http://openproblemgarden.org/op/partitioning_the_projective_plane Comments for "Partitioning the Projective Plane" en http://openproblemgarden.org/op/partitioning_the_projective_plane <table cellspacing="10"> <tr> <td> Author(s): <a href="/category/noel_jonathan_a">Noel</a>&nbsp;&nbsp; </td> <td align=right> Subject: <a href="/category/geometry">Geometry</a>&nbsp;&nbsp; </td> </tr> <tr> <td colspan=2> <table border=1 cellspacing="5"> <tr><td> <p>Throughout this post, by <em>projective plane</em> we mean the set of all lines through the origin in <img class="teximage" src="/files/tex/931fdac9bef64e03768ea90b88f8428005db887d.png" alt="$ \mathbb{R}^3 $" />.</p> <div class="envsimple"><b>Definition</b>&nbsp;&nbsp; Say that a subset <img class="teximage" src="/files/tex/d2b76a0ee5465d3e3ecc846c8e3d632edd8b2bbf.png" alt="$ S $" /> of the projective plane is <em>octahedral</em> if all lines in <img class="teximage" src="/files/tex/d2b76a0ee5465d3e3ecc846c8e3d632edd8b2bbf.png" alt="$ S $" /> pass through the closure of two opposite faces of a regular octahedron centered at the origin. </div> <div class="envsimple"><b>Definition</b>&nbsp;&nbsp; Say that a subset <img class="teximage" src="/files/tex/d2b76a0ee5465d3e3ecc846c8e3d632edd8b2bbf.png" alt="$ S $" /> of the projective plane is <em>weakly octahedral</em> if every set <img class="teximage" src="/files/tex/e8109a0f5c1dcb11fe3245b53b8bd2bc9d6418d1.png" alt="$ S&#039;\subseteq S $" /> such that <img class="teximage" src="/files/tex/e1b8bc4df405ab37c6b4aa2f638a5c2df882f7a9.png" alt="$ |S&#039;|=3 $" /> is octahedral. </div> <div class="envtheorem"><b>Conjecture</b>&nbsp;&nbsp; Suppose that the projective plane can be partitioned into four sets, say <img class="teximage" src="/files/tex/6f2284a502ee75f719fa3d5c2430c467e11df0c4.png" alt="$ S_1,S_2,S_3 $" /> and <img class="teximage" src="/files/tex/a0fc8ce0b0dfbf88309c7c045fff90a5cadd5117.png" alt="$ S_4 $" /> such that each set <img class="teximage" src="/files/tex/110ae457d97eebe47aa4d2e8c6237fdb9317f11e.png" alt="$ S_i $" /> is weakly octahedral. Then each <img class="teximage" src="/files/tex/110ae457d97eebe47aa4d2e8c6237fdb9317f11e.png" alt="$ S_i $" /> is octahedral. </div> </tr></td> </table> </td> </tr> </table> Noel, Jonathan A. Partitioning projective plane Geometry http://openproblemgarden.org/op/partitioning_the_projective_plane#comment Tue, 27 Aug 2013 07:41:58 +0200 Jon Noel 56328 at http://openproblemgarden.org