $D$ being mapped to $C$, and those are two adjacent unit squares, by the above argument two collinear incident creases must be folded by $\pm 180^\circ$.
\end{proof}

\begin{figure}[htb]
	\centering
	\includegraphics[page=1]{v-crease-pattern-color.pdf}
	\caption{Illustration of \cref{le:4by4,le:4by4_2}. The thick dotted/purple lines represent creases fold by $\pm 180^{\circ}$; they could be mountain or valley.}
	\label{fig:180creases}
\end{figure}

\begin{lemma}\label{le:4by4_2}
Consider a grid point $v$ that is not on the boundary of a polyomino $P$ that folds into \ensuremath{\mathcal{C}}\xspace. If one crease of $v$ is folded by $\pm 180^{\circ}$, then the incident collinear crease is also folded by $\pm 180^{\circ}$.
\end{lemma}
\begin{proof}

Without loss of generality, we show that if  the left horizontal crease of $v$ is folded by $\pm 180^{\circ}$, the same holds for the right horizontal crease. We denote the left and right adjacent grid points of $v$ by $a$ and $b$, respectively, as indicated in \cref{fig:180creases}, right.

Suppose for a contradiction, that the right crease is not folded by $\pm 180^{\circ}$. Then, by  \cref{le:4by4}, both vertical creases are folded by $\pm 180^{\circ}$. In particular, $a$ and $b$ are mapped to the same corner of \ensuremath{\mathcal{C}}\xspace and thus the edges $av$ and $bv$ coincide. Hence, because $av$ is folded by $\pm 180^{\circ}$, $bv$ is also folded by $\pm 180^{\circ}$.
\end{proof}

\cref{le:4by4,le:4by4_2} imply that:
\begin{corollary}\label{cor:rows}
Let $k,n\geq2$ and let $P$ be a polyomino containing a rectangular $k\times n$-subpolyomino $P'$ whose interior does not contain any boundary of $P$. Then, in every folding of $P$ into \ensuremath{\mathcal{C}}\xspace, all collinear creases of $P'$ are either folded by  $\pm 90^{\circ}$ or by $\pm 180^{\circ}$. Moreover, either all horizontal or all vertical creases of $P'$ are folded by  $\pm 180^{\circ}$; see \cref{fig:reducing}.
\end{corollary}
\begin{proof}
First, suppose for a contradiction that there exist two collinear creases, one of which is folded by $\pm 90^{\circ}$ and the other by $\pm 180^{\circ}$. Then there also exists an interior grid point of $P'$ where the crease type of the two collinear edges changes from $\pm 90^{\circ}$ to $\pm 180^{\circ}$. However, by \cref{le:4by4_2}, if one is folded by $\pm 180^{\circ}$, then both are. A contradiction.

Second, suppose that not all horizontal creases are folded by $\pm 180^{\circ}$. Then, by the first statement, there exists a row in which no grid point has a horizontal edge that is folded by $\pm 180^{\circ}$. By \cref{le:4by4}, all vertical creases incident to the grid points of this row are folded by $\pm 180^{\circ}$. Because all collinear edges behave alike, it follows that all vertical creases are folded by $\pm 180^{\circ}$.
\end{proof}

\begin{figure}[htb]
	\centering
	\includegraphics[page=2]{v-crease-pattern-color.pdf}
	\caption{Illustration of \cref{cor:rows}.}
	\label{fig:reducing}
\end{figure}

\begin{corollary}\label{cor:rect}
Let $P$ be a rectangular $k\times n$-polyomino without any holes, then $P$ does not fold into $\ensuremath{\mathcal{C}}\xspace$.
\end{corollary}

\subsection{Polyominoes with Unit Square, L-Shaped, and U-Shaped Holes}\label{subsec:pol-ush-L-U}

We begin by examining the possible foldings of a polyomino containing a unit-square hole.
Suppose that a given polyomino $P$ with a unit-square hole~$h$ folds into a cube.
Furthermore, let the shape of $h$ no longer be a square in the folded state; we say hole $h$ is folded in a \emph{non-trivial} way. For an example consider Figures~\ref{fig:non-trivial} and \ref{fig:one-edge-fold}. Then, in the folded state, either all edges of $h$ are mapped to the same edge of \ensuremath{\mathcal{C}}\xspace, or two pairs of edges are glued forming an L-shape.
In the following, we show that if $P$ folds into \ensuremath{\mathcal{C}}\xspace, the first case is impossible, while the second produces a  specific crease pattern around~$h$.

\begin{figure}[h]
\centering
\includegraphics{non-trivial-fold.pdf}
\caption{An example of a non-trivial fold of a $3\times 3$ square with a unit square hole. The crease pattern is a special case of the one in \cref{fig:hole-crease-pattern}.}
\label{fig:non-trivial}
\end{figure}

\begin{lemma}\label{lem:SquareHole1}
The four edges of a unit-square hole $h$ of a polyomino $P$ that folds into~$\ensuremath{\mathcal{C}}\xspace$ are not mapped to the same edge of \ensuremath{\mathcal{C}}\xspace in the folded state.
\end{lemma}
\begin{proof}
We denote the four unit squares of the polyomino edge-adjacent to $h$ by $A$, $B$, $C$, and $D$, and the four unit squares adjacent to $h$ via a grid point as $F_{1}$, $F_{2}$, $F_{3}$, and $F_{4}$, as illustrated in \cref{fig:one-edge-fold}. Assume for a contradiction that all edges of~$h$ are mapped to the same edge of $\ensuremath{\mathcal{C}}\xspace$. Consider $A$, $F_{1}$, and $B$ in the folded state. As the two corresponding edges of $h$ are glued together, the three faces must be pairwise perpendicular. The similar statement holds for the triples $\{B,F_{2},C\}$, $\{C,F_{3},D\}$, and $\{D,F_{4},A\}$. This results in a configuration as illustrated in the right of \cref{fig:one-edge-fold}.

\begin{figure}[htb]
	\centering
	\includegraphics[page=1]{one-edge-fold-color.pdf}
	\caption{Four edges of a square hole glued together.}
	\label{fig:one-edge-fold}
\end{figure}

Because the faces $A,B,C$ share an edge of \ensuremath{\mathcal{C}}\xspace in the folded state such that $A$ and~$B$, as well as $B$ and $C$ are perpendicular, $A$ and $C$ must cover the same face of \ensuremath{\mathcal{C}}\xspace. Likewise, $B$ and $D$ cover the same face of \ensuremath{\mathcal{C}}\xspace. If $P$ folds into \ensuremath{\mathcal{C}}\xspace, then $F_{1}$ and $F_{3}$, as well as $F_{2}$ and $F_{4}$ are mapped to the same faces of \ensuremath{\mathcal{C}}\xspace.
Suppose, without loss of generality, that in the folded state $A$ lies in a more outer layer than $C$. Then, $F_{1}$ and $F_{4}$ are in a more outer layer than $F_{3}$ and $F_{2}$, respectively. Thus, $B$ connects the more inner layer of $F_{2}$ to the more outer layer of $F_{1}$, and at the same time $D$ connects the inner layer of $F_{3}$ to the outer layer of $F_{4}$. Hence, $B$ and $D$ intersect, which is impossible. Therefore, if the polyomino folds into a cube, the four edges of a square hole cannot all be mapped to the same edge of \ensuremath{\mathcal{C}}\xspace.
\end{proof}

It follows that the only non-trivial way to glue the edges of a square hole $h$ of a polyomino folded into a cube is to form an L-shape. We use this to show the following fact:

\begin{lemma}\label{lem:SquareHoleCreasePattern}
Let $P$ be a polyomino with a unit-square hole that folds into~\ensuremath{\mathcal{C}}\xspace.
 In every folding of $P$ into~\ensuremath{\mathcal{C}}\xspace where $h$ is folded non-trivially (i.e., $h$ is not mapped to a square), the crease pattern of the unit squares incident to $h$ is as illustrated in the right image of Figure~\ref{fig:hole-crease-pattern} (up to rotation and reflection).
\end{lemma}
\begin{proof}
Suppose the four edges of $h$ are not mapped to distinct edges of~\ensuremath{\mathcal{C}}\xspace. Then, by \cref{lem:SquareHole1}, the four edges are not mapped to the same edge, but to two edges forming an L-shape. This effectively amounts to gluing a pair of diagonal grid points of the hole.

Let $a$, $b$, $c$, and $d$ be the four grid points of $h$, and suppose $a$ and $c$ are mapped to the same corner of \ensuremath{\mathcal{C}}\xspace when folding $P$ into \ensuremath{\mathcal{C}}\xspace; see also the left image of \cref{fig:hole-crease-pattern}.
\begin{figure}[t]
	\centering
	\includegraphics[page=1]{hole-crease-pattern-color.pdf}
	\hfil
	\includegraphics[page=2]{hole-crease-pattern-color.pdf}
	\caption{Thinner dotted/purple lines indicate creases folded by $\pm 90^{\circ}$, while thicker dotted/purple lines indicate creases folded by $\pm 180^{\circ}$. Left: crease pattern around a unit-square hole folding into an L-shape when grid points $a$ and $c$ are mapped to the same corner of \ensuremath{\mathcal{C}}\xspace; creases shown in purple can be both mountain or valley. Right: numbers indicate the face of the cube in the folded state; mountain folds are shown as red solid lines, and valley folds as blue dashed lines.}
	\label{fig:hole-crease-pattern}
\end{figure}

Consider the crease pattern around $h$.
We shall only focus on the angles of the creases and not the type of the fold, as there may be (and will be) other creases in $P$ affecting the type of the creases under our consideration.
Observe that the three faces incident to each of the grid points $b$ and $d$ are pairwise perpendicular, they form a corner of a cube.
Thus, the creases emanating from $b$ and $d$ are all folded by $\pm 90^{\circ}$.
Further observe that the three unit squares around each of the grid points $a$ and $c$ fold into two faces of a cube, thus, leading to one of the creases being folded by $\pm 90^{\circ}$ and the other folded by $\pm 180^{\circ}$.
Finally, the two creases folded by $\pm 180^{\circ}$ are parallel to each other.
Indeed, consider the right side of \cref{fig:one-edge-fold}.
For a crease to form an L-shape one of the two dashed blue lines must fold by $\pm 180^{\circ}$, which corresponds to two parallel creases in the unfolded state.
Therefore, the crease pattern in \cref{fig:hole-crease-pattern} (left) is the only pattern of creases (up to rotation and reflection) around a non-trivially folded square hole.
\cref{fig:hole-crease-pattern} (right) shows the faces of the corresponding crease pattern, and \cref{fig:non-trivial} shows the folding process of this crease pattern.
\end{proof}

With the help of \cref{lem:SquareHoleCreasePattern}, we can show that several types of polyominoes with unit-square holes do not fold into \ensuremath{\mathcal{C}}\xspace.

\begin{theorem}\label{thm:RectangleSquareHole}
If $P$ is a rectangle with exactly one unit-square hole~$h$, then $P$ does not fold into $\ensuremath{\mathcal{C}}\xspace$.
\end{theorem}
\begin{proof}
First note that $h$ is folded non-trivially, otherwise $P$ corresponds to a rectangle which does not fold into \ensuremath{\mathcal{C}}\xspace (Corollary~\ref{cor:rect}). Therefore, by \cref{lem:SquareHoleCreasePattern}, the crease pattern around $h$ is as depicted in \cref{fig:hole-crease-pattern}. Note that, on each side of~$h$, there exists a fold by $\pm 90^\circ$.

Consider the rectangle $R$ obtained by cutting $P$ by the top edge of $h$ and deleting the part below. If $R$ has a height of at least 2, then by \cref{cor:rows}, either all vertical or all horizontal creases are folded by $\pm 180^\circ$.
In the first case, in particular the creases incident to $h$ are folded by  $\pm 180^\circ$. However, this is a contradiction to the crease pattern around $h$ in which each side of $h$ has fold by $\pm 90^\circ$.
Consequently, all horizontal edges are folded by $\pm 180^\circ$. This corresponds to folding $R$ on top of the row above $h$. In particular, $P$ is foldable into \ensuremath{\mathcal{C}}\xspace if and only if the polyomino $P'$ obtained from $P$ by cutting-off all rows above $h$ is foldable. Hence, we consider $P'$.

Likewise, we treat all other sides of $P'$ and obtain the polyomino $P''$ consisting of a $3\times 3$-rectangle with a central unit-square hole; see also \cref{fig:hole-crease-pattern} (right). In particular,
$P$ is foldable (if and) only if $P''$ is foldable into \ensuremath{\mathcal{C}}\xspace.

Because $h$ is folded non-trivially, the crease pattern of $P''$ is given by \cref{fig:hole-crease-pattern}. Note that in the folded state $P''$  covers only 5 faces and, hence,  $P''$ does not fold into~\ensuremath{\mathcal{C}}\xspace.
\end{proof}

A similar result holds for rectangular polyominoes with two unit-square holes.

\begin{theorem}\label{thm:twoholeseven}
A rectangle with exactly two unit-square holes in the same row does not fold into \ensuremath{\mathcal{C}}\xspace if the number of columns between the holes is even.
\end{theorem}

\begin{proof}
Note that if the polyomino can be folded into~\ensuremath{\mathcal{C}}\xspace, both holes must be folded non-trivially: If one hole behaves as a square in the folded state, i.e., is folded trivially, the polyomino is effectively reduced to a rectangle with one basic~hole. However, by  \cref{thm:RectangleSquareHole}, this does not fold into~\ensuremath{\mathcal{C}}\xspace. Consequently, both holes are folded non-trivially.

Therefore, by \cref{lem:SquareHoleCreasePattern}, the crease pattern around the two holes is as depicted in \cref{fig:hole-crease-pattern}.
Consider the $3\times 2k$-rectangle $R$ between the two holes (with $k\ge 1$).
By the above observation, at least one horizontal edge of $R$ is folded by $\pm 90^{\circ}$. Consequently, \cref{cor:rows} implies that all vertical edges are folded by $\pm 180^{\circ}$.
In particular, every unit square of $R$ is mapped to the same face of \ensuremath{\mathcal{C}}\xspace as the leftmost (or rightmost) unit square in the same row of $R$. This reduces the polyomino to one with~$R$ being a $3\times 2$-rectangle.
We will show that the squares of $P$ neighbouring the two holes are not able to cover~\ensuremath{\mathcal{C}}\xspace,
that is, it remains to show that the polyomino $P$, depicted in \cref{fig:two-holes-even}, does not fold into~\ensuremath{\mathcal{C}}\xspace.

\begin{figure}[tb]
	\centering
	\includegraphics[page=1]{two-holes-even-color.pdf}
	\hfil
	\includegraphics[page=2]{two-holes-even-color.pdf}
	\caption{A polyomino that does not fold into a cube.}
	\label{fig:two-holes-even}
\end{figure}

 Consider the left $3\times 3$ block of $P$. If the two parallel creases folded by $\pm 90^{\circ}$ are vertical, then the right $3\times 3$ block will also have the two parallel creases folded by $\pm 90^{\circ}$ run vertical; see \cref{fig:two-holes-even} (left). Then, the four unit squares above and below the two holes match to the same face on $\ensuremath{\mathcal{C}}\xspace$. Denote it as~`1'. Observe that the rest of the unit squares share a grid point with `1' and thus cannot cover the face on $\ensuremath{\mathcal{C}}\xspace$ opposite to `1'.

 In the second case, when the two parallel creases folded by $\pm 90^{\circ}$ of the left block are horizontal, then they extend into the right $3\times 3$ block by \cref{cor:rows}. Refer to \cref{fig:two-holes-even} (right). Then, the four unit squares to the left and to the right of the two holes match to the same face on $\ensuremath{\mathcal{C}}\xspace$, which we denote by `1'. As before, every unit square of $P$ shares a grid point with `1' and thus the face opposite to `1' on $\ensuremath{\mathcal{C}}\xspace$ cannot be covered.
\end{proof}

\noindent \textbf{\emph{ Remark.}}
Note that the arguments of \cref{lem:SquareHoleCreasePattern} and \cref{thm:RectangleSquareHole,thm:twoholeseven} extend to an L-slit of size $2$, and a U-slit of size $3$. The resulting crease patterns are illustrated in \cref{fig:hole-crease-pattern2}.
\begin{figure}[htb]
	\centering
	\includegraphics[page=3]{hole-crease-pattern-color.pdf}
	\hfil
	\includegraphics[page=4]{hole-crease-pattern-color.pdf}
	\caption{Crease pattern around an L-slit (left) and a U-slit (right).
	Numbers indicate the face of the six-sided die in the folded state; thinner lines denote creases folded by $\pm 90^{\circ}$; thicker lines denote creases folded by $\pm 180^{\circ}$; mountain folds are drawn solid/red; and valley folds are drawn dashed/blue.

	}
	\label{fig:hole-crease-pattern2}
\end{figure}
\medskip

These insights help to obtain the following result:

\begin{theorem}
\label{thm:two-holes-far}
Let $P$ be polyomino with two holes, which are both either a unit square, or an L-slit of size $2$, or a U-slit of size $3$, such that (1) $P$ contains all the other cells of the bounding box of the two holes and (2) the number of rows and the number of columns between the holes is at least $1$. In every folding of $P$ into~\ensuremath{\mathcal{C}}\xspace, the two holes are not both folded non-trivially.
\end{theorem}
\begin{proof}
If $P$ contains a unit-square hole that is not folded non-trivially, then, by \cref{lem:SquareHoleCreasePattern}, the crease pattern in the neighborhood of the hole is as depicted in  \cref{fig:hole-crease-pattern}. Likewise, if $P$ contains an L-slit of size 2 or a U-slit of size 3 that is folded non-trivially, the crease pattern in the neighborhood of the hole is as depicted in  \cref{fig:hole-crease-pattern2}. Note that on each side of the crease patterns in the neighborhood of the holes, there exists a crease folded by $\pm 90^{\circ}$.

We turn the paper such that the left hole is above the right hole as in \cref{fig:two-hole-crease-pattern} and consider the rectangular region $R$ to the right of the left hole and above the right hole.

\begin{figure}[htb]
	\centering
	\includegraphics{two-holes-crease-pattern-color.pdf}
	\caption{Two unit-square holes with at least one row and column in between, if folded non-trivially imply two perpendicular creases folded by $\pm 90^{\circ}$ (drawn as thin dotted lines).}
	\label{fig:two-hole-crease-pattern}
\end{figure}

Because each side of the crease patterns in the neighborhood of the holes has a crease folded by $\pm 90^{\circ}$ (Lemma~\ref{lem:SquareHoleCreasePattern}), $R$ contains a vertical and a horizontal crease folded by $\pm 90^{\circ}$. By \cref{cor:rows}, all collinear creases are also folded by $\pm 90^{\circ}$. Hence, there exists a grid point in $R$ for which all incident creases are folded by $\pm 90^{\circ}$, yielding a contradiction to \cref{le:4by4}.
\end{proof}

\subsection{Polyominoes with a Single Slit of Size 1}\label{subsec:not-fold-slit}

\iffalse
The following Lemmas show how the paper is constrained around an interior grid point.

\begin{lemma}\label{le:43}
Four unit squares around a polyomino grid point $v$ for which the dual graph is connected cannot cover more than three faces of $\ensuremath{\mathcal{C}}\xspace$.
\end{lemma}
\begin{proof}
 $v$ is incident to 4 unit squares in $P$. As grid points of $P$ are mapped to corners of $\ensuremath{\mathcal{C}}\xspace$ and all corners of $\ensuremath{\mathcal{C}}\xspace$ are incident to 3 faces, $v$ is incident to only 3 faces in $\ensuremath{\mathcal{C}}\xspace$.
\end{proof}

\begin{lemma}\label{le:4by4}
Four unit squares around a grid point $v$ not on the boundary of $P$ cannot cover more than two faces of $\ensuremath{\mathcal{C}}\xspace$. In particular, at least two collinear incident creases must be folded by $\pm 180^\circ$.
\end{lemma}
\begin{proof}
Let A, B, $C$,and $D$ be the unit squares around $v$ in circular order.
By \cref{le:43}, at most three faces of $\ensuremath{\mathcal{C}}\xspace$ are covered by A, B, $C$,and D. Hence, at least two unit squares map to the same face of $\ensuremath{\mathcal{C}}\xspace$. These can either be edge-adjacent of diagonal. For the first case, let this without loss of generality be $A$ and B. Hence, the crease between them must be folded by $\pm 180^\circ$. Then $C$ and $D$ must also map to the same face of $\ensuremath{\mathcal{C}}\xspace$ to maintain the paper connected. The crease between $C$ and $D$ must also be folded by $\pm 180^\circ$. In the latter case, without loss of generality $A$ and $C$ map to the same face of $\ensuremath{\mathcal{C}}\xspace$. As they are both incident to $v$, only two options of folding those two unit squares on top of each other exist. Either the edge between $A$ and $B$ gets folded on top of the edge between $B$ and $C$,this leaves a diagonal fold on B, a contradiction, or the edge between $A$ and $D$ gets folded on top of the edge between $B$ and $C$,which results in $D$ being mapped to $C$,and those are two adjacent unit squares, in which case we already argued that two collinear incident creases must be folded by $\pm 180^\circ$.
\iffalse
By \cref{le:43}, without loss of generality, $A$ and $B$ maps to the same face of $\ensuremath{\mathcal{C}}\xspace$ and, hence, the crease between them must be folded by $\pm 180^\circ$.
Then $C$ and $D$ must also map to the same face of $\ensuremath{\mathcal{C}}\xspace$ to maintain the paper connected. The crease between $C$ and $D$ must also be folded by $\pm 180^\circ$.
\textcolor{blue}{Alternatively, without loss of generality, $A$ and $C$ map to the same face. ...two collinear incident creases also follow.}\fi
\end{proof}
\fi

In the following, we show that a slit hole of size~1 does not help in folding a rectangular polyomino into $\ensuremath{\mathcal{C}}\xspace$.
We start with a lemma:

\begin{lemma}\label{lem:slitNEW1}
In every folding of a polyomino $P$ with a slit hole of size 1, the crease pattern behaves as if the slit hole was nonexistent.
\end{lemma}
\begin{proof}
To prove the lemma we examine the local neighborhood of the slit and analyze the possible folding patterns we can obtain between adjacent faces. More specifically, we consider the six unit squares $A$, $B$, $C$, $D$, $E$ and $F$ of $P$ that are incident to the slit hole of size 1 as illustrated in \cref{fig:slit1}.
We distinguish two cases: The crease between $A$ and $F$ is folded by $\pm 90^\circ$ or $\pm 180^\circ$.

\begin{figure}[ht]
	\centering

	\includegraphics[page=2]{one-edge-fold-color.pdf}
	\caption{A polyomino with a slit hole of size one.}
	\label{fig:slit1}
\end{figure}

If the $AF$ crease is folded by $\pm 90^\circ$, we must further distinguish if the $EF$ crease is folded by $\pm 90^\circ$ or by $\pm 180^\circ$.
If the $EF$ crease is folded by $\pm 180^\circ$, then the slit edge is mapped to the edge between $AF$, fixing that $B$ maps to $A$. Hence, this corresponds to a crease folded by $\pm 90^\circ$ of the slit edge.

By symmetry, we may assume that both the $AB$ crease and the $EF$ crease are folded by $\pm 180^\circ$. This implies that $B$ and $E$ cover the same face in such a way that the top edge of $B$ is mapped to the left edge of $E$. However, then the bottom left corner of $D$ is also mapped to the top left corner of $E$. A contradiction. Consequently, this is impossible.

If the $AF$ crease is folded by $\pm 180^\circ$, then $A$ and $F$ cover the same face and, in particular, their left edges are mapped to the same edge such that the top edge of $F$ and the bottom edge of $A$ coincide. This implies that the left edge of $E$ and the left edge of $B$ also coincide such that the top edge of E and the bottom edge of $B$ coincide. This corresponds to crease folded by $\pm 180^\circ$ of the slit edge.

This shows that the slit edge is a crease folded by $\pm 90^\circ$ or by $\pm 180^\circ$. Hence, the crease pattern behaves as if the slit hole was nonexistent.
\end{proof}

\begin{theorem}\label{thm:slit1}
If $P$ is a rectangle with exactly one slit of size~1, then $P$ does not fold into \ensuremath{\mathcal{C}}\xspace.
\end{theorem}
\begin{proof}
By \cref{lem:slitNEW1}, the crease pattern behaves as if the slit was nonexistent, i.e., as if $P$ was a rectangle. By \cref{cor:rows}, all horizontal or vertical creases are folded by $\pm 180^\circ$, reducing $P$ to a rectangle of height or width 1, which does not fold into \ensuremath{\mathcal{C}}\xspace.
\end{proof}

Furthermore, we conjecture that the slit of size~1 never is the deciding factor for foldability.
\begin{conjecture}\label{conj:slit}
Let polyomino $P'$ be obtained from a polyomino $P$ by adding a slit~$s$ of size 1. If $P'$ folds into $\ensuremath{\mathcal{C}}\xspace$, then $P$ folds into $\ensuremath{\mathcal{C}}\xspace$ as well.
\end{conjecture}

We note that this is not true for arbitrarily large \emph{polycubes} (connected three-dimensional polyhedron that are formed by a union of face-adjacent unit cubes on the cube lattice):
\begin{lemma}\label{le:slit-polycube}
There exists a polyomino $P$ with a slit $s$ of size 1 and a polycube $Q$, such that $P$ can be folded into $Q$, but the polyomino $P'$ without $s$ cannot be folded into $Q$. That is, for larger polycubes, a slit of size 1 can be the deciding factor for foldability.
\end{lemma}
\begin{proof}

\begin{figure}[htb]
\centering
\includegraphics[scale=1]{polycube-slit.pdf}
\caption{Left: Polyomino $P$ (with slit $s$), right: polycube $Q$. Colors of $P$ coincide with the parts of the same color in $Q$, the light gray unit squares are not mapped to outer faces of $Q$.}
\label{fig:polycube-slit}
\end{figure}
We consider the polyomino $P$ and the polycube $Q$ from Figure~\ref{fig:polycube-slit}. $P$ has 40 unit squares, and $Q$ has 38 faces. Therefore, 38 out of the 40 unit squares will be the faces of $Q$ when folded, hence, at most two unit squares of $P$ may be folded on top of other unit squares to obtain $Q$.

$P$ contains two rectangular $k\times n$-subpolyominoes with $k,n\geq 2$ that do not contain $s$: a rectangular $3\times 4$-subpolyomino in the lower left (green) and a rectangular $5\times 2$-subpolyomino (pink). Similar to the proof for~\cref{cor:rows}, we know that there do not exist two collinear creases in these rectangular subpolyominoes, one of which is folded by $\pm 90^{\circ}$ and the other by $\pm 180^{\circ}$. Hence, if we fold a crease in those rectangular subpolyominoes by some angle, all other collinear creases (in the same row or column) are also folded by the same angle.
Observe that the surface of $Q$ does not contain any $2\times2$ flat squares. Hence, for every grid point contained in a rectangle at least the vertical or horizontal creases are folded by some angle.

Assume that we fold the vertical crease of length 5 in the pink rectangular $5\times 2$-subpolyomino by $\pm 180^{\circ}$. Then 5 unit squares would be folded on top of other unit squares in $Q$, again a contradiction. If, on the other hand, we fold a horizontal crease of length 2 in the rectangular $5\times 2$-subpolyomino by $\pm 180^{\circ}$, then all other unit squares need to appear as a face of $Q$. Similarly, if all of these creases would be $\pm 90^{\circ}$, again 2 unit squares would be folded on top of other unit squares. However, there are unit squares attached at the bottom and the top of the $5\times 2$-subpolyomino, which in that case cannot cover separate faces (5 unit squares from the pink rectangular $5\times 2$-subpolyomino plus these two adjacent unit squares can cover at most 4 faces of $Q$), which would yield further overlap, a contradiction.
Hence, the crease of length 5 must be $\pm 90^{\circ}$, thus, this will constitute part of the row of eight unit cubes in $Q$.

Analogously, assume that we fold any of the horizontal or vertical creases of the green rectangular $3\times 4$-subpolyomino by $\pm 180^{\circ}$. Hence, 3 or 4 of the unit squares would be folded on top of other unit squares in $Q$, a contradiction. Consequently, all existing folds must be $\pm 90^{\circ}$.

Assume that we fold all vertical creases in the green rectangular $3\times 4$ subpolyomino $\pm 90^{\circ}$. This would yield 3 faces of a tube-like $4\times 1 \times 1$-polycube for which the $1\times 1$ top and bottom faces and one of the $4\times 1$ faces are missing. However, together with (part of) the pink $5\times 1\times1$ polycube, this would yield a row of nine unit cubes, which cannot be combined for $Q$. Hence, all horizontal creases must be $\pm 90^{\circ}$.

Consequently, the green rectangular $3\times 4$-subpolyomino can be folded in a tube-like $3\times 1 \times 1$-polycube for which the $1\times 1$ top and bottom faces are missing. If we did not use the dark-green leftmost bottom unit square to cover one of these faces, this closing face would need to be a unit square of the remaining $27 (=40-(3\times 4+1))$ unit squares, however, then three unit squares must be folded on top of unit squares of the folded $3\times 4$-subpolyomino, a contradiction to the number of faces of $Q$ again.

Hence the $3\times 1 \times 1$ polycube with one $1\times 1$-face missing (obtained from the green rectangular $3\times 4$-subpolyomino and the adjacent dark-green unit square), must cover the left $3\times 1 \times 1$-subpolycube of $Q$.

Then, the only part of $P$ that can be folded into the blue attached unit cube is the blue T-shape.

The vertical unit-square row on top and below that T has length 5, hence, it must cover a part of the right $5\times 1\times 1$ of $Q$ (again, otherwise too many unit squares would be folded on top of each other).

We obtain this only when using the slit of size 1 (we push the green $3\times 4$-subpolyomino and the adjacent dark-green unit square through the slit and unfold then again).
\end{proof}

\subsection{An Algorithm to Check a Necessary Local Condition for Foldability}\label{subsec:alg}
Consider the following local condition: let $s$ be a unit square in a
polyomino $P$ such that the mapping between grid points of $s$ and
corners of a face of $\ensuremath{\mathcal{C}}\xspace$ has been fixed. Then, for every unit
square $s'$ adjacent to $s$, there are two possibilities on how to map its four
grid points onto $\ensuremath{\mathcal{C}}\xspace$: the two grid points shared by $s$ and $s'$ must
be mapped consistently and for the other two grid points there are two
options depending on whether $s'$ is folded by $\pm 90^{\circ}$
to an adjacent face of $\ensuremath{\mathcal{C}}\xspace$, or whether it is folded by $\pm 180^{\circ}$
to the same face of $\ensuremath{\mathcal{C}}\xspace$.

The algorithm below checks whether there exists a mapping between
all grid points of unit squares of $P$ to corners of $\ensuremath{\mathcal{C}}\xspace$ such that the
above condition holds for every pair of adjacent polyomino squares
of $P$.
\begin{compactenum}
\item Run a breadth-first search on the polyomino unit squares, starting with
the leftmost unit square in the top row of $P$ and continue via adjacent
unit squares. This produces a numbering of polyomino unit squares in which each
but the first unit square is adjacent to at least one unit square with smaller
number. See Figure~\ref{fig:algo-step1} for an example.
\item Map grid points of the first unit square to the bottom face of $\ensuremath{\mathcal{C}}\xspace$. Extend
the mapping one unit square at a time according to the numbering, respecting
the local condition (that is, in up to two ways). Track all such partial
mappings.
\end{compactenum}

\begin{figure}[htb]
\centering
\includegraphics[scale=1]{cube-algo.pdf}
\caption{Example of Step 1 of the algorithm. It shows the numbering of polyomino squares produced by the breadth-first search.}
\label{fig:algo-step1}
\end{figure}

The algorithm is exponential, because unless inconsistencies are produced, the number of possible partial mappings doubles with every polyomino unit
square. Nevertheless, it can be used to show non-foldability for small
polyominoes: if no consistent mapping exists for a polyomino, then
the polyomino cannot be folded onto $\ensuremath{\mathcal{C}}\xspace$. On the other hand, any consistent
grid-point mapping covering all faces of $\ensuremath{\mathcal{C}}\xspace$ obtained by the algorithm
that we tried could in practice be turned into a folding. However,
we have not been able to prove that this is always the case.

The algorithm above was used to prove that polyominoes in \cref{fig:simple_slits_cases}
do not fold, as well as it aided us to find the foldings of polyominoes in  \cref{fig:twoholes1}. An implementation of the algorithm is available at the following site \url{http://github.com/zuzana-masarova/cube-folding}.

\begin{figure}[htb]
\centering
\includegraphics{simple_holes_dont_fold-color.pdf}
\caption{These polyominoes with single L-, U- and straight size-2 slits do not
fold into a cube. }
\label{fig:simple_slits_cases}
\end{figure}

\section{Conclusion and Open Problems}\label{sec:concl}

We showed that, if a polyomino $P$ does contain a non-basic~hole, then $P$ folds into $\ensuremath{\mathcal{C}}\xspace$. Moreover, we showed that a unit-square hole, size-2 slits (straight or L), and a size-3 U-slit sometimes allow for foldability.

Based on the presented results, we created a font of 26 polyominoes with slits that look like each letter of the alphabet,
and each fold into $\ensuremath{\mathcal{C}}\xspace$.
See \cref{fig:font}, and \url{http://erikdemaine.org/fonts/cubefolding/} for a web app.

\begin{figure}[htbp]
  \centering
  \def0.22{0.22}
  \parskip=3pt

  \includegraphics[scale=0.22]{font_a}
  \includegraphics[scale=0.22]{font_b}
  \includegraphics[scale=0.22]{font_c}
  \includegraphics[scale=0.22]{font_d}
  \includegraphics[scale=0.22]{font_e}
  \includegraphics[scale=0.22]{font_f}

  \includegraphics[scale=0.22]{font_g}
  \includegraphics[scale=0.22]{font_h}
  \includegraphics[scale=0.22]{font_i}
  \includegraphics[scale=0.22]{font_j}
  \includegraphics[scale=0.22]{font_k}
  \includegraphics[scale=0.22]{font_l}

  \includegraphics[scale=0.22]{font_m}
  \includegraphics[scale=0.22]{font_n}
  \includegraphics[scale=0.22]{font_o}
  \includegraphics[scale=0.22]{font_p}
  \includegraphics[scale=0.22]{font_q}

  \includegraphics[scale=0.22]{font_r}
  \includegraphics[scale=0.22]{font_s}
  \includegraphics[scale=0.22]{font_t}
  \includegraphics[scale=0.22]{font_u}
  \includegraphics[scale=0.22]{font_v}

  \includegraphics[scale=0.22]{font_w}
  \includegraphics[scale=0.22]{font_x}
  \includegraphics[scale=0.22]{font_y}
  \includegraphics[scale=0.22]{font_z}

  \caption{Cube-folding font: the slits representing each letter enable each rectangular puzzle to fold into a cube.}
  \label{fig:font}
\end{figure}

\medskip
We conclude with a list of interesting open problems:
\begin{compactitem}
\item Does a consistent grid-point mapping output by the algorithm in \cref{subsec:alg} imply that the polyomino is foldable? If so, is the folding uniquely determined?
\item Is any rectangular polyomino with one L-slit, U-slit, or straight slit of size~2  foldable? Currently, we only know that the small polyominoes in  \cref{fig:simple_slits_cases} do not fold.
\item We considered the existence of only a folded state in the shape of~$\ensuremath{\mathcal{C}}\xspace$,
  but what if we require a continuous folding motion from the unfolded
  polyomino into~$\ensuremath{\mathcal{C}}\xspace$?  These two models are known to be equivalent for
  polygons without holes \cite{PaperReachability_CCCG2001,GFALOP}, but
  equivalence remains an open problem for polygons with holes as in our case.
\end{compactitem}

\section*{Acknowledgments}
This research was performed in part at the 33rd Bellairs Winter Workshop on Computational Geometry. We thank all other participants for a fruitful atmosphere.
H. Akitaya was supported by NSF CCF-1422311 \& 1423615. Z. Mas\'arov\'a was partially funded by Wittgenstein Prize, Austrian
Science Fund (FWF), grant no. Z 342-N31.

\newpage
\bibliographystyle{elsarticle-num}

redpajama_set_name: RedPajamaArXiv


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The real benefit of alternative cannulation technique for repair of acute type A aortic dissection remains controversial.
We evaluated our results of ascending aortic cannulation compared with femoral cannulation in acute type A aortic dissection.
From January 2000 to December 2014, 121 patients were operated on for acute type A aortic dissection. Cannulation was accomplished in 79 patients through the ascending aorta and 42 patients through the femoral artery.
Patient characteristics such as preoperative complications and shock status were almost similar between the groups. Although intraoperative parameters such as total cardiopulmonary bypass time, hypothermic circulatory arrest time, minimum bladder temperature and percentage of total arch replacement were also similar between the groups, ascending aortic cannulation significantly reduced arterial pressure on cardiopulmonary bypass circuit (156+39 vs 169+40 mmHg, p < 0.05), core-cooling time (33+10 vs 39+7 min., p < 0.01) and operation time (325+72 vs 357+96 min., p < 0.05). There were no significant differences in postoperative 30-day mortality and in-hospital mortality between the groups. However, incidence of temporary neurological dysfunction (4.0 vs 21.4 %, p < 0.05) and respiratory failure (4.0 vs 21.4 %, p < 0.05) as postoperative complications were significantly lower in the aortic cannulation group, although the incidence of stroke and renal failure did disclose any differences between the groups. Moreover, ascending aortic cannulation significantly reduced the hospital stay (26+10 vs 41+32 days, p < 0.01).
The ascending aortic cannulation for repair of acute type A aortic dissection can be advantageous to rapid cooling after the establishment of cardiopulmonary bypass. It might offer benefit on the incidence of postoperative complications and the length of hospitalization in patients with of acute type A aortic dissection.


text:
Weekly Links of Interest
Life In Taiwan writes about a rainstorm and flying termites.
Taiwanese Secrets Travel Guide explains to manners-impaired westerners how to get on well with the Chinese people of Taiwan.
Alton's Images has photographs that range from the bizarre to the beautiful.
Andre In Taiwan takes his daughter to the Taipei Zoo.
Taiwan News reports that a group of retired Taiwan military officers are on a golf tour with their former adversaries in The People's Republic. As Bob Dylan wrote, "The times, they are a'changin'." Yeah, well...maybe.
Neil Wade's Photography Blog tours Yangming Shan and Taiwanese volcanoes. Awesome!
Labels: Links
Taipei MAAG Officers Open Mess
I've been corresponding with Bob Loiselle, the son of Air Force Chief Warrant Officer V.W.Loiselle, who was the club manager of the MAAG Officers Club from 1959 until 1961.
He sent along a couple of items, including this invitation to the inaugural celebration of the new Civil Air Transport aircraft, "The Mandarin Jet," which was a Convair 880-M.
The second item was a photo of a pool patch, which was sewn onto swim trunks. Bob thinks this one was his sister's.
As always, you can click on either image to see a larger version.
Labels: MAAG, Memorabilia, Recreation
I received some Corona satellite images of Taipei via TaiwanAirPower.org. Corona was a US military reconnaissance satellite system operated by the CIA Directorate of Science and Technology with substantial assistance from the US Air Force, used for photographic surveillance of the Soviet Union, China and other areas from June 1959 until May 1972. The images were declassified during the Clinton administration.
The first image here is a diagram of buildings in the HSA East Compound as they appeared around the 1970s. If you click on this same image down the right-hand column of this page, you'll be able to view a much larger version in which you can read the purpose of each building. USTDC is at the top of the page.
The next image is from 1966 and shows the same general area. You can just make out some of the buildings, as well as the river at the top of the page. The three "H" shaped buildings to the right should help you get oriented.
The next image is from 1969, three years after the one above was taken. The image is noticeably clearer, showing the technical improvement in spy satellites in that relatively brief period of time. The large white building toward the north and slightly to the east was the TACF and the main USTDC building was to its left.
Finally, here is a modern satellite photo of the area. The structure toward the upper left -- the one that looks like it was constructed with Lego blocks -- is the Museum of Modern Art. To the east of that was the approximate location of the US Taiwan Defense Command.
Labels: Compound, History, TDC Building
More TDC Memorabilia
Vic Gerlach (USTDC, '74-'75) sent in these photos today. Here's his description:
The photo you posted of the brass ashtray from the 63 Club reminded me that I have a brass ashtray also. Mine was presented during my departure ceremony/party and is engraved with signatures of some of the personnel assigned there at the time. Some of the legible names are VAdm Snyder, BGen Williams, Col Peters, Lt Col Blanton, Major Delano, CPO M Johnson and others.
We were also presented with a USTDC necklace. It's the USTDC metal logo badge mounted on a piece of green jade. I am also enclosing a photo of the necklace.
I try to check out Taiwan-related blogs from time to time, and I thought that each week I'd post a few links to some of the ones I thought might be of interest to the folks who drop by here.
Here are three to get things rolling and I'll try to post more around the end of each week.
TAIWANAIRBLOG has a nice set of aerial photos of the former Shu Lin Kou Air Station.
A Hungry Girl's Guide To Taipei highly recommends DING WANG MALA GUO for spicy mala hotpot and other delights.
Life In Taiwan has a very nice piece about overseas (including American) teachers finding respect and opportunities teaching in Taiwan.
The Birth of USTDC -- November 1, 1955
Bill Thayer sent me this Associated Press article from October 23, 1955. It shows that Vice Admiral Alfred M. Pride, who was Commander of the US 7th Fleet, announced that the Formosa Liaison Center was being changed to the United States Defense Command on November 1st of that year. VADM Pride was Commander of the US Seventh Fleet from December 1, 1953 until December 19, 1955. This link goes to the history of the Seventh Fleet and it shows VADM Ingersoll holding that position after VADM Pride.
Also, note in the article that when they created USTDC, they renamed the Military Assistance Advisory Group from MAAG Formosa to MAAG Taiwan and the name of the radio station from Voice of MAAG Formosa to Voice of MAAG Taiwan.
From the second article displayed here -- also from Bill -- it appears that the Commander of the US Seventh Fleet was concurrently Commander of USTDC because those two commands were not separated until early in 1957. That may mean that VADM Pride was technically the first COMUSTDC, at least for about 19 days until the arrival of VADM Ingersoll, though I don't know if VADM Pride was actually designated as such.
Click on either article to view the full text of each.
Labels: Characters, History, MAAG
Club 63 in 65 and 66
Len dropped by here at the TDC quarterdeck yesterday and left this note about the Club 63:
From some of the comments on your page, sure am glad I was there before the Navy pulled the slots (1965 – 66). They provided the revenue that paid for a lot of 10¢ drinks and great acts plus monthly non-family entertainment!
Here's a picture of a brass ashtray from the day! I lived in the Signal Compound right behind TDC – the good old days waking up at 0200 covered by termites twice a year when they swarmed!
4799th Signal Communications Agency, Taiwan
Thanks, Len. I've seen photos of these ashtrays on eBay before and have always wondered if they were given to departing club members or if they were just "liberated" from time to time. Of course back in those days ashtrays were everywhere. When I was a kid, some would make them in school as a gift for their dads. That's hard to believe today.
If my memory is accurate, it seems to me that when I was at TDC during 1973-74, the Club 63 had a MAAG logo on the outside, but I'm not sure who ran it (Army? Air Force?). All I know is that the Navy took over its operation during my stay, removed the MAAG logo and changed the name to the China Seas Club. Among other changes, they installed slot machines. I think both the Army and Air Force banned slots from their clubs sometime in the late 1960s because a club manager or two started raking off profits for personal gain. The Navy already had slot machines in the Headquarters Support Activity (HSA) west compound when I arrived at TDC.
I remember plenty of cockroaches and small lizards (geckos?) but this is the first I've heard about swarming termites.
Labels: China Seas, Housing, Recreation
Request for Information -- Life in Taiwan During The 1950s
I received a note this weekend from Dr. Stephen Craft, a professor at Embry-Riddle Aeronautical University regarding a book project he's been working on. He explains it here:
I am a professor at Embry-Riddle Aeronautical University where I teach military and aviation history. I lived in Taiwan for many years in the 1980s and 1990s, and go back there every two years. Over the last several years, I have published a book on V.K. Wellington Koo and another on Embry-Riddle's role during WWII. Over the last ten years or so, I have been working on a third book project that deals with MAAG-Taiwan and the Reynolds Riot of 1957. I am really interested in interviewing Americans who lived in Taiwan in the 1950s to get a feel for expatriate life at the height of the Cold War. I can always be contacted at crafts@erau.edu.
Steve Craft
Stephen G. Craft, Ph.D.
HU/SS Department
If you were in Taiwan during the 1950s, and would be interested in contributing to this work, please drop Steve an e-mail.
Labels: Duty, History
TDC Chiefs of Staff -- Update
During the past three years I've tried to assemble an accurate list of all the Air Force brigadier generals who served as USTDC Chief of Staff. Some of them were fairly easy because their official Air Force biographies showed when they assumed that position. But a few of them were just my best guess, usually because their biographies mentioned only that they were assigned there, but didn't specify when. That was the case with Brigadier General William G. Lee Jr., and I listed him as being assigned there in 1957. It now appears that I guessed wrong.
Today I found a biography for Major General Harold Huntley Bassett (photo at left), which shows he was assigned as USTDC Chief of Staff in 1957, so I replaced General Lee with General Bassett. Bill Thayer recently provided confirmation that General Lee departed TDC during 1960.
My updated list of USTDC Commanders and Chiefs of Staff can be found here. I'm sure the list of admirals is incomplete but I've never been able to locate an on-line archive of flag officer biographies like the Air Force has. If you spot any other errors or omissions, please let me know.
Labels: Characters, History
US Military Housing In Taipei
I've written several times previously about the American military housing areas in the Taipei area. Today I received an interesting link from Klaus Bardenhagen, a German reporter who lives and works in Taipei. You can view his website (with a link to his blog) at http://www.taiwanreporter.com.
The link he provided is to an article in Taipei City's tourism magazine titled, "The US Military's Cold War Housing in Shanzihou & Tianmu." It goes into some detail about the architecture of the old structures and how they were perceived by local citizens.
Labels: Housing
Taiwan Furniture
Many of us purchased locally made furniture when we were in Taiwan. A couple of years ago, I wrote about the coffee table and end table set I bought during 1974. I can't recall the name of the shop, but I know it was within easy walking distance of the compound. As I wrote earlier, I started looking at these shortly after my arrival in 1973 and finally bought this set a few weeks before my departure.
When we decided to downsize about four years ago, one of our daughters asked for the set, which she and her husband hauled up to their Minnesota home. We drove up to the Frozen North a few days ago for a visit and I took a couple of pictures to post here.
This is an example of what was called an "APO set" because the coffee table could be disassembled into three sections and mailed through the post office back to the States. The set is made of teak and camphor woods with marble set into the tops and solid brass pulls on all doors. It is quite heavy and I tried to avoid moving it around the house as much as possible. After several household moves and exposure to numerous kids and grandkids, it has a few small nicks and scratches, but it still looks almost as good as the day it was delivered.
I don't have any idea what a similar set would cost today -- if you could find one -- but I think I paid about $120 for it in 1974. That was still a princely sum in those days, but very reasonable for something of this quality.
April 30, 1979 -- Zàijiàn
There's not much new in this United Press International article from April 30, 1979, but I thought I'd post it as a matter of general interest.
Admiral and Mrs. James Linder had already departed for Hawaii when this article was written. It says that there were a few military officers who stayed until the end of April, the mandatory departure date ordered by President James Earl Carter.
I have often thought about what it was like for the final few American military people to depart Taiwan. Speaking at a conference twenty years later, Admiral Linder said that the people of Taiwan felt that we were going away and they were in deep trouble. He went on to say, "We sort of felt the same way."
Click on the article for a larger view.
Labels: Withdrawal
TDC Nooners League
I've written once or twice about the TDC Nooners League. We bowled one day a week (Wednesdays, I think) as part of our physical training program (honest!). During my time it was common for several of us to stop off at the China Seas (Club 63) after bowling. Some at TDC who weren't bowlers just avoided the hassle of organized exercise altogether and went straight to the club for their PT.
My team was called Granny's Guys, named after that old cartoon character in Playboy, and our shirts had her image embroidered on the back -- very tastefully done, of course. Office mate John Cranford (Army) was on my team, along with DOD civilian Larry Marcum. I believe there was an Air Force major whose name escapes me, and probably one or two other guys.
I just received these two photos from Vic Gerlach, who arrived at TDC around the time I left. He was also in J-1 (Personnel) and became a Nooners League member. His team was called the Combines and the shirt was made by Mr. Loo the tailor, as I think most of them were at the time It's exactly like mine...except for Granny of course.
Request for Information -- Life in Taiwan During T...

redpajama_set_name: RedPajamaCommonCrawl


This weekend is so busy, you'll barely have time to stop eating.
The 3rd Annual CLASH (Cape Land and Sea Harvest) kicks off tomorrow. For the first time, CLASH will be centered behind the Cape Cod Maritime Museum on South Street in Hyannis. Things get started from 6 to 8 p.m. with a tasting party hosted by local chefs, and complete with organic wines, local craft beer and jazz. Opening gala tickets are $35. CLASH wraps up Sunday with the Mid-Cape Season's Finale Farmers Market, and the Cape Cod Oyster Festival on the waterfront at Aselton Park in Hyannis. Oyster tickets are $45 to benefit the Cape and Islands United Way. In between are classes, demonstrations and dinners. Info: www.capecodclash.org.
The Feasts of Falmouth and Beyond, to raise money for VIPS, or Volunteers in Public Schools, is from 11 a.m. to 2 p.m. Saturday at the Marina Park Bandshell at Falmouth Harbor. Last year, 25 Falmouth restaurants and caterers served up appetizer-size portions of their culinary specialties to 1,500 people. Each $20 ticket is good for 10 "tastes." Info: www.falmouthvips.org.
In the mood for German food? Check out Oktoberfest from 3 to 9 p.m. at the Orleans-Eastham Elks Lodge on McKoy Road in North Eastham. Music is by Alpenblumen Musikanten. Admission is $3; dinner is $10 or buy a la carte.
If you're feeling hungry and lucky Thursday, the Patio American Grill and Blue Bar in Provincetown has a new fall menu and will flip you for the cost of dinner. That's right, they're back doing Toss-up Thursdays for the shoulder season.
Tickets are available for the Oct. 3 "7th Annual Taste & See Cape Cod," a gala to benefit three organizations that serve the Cape's homeless: the NOAH Center, Champ Homes and Pilot House. Details and reservations: Nancy Ronayne, 508-775-5744, ext 101, or Rick Brigham, 508-364-7590; rbrigham@haconcapecod.org.
Also, on Oct. 5, a $10 spaghetti dinner at Jack's Outback II in Yarmouthport, will benefit the Education Foundation for Dennis and Yarmouth. Tickets: Joan Shostak at 508-362-2335.


Don't just tell your MP it's important to take action on climate change, let's show them why it matters to us.
So, you've mastered the central heating timer and stopped the sofa from blocking the radiator. Still wearing ski gloves in the living room? Let's take this to the next level!


A global manufacturer of steel and specialty alloys has purchased a SECO/WARWICK nitriding system with vacuum purge and two tempering units to support their expansion in Mexico.
The tempering furnaces will be paired with two existing systems, while the large nitriding unit will stand alone, equipped with ZeroFlow® gas nitriding technology.
The two SECO/WARWICK standard box (chamber) tempering furnaces feature a rugged, heavy duty industrial design for uniform heat treating and tempering a wide variety of parts. Also included in this package were comprehensive Professional Technical Services including pairing the new equipment with existing systems, start up, and commissioning of the completed heat treating lines.
"Serving our global partners is the primary reason that SECO/WARWICK has introduced both the ZeroFlow technology and the new Professional Technical Service program. This is a good example of our Service program in action, supporting our customers by pairing new equipment with their existing manufacturing cell enabling them to get the most value from their capital purchase," commented Jonathan Markley, Managing Director, SECO/WARWICK, USA.


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Film Review: Searching (USA, 2018) surprises with inventive storytelling, timely themes and strong performances
Harris Dang
It is quite amusing to think that we have many films released over the years, regardless of genre, that span across many imaginative worlds, planets, fantasy settings and so on. With the vast amount of superhero films and blockbusters, it's hard not to see why. But, the world that has not been mined more than enough, despite the great films that have been made from it, is the World Wide Web. Such ingenuity can be extracted from the setting that we can have great films like The Social Network, Catfish, Unfriended, and The Den. It is mind-boggling to think that this isn't done more often.
But what's even more mind-boggling is that new upcoming thriller, Searching, is the first Hollywood mainstream thriller to feature an Asian-American in the lead role. To me, it's puzzling enough that we rarely have focus on Asian-American families on-screen, but the fact above… wow! The film, a new technological cyber-thriller from feature debut director Aneesh Chaganty, has been gathering some critical buzz since its premiere at the Sundance Film Festival. With a host of established actors (John Cho, Debra Messing) and rising talent (Michelle La), the question is, will Searching hit that wow factor?
After David Kim's (John Cho) 16-year-old daughter, Margot (Michelle La) goes missing, a local investigation is opened, with Detective Rosemary Vick (Debra Messing) assigned to the case. But, 37 hours later and without a single lead, David decides to search the one place no one has looked yet. The place where all secrets are kept these days: his daughter's laptop. With her social life on the internet becomes an illuminating rabbit hole that goes deeper and deeper, and with a limited amount of time, David must trace his daughter's digital footprints before she disappears forever.
One of the major positives that makes Searching work is how writer and director Aneesh Chaganty, and co-writer Sev Ohanian really attacked the material with a high degree of verisimilitude; using the exact sounds and layouts of the programs like Skype, YouTube and Mac programs. It really added to the gripping storytelling. Even the video freezes and sound lags are used to great effect, adding to the atmosphere and immersion.
Also adding to the storytelling is the depiction of the characters' use of technology. For example, a character would start typing a message and then erase it and change it completely. It's a clever bit of storytelling, and adds much-needed character development, whilst also a realistic look at people's use of technology. There is even some very ingenious foreshadowing for eagle-eyed viewers, which all adds to the film's replay value. Another example, is how these characters think they are invincible behind a shield of anonymity, thinking they can get away with their bad deeds. But when their secrets are revealed, we know more about the characters. That progression between anonymity and clarity is scary, since again, it stems from reality. The film even drives the idea of internet addiction into the tale, not only selling the premise quite well, but also alleviating supposed plot holes.
Next comes the great performances. John Cho, who has always been an understated actor in indie dramas like Columbus, Gemini and even the Harold and Kumar films. But in the case of Searching, he has his first leading role in a Hollywood thriller, and he does a fantastic job. Cho makes the progression from grieving single father, to obsessive investigator, to a man driven with simmering rage, look smooth and effortless. In one particular scene, his character goes from being aggressive to conflicted and eventually collapsing to the fetal position, and it is a compelling gut-punch to witness. Cho really nails it.
The other two leads, Debra Messing and newcomer Michelle La also give great performances that are nuanced and convincing, and reveal the hidden depths of their characters. La, in particular, has the harder task, mainly due to her limited screen-time and Chaganty's attempts to skew the audience's perception of her character. But, the moments when she's on-screen (particularly during the live-broadcast moments) she positively stands out.
As for the film's glitches flaws, Chaganty does veer towards sentimentality at times. This particularly becomes prevalent when he relies on Torin Borrowdale's musical score, which is quite jarring considering that the film takes place on technological screens. Some restraint would have also been beneficial as to how much Chaganty and cinematographer Juan Sebastian Baron zoom in & out of the screens to telegraph the drama or hint towards revelations. Although this is perhaps understandable given that it is likely done to cater to those in the audience that are not computer-literate. There are also the plot contrivances that pile up during the third act when all the revelations and character reveals come into place, which do detract from the plausibility of the situation. I mean, how did that character accomplish all that, in that short amount of time?
However, overall, with clever and immersive storytelling (thanks to its creative use of the technology) great performances, rich characterizations, surprising twists and ample amounts of food for thought, director Aneesh Chaganty has made a great feature debut with Searching. I can't wait to see what he'll come up with next.
FOUR STARS (OUT OF FIVE)
Searching hits cinemas this Thursday 13th September
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Tags: Aneesh Chaganty, Debra Messing, Four Stars, John Cho, Joseph Lee, Michelle La, Sara Sohn


text:
Rising Sun Country Park
An Interactive Guide To Birdwatching Sites in the North East of England
A very popular and well visited country park, which is rich in wildlife. There is an extensive network of paths; great facilities and free parking. Highlights include the Swallow Pond nature reserve and the Rising Sun CP visitor centre, which has toilets and a great cafe. From the main path, a public hide with wheelchair access overlooks the pond, whilst to the west there is a screen hide. Towering to the side of Swallow Pond, there is a pit heap, which is a reminder of the sites mining past. Skylark's can be heard in the sky above, and in some year's, Ring Ouzel can be found on the pit heap as they stop off for a break, on their way inland during migration.
Swallow Pond was great for waders in the 80's, when water levels were much lower in the autumn, attracting rarities such as Black winged Stilt, Pectoral Sandpiper and White-rumped Sandpiper. Other waders such as Common, Curlew, Green and Wood Sandpipers, Spotted Redshank, Little Stint and Ruff were regular visitors. Nowadays water levels are much higher, as nearby trees and vegetation have increased their range and such waders are not now common. The site remains popular with a wide range of wildfowl all year round and has over the past couple of years been a popular stopping off point for Iceland Gulls in the winter. Great Crested Grebes which were once regulars every year, are no longer present very much. Little Grebes remain faithful however as do Pochards. Swallow Pond has always been a favourite for Shoveler, but in recent years numbers have increased with peaks of 56 recorded in 2016.
A Shoveler
If your lucky you might see or hear a nearby Water Rail, a local Bullfinch, Willow Tit or even a visiting Crossbill. The site boasts some well established coniferous plantations which are great for Goldcrests and Siskin.
During the spring and summer months Common Terns can be heard, announcing their arrival and the calls of Willow Warblers and Chiffchaff are everywhere. As you explore the nearby woodland, Blackcaps can also be found. The nearby farmland and fields are popular with Fieldfares and Redwings in the winter and large groups of Woodpigeons all year round. Other key species to watch out for include Stock Doves, which can usually be found on the fields to the back of the pond, close to the visitor centre and Whitethroat and Lesser Whitethroat, which favour the trees close to the farm and Pit heap during the spring/summer months.
The Rising Sun Country Park has been increasing in popularity and the variety of birds recorded has been increasing again. Jay's, which were once a rare visitor are now resident and can be easily found. Little Egrets have also started to visit Swallow Pond over the last two years. It would be great to see these as regulars alongside the local Herons.
A Red Deer
In the winter of 2013-14 birders gathered in the pubic hide, hoping to catch a glimpse of a Bittern, or some noisy Water Rails. Both usually secretive but showing very well for many on occasions for over a month.
Another highlight for many is the presence of a Red Deer, which can usually be found close to the visitors centre. If your lucky you might also see a Roe Deer or a Fox. Long gone are the Water Voles however, which were once a common sight close to what now is the 'screen hide' to the West, which was originally the site of a NWT hide in the 80's and 90's. Weasels are no longer as common either and Hares have pretty much vanished from the landscape. The resurgence of Otters in the county has not brought any Otters to Swallow yet.
To the right of the visitor centre and the back of Swallow Pond, another pond can be enjoyed in modern times. Duke's Pond, which is gradually developing into a great wildlife habitat. This may be small, but it's always worth a look, as a good selection of birds have been noted.
A Lapwing
Visitors are also recommended to visit the nearby birding sites of Backworth Pond, Killingworth Lake and Wallsend Dene & Parks.
Tyne & Wear, North Tyneside.
Map Reference
NZ 299 693
There is free public access to the country park. The park is accessible to wheelchair users. Paths can be muddy at times, so appropriate outdoor footwear is recommended.
There are toilets and cafes available at the Rising Sun Countryside Centre.
There is parking available adjacent to the Rising Sun Countryside Centre.
The site is maintained by North Tyneside Council. For more information please visit www.northtyneside.gov.uk
Swallow Pond and the adjacent plantation has been a Local Nature Reserve since 1992.
Email: mail@birdwatchingsites.co.uk
© 2019 BirdWatching Sites
Web Design by: Paul Buskin


text:
Юрьевцево — деревня в Вологодском районе Вологодской области.

Входит в состав Подлесного сельского поселения, с точки зрения административно-территориального деления — в Подлесный сельсовет.

Расстояние по автодороге до районного центра Вологды — 18 км, до центра муниципального образования Огарково — 5 км. Ближайшие населённые пункты — Невинниково, Конюхово, Снасудово, Реброво, Лисицыно.

По переписи 2002 года население — 3 человека.

Примечания

Ссылки
 Юрьевцево в реестре населённых пунктов Вологодской области

Населённые пункты Вологодского района

redpajama_set_name: RedPajamaWikipedia


(c) Operating cash flow less capital expenditure, disposal of tangible assest, purchase and disposal of intangible assets, movement with assiciates as well as with non-controlling interests.
(b) Profit for the period attributable to shareholders of the parent from continuing operations before impairments, restructuring costs, results on disposals and significant one-off itmes. The tax impact from the adjusted items is also adjusted for.
CARAMEL HONEY MACADAMIA: One of the three ice cream recipes to star in the advertisements.
Three products - Classic Vanilla, Caramel Honey Macadamia, and Café Grande- star in the advertisements, which show ingredients arranged in precarious positions to resemble modern art installations.
In the first, a flurry of feather-light vanilla beans appear to cascade over splashes of split fresh cream, while in the second, a tilting tower of caramel and macadamia nuts teeters on a table's edge.
Finally, meticulous measures of Grand Marnier liqueur and cream are suspended tantalisingly close to painstakingly piled coffee beans and chocolate-coated almonds.
Camille Meylan, Head of Marketing, Nestlé Peters, explained the purpose of the campaign.
He said: "We wanted to bring the artistry of Connoisseur ice cream to life by celebrating the skill and care we take in selecting each of the ingredients.


text:
Huawei's Hongmeng May Not Replace Android On Smartphones After All
Technode
August 9, 2019 — 22:05
Attendees try Huawei's Mate 20 Pro at CES Asia 2019 in Shanghai, China on June 11, 2019. (Image credit: TechNode/Shi Jiayi)
This article by WEI SHENG originally appeared on TechNode, the leading English authority on technology in China.
Huawei on Friday unveiled its long-awaited self-developed operating system HarmonyOS on its smart television product, but it may not be an Android alternative as previously rumored.
Why it matters: HarmonyOS, also known as Hongmeng OS, was deemed to be Huawei's alternative to Google's Android after the Chinese firm was cut off from US technology. The debut of the operating system was on Huawei's TV set, but it is not yet available on smartphones.
The current version of the operating system, or the HarmonyOS 1.0, runs on the company's TV product, which was released last month.
The future HarmonyOS will support a wide range of devices from personal computers to smartwatches, as well as virtual reality glasses, said Yu Chengdong, CEO of Huawei's consumer business group, at the Huawei Developer Conference in Dongguan on Friday, without mentioning any plan to install it on smartphones.
"HarmonyOS is completely different from Android and iOS. It is a microkernel-based, distributed OS that delivers a smooth experience across all scenarios."
— Yu Chengdong, at the Huawei Developer Conference on Friday
Details: Huawei said HarmonyOS will be open source and the firm will establish an open-source foundation and community to support developers.
HarmonyOS is ready to run on phones, but "for the consideration of partnerships and the ecosystem," Huawei won't be using it on handsets just yet, said Yu, adding that migrating from Android to HarmonyOS would only take a few days.
The current version of HarmonyOS is based on open-sourced frameworks and some self-developed modules, but future versions will be entirely developed in-house, said Yu.
Huawei said in a statement that it will lay the foundation for operating systems in the Chinese market, and then expand overseas.
Context: The mysterious operating system has been at the center of rumors regarding Huawei's so-called 'plan B' against US sanctions. The company's executives have given inconsistent statements about the OS in recent months.
In a March interview with Die Welt, Yu said Huawei had prepared the operating system as an alternative to Google's Android and Microsoft's Windows.
Huawei communications VP Andrew Williamson told Reuters in June that the company was in the process of potentially launching an Android replacement, adding that it would be ready "in months."
Then in July, chairman Liang Hua said the operating system was developed primarily for the internet of things (IoT) devices instead of smartphones, and the company was still using Android as a "first choice."
In a late-July news conference, Liang reaffirmed Hongmeng OS was not developed for smartphones, and the company still preferred to continue to use Google's Android OS for future phones.
When asked by a reporter about the inconsistent statements, Liang explained that the operating system was part of the company's long-term strategy, and it could be used on smartphones. "We are definitely not bluffing," he said.
HarmonyOS
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