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\centerline{A proposal for}\vs

\centerline{\large \bf
 A Penrose Tiling for the Simons Center Plaza-level Floor}\vs

\centerline{Tony Phillips, October 2009}
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\noindent{\bf The Penrose Tiling}\vs

\noindent
The Penrose tiling in its most elegant form is the projection onto
the plane of a 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, and in our projection these
give two tile shapes, with
5 possible orientations for each of them. The shapes are rhombi
(parallelograms), a ``fat'' rhombus with angles $108^{\circ}$ and 
$72^{\circ}$,
a ``thin'' rhombus with angles $144^{\circ}$ and $36^{\circ}$. 

\begin{figure}[htp]
\centering \includegraphics[width=5in]{penrose-demo.eps}
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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, would 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.
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\noindent
The next page shows a proposed Penrose tiling of the Plaza-level
floor in the Simons Center for Geometry and Physics. The tile size 
chosen here is arbitrary, but it would be desirable for the tiles
to be large enough for the perspective effect to be enjoyed by
spectators looking down from the gallery.
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\noindent
Please note that this  is not an authentic Penrose tiling, but
has been cobbled together from two large pieces, with some extra
bits along the edges, for the purposes of this proposal. 
A real Penrose tiling never repeats itself exactly.


  
\begin{figure}[htp]
\centering \includegraphics[width=2.8in]{penrose-plaza-vert.eps}
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