Perles configuration

<p>In geometry, the Perles configuration is a system of nine points and nine lines in the <a href="/facts/Euclidean_plane/tAUtRFcn">Euclidean plane</a> for which every combinatorially equivalent realization has at least one <a href="/facts/Irrational_number/Psv19TET">irrational number</a> as one of its coordinates. It can be constructed from the diagonals and symmetry lines of a <a href="/facts/Regular_pentagon/xoq5oJeQ">regular pentagon</a>, and their crossing points. All of the realizations of the Perles configuration in the <a href="/facts/Projective_plane/p6ot8oNv">projective plane</a> are equivalent to each other under <a href="/facts/Projective_transformation/wCkNzUjd">projective transformations</a>.
</p><p>The Perles configuration is the smallest configuration of points and lines that cannot be realized with rational coordinates. It is named after <a href="/facts/Micha_Perles/BzxGwD6K">Micha Perles</a>, who used it to construct an eight-dimensional <a href="/facts/Convex_polytope/2pRHcRgC">convex polytope</a> that cannot be given rational number coordinates and that have the fewest vertices (twelve) of any known irrational polytope.
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Perles configuration open-in-new

Perles configuration