\documentclass{amsart}
\usepackage[all]{xy} 
\newcommand{\OriSquare}[5]
{
\begin{xy}
(0,0)="Pos";
"Pos"+(1,0) **@{-}; ?*h!U!/^2pt/{\text{\scriptsize #5}}, 
"Pos"+(1,1) **@{-}; ?*h!L!/^1pt/{\text{\scriptsize #2}},
"Pos"+(0,1) **@{-}; ?*h!D!/^1pt/{\text{\scriptsize #3}},
"Pos" **@{-}; ?*h!R!/^1pt/{\text{\scriptsize #4}},
"Pos"+(0.5,0.5) *h{#1};
\end{xy}
}

\begin{document}
% Origami of size 8
\begin{xy}
<0.8cm,0cm>:
(0,0)*{\OriSquare{1}{}{8}{2}{8}};
(1,0)*{\OriSquare{2}{1}{}{}{3}};
(1,1)*{\OriSquare{3}{}{2}{4}{}};
(2,1)*{\OriSquare{4}{3}{}{}{5}};
(2,2)*{\OriSquare{5}{}{4}{6}{}};
(3,2)*{\OriSquare{6}{5}{}{}{7}};
(3,3)*{\OriSquare{7}{}{6}{8}{}};
(4,3)*{\OriSquare{8}{7}{1}{}{1}};
\end{xy}
\end{document}
