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• Real_projective_plane abstract "In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold, that is, a one-sided surface. It cannot be embedded in standard three-dimensional space without intersecting itself. It has basic applications to geometry, since the common construction of the real projective plane is as the space of lines in R3 passing through the origin.The plane is also often described topologically, in terms of a construction based on the Möbius strip: if one could glue the (single) edge of the Möbius strip to itself in the correct direction, one would obtain the projective plane. (This cannot be done in three-dimensional space.) Equivalently, gluing a disk along the boundary of the Möbius strip gives the projective plane. Topologically, it has Euler characteristic 1, hence a demigenus (non-orientable genus, Euler genus) of 1.Since the Möbius strip, in turn, can be constructed from a square by gluing two of its sides together, the real projective plane can thus be represented as a unit square (that is, [0,1] × [0,1] ) with its sides identified by the following equivalence relations:(0, y) ~ (1, 1 − y) for 0 ≤ y ≤ 1and(x, 0) ~ (1 − x, 1) for 0 ≤ x ≤ 1,as in the leftmost diagram on the right.".
• Real_projective_plane thumbnail ProjectivePlaneAsSquare.svg?width=300.
• Real_projective_plane wikiPageExternalLink rptwo.htm.
• Real_projective_plane wikiPageExternalLink Demiralp-2009-CLF.pdf.
• Real_projective_plane wikiPageExternalLink watch?v=lDqmaPEjJpk.
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• Real_projective_plane wikiPageRevisionID "675757760".
• Real_projective_plane wikiPageWikiLink Abstract_polytope.
• Real_projective_plane wikiPageWikiLink Abstract_regular_polytope.
• Real_projective_plane wikiPageWikiLink Antipodal_point.
• Real_projective_plane wikiPageWikiLink Boys_surface.
• Real_projective_plane wikiPageWikiLink Cartesian_product.
• Real_projective_plane wikiPageWikiLink Category:Geometric_topology.
• Real_projective_plane wikiPageWikiLink Category:Surfaces.
• Real_projective_plane wikiPageWikiLink Column_vector.
• Real_projective_plane wikiPageWikiLink Conditional_clause.
• Real_projective_plane wikiPageWikiLink Conditional_sentence.
• Real_projective_plane wikiPageWikiLink Covering_map.
• Real_projective_plane wikiPageWikiLink Covering_space.
• Real_projective_plane wikiPageWikiLink Cross-cap.
• Real_projective_plane wikiPageWikiLink Cross_product.
• Real_projective_plane wikiPageWikiLink Data_plot.
• Real_projective_plane wikiPageWikiLink Disk_(mathematics).
• Real_projective_plane wikiPageWikiLink Dot_product.
• Real_projective_plane wikiPageWikiLink Embedding.
• Real_projective_plane wikiPageWikiLink Equivalence_relation.
• Real_projective_plane wikiPageWikiLink Euclidean_geometry.
• Real_projective_plane wikiPageWikiLink Euclidean_space.
• Real_projective_plane wikiPageWikiLink Euler_characteristic.
• Real_projective_plane wikiPageWikiLink File:Proj_geom1.PNG.
• Real_projective_plane wikiPageWikiLink Fundamental_group.
• Real_projective_plane wikiPageWikiLink Fundamental_polygon.
• Real_projective_plane wikiPageWikiLink Genus_(mathematics).
• Real_projective_plane wikiPageWikiLink Geometry.
• Real_projective_plane wikiPageWikiLink Great_circle.
• Real_projective_plane wikiPageWikiLink Hemi-dodecahedron.
• Real_projective_plane wikiPageWikiLink Hemi-icosahedron.
• Real_projective_plane wikiPageWikiLink Hemicube_(geometry).
• Real_projective_plane wikiPageWikiLink Homeomorphic.
• Real_projective_plane wikiPageWikiLink Homeomorphism.
• Real_projective_plane wikiPageWikiLink Homogeneous_coordinates.
• Real_projective_plane wikiPageWikiLink Immersion_(mathematics).
• Real_projective_plane wikiPageWikiLink Jordan_curve_theorem.
• Real_projective_plane wikiPageWikiLink Klein_bottle.
• Real_projective_plane wikiPageWikiLink Line_at_infinity.
• Real_projective_plane wikiPageWikiLink Linear_subspace.
• Real_projective_plane wikiPageWikiLink Manifold.
• Real_projective_plane wikiPageWikiLink Mathematics.
• Real_projective_plane wikiPageWikiLink Möbius_strip.
• Real_projective_plane wikiPageWikiLink Orientability.
• Real_projective_plane wikiPageWikiLink Orientation_(mathematics).
• Real_projective_plane wikiPageWikiLink Orientation_(vector_space).
• Real_projective_plane wikiPageWikiLink Origin_(mathematics).
• Real_projective_plane wikiPageWikiLink Orthogonal.
• Real_projective_plane wikiPageWikiLink Orthogonality.
• Real_projective_plane wikiPageWikiLink Plane_of_symmetry.
• Real_projective_plane wikiPageWikiLink Plot_(graphics).
• Real_projective_plane wikiPageWikiLink Polyhedron.
• Real_projective_plane wikiPageWikiLink Projective_plane.
• Real_projective_plane wikiPageWikiLink Projective_polyhedra.
• Real_projective_plane wikiPageWikiLink Projective_polyhedron.
• Real_projective_plane wikiPageWikiLink Projective_space.
• Real_projective_plane wikiPageWikiLink Pus_inequality.
• Real_projective_plane wikiPageWikiLink Quotient_space_(topology).
• Real_projective_plane wikiPageWikiLink Real_coordinate_space.
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• Real_projective_plane wikiPageWikiLink Real_projective_space.
• Real_projective_plane wikiPageWikiLink Reflection_(mathematics).
• Real_projective_plane wikiPageWikiLink Reflection_symmetry.
• Real_projective_plane wikiPageWikiLink Roman_surface.
• Real_projective_plane wikiPageWikiLink Row_and_column_vectors.
• Real_projective_plane wikiPageWikiLink Sphere.
• Real_projective_plane wikiPageWikiLink Square.
• Real_projective_plane wikiPageWikiLink Square_(geometry).
• Real_projective_plane wikiPageWikiLink Surface.
• Real_projective_plane wikiPageWikiLink Tetrahemihexahedron.
• Real_projective_plane wikiPageWikiLink Torus.
• Real_projective_plane wikiPageWikiLink File:CrossCapSlicedOpen.PNG.
• Real_projective_plane wikiPageWikiLink File:CrossCapTwoViews.PNG.
• Real_projective_plane wikiPageWikiLink File:KleinBottleAsSquare.svg.
• Real_projective_plane wikiPageWikiLink File:MöbiusStripAsSquare.svg.
• Real_projective_plane wikiPageWikiLink File:Prj_geom.svg.
• Real_projective_plane wikiPageWikiLink File:ProjectivePlaneAsSquare.svg.
• Real_projective_plane wikiPageWikiLink File:Projective_geometry_diagram_2.svg.
• Real_projective_plane wikiPageWikiLink File:SelfIntersectingDisk.PNG.
• Real_projective_plane wikiPageWikiLink File:Steiners_Roman_Surface.gif.
• Real_projective_plane wikiPageWikiLink File:Tetrahemihexahedron.png.
• Real_projective_plane wikiPageWikiLinkText "Flat projective plane".
• Real_projective_plane wikiPageWikiLinkText "Projective plane".
• Real_projective_plane wikiPageWikiLinkText "Real projective plane".
• Real_projective_plane wikiPageWikiLinkText "infinity".
• Real_projective_plane wikiPageWikiLinkText "projective plane".
• Real_projective_plane wikiPageWikiLinkText "real projective plane".
• Real_projective_plane hasPhotoCollection Real_projective_plane.
• Real_projective_plane title "Real Projective Plane".
• Real_projective_plane urlname "RealProjectivePlane".
• Real_projective_plane wikiPageUsesTemplate Template:Citation.
• Real_projective_plane wikiPageUsesTemplate Template:Clear.

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