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\centerline{\bf The Penrose Tiling}\vs

\noindent The Penrose tiling on this pavement is part of a scheme
for filling up the entire plane with copies of the ``fat rhombus'' 
(angles $108^{\circ}$ and $72^{\circ}$),
and the ``thin rhombus'' (angles $144^{\circ}$ 
and $36^{\circ}$). Its two most important properties are
\begin{itemize} 
\item The tiling is not periodic. There is no way to slide the
picture, in any direction,  so that it matches itself perfectly.
\item Nevertheless, any finite patch of the tiling, no matter how
large, will appear in exact copies of itself over and over again 
as we look farther and farther out in the plane.
\end{itemize}
This tiling is one of a set of aperiodic tilings investigated by
the British mathematician Roger Penrose (now Sir Roger) in the 1970s.
\vs  

\noindent
The most elegant way to describe the Penrose tiling is as the projection onto
the plane of a certain specific
surface in 5-dimensional space. This surface is made
up of squares parallel to the coordinate planes, with adjoining squares 
perpendicular to each other (all this in 5-space). There are  10
possible orientations for such a square; projected these
give the two rhombic shapes, with
5 possible orientations for each of them. 

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\noindent
In the surface in 5-space, whenever a vertex meets exactly three
squares, these must be relatively positioned like three faces of a cube.
When the three are projected into the plane, they look like the
isometric perspective image of an ordinary 3-dimensional cube.
But this interpretation cannot be extended to neighboring tiles.
Even though those may be part of another cube, the cubes, interpreted
in 3-space, form an ``impossible'' configuration. Some
instances of this phenomenon, which forces our perception of the
the tiling to continually shift from one interpretation to another,
are highlighted in blue in the figure above.
\vs

\noindent
{\it Tony Phillips}\\
{\it May, 2013}



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