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We explain how the ``non-commutative algebraic variety''
having homogeneous coordinate
ring the free algebra k modulo the ideal generated by y^2
can be viewed as the space of Penrose tilings of the plane.
Each Penrose tiling yields a ``point'' on this non-commutative
variety and two tilings yield the same ``point'' if and only if
one tiling can be obtained from the other by an isometry of the plane.
This approach is entirely compatible with the approximately finite
non-commutative C^*-algebra that Alain Connes associates to
the space of Penrose tilings of the plane.
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