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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Projective geometry</span></span>
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</style><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist"><tbody><tr><th class="sidebar-title"><a href="Geometry" title="Geometry">Geometry</a></th></tr><tr><td class="sidebar-image"><div class="sidebar-caption"> a <a href="Sphere" title="Sphere">sphere</a> to a <a href="Plane_(geometry)" class="mw-redirect" title="Plane (geometry)">plane</a></div></td></tr><tr><td class="sidebar-content">
<div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="List_of_geometry_topics" class="mw-redirect" title="List of geometry topics">Branches</a></div></div><div class="sidebar-list-content mw-collapsible-content hlist">
<ul><li><a href="Euclidean_geometry" title="Euclidean geometry">Euclidean</a></li>
<li><a href="Non-Euclidean_geometry" title="Non-Euclidean geometry">Non-Euclidean</a>
<ul><li><a href="Elliptic_geometry" title="Elliptic geometry">Elliptic</a>
<ul><li><a href="Spherical_geometry" title="Spherical geometry">Spherical</a></li></ul></li>
<li><a href="Hyperbolic_geometry" title="Hyperbolic geometry">Hyperbolic</a></li></ul></li>
<li><a href="Non-Archimedean_geometry" title="Non-Archimedean geometry">Non-Archimedean geometry</a></li>
<li><a href="Affine_geometry" title="Affine geometry">Affine</a></li>
<li><a href="Synthetic_geometry" title="Synthetic geometry">Synthetic</a></li>
<li><a href="Analytic_geometry" title="Analytic geometry">Analytic</a></li>
<li><a href="Algebraic_geometry" title="Algebraic geometry">Algebraic</a>
<ul><li><a href="Arithmetic_geometry" title="Arithmetic geometry">Arithmetic</a></li>
<li><a href="Diophantine_geometry" title="Diophantine geometry">Diophantine</a></li></ul></li>
<li><a href="Differential_geometry" title="Differential geometry">Differential</a>
<ul><li><a href="Riemannian_geometry" title="Riemannian geometry">Riemannian</a></li>
<li><a href="Symplectic_geometry" title="Symplectic geometry">Symplectic</a></li>
<li><a href="Discrete_differential_geometry" title="Discrete differential geometry">Discrete differential</a></li></ul></li>
<li><a href="Complex_geometry" title="Complex geometry">Complex</a></li>
<li><a href="Finite_geometry" title="Finite geometry">Finite</a></li>
<li><a href="Discrete_geometry" title="Discrete geometry">Discrete/Combinatorial</a>
<ul><li><a href="Digital_geometry" title="Digital geometry">Digital</a></li></ul></li>
<li><a href="Convex_geometry" title="Convex geometry">Convex</a></li>
<li><a href="Computational_geometry" title="Computational geometry">Computational</a></li>
<li><a href="Fractal" title="Fractal">Fractal</a></li>
<li><a href="Incidence_geometry" title="Incidence geometry">Incidence </a></li>
<li><a href="Noncommutative_geometry" title="Noncommutative geometry">Noncommutative geometry</a>
<ul><li><a href="Noncommutative_algebraic_geometry" title="Noncommutative algebraic geometry">Noncommutative algebraic geometry</a></li></ul></li></ul></div></div></td>
</tr><tr><td class="sidebar-content">
<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><div class="hlist"><ul><li>Concepts</li><li>Features</li></ul></div></div></div><div class="sidebar-list-content mw-collapsible-content hlist"><a href="Dimension_(geometry)" class="mw-redirect" title="Dimension (geometry)">Dimension</a>
<ul><li><a href="Straightedge_and_compass_construction" title="Straightedge and compass construction">Straightedge and compass constructions</a></li></ul>
<ul><li><a href="Angle" title="Angle">Angle</a></li>
<li><a href="Curve" title="Curve">Curve</a></li>
<li><a href="Diagonal" title="Diagonal">Diagonal</a></li>
<li><a href="Orthogonality" title="Orthogonality">Orthogonality</a> (<a href="Perpendicular" title="Perpendicular">Perpendicular</a>)</li>
<li><a href="Parallel_(geometry)" title="Parallel (geometry)">Parallel</a></li>
<li><a href="Vertex_(geometry)" title="Vertex (geometry)">Vertex</a></li></ul>
<ul><li><a href="Congruence_(geometry)" title="Congruence (geometry)">Congruence</a></li>
<li><a href="Similarity_(geometry)" title="Similarity (geometry)">Similarity</a></li>
<li><a href="Symmetry" title="Symmetry">Symmetry</a></li></ul></div></div></td>
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<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="Zero-dimensional_space" title="Zero-dimensional space">Zero-dimensional</a></div></div><div class="sidebar-list-content mw-collapsible-content hlist">
<ul><li><a href="Point_(geometry)" title="Point (geometry)">Point</a></li></ul></div></div></td>
</tr><tr><td class="sidebar-content">
<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="One-dimensional_space" title="One-dimensional space">One-dimensional</a></div></div><div class="sidebar-list-content mw-collapsible-content hlist">
<ul><li><a href="Line_(geometry)" title="Line (geometry)">Line</a>
<ul><li><a href="Line_segment" title="Line segment">segment</a></li>
<li><a href="Line_(geometry)#Ray" title="Line (geometry)">ray</a></li></ul></li>
<li><a href="Length" title="Length">Length</a></li></ul></div></div></td>
</tr><tr><td class="sidebar-content-with-subgroup">
<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="Two-dimensional_space" title="Two-dimensional space">Two-dimensional</a></div></div><div class="sidebar-list-content mw-collapsible-content hlist" style="padding-bottom:0;"><table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content hlist">
<ul><li><a href="Plane_(geometry)" class="mw-redirect" title="Plane (geometry)">Plane</a></li>
<li><a href="Area" title="Area">Area</a></li>
<li><a href="Polygon" title="Polygon">Polygon</a></li></ul></td>
</tr><tr><th class="sidebar-heading">
<a href="Triangle" title="Triangle">Triangle</a></th></tr><tr><td class="sidebar-content hlist">
<ul><li><a href="Altitude_(triangle)" title="Altitude (triangle)">Altitude</a></li>
<li><a href="Hypotenuse" title="Hypotenuse">Hypotenuse</a></li>
<li><a href="Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a></li></ul></td>
</tr><tr><th class="sidebar-heading">
<a href="Parallelogram" title="Parallelogram">Parallelogram</a></th></tr><tr><td class="sidebar-content hlist">
<ul><li><a href="Square" title="Square">Square</a></li>
<li><a href="Rectangle" title="Rectangle">Rectangle</a></li>
<li><a href="Rhombus" title="Rhombus">Rhombus</a></li>
<li><a href="Rhomboid" title="Rhomboid">Rhomboid</a></li></ul></td>
</tr><tr><th class="sidebar-heading">
<a href="Quadrilateral" title="Quadrilateral">Quadrilateral</a></th></tr><tr><td class="sidebar-content hlist">
<ul><li><a href="Trapezoid" title="Trapezoid">Trapezoid</a></li>
<li><a href="Kite_(geometry)" title="Kite (geometry)">Kite</a></li></ul></td>
</tr><tr><th class="sidebar-heading">
<a href="Circle" title="Circle">Circle</a></th></tr><tr><td class="sidebar-content hlist">
<ul><li><a href="Diameter" title="Diameter">Diameter</a></li>
<li><a href="Circumference" title="Circumference">Circumference</a></li>
<li><a href="Area_of_a_circle" title="Area of a circle">Area</a></li></ul></td>
</tr></tbody></table></div></div></td>
</tr><tr><td class="sidebar-content">
<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="Three-dimensional_space" title="Three-dimensional space">Three-dimensional</a></div></div><div class="sidebar-list-content mw-collapsible-content hlist">
<ul><li><a href="Volume" title="Volume">Volume</a></li></ul>
<ul><li><a href="Cube" title="Cube">Cube</a>
<ul><li><a href="Cuboid" title="Cuboid">cuboid</a></li></ul></li>
<li><a href="Cylinder_(geometry)" class="mw-redirect" title="Cylinder (geometry)">Cylinder</a></li>
<li><a href="Dodecahedron" title="Dodecahedron">Dodecahedron</a></li>
<li><a href="Icosahedron" title="Icosahedron">Icosahedron</a></li>
<li><a href="Octahedron" title="Octahedron">Octahedron</a></li>
<li><a href="Pyramid_(geometry)" title="Pyramid (geometry)">Pyramid</a></li>
<li><a href="Platonic_Solid" class="mw-redirect" title="Platonic Solid">Platonic Solid</a></li>
<li><a href="Sphere" title="Sphere">Sphere</a></li>
<li><a href="Tetrahedron" title="Tetrahedron">Tetrahedron</a></li></ul></div></div></td>
</tr><tr><td class="sidebar-content">
<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="Four-dimensional_space" title="Four-dimensional space">Four</a>-/other-dimensional</div></div><div class="sidebar-list-content mw-collapsible-content hlist">
<ul><li><a href="Tesseract" title="Tesseract">Tesseract</a></li>
<li><a href="Hypersphere" class="mw-redirect" title="Hypersphere">Hypersphere</a></li></ul></div></div></td>
</tr><tr><th class="sidebar-heading" style="padding-bottom:0.2em;">
<a href="List_of_geometers" title="List of geometers">Geometers</a></th></tr><tr><td class="sidebar-content">
<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c">by name</div></div><div class="sidebar-list-content mw-collapsible-content hlist">
<ul><li><a href="Yasuaki_Aida" class="mw-redirect" title="Yasuaki Aida">Aida</a></li>
<li><a href="Aryabhata" title="Aryabhata">Aryabhata</a></li>
<li><a href="Ahmes" title="Ahmes">Ahmes</a></li>
<li><a href="Alhazen" class="mw-redirect" title="Alhazen">Alhazen</a></li>
<li><a href="Apollonius_of_Perga" title="Apollonius of Perga">Apollonius</a></li>
<li><a href="Archimedes" title="Archimedes">Archimedes</a></li>
<li><a href="Michael_Atiyah" title="Michael Atiyah">Atiyah</a></li>
<li><a href="Baudhayana" title="Baudhayana">Baudhayana</a></li>
<li><a href="J%C3%A1nos_Bolyai" title="János Bolyai">Bolyai</a></li>
<li><a href="Brahmagupta" title="Brahmagupta">Brahmagupta</a></li>
<li><a href="%C3%89lie_Cartan" title="Élie Cartan">Cartan</a></li>
<li><a href="Shiing-Shen_Chern" title="Shiing-Shen Chern">Chern</a></li>
<li><a href="Harold_Scott_MacDonald_Coxeter" title="Harold Scott MacDonald Coxeter">Coxeter</a></li>
<li><a href="Ren%C3%A9_Descartes" title="René Descartes">Descartes</a></li>
<li><a href="Euclid" title="Euclid">Euclid</a></li>
<li><a href="Leonhard_Euler" title="Leonhard Euler">Euler</a></li>
<li><a href="Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Gauss</a></li>
<li><a href="Mikhail_Leonidovich_Gromov" class="mw-redirect" title="Mikhail Leonidovich Gromov">Gromov</a></li>
<li><a href="David_Hilbert" title="David Hilbert">Hilbert</a></li>
<li><a href="Christiaan_Huygens" title="Christiaan Huygens">Huygens</a></li>
<li><a href="Jye%E1%B9%A3%E1%B9%ADhadeva" title="Jyeṣṭhadeva">Jyeṣṭhadeva</a></li>
<li><a href="K%C4%81ty%C4%81yana" title="Kātyāyana">Kātyāyana</a></li>
<li><a href="Omar_Khayy%C3%A1m" class="mw-redirect" title="Omar Khayyám">Khayyám</a></li>
<li><a href="Felix_Klein" title="Felix Klein">Klein</a></li>
<li><a href="Nikolai_Lobachevsky" title="Nikolai Lobachevsky">Lobachevsky</a></li>
<li><a href="Manava" title="Manava">Manava</a></li>
<li><a href="Hermann_Minkowski" title="Hermann Minkowski">Minkowski</a></li>
<li><a href="Minggatu" title="Minggatu">Minggatu</a></li>
<li><a href="Blaise_Pascal" title="Blaise Pascal">Pascal</a></li>
<li><a href="Pythagoras" title="Pythagoras">Pythagoras</a></li>
<li><a href="Parameshvara" class="mw-redirect" title="Parameshvara">Parameshvara</a></li>
<li><a href="Henri_Poincar%C3%A9" title="Henri Poincaré">Poincaré</a></li>
<li><a href="Bernhard_Riemann" title="Bernhard Riemann">Riemann</a></li>
<li><a href="Sakabe_K%C5%8Dhan" title="Sakabe Kōhan">Sakabe</a></li>
<li><a href="Sijzi" class="mw-redirect" title="Sijzi">Sijzi</a></li>
<li><a href="Nasir_al-Din_al-Tusi" title="Nasir al-Din al-Tusi">al-Tusi</a></li>
<li><a href="Oswald_Veblen" title="Oswald Veblen">Veblen</a></li>
<li><a href="Virasena" title="Virasena">Virasena</a></li>
<li><a href="Yang_Hui" title="Yang Hui">Yang Hui</a></li>
<li><a href="Ibn_al-Yasamin" title="Ibn al-Yasamin">al-Yasamin</a></li>
<li><a href="Zhang_Heng" title="Zhang Heng">Zhang</a></li>
<li><a href="List_of_geometers" title="List of geometers">List of geometers</a></li></ul></div></div></td>
</tr><tr><td class="sidebar-content-with-subgroup">
<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c">by period</div></div><div class="sidebar-list-content mw-collapsible-content hlist" style="padding-bottom:0;"><table class="sidebar-subgroup"><tbody><tr><th class="sidebar-heading">
<a href="Before_Common_Era" class="mw-redirect" title="Before Common Era">BCE</a></th></tr><tr><td class="sidebar-content hlist">
<ul><li><a href="Ahmes" title="Ahmes">Ahmes</a></li>
<li><a href="Baudhayana" title="Baudhayana">Baudhayana</a></li>
<li><a href="Manava" title="Manava">Manava</a></li>
<li><a href="Pythagoras" title="Pythagoras">Pythagoras</a></li>
<li><a href="Euclid" title="Euclid">Euclid</a></li>
<li><a href="Archimedes" title="Archimedes">Archimedes</a></li>
<li><a href="Apollonius_of_Perga" title="Apollonius of Perga">Apollonius</a></li></ul></td>
</tr><tr><th class="sidebar-heading">
1–1400s</th></tr><tr><td class="sidebar-content hlist">
<ul><li><a href="Zhang_Heng" title="Zhang Heng">Zhang</a></li>
<li><a href="K%C4%81ty%C4%81yana" title="Kātyāyana">Kātyāyana</a></li>
<li><a href="Aryabhata" title="Aryabhata">Aryabhata</a></li>
<li><a href="Brahmagupta" title="Brahmagupta">Brahmagupta</a></li>
<li><a href="Virasena" title="Virasena">Virasena</a></li>
<li><a href="Alhazen" class="mw-redirect" title="Alhazen">Alhazen</a></li>
<li><a href="Sijzi" class="mw-redirect" title="Sijzi">Sijzi</a></li>
<li><a href="Omar_Khayy%C3%A1m" class="mw-redirect" title="Omar Khayyám">Khayyám</a></li>
<li><a href="Ibn_al-Yasamin" title="Ibn al-Yasamin">al-Yasamin</a></li>
<li><a href="Nasir_al-Din_al-Tusi" title="Nasir al-Din al-Tusi">al-Tusi</a></li>
<li><a href="Yang_Hui" title="Yang Hui">Yang Hui</a></li>
<li><a href="Parameshvara" class="mw-redirect" title="Parameshvara">Parameshvara</a></li></ul></td>
</tr><tr><th class="sidebar-heading">
1400s–1700s</th></tr><tr><td class="sidebar-content hlist">
<ul><li><a href="Jye%E1%B9%A3%E1%B9%ADhadeva" title="Jyeṣṭhadeva">Jyeṣṭhadeva</a></li>
<li><a href="Ren%C3%A9_Descartes" title="René Descartes">Descartes</a></li>
<li><a href="Blaise_Pascal" title="Blaise Pascal">Pascal</a></li>
<li><a href="Christiaan_Huygens" title="Christiaan Huygens">Huygens</a></li>
<li><a href="Minggatu" title="Minggatu">Minggatu</a></li>
<li><a href="Leonhard_Euler" title="Leonhard Euler">Euler</a></li>
<li><a href="Sakabe_K%C5%8Dhan" title="Sakabe Kōhan">Sakabe</a></li>
<li><a href="Yasuaki_Aida" class="mw-redirect" title="Yasuaki Aida">Aida</a></li></ul></td>
</tr><tr><th class="sidebar-heading">
1700s–1900s</th></tr><tr><td class="sidebar-content hlist">
<ul><li><a href="Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Gauss</a></li>
<li><a href="Nikolai_Lobachevsky" title="Nikolai Lobachevsky">Lobachevsky</a></li>
<li><a href="J%C3%A1nos_Bolyai" title="János Bolyai">Bolyai</a></li>
<li><a href="Bernhard_Riemann" title="Bernhard Riemann">Riemann</a></li>
<li><a href="Felix_Klein" title="Felix Klein">Klein</a></li>
<li><a href="Henri_Poincar%C3%A9" title="Henri Poincaré">Poincaré</a></li>
<li><a href="David_Hilbert" title="David Hilbert">Hilbert</a></li>
<li><a href="Hermann_Minkowski" title="Hermann Minkowski">Minkowski</a></li>
<li><a href="%C3%89lie_Cartan" title="Élie Cartan">Cartan</a></li>
<li><a href="Oswald_Veblen" title="Oswald Veblen">Veblen</a></li>
<li><a href="Harold_Scott_MacDonald_Coxeter" title="Harold Scott MacDonald Coxeter">Coxeter</a></li>
<li><a href="Shiing-Shen_Chern" title="Shiing-Shen Chern">Chern</a></li></ul></td>
</tr><tr><th class="sidebar-heading">
Present day</th></tr><tr><td class="sidebar-content hlist">
<ul><li><a href="Michael_Atiyah" title="Michael Atiyah">Atiyah</a></li>
<li><a href="Mikhail_Leonidovich_Gromov" class="mw-redirect" title="Mikhail Leonidovich Gromov">Gromov</a></li></ul></td>
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<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, <b>projective geometry</b> is the study of geometric properties that are invariant with respect to <a href="Projective_transformation" class="mw-redirect" title="Projective transformation">projective transformations</a>. This means that, compared to elementary <a href="Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a>, projective geometry has a different setting (<i><a href="Projective_space" title="Projective space">projective space</a></i>) and a selective set of basic geometric concepts. The basic intuitions are that projective space has more points than <a href="Euclidean_space" title="Euclidean space">Euclidean space</a>, for a given dimension, and that <a href="Geometric_transformation" title="Geometric transformation">geometric transformations</a> are permitted that transform the extra points (called "<a href="Point_at_infinity" title="Point at infinity">points at infinity</a>") to Euclidean points, and vice versa.
</p><p>Properties meaningful for projective geometry are respected by this new idea of transformation, which is more radical in its effects than can be expressed by a <a href="Transformation_matrix" title="Transformation matrix">transformation matrix</a> and <a href="Translation_(geometry)" title="Translation (geometry)">translations</a> (the <a href="Affine_transformation" title="Affine transformation">affine transformations</a>). The first issue for geometers is what kind of geometry is adequate for a novel situation. Unlike in <a href="Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a>, the concept of an <a href="Angle" title="Angle">angle</a> does not apply in projective geometry, because no measure of angles is invariant with respect to projective transformations, as is seen in <a href="Perspective_drawing" class="mw-redirect" title="Perspective drawing">perspective drawing</a> from a changing perspective. One source for projective geometry was indeed the theory of perspective. Another difference from elementary geometry is the way in which <a href="Parallel_(geometry)" title="Parallel (geometry)">parallel lines</a> can be said to meet in a <a href="Point_at_infinity" title="Point at infinity">point at infinity</a>, once the concept is translated into projective geometry's terms. Again this notion has an intuitive basis, such as railway tracks meeting at the horizon in a perspective drawing. See <i><a href="Projective_plane" title="Projective plane">Projective plane</a></i> for the basics of projective geometry in two dimensions.
</p><p>While the ideas were available earlier, projective geometry was mainly a development of the 19th century. This included the theory of <a href="Complex_projective_space" title="Complex projective space">complex projective space</a>, the coordinates used (<a href="Homogeneous_coordinates" title="Homogeneous coordinates">homogeneous coordinates</a>) being complex numbers. Several major types of more abstract mathematics (including <a href="Invariant_theory" title="Invariant theory">invariant theory</a>, the <a href="Italian_school_of_algebraic_geometry" title="Italian school of algebraic geometry">Italian school of algebraic geometry</a>, and <a href="Felix_Klein" title="Felix Klein">Felix Klein</a>'s <a href="Erlangen_programme" class="mw-redirect" title="Erlangen programme">Erlangen programme</a> resulting in the study of the <a href="Classical_groups" class="mw-redirect" title="Classical groups">classical groups</a>) were motivated by projective geometry. It was also a subject with many practitioners for its own sake, as <a href="Synthetic_geometry" title="Synthetic geometry">synthetic geometry</a>. Another topic that developed from axiomatic studies of projective geometry is <a href="Finite_geometry" title="Finite geometry">finite geometry</a>.
</p><p>The topic of projective geometry is itself now divided into many research subtopics, two examples of which are projective algebraic geometry (the study of <a href="Algebraic_variety#Projective_varieties" title="Algebraic variety">projective varieties</a>) and <a href="Projective_differential_geometry" title="Projective differential geometry">projective differential geometry</a> (the study of <a href="Differential_geometry" title="Differential geometry">differential invariants</a> of the projective transformations).
</p>
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<div class="mw-heading mw-heading2"><h2 id="Overview">Overview</h2></div>
<p>Projective geometry is an elementary non-<a href="Metric_(mathematics)" class="mw-redirect" title="Metric (mathematics)">metrical</a> form of geometry, meaning that it does not support any concept of distance. In two dimensions it begins with the study of <a href="Configuration_(geometry)" title="Configuration (geometry)">configurations</a> of <a href="Point_(geometry)" title="Point (geometry)">points</a> and <a href="Line_(geometry)" title="Line (geometry)">lines</a>. That there is indeed some geometric interest in this sparse setting was first established by <a href="Girard_Desargues" title="Girard Desargues">Desargues</a> and others in their exploration of the principles of <a href="Perspective_(graphical)" title="Perspective (graphical)">perspective art</a>.<sup id="cite_ref-FOOTNOTERamanan199788_1-0" class="reference"><a href="#cite_note-FOOTNOTERamanan199788-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> In <a href="Higher_dimension" class="mw-redirect" title="Higher dimension">higher dimensional</a> spaces there are considered <a href="Hyperplane" title="Hyperplane">hyperplanes</a> (that always meet), and other linear subspaces, which exhibit <a href="#Duality">the principle of duality</a>. The simplest illustration of duality is in the projective plane, where the statements "two distinct points determine a unique line" (i.e. the line through them) and "two distinct lines determine a unique point" (i.e. their point of intersection) show the same structure as propositions. Projective geometry can also be seen as a geometry of constructions with a <a href="Straightedge" title="Straightedge">straight-edge</a> alone, excluding <a href="Compass_(drafting)" class="mw-redirect" title="Compass (drafting)">compass</a> constructions, common in <a href="Straightedge_and_compass_construction" title="Straightedge and compass construction">straightedge and compass constructions</a>.<sup id="cite_ref-FOOTNOTECoxeter2003v_2-0" class="reference"><a href="#cite_note-FOOTNOTECoxeter2003v-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> As such, there are no circles, no angles, no measurements, no parallels, and no concept of <a href="https://en.wiktionary.org/wiki/intermediacy" class="extiw external" title="wikt:intermediacy">intermediacy</a> (or "betweenness").<sup id="cite_ref-FOOTNOTECoxeter1969229_3-0" class="reference"><a href="#cite_note-FOOTNOTECoxeter1969229-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> It was realised that the theorems that do apply to projective geometry are simpler statements. For example, the different <a href="Conic_section" title="Conic section">conic sections</a> are all equivalent in (complex) projective geometry, and some theorems about circles can be considered as special cases of these general theorems.
</p><p>During the early 19th century the work of <a href="Jean-Victor_Poncelet" title="Jean-Victor Poncelet">Jean-Victor Poncelet</a>, <a href="Lazare_Carnot" title="Lazare Carnot">Lazare Carnot</a> and others established projective geometry as an independent field of <a href="Mathematics" title="Mathematics">mathematics</a>.<sup id="cite_ref-FOOTNOTECoxeter1969229_3-1" class="reference"><a href="#cite_note-FOOTNOTECoxeter1969229-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Its rigorous foundations were addressed by <a href="Karl_von_Staudt" class="mw-redirect" title="Karl von Staudt">Karl von Staudt</a> and perfected by Italians <a href="Giuseppe_Peano" title="Giuseppe Peano">Giuseppe Peano</a>, <a href="Mario_Pieri" title="Mario Pieri">Mario Pieri</a>, <a href="Alessandro_Padoa" title="Alessandro Padoa">Alessandro Padoa</a> and <a href="Gino_Fano" title="Gino Fano">Gino Fano</a> during the late 19th century.<sup id="cite_ref-FOOTNOTECoxeter200314_4-0" class="reference"><a href="#cite_note-FOOTNOTECoxeter200314-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Projective geometry, like <a href="Affine_geometry" title="Affine geometry">affine</a> and <a href="Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a>, can also be developed from the <a href="Erlangen_program" title="Erlangen program">Erlangen program</a> of Felix Klein; projective geometry is characterized by <a href="Invariant_(mathematics)" title="Invariant (mathematics)">invariants</a> under <a href="Transformation_(geometry)" class="mw-redirect" title="Transformation (geometry)">transformations</a> of the <a href="Projective_group" class="mw-redirect" title="Projective group">projective group</a>.
</p><p>After much work on the very large number of theorems in the subject, therefore, the basics of projective geometry became understood. The <a href="Incidence_structure" title="Incidence structure">incidence structure</a> and the <a href="Cross-ratio" title="Cross-ratio">cross-ratio</a> are fundamental invariants under projective transformations. Projective geometry can be modeled by the <a href="Affine_geometry" title="Affine geometry">affine plane</a> (or affine space) plus a line (hyperplane) "at infinity" and then treating that line (or hyperplane) as "ordinary".<sup id="cite_ref-FOOTNOTECoxeter196993,_261_5-0" class="reference"><a href="#cite_note-FOOTNOTECoxeter196993,_261-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> An algebraic model for doing projective geometry in the style of <a href="Analytic_geometry" title="Analytic geometry">analytic geometry</a> is given by homogeneous coordinates.<sup id="cite_ref-FOOTNOTECoxeter1969234–238_6-0" class="reference"><a href="#cite_note-FOOTNOTECoxeter1969234–238-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTECoxeter2003111–132_7-0" class="reference"><a href="#cite_note-FOOTNOTECoxeter2003111–132-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> On the other hand, axiomatic studies revealed the existence of <a href="Non-Desarguesian_plane" title="Non-Desarguesian plane">non-Desarguesian planes</a>, examples to show that the axioms of incidence can be modelled (in two dimensions only) by structures not accessible to reasoning through homogeneous coordinate systems.
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<p>In a foundational sense, projective geometry and <a href="Ordered_geometry" title="Ordered geometry">ordered geometry</a> are elementary since they each involve a minimal set of <a href="Axioms" class="mw-redirect" title="Axioms">axioms</a> and either can be used as the foundation for <a href="Affine_geometry" title="Affine geometry">affine</a> and <a href="Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a>.<sup id="cite_ref-FOOTNOTECoxeter1969175–262_8-0" class="reference"><a href="#cite_note-FOOTNOTECoxeter1969175–262-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTECoxeter2003102–110_9-0" class="reference"><a href="#cite_note-FOOTNOTECoxeter2003102–110-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> Projective geometry is not "ordered"<sup id="cite_ref-FOOTNOTECoxeter1969229_3-2" class="reference"><a href="#cite_note-FOOTNOTECoxeter1969229-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> and so it is a distinct foundation for geometry.
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<div class="mw-heading mw-heading2"><h2 id="Description">Description</h2></div>
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<p>Projective geometry is less restrictive than either <a href="Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a> or <a href="Affine_geometry" title="Affine geometry">affine geometry</a>. It is an intrinsically non-<a href="Metric_(mathematics)" class="mw-redirect" title="Metric (mathematics)">metrical</a> geometry, meaning that facts are independent of any metric structure. Under the projective transformations, the <a href="Incidence_structure" title="Incidence structure">incidence structure</a> and the relation of <a href="Projective_harmonic_conjugate" title="Projective harmonic conjugate">projective harmonic conjugates</a> are preserved. A <a href="Projective_range" title="Projective range">projective range</a> is the one-dimensional foundation. Projective geometry formalizes one of the central principles of perspective art: that <a href="Parallel_(geometry)" title="Parallel (geometry)">parallel</a> lines meet at <a href="Infinity" title="Infinity">infinity</a>, and therefore are drawn that way. In essence, a projective geometry may be thought of as an extension of Euclidean geometry in which the "direction" of each line is subsumed within the line as an extra "point", and in which a "horizon" of directions corresponding to coplanar lines is regarded as a "line". Thus, two parallel lines meet on a horizon line by virtue of their incorporating the same direction.
</p><p>Idealized directions are referred to as points at infinity, while idealized horizons are referred to as lines at infinity. In turn, all these lines lie in the plane at infinity. However, infinity is a metric concept, so a purely projective geometry does not single out any points, lines or planes in this regard—those at infinity are treated just like any others.
</p><p>Because a <a href="Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a> is contained within a projective geometry—with projective geometry having a simpler foundation—general results in Euclidean geometry may be derived in a more transparent manner, where separate but similar theorems of Euclidean geometry may be handled collectively within the framework of projective geometry. For example, parallel and nonparallel lines need not be treated as separate cases; rather an arbitrary projective plane is singled out as the ideal plane and located "at infinity" using <a href="Homogeneous_coordinates" title="Homogeneous coordinates">homogeneous coordinates</a>.
</p><p>Additional properties of fundamental importance include <a href="Desargues'_Theorem" class="mw-redirect" title="Desargues' Theorem">Desargues' Theorem</a> and the <a href="Pappus's_hexagon_theorem" title="Pappus's hexagon theorem">Theorem of Pappus</a>. In projective spaces of dimension 3 or greater there is a construction that allows one to prove <a href="Desargues'_Theorem" class="mw-redirect" title="Desargues' Theorem">Desargues' Theorem</a>. But for dimension 2, it must be separately postulated.
</p><p>Using <a href="Desargues'_Theorem" class="mw-redirect" title="Desargues' Theorem">Desargues' Theorem</a>, combined with the other axioms, it is possible to define the basic operations of arithmetic, geometrically. The resulting operations satisfy the axioms of a field – except that the commutativity of multiplication requires <a href="Pappus's_hexagon_theorem" title="Pappus's hexagon theorem">Pappus's hexagon theorem</a>. As a result, the points of each line are in one-to-one correspondence with a given field, <span class="texhtml mvar" style="font-style:italic;">F</span>, supplemented by an additional element, ∞, such that <span class="texhtml"><var style="padding-right: 1px;">r</var> ⋅ ∞ = ∞</span>, <span class="texhtml">−∞ = ∞</span>, <span class="texhtml"><var style="padding-right: 1px;">r</var> + ∞ = ∞</span>, <span class="texhtml"><var style="padding-right: 1px;">r</var> / 0 = ∞</span>, <span class="texhtml"><var style="padding-right: 1px;">r</var> / ∞ = 0</span>, <span class="texhtml">∞ − <var style="padding-right: 1px;">r</var> = <var style="padding-right: 1px;">r</var> − ∞ = ∞</span>, except that <span class="texhtml">0 / 0</span>, <span class="texhtml">∞ / ∞</span>, <span class="texhtml">∞ + ∞</span>, <span class="texhtml">∞ − ∞</span>, <span class="texhtml">0 ⋅ ∞</span> and <span class="texhtml">∞ ⋅ 0</span> remain undefined.
</p><p>Projective geometry also includes a full theory of <a href="Conic_sections" class="mw-redirect" title="Conic sections">conic sections</a>, a subject also extensively developed in Euclidean geometry. There are advantages to being able to think of a <a href="Hyperbola" title="Hyperbola">hyperbola</a> and an <a href="Ellipse" title="Ellipse">ellipse</a> as distinguished only by the way the hyperbola <i>lies across the line at infinity</i>; and that a <a href="Parabola" title="Parabola">parabola</a> is distinguished only by being tangent to the same line. The whole family of circles can be considered as <i>conics passing through two given points on the line at infinity</i> — at the cost of requiring <a href="Complex_number" title="Complex number">complex</a> coordinates. Since coordinates are not "synthetic", one replaces them by fixing a line and two points on it, and considering the <i>linear system</i> of all conics passing through those points as the basic object of study. This method proved very attractive to talented geometers, and the topic was studied thoroughly. An example of this method is the multi-volume treatise by <a href="H._F._Baker" class="mw-redirect" title="H. F. Baker">H. F. Baker</a>.
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<div class="mw-heading mw-heading2"><h2 id="History">History</h2></div>
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</style><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="Mathematics_and_art" title="Mathematics and art">Mathematics and art</a></div>
<p>The first geometrical properties of a projective nature were discovered during the 3rd century by <a href="Pappus_of_Alexandria" title="Pappus of Alexandria">Pappus of Alexandria</a>.<sup id="cite_ref-FOOTNOTECoxeter1969229_3-3" class="reference"><a href="#cite_note-FOOTNOTECoxeter1969229-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> <a href="Filippo_Brunelleschi" title="Filippo Brunelleschi">Filippo Brunelleschi</a> (1404–1472) started investigating the geometry of perspective during 1425<sup id="cite_ref-FOOTNOTECoxeter20032_10-0" class="reference"><a href="#cite_note-FOOTNOTECoxeter20032-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> (see <i><a href="Perspective_(graphical)#History" title="Perspective (graphical)">Perspective (graphical) § History</a></i> for a more thorough discussion of the work in the fine arts that motivated much of the development of projective geometry). <a href="Johannes_Kepler" title="Johannes Kepler">Johannes Kepler</a> (1571–1630) and <a href="Girard_Desargues" title="Girard Desargues">Girard Desargues</a> (1591–1661) independently developed the concept of the "point at infinity".<sup id="cite_ref-FOOTNOTECoxeter20033_11-0" class="reference"><a href="#cite_note-FOOTNOTECoxeter20033-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> Desargues developed an alternative way of constructing perspective drawings by generalizing the use of vanishing points to include the case when these are infinitely far away. He made <a href="Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a>, where parallel lines are truly parallel, into a special case of an all-encompassing geometric system. Desargues's study on conic sections drew the attention of 16-year-old <a href="Blaise_Pascal" title="Blaise Pascal">Blaise Pascal</a> and helped him formulate <a href="Pascal's_theorem" title="Pascal's theorem">Pascal's theorem</a>. The works of <a href="Gaspard_Monge" title="Gaspard Monge">Gaspard Monge</a> at the end of 18th and beginning of 19th century were important for the subsequent development of projective geometry. The work of Desargues was ignored until <a href="Michel_Chasles" title="Michel Chasles">Michel Chasles</a> chanced upon a handwritten copy during 1845. Meanwhile, <a href="Jean-Victor_Poncelet" title="Jean-Victor Poncelet">Jean-Victor Poncelet</a> had published the foundational treatise on projective geometry during 1822. Poncelet examined the projective properties of objects (those invariant under central projection) and, by basing his theory on the concrete <a href="Pole_and_polar" title="Pole and polar">pole and polar</a> relation with respect to a circle, established a relationship between metric and projective properties. The <a href="Non-Euclidean_geometry" title="Non-Euclidean geometry">non-Euclidean geometries</a> discovered soon thereafter were eventually demonstrated to have models, such as the <a href="Klein_model" class="mw-redirect" title="Klein model">Klein model</a> of <a href="Hyperbolic_space" title="Hyperbolic space">hyperbolic space</a>, relating to projective geometry.
</p><p>In 1855 <a href="A._F._M%C3%B6bius" class="mw-redirect" title="A. F. Möbius">A. F. Möbius</a> wrote an article about permutations, now called <a href="M%C3%B6bius_transformation" title="Möbius transformation">Möbius transformations</a>, of <a href="Generalised_circle" title="Generalised circle">generalised circles</a> in the <a href="Complex_plane" title="Complex plane">complex plane</a>. These transformations represent projectivities of the <a href="Complex_projective_line" class="mw-redirect" title="Complex projective line">complex projective line</a>. In the study of lines in space, <a href="Julius_Pl%C3%BCcker" title="Julius Plücker">Julius Plücker</a> used <a href="Homogeneous_coordinates" title="Homogeneous coordinates">homogeneous coordinates</a> in his description, and the set of lines was viewed on the <a href="Klein_quadric" title="Klein quadric">Klein quadric</a>, one of the early contributions of projective geometry to a new field called <a href="Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>, an offshoot of <a href="Analytic_geometry" title="Analytic geometry">analytic geometry</a> with projective ideas.
</p><p>Projective geometry was instrumental in the validation of speculations of Lobachevski and Bolyai concerning <a href="Hyperbolic_geometry" title="Hyperbolic geometry">hyperbolic geometry</a> by providing <a href="Model_(logic)" class="mw-redirect" title="Model (logic)">models</a> for the <a href="Coordinate_systems_for_the_hyperbolic_plane" title="Coordinate systems for the hyperbolic plane">hyperbolic plane</a>:<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> for example, the <a href="Poincar%C3%A9_disc_model" class="mw-redirect" title="Poincaré disc model">Poincaré disc model</a> where generalised circles perpendicular to the <a href="Unit_circle" title="Unit circle">unit circle</a> correspond to "hyperbolic lines" (<a href="Geodesic" title="Geodesic">geodesics</a>), and the "translations" of this model are described by Möbius transformations that map the <a href="Unit_disc" class="mw-redirect" title="Unit disc">unit disc</a> to itself. The distance between points is given by a <a href="Cayley%E2%80%93Klein_metric" title="Cayley–Klein metric">Cayley–Klein metric</a>, known to be invariant under the translations since it depends on <a href="Cross-ratio" title="Cross-ratio">cross-ratio</a>, a key projective invariant. The translations are described variously as <a href="Isometries" class="mw-redirect" title="Isometries">isometries</a> in <a href="Metric_space" title="Metric space">metric space</a> theory, as <a href="Linear_fractional_transformation" title="Linear fractional transformation">linear fractional transformations</a> formally, and as projective linear transformations of the <a href="Projective_linear_group" title="Projective linear group">projective linear group</a>, in this case <span class="nowrap"><a href="SU(1%2C_1)" class="mw-redirect" title="SU(1, 1)">SU(1, 1)</a></span>.
</p><p>The work of <a href="Jean-Victor_Poncelet" title="Jean-Victor Poncelet">Poncelet</a>, <a href="Jakob_Steiner" title="Jakob Steiner">Jakob Steiner</a> and others was not intended to extend analytic geometry. Techniques were supposed to be <i><a href="Synthetic_geometry" title="Synthetic geometry">synthetic</a></i>: in effect <a href="Projective_space" title="Projective space">projective space</a> as now understood was to be introduced axiomatically. As a result, reformulating early work in projective geometry so that it satisfies current standards of rigor can be somewhat difficult. Even in the case of the <a href="Projective_plane" title="Projective plane">projective plane</a> alone, the axiomatic approach can result in <a href="Model_theory" title="Model theory">models</a> not describable via <a href="Linear_algebra" title="Linear algebra">linear algebra</a>.
</p><p>This period in geometry was overtaken by research on the general <a href="Algebraic_curve" title="Algebraic curve">algebraic curve</a> by <a href="Clebsch" class="mw-redirect" title="Clebsch">Clebsch</a>, <a href="Bernhard_Riemann" title="Bernhard Riemann">Riemann</a>, <a href="Max_Noether" title="Max Noether">Max Noether</a> and others, which stretched existing techniques, and then by <a href="Invariant_theory" title="Invariant theory">invariant theory</a>. Towards the end of the century, the <a href="Italian_school_of_algebraic_geometry" title="Italian school of algebraic geometry">Italian school of algebraic geometry</a> (<a href="Federigo_Enriques" title="Federigo Enriques">Enriques</a>, <a href="Corrado_Segre" title="Corrado Segre">Segre</a>, <a href="Francesco_Severi" title="Francesco Severi">Severi</a>) broke out of the traditional subject matter into an area demanding deeper techniques.
</p><p>During the later part of the 19th century, the detailed study of projective geometry became less fashionable, although the literature is voluminous. Some important work was done in <a href="Enumerative_geometry" title="Enumerative geometry">enumerative geometry</a> in particular, by Schubert, that is now considered as anticipating the theory of <a href="Chern_class" title="Chern class">Chern classes</a>, taken as representing the <a href="Algebraic_topology" title="Algebraic topology">algebraic topology</a> of <a href="Grassmannian" title="Grassmannian">Grassmannians</a>.
</p><p>Projective geometry later proved key to <a href="Paul_Dirac" title="Paul Dirac">Paul Dirac</a>'s invention of <a href="Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>. At a foundational level, the discovery that <a href="Quantum_measurement" class="mw-redirect" title="Quantum measurement">quantum measurements</a> could fail to commute had disturbed and dissuaded <a href="Werner_Heisenberg" title="Werner Heisenberg">Heisenberg</a>, but past study of projective planes over noncommutative rings had likely desensitized Dirac. In more advanced work, Dirac used extensive drawings in projective geometry to understand the intuitive meaning of his equations, before writing up his work in an exclusively algebraic formalism.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup>
</p>
<div class="mw-heading mw-heading2"><h2 id="Classification">Classification</h2></div>
<p>There are many projective geometries, which may be divided into discrete and continuous: a <i>discrete</i> geometry comprises a set of points, which may or may not be <i>finite</i> in number, while a <i>continuous</i> geometry has infinitely many points with no gaps in between.
</p><p>The only projective geometry of dimension 0 is a single point. A projective geometry of dimension 1 consists of a single line containing at least 3 points. The geometric construction of arithmetic operations cannot be performed in either of these cases. For dimension 2, there is a rich structure in virtue of the absence of <a href="Desargues'_Theorem" class="mw-redirect" title="Desargues' Theorem">Desargues' Theorem</a>.
</p>
<p>The smallest 2-dimensional projective geometry (that with the fewest points) is the <a href="Fano_plane" title="Fano plane">Fano plane</a>, which has 3 points on every line, with 7 points and 7 lines in all, having the following collinearities:
</p>
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<ul><li>[ABC]</li>
<li>[ADE]</li>
<li>[AFG]</li>
<li>[BDG]</li>
<li>[BEF]</li>
<li>[CDF]</li>
<li>[CEG]</li></ul>
</div>
<p>with <a href="Homogeneous_coordinates" title="Homogeneous coordinates">homogeneous coordinates</a> <span class="texhtml">A = (0,0,1)</span>, <span class="texhtml">B = (0,1,1)</span>, <span class="texhtml">C = (0,1,0)</span>, <span class="texhtml">D = (1,0,1)</span>, <span class="texhtml">E = (1,0,0)</span>, <span class="texhtml">F = (1,1,1)</span>, <span class="texhtml">G = (1,1,0)</span>, or, in affine coordinates, <span class="texhtml">A = (0,0)</span>, <span class="texhtml">B = (0,1)</span>, <span class="texhtml">C = (∞)</span>, <span class="texhtml">D = (1,0)</span>, <span class="texhtml">E = (0)</span>, <span class="texhtml">F = (1,1)</span>and <span class="texhtml">G = (1)</span>. The affine coordinates in a Desarguesian plane for the points designated to be the points at infinity (in this example: C, E and G) can be defined in several other ways.
</p><p>In standard notation, a <a href="Finite_projective_geometry" class="mw-redirect" title="Finite projective geometry">finite projective geometry</a> is written <span class="texhtml">PG(<i>a</i>, <i>b</i>)</span> where:
</p>
<dl><dd><span class="texhtml mvar" style="font-style:italic;">a</span> is the projective (or geometric) dimension, and</dd>
<dd><span class="texhtml mvar" style="font-style:italic;">b</span> is one less than the number of points on a line (called the <i>order</i> of the geometry).</dd></dl>
<p>Thus, the example having only 7 points is written <span class="texhtml">PG(2, 2)</span>.
</p><p>The term "projective geometry" is used sometimes to indicate the generalised underlying abstract geometry, and sometimes to indicate a particular geometry of wide interest, such as the metric geometry of flat space which we analyse through the use of <a href="Homogeneous_coordinates" title="Homogeneous coordinates">homogeneous coordinates</a>, and in which <a href="Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a> may be embedded (hence its name, <a href="Projective_plane#Some_examples" title="Projective plane">Extended Euclidean plane</a>).
</p><p>The fundamental property that singles out all projective geometries is the <i>elliptic</i> <a href="Incidence_(mathematics)" class="mw-redirect" title="Incidence (mathematics)">incidence</a> property that any two distinct lines <span class="texhtml mvar" style="font-style:italic;">L</span> and <span class="texhtml mvar" style="font-style:italic;">M</span> in the <a href="Projective_plane" title="Projective plane">projective plane</a> intersect at exactly one point <span class="texhtml mvar" style="font-style:italic;">P</span>. The special case in <a href="Analytic_geometry" title="Analytic geometry">analytic geometry</a> of <i>parallel</i> lines is subsumed in the smoother form of a line <i>at infinity</i> on which <span class="texhtml mvar" style="font-style:italic;">P</span> lies. The <i>line at infinity</i> is thus a line like any other in the theory: it is in no way special or distinguished. (In the later spirit of the <a href="Erlangen_programme" class="mw-redirect" title="Erlangen programme">Erlangen programme</a> one could point to the way the <a href="Group_(mathematics)" title="Group (mathematics)">group</a> of transformations can move any line to the <i>line at infinity</i>).
</p><p>The parallel properties of elliptic, Euclidean and hyperbolic geometries contrast as follows:
</p>
<dl><dd>Given a line <span class="texhtml mvar" style="font-style:italic;">l</span> and a point <span class="texhtml mvar" style="font-style:italic;">P</span> not on the line,
<dl><dd><dl><dt><i><a href="Elliptic_geometry" title="Elliptic geometry">Elliptic</a></i></dt>
<dd>there exists no line through <span class="texhtml mvar" style="font-style:italic;">P</span> that does not meet <span class="texhtml mvar" style="font-style:italic;">l</span></dd>
<dt><i><a href="Euclidean_geometry" title="Euclidean geometry">Euclidean</a></i></dt>
<dd>there exists exactly one line through <span class="texhtml mvar" style="font-style:italic;">P</span> that does not meet <span class="texhtml mvar" style="font-style:italic;">l</span></dd>
<dt><i><a href="Hyperbolic_geometry" title="Hyperbolic geometry">Hyperbolic</a></i></dt>
<dd>there exists more than one line through <span class="texhtml mvar" style="font-style:italic;">P</span> that does not meet <span class="texhtml mvar" style="font-style:italic;">l</span></dd></dl></dd></dl></dd></dl>
<p>The parallel property of elliptic geometry is the key idea that leads to the principle of projective duality, possibly the most important property that all projective geometries have in common.
</p>
<div class="mw-heading mw-heading2"><h2 id="Duality">Duality</h2></div>
<div role="note" class="hatnote navigation-not-searchable">Further information: <a href="Duality_(projective_geometry)" title="Duality (projective geometry)">Duality (projective geometry)</a></div>
<p>In 1825, <a href="Joseph_Gergonne" class="mw-redirect" title="Joseph Gergonne">Joseph Gergonne</a> noted the principle of <a href="Duality_(projective_geometry)" title="Duality (projective geometry)">duality</a> characterizing projective plane geometry: given any theorem or definition of that geometry, substituting <i>point</i> for <i>line</i>, <i>lie on</i> for <i>pass through</i>, <i>collinear</i> for <i>concurrent</i>, <i>intersection</i> for <i>join</i>, or vice versa, results in another theorem or valid definition, the "dual" of the first. Similarly in 3 dimensions, the duality relation holds between points and planes, allowing any theorem to be transformed by swapping <i>point</i> and <i>plane</i>, <i>is contained by</i> and <i>contains</i>. More generally, for projective spaces of dimension N, there is a duality between the subspaces of dimension <i>R</i> and dimension <span class="nowrap"><i>N</i> − <i>R</i> − 1</span>. For <span class="nowrap"><i>N</i> = 2</span>, this specializes to the most commonly known form of duality—that between points and lines.
The duality principle was also discovered independently by <a href="Jean-Victor_Poncelet" title="Jean-Victor Poncelet">Jean-Victor Poncelet</a>.
</p><p>To establish duality only requires establishing theorems which are the dual versions of the axioms for the dimension in question. Thus, for 3-dimensional spaces, one needs to show that (1*) every point lies in 3 distinct planes, (2*) every two planes intersect in a unique line and a dual version of (3*) to the effect: if the intersection of plane P and Q is coplanar with the intersection of plane R and S, then so are the respective intersections of planes P and R, Q and S (assuming planes P and S are distinct from Q and R).
</p><p>In practice, the principle of duality allows us to set up a <i>dual correspondence</i> between two geometric constructions. The most famous of these is the polarity or reciprocity of two figures in a <a href="Conic" class="mw-redirect" title="Conic">conic</a> curve (in 2 dimensions) or a quadric surface (in 3 dimensions). A commonplace example is found in the reciprocation of a symmetrical <a href="Polyhedron" title="Polyhedron">polyhedron</a> in a concentric sphere to obtain the dual polyhedron.
</p><p>Another example is <a href="Brianchon's_theorem" title="Brianchon's theorem">Brianchon's theorem</a>, the dual of the already mentioned <a href="Pascal's_theorem" title="Pascal's theorem">Pascal's theorem</a>, and one of whose proofs simply consists of applying the principle of duality to Pascal's. Here are comparative statements of these two theorems (in both cases within the framework of the projective plane):
</p>
<ul><li><b>Pascal:</b> If all six vertices of a hexagon lie on a <a href="Conic_section#In_the_real_projective_plane" title="Conic section">conic</a>, then the intersections of its opposite sides <i>(regarded as full lines, since in the projective plane there is no such thing as a "line segment")</i> are three collinear points. The line joining them is then called the <b>Pascal line</b> of the hexagon.</li>
<li><b>Brianchon:</b> If all six sides of a hexagon are tangent to a conic, then its diagonals (i.e. the lines joining opposite vertices) are three concurrent lines. Their point of intersection is then called the <b>Brianchon point</b> of the hexagon.</li></ul>
<dl><dd>(If the conic degenerates into two straight lines, Pascal's becomes <a href="Pappus's_hexagon_theorem" title="Pappus's hexagon theorem">Pappus's theorem</a>, which has no interesting dual, since the Brianchon point trivially becomes the two lines' intersection point.)</dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Axioms_of_projective_geometry">Axioms of projective geometry</h2></div>
<p>Any given geometry may be deduced from an appropriate set of <a href="Axiom" title="Axiom">axioms</a>. Projective geometries are characterised by the "elliptic parallel" axiom, that <i>any two planes always meet in just one line</i>, or in the plane, <i>any two lines always meet in just one point</i>. In other words, there are no such things as parallel lines or planes in projective geometry.
</p><p>Many alternative sets of axioms for projective geometry have been proposed (see for example Coxeter 2003, Hilbert & Cohn-Vossen 1999, Greenberg 1980).
</p>
<div class="mw-heading mw-heading3"><h3 id="Whitehead's_axioms">Whitehead's axioms</h3></div>
<p>These axioms are based on <a href="Alfred_North_Whitehead" title="Alfred North Whitehead">Whitehead</a>, "The Axioms of Projective Geometry". There are two types, points and lines, and one "incidence" relation between points and lines. The three axioms are:
</p>
<ul><li>G1: Every line contains at least 3 points</li>
<li>G2: Every two distinct points, A and B, lie on a unique line, AB.</li>
<li>G3: If lines AB and CD intersect, then so do lines AC and BD (where it is assumed that A and D are distinct from B and C).</li></ul>
<p>The reason each line is assumed to contain at least 3 points is to eliminate some degenerate cases. The spaces satisfying these
three axioms either have at most one line, or are projective spaces of some dimension over a <a href="Division_ring" title="Division ring">division ring</a>, or are <a href="Non-Desarguesian_plane" title="Non-Desarguesian plane">non-Desarguesian planes</a>.
</p>
<div class="mw-heading mw-heading3"><h3 id="Additional_axioms">Additional axioms</h3></div>
<p>One can add further axioms restricting the dimension or the coordinate ring. For example, Coxeter's <i>Projective Geometry</i>,<sup id="cite_ref-FOOTNOTECoxeter200314–15_14-0" class="reference"><a href="#cite_note-FOOTNOTECoxeter200314–15-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> references Veblen<sup id="cite_ref-FOOTNOTEVeblenYoung193816,_18,_24,_45_15-0" class="reference"><a href="#cite_note-FOOTNOTEVeblenYoung193816,_18,_24,_45-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> in the three axioms above, together with a further 5 axioms that make the dimension 3 and the coordinate ring a commutative field of characteristic not 2.
</p>
<div class="mw-heading mw-heading3"><h3 id="Axioms_using_a_ternary_relation">Axioms using a ternary relation</h3></div>
<p>One can pursue axiomatization by postulating a ternary relation, [ABC] to denote when three points (not all necessarily distinct) are collinear. An axiomatization may be written down in terms of this relation as well:
</p>
<ul><li>C0: [ABA]</li>
<li>C1: If A and B are distinct points such that [ABC] and [ABD] then [BDC]</li>
<li>C2: If A and B are distinct points then there exists a third distinct point C such that [ABC]</li>
<li>C3: If A and C are distinct points, and B and D are distinct points, with [BCE] and [ADE] but not [ABE], then there is a point F such that [ACF] and [BDF].</li></ul>
<p>For two distinct points, A and B, the line AB is defined as consisting of all points C for which [ABC]. The axioms C0 and C1 then provide a formalization of G2; C2 for G1 and C3 for G3.
</p><p>The concept of line generalizes to planes and higher-dimensional subspaces. A subspace, AB...XY may thus be recursively defined in terms of the subspace AB...X as that containing all the points of all lines YZ, as Z ranges over AB...X. Collinearity then generalizes to the relation of "independence". A set <span class="nowrap">{A, B, ..., Z}</span> of points is independent, [AB...Z] if <span class="nowrap">{A, B, ..., Z}</span> is a minimal generating subset for the subspace AB...Z.
</p><p>The projective axioms may be supplemented by further axioms postulating limits on the dimension of the space. The minimum dimension is determined by the existence of an independent set of the required size. For the lowest dimensions, the relevant conditions may be stated in equivalent
form as follows. A projective space is of:
</p>
<ul><li>(L1) at least dimension 0 if it has at least 1 point,</li>
<li>(L2) at least dimension 1 if it has at least 2 distinct points (and therefore a line),</li>
<li>(L3) at least dimension 2 if it has at least 3 non-collinear points (or two lines, or a line and a point not on the line),</li>
<li>(L4) at least dimension 3 if it has at least 4 non-coplanar points.</li></ul>
<p>The maximum dimension may also be determined in a similar fashion. For the lowest dimensions, they take on the following forms. A projective space is of:
</p>
<ul><li>(M1) at most dimension 0 if it has no more than 1 point,</li>
<li>(M2) at most dimension 1 if it has no more than 1 line,</li>
<li>(M3) at most dimension 2 if it has no more than 1 plane,</li></ul>
<p>and so on. It is a general theorem (a consequence of axiom (3)) that all coplanar lines intersect—the very principle that projective geometry was originally intended to embody. Therefore, property (M3) may be equivalently stated that all lines intersect one another.
</p><p>It is generally assumed that projective spaces are of at least dimension 2. In some cases, if the focus is on projective planes, a variant of M3 may be postulated. The axioms of (Eves 1997: 111), for instance, include (1), (2), (L3) and (M3). Axiom (3) becomes vacuously true under (M3) and is therefore not needed in this context.
</p>
<div class="mw-heading mw-heading3"><h3 id="Axioms_for_projective_planes">Axioms for projective planes</h3></div>
<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Projective_plane" title="Projective plane">Projective plane</a></div>
<p>In <a href="Incidence_geometry" title="Incidence geometry">incidence geometry</a>, most authors<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> give a treatment that embraces the <a href="Fano_plane" title="Fano plane">Fano plane</a> <span class="nowrap">PG(2, 2)</span> as the smallest finite projective plane. An axiom system that achieves this is as follows:
</p>
<ul><li>(P1) Any two distinct points lie on a line that is unique.</li>
<li>(P2) Any two distinct lines meet at a point that is unique.</li>
<li>(P3) There exist at least four points of which no three are collinear.</li></ul>
<p>Coxeter's <i>Introduction to Geometry</i><sup id="cite_ref-FOOTNOTECoxeter1969229–234_17-0" class="reference"><a href="#cite_note-FOOTNOTECoxeter1969229–234-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> gives a list of five axioms for a more restrictive concept of a projective plane that is attributed to Bachmann, adding <a href="Pappus's_hexagon_theorem" title="Pappus's hexagon theorem">Pappus's theorem</a> to the list of axioms above (which eliminates <a href="Non-Desarguesian_plane" title="Non-Desarguesian plane">non-Desarguesian planes</a>) and excluding projective planes over fields of characteristic 2 (those that do not satisfy <a href="Fano's_axiom" class="mw-redirect" title="Fano's axiom">Fano's axiom</a>). The restricted planes given in this manner more closely resemble the <a href="Real_projective_plane" title="Real projective plane">real projective plane</a>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Perspectivity_and_projectivity">Perspectivity and projectivity</h2></div>
<p>Given three non-<a href="Collinear" class="mw-redirect" title="Collinear">collinear</a> points, there are three lines connecting them, but with four points, no three collinear, there are six connecting lines and three additional "diagonal points" determined by their intersections. The science of projective geometry captures this surplus determined by four points through a quaternary relation and the projectivities which preserve the <a href="Complete_quadrangle" title="Complete quadrangle">complete quadrangle</a> configuration.
</p><p>An <a href="Harmonic_quadruple" class="mw-redirect" title="Harmonic quadruple">harmonic quadruple</a> of points on a line occurs when there is a complete quadrangle two of whose diagonal points are in the first and third position of the quadruple, and the other two positions are points on the lines joining two quadrangle points through the third diagonal point.<sup id="cite_ref-FOOTNOTEHalsted190615,_16_18-0" class="reference"><a href="#cite_note-FOOTNOTEHalsted190615,_16-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup>
</p><p>A spatial <a href="Perspectivity" title="Perspectivity">perspectivity</a> of a <a href="Projective_configuration" class="mw-redirect" title="Projective configuration">projective configuration</a> in one plane yields such a configuration in another, and this applies to the configuration of the complete quadrangle. Thus harmonic quadruples are preserved by perspectivity. If one perspectivity follows another the configurations follow along. The composition of two perspectivities is no longer a perspectivity, but a <b>projectivity</b>.
</p><p>While corresponding points of a perspectivity all converge at a point, this convergence is <i>not</i> true for a projectivity that is <i>not</i> a perspectivity. In projective geometry the intersection of lines formed by corresponding points of a projectivity in a plane are of particular interest. The set of such intersections is called a <b>projective conic</b>, and in acknowledgement of the work of <a href="Jakob_Steiner" title="Jakob Steiner">Jakob Steiner</a>, it is referred to as a <a href="Steiner_conic" title="Steiner conic">Steiner conic</a>.
</p><p>Suppose a projectivity is formed by two perspectivities centered on points <i>A</i> and <i>B</i>, relating <i>x</i> to <i>X</i> by an intermediary <i>p</i>:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\ {\overset {A}{\doublebarwedge }}\ p\ {\overset {B}{\doublebarwedge }}\ X.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
<mtext> </mtext>
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mo>⩞<!-- ⩞ --></mo>
<mi>A</mi>
</mover>
</mrow>
<mtext> </mtext>
<mi>p</mi>
<mtext> </mtext>
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mo>⩞<!-- ⩞ --></mo>
<mi>B</mi>
</mover>
</mrow>
<mtext> </mtext>
<mi>X</mi>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x\ {\overset {A}{\doublebarwedge }}\ p\ {\overset {B}{\doublebarwedge }}\ X.}</annotation>
</semantics>
</math></span><img src="./2e7a2e48199993c482d0f25b04b7d6fbb932ac0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.289ex; height:4.676ex;" alt="{\displaystyle x\ {\overset {A}{\doublebarwedge }}\ p\ {\overset {B}{\doublebarwedge }}\ X.}" loading="lazy"></span></dd></dl>
<p>The projectivity is then <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\ \barwedge \ X.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
<mtext> </mtext>
<mo>⊼<!-- ⊼ --></mo>
<mtext> </mtext>
<mi>X</mi>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x\ \barwedge \ X.}</annotation>
</semantics>
</math></span><img src="./7c8ad5bc35106903ce442079c36981c98a871350.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.57ex; height:2.176ex;" alt="{\displaystyle x\ \barwedge \ X.}" loading="lazy"></span> Then given the projectivity <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \barwedge }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>⊼<!-- ⊼ --></mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \barwedge }</annotation>
</semantics>
</math></span><img src="./594efe4b2136418958bd9587ce8a583e93e024a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\displaystyle \barwedge }" loading="lazy"></span> the induced conic is
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C(\barwedge )\ =\ \bigcup \{xX\cdot yY:x\barwedge X\ \ \land \ \ y\barwedge Y\}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>C</mi>
<mo stretchy="false">(</mo>
<mo>⊼<!-- ⊼ --></mo>
<mo stretchy="false">)</mo>
<mtext> </mtext>
<mo>=</mo>
<mtext> </mtext>
<mo>⋃<!-- ⋃ --></mo>
<mo fence="false" stretchy="false">{</mo>
<mi>x</mi>
<mi>X</mi>
<mo>⋅<!-- ⋅ --></mo>
<mi>y</mi>
<mi>Y</mi>
<mo>:</mo>
<mi>x</mi>
<mo>⊼<!-- ⊼ --></mo>
<mi>X</mi>
<mtext> </mtext>
<mtext> </mtext>
<mo>∧<!-- ∧ --></mo>
<mtext> </mtext>
<mtext> </mtext>
<mi>y</mi>
<mo>⊼<!-- ⊼ --></mo>
<mi>Y</mi>
<mo fence="false" stretchy="false">}</mo>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle C(\barwedge )\ =\ \bigcup \{xX\cdot yY:x\barwedge X\ \ \land \ \ y\barwedge Y\}.}</annotation>
</semantics>
</math></span><img src="./cb0d3a8a8fb958ea3c3b9d5ac435db05a64207fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:41.1ex; height:3.843ex;" alt="{\displaystyle C(\barwedge )\ =\ \bigcup \{xX\cdot yY:x\barwedge X\ \ \land \ \ y\barwedge Y\}.}" loading="lazy"></span></dd></dl>
<p>Given a conic <i>C</i> and a point <i>P</i> not on it, two distinct <a href="Secant_line" title="Secant line">secant lines</a> through <i>P</i> intersect <i>C</i> in four points. These four points determine a quadrangle of which <i>P</i> is a diagonal point. The line through the other two diagonal points is called the <a href="Pole_and_polar" title="Pole and polar">polar of <i>P</i></a> and <i>P </i>is the <b>pole</b> of this line.<sup id="cite_ref-FOOTNOTEHalsted190625_19-0" class="reference"><a href="#cite_note-FOOTNOTEHalsted190625-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> Alternatively, the polar line of <i>P</i> is the set of <a href="Projective_harmonic_conjugate" title="Projective harmonic conjugate">projective harmonic conjugates</a> of <i>P</i> on a variable secant line passing through <i>P</i> and <i>C</i>.
</p>
<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
<div class="div-col" style="column-width: 22em;">
<ul><li><a href="Projective_line" title="Projective line">Projective line</a></li>
<li><a href="Projective_plane" title="Projective plane">Projective plane</a></li>
<li><a href="Incidence_(mathematics)" class="mw-redirect" title="Incidence (mathematics)">Incidence (mathematics)</a></li>
<li><a href="Fundamental_theorem_of_projective_geometry" class="mw-redirect" title="Fundamental theorem of projective geometry">Fundamental theorem of projective geometry</a></li>
<li><a href="Projective_line_over_a_ring" title="Projective line over a ring">Projective line over a ring</a></li>
<li><a href="Grassmann%E2%80%93Cayley_algebra" title="Grassmann–Cayley algebra">Grassmann–Cayley algebra</a></li></ul>
</div>
<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>
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<ol class="references">
<li id="cite_note-FOOTNOTERamanan199788-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTERamanan199788_1-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFRamanan1997">Ramanan 1997</a>, p. 88.</span>
</li>
<li id="cite_note-FOOTNOTECoxeter2003v-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTECoxeter2003v_2-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFCoxeter2003">Coxeter 2003</a>, p. v.</span>
</li>
<li id="cite_note-FOOTNOTECoxeter1969229-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTECoxeter1969229_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTECoxeter1969229_3-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-FOOTNOTECoxeter1969229_3-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-FOOTNOTECoxeter1969229_3-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFCoxeter1969">Coxeter 1969</a>, p. 229.</span>
</li>
<li id="cite_note-FOOTNOTECoxeter200314-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTECoxeter200314_4-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFCoxeter2003">Coxeter 2003</a>, p. 14.</span>
</li>
<li id="cite_note-FOOTNOTECoxeter196993,_261-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTECoxeter196993,_261_5-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFCoxeter1969">Coxeter 1969</a>, pp. 93, 261.</span>
</li>
<li id="cite_note-FOOTNOTECoxeter1969234–238-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTECoxeter1969234–238_6-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFCoxeter1969">Coxeter 1969</a>, pp. 234–238.</span>
</li>
<li id="cite_note-FOOTNOTECoxeter2003111–132-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTECoxeter2003111–132_7-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFCoxeter2003">Coxeter 2003</a>, pp. 111–132.</span>
</li>
<li id="cite_note-FOOTNOTECoxeter1969175–262-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTECoxeter1969175–262_8-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFCoxeter1969">Coxeter 1969</a>, pp. 175–262.</span>
</li>
<li id="cite_note-FOOTNOTECoxeter2003102–110-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTECoxeter2003102–110_9-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFCoxeter2003">Coxeter 2003</a>, pp. 102–110.</span>
</li>
<li id="cite_note-FOOTNOTECoxeter20032-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTECoxeter20032_10-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFCoxeter2003">Coxeter 2003</a>, p. 2.</span>
</li>
<li id="cite_note-FOOTNOTECoxeter20033-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTECoxeter20033_11-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFCoxeter2003">Coxeter 2003</a>, p. 3.</span>
</li>
<li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><a href="John_Milnor" title="John Milnor">John Milnor</a> (1982) <a rel="nofollow" class="external text" href="https://projecteuclid.org/euclid.bams/1183548588">Hyperbolic geometry: The first 150 years</a>, <a href="Bulletin_of_the_American_Mathematical_Society" title="Bulletin of the American Mathematical Society">Bulletin of the American Mathematical Society</a> via <a href="Project_Euclid" title="Project Euclid">Project Euclid</a></span>
</li>
<li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">
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</style><cite id="CITEREFFarmelo2005" class="citation journal cs1">Farmelo, Graham (September 15, 2005). <a rel="nofollow" class="external text" href="https://www.nature.com/articles/437323a.pdf">"Dirac's hidden geometry"</a> <span class="cs1-format">(PDF)</span>. Essay. <i><a href="Nature_(journal)" title="Nature (journal)">Nature</a></i>. <b>437</b> (7057). Nature Publishing Group: 323. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2005Natur.437..323F">2005Natur.437..323F</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2F437323a">10.1038/437323a</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/16163331">16163331</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:34940597">34940597</a>.</cite></span>
</li>
<li id="cite_note-FOOTNOTECoxeter200314–15-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTECoxeter200314–15_14-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFCoxeter2003">Coxeter 2003</a>, pp. 14–15.</span>
</li>
<li id="cite_note-FOOTNOTEVeblenYoung193816,_18,_24,_45-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEVeblenYoung193816,_18,_24,_45_15-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFVeblenYoung1938">Veblen & Young 1938</a>, pp. 16, 18, 24, 45.</span>
</li>
<li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><a href="#CITEREFBennett1995">Bennett 1995</a>, p. 4, <a href="#CITEREFBeutelspacherRosenbaum1998">Beutelspacher & Rosenbaum 1998</a>, p. 8, <a href="#CITEREFCasse2006">Casse 2006</a>, p. 29, <a href="#CITEREFCederberg2001">Cederberg 2001</a>, p. 9, <a href="#CITEREFGarner1981">Garner 1981</a>, p. 7, <a href="#CITEREFHughesPiper1973">Hughes & Piper 1973</a>, p. 77, <a href="#CITEREFMihalek1972">Mihalek 1972</a>, p. 29, <a href="#CITEREFPolster1998">Polster 1998</a>, p. 5 and <a href="#CITEREFSamuel1988">Samuel 1988</a>, p. 21 among the references given.</span>
</li>
<li id="cite_note-FOOTNOTECoxeter1969229–234-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTECoxeter1969229–234_17-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFCoxeter1969">Coxeter 1969</a>, pp. 229–234.</span>
</li>
<li id="cite_note-FOOTNOTEHalsted190615,_16-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHalsted190615,_16_18-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHalsted1906">Halsted 1906</a>, pp. 15, 16.</span>
</li>
<li id="cite_note-FOOTNOTEHalsted190625-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHalsted190625_19-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHalsted1906">Halsted 1906</a>, p. 25.</span>
</li>
</ol></div>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
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<ul><li><cite id="CITEREFBachmann2013" class="citation book cs1">Bachmann, F. (2013) [1959]. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=skGoBgAAQBAJ&pg=PR2"><i>Aufbau der Geometrie aus dem Spiegelungsbegriff</i></a> (2nd ed.). Springer-Verlag. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-642-65537-1</bdi>.</cite></li>
<li><cite id="CITEREFBaer2005" class="citation book cs1">Baer, Reinhold (2005). <i>Linear Algebra and Projective Geometry</i>. Mineola NY: Dover. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-486-44565-8</bdi>.</cite></li>
<li><cite id="CITEREFBennett1995" class="citation book cs1">Bennett, M.K. (1995). <i>Affine and Projective Geometry</i>. New York: Wiley. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-471-11315-8</bdi>.</cite></li>
<li><cite id="CITEREFBeutelspacherRosenbaum1998" class="citation book cs1">Beutelspacher, Albrecht; Rosenbaum, Ute (1998). <i>Projective Geometry: From Foundations to Applications</i>. Cambridge: Cambridge University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-48277-1</bdi>.</cite></li>
<li><cite id="CITEREFCasse2006" class="citation book cs1">Casse, Rey (2006). <i>Projective Geometry: An Introduction</i>. Oxford University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-19-929886-6</bdi>.</cite></li>
<li><cite id="CITEREFCederberg2001" class="citation book cs1">Cederberg, Judith N. (2001). <i>A Course in Modern Geometries</i>. Springer-Verlag. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-98972-2</bdi>.</cite></li>
<li><cite id="CITEREFCoxeter2013" class="citation book cs1"><a href="H._S._M._Coxeter" class="mw-redirect" title="H. S. M. Coxeter">Coxeter, H.S.M.</a> (2013) [1993]. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=uz3aBwAAQBAJ"><i>The Real Projective Plane</i></a> (3rd ed.). Springer Verlag. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9781461227342</bdi>.</cite></li>
<li><cite id="CITEREFCoxeter2003" class="citation book cs1">Coxeter, H.S.M. (2003). <i>Projective Geometry</i> (2nd ed.). Springer Verlag. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-40623-7</bdi>.</cite></li>
<li><cite id="CITEREFCoxeter1969" class="citation book cs1">Coxeter, H.S.M. (1969). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoge0002coxe"><i>Introduction to Geometry</i></a></span>. Wiley. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-471-50458-0</bdi>.</cite></li>
<li><cite id="CITEREFDembowski1968" class="citation book cs1">Dembowski, Peter (1968). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/finitegeometries0000demb"><i>Finite Geometries</i></a></span>. <a href="Ergebnisse_der_Mathematik_und_ihrer_Grenzgebiete" title="Ergebnisse der Mathematik und ihrer Grenzgebiete">Ergebnisse der Mathematik und ihrer Grenzgebiete</a>, Band 44. Berlin, New York: <a href="Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-540-61786-8</bdi>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0233275">0233275</a>.</cite></li>
<li><cite id="CITEREFEves2012" class="citation book cs1"><a href="Howard_Eves" title="Howard Eves">Eves, Howard</a> (2012) [1997]. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=J9QcmFHj8EwC&pg=PP1"><i>Foundations and Fundamental Concepts of Mathematics</i></a> (3rd ed.). Courier Corporation. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-486-13220-4</bdi>.</cite></li>
<li><cite id="CITEREFGarner1981" class="citation book cs1">Garner, Lynn E. (1981). <i>An Outline of Projective Geometry</i>. North Holland. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-444-00423-8</bdi>.</cite></li>
<li><cite id="CITEREFGreenberg2008" class="citation book cs1">Greenberg, M.J. (2008). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=4uw0dwi7bmQC"><i>Euclidean and Non-Euclidean Geometries: Development and History</i></a> (4th ed.). W. H. Freeman. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4292-8133-1</bdi>.</cite></li>
<li><cite id="CITEREFHalsted1906" class="citation book cs1"><a href="G._B._Halsted" title="G. B. Halsted">Halsted, G. B.</a> (1906). <a rel="nofollow" class="external text" href="https://archive.org/details/syntheticproject00halsuoft/page/n3/mode/2up"><i>Synthetic Projective Geometry</i></a>. New York Wiley.</cite></li>
<li><cite id="CITEREFHartleyZisserman2003" class="citation book cs1">Hartley, Richard; Zisserman, Andrew (2003). <i>Multiple view geometry in computer vision</i> (2nd ed.). Cambridge University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-54051-8</bdi>.</cite></li>
<li><cite id="CITEREFHartshorne2009" class="citation book cs1"><a href="Robin_Hartshorne" title="Robin Hartshorne">Hartshorne, Robin</a> (2009). <i>Foundations of Projective Geometry</i> (2nd ed.). Ishi Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-4-87187-837-1</bdi>.</cite></li>
<li><cite id="CITEREFHartshorne2013" class="citation book cs1">Hartshorne, Robin (2013) [2000]. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=C5fSBwAAQBAJ"><i>Geometry: Euclid and Beyond</i></a>. Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-22676-7</bdi>.</cite></li>
<li><cite id="CITEREFHilbertCohn-Vossen1999" class="citation book cs1"><a href="David_Hilbert" title="David Hilbert">Hilbert, D.</a>; Cohn-Vossen, S. (1999). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=7WY5AAAAQBAJ"><i>Geometry and the Imagination</i></a> (2nd ed.). American Mathematical Society. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8218-1998-2</bdi>.</cite></li>
<li><cite id="refHughes1973"><cite id="CITEREFHughesPiper1973" class="citation book cs1">Hughes, D.R.; Piper, F.C. (1973). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=bKh6QgAACAAJ"><i>Projective Planes</i></a>. Springer-Verlag. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-90044-3</bdi>.</cite></cite></li>
<li><cite id="CITEREFMihalek1972" class="citation book cs1">Mihalek, R.J. (1972). <i>Projective Geometry and Algebraic Structures</i>. New York: Academic Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-12-495550-9</bdi>.</cite></li>
<li><cite id="CITEREFPolster1998" class="citation book cs1">Polster, Burkard (1998). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/geometricalpictu0000pols"><i>A Geometrical Picture Book</i></a></span>. Springer-Verlag. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-98437-2</bdi>.</cite></li>
<li><cite id="CITEREFRamanan1997" class="citation journal cs1">Ramanan, S. (August 1997). "Projective geometry". <i>Resonance</i>. <b>2</b> (8). Springer India: <span class="nowrap">87–</span>94. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02835009">10.1007/BF02835009</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0971-8044">0971-8044</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:195303696">195303696</a>.</cite></li>
<li><cite id="CITEREFSamuel1988" class="citation book cs1">Samuel, Pierre (1988). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/projectivegeomet0000samu"><i>Projective Geometry</i></a></span>. Springer-Verlag. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-96752-4</bdi>.</cite></li>
<li><a href="Luis_Santal%C3%B3" title="Luis Santaló">Santaló, Luis</a> (1966) <i>Geometría proyectiva</i>, Editorial Universitaria de Buenos Aires</li>
<li><cite id="CITEREFVeblenYoung1938" class="citation book cs1">Veblen, Oswald; Young, J. W. A. (1938). <a rel="nofollow" class="external text" href="https://archive.org/details/117714799_001"><i>Projective Geometry</i></a>. Boston: Ginn & Co. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4181-8285-4</bdi>.</cite> <span class="cs1-hidden-error citation-comment"><code class="cs1-code">{{cite book}}</code>: </span><span class="cs1-hidden-error citation-comment">ISBN / Date incompatibility (help)</span></li></ul>
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<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>
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<div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Projective_geometry" class="extiw external" title="commons:Category:Projective geometry">Projective geometry</a></span>.</div></div>
</div>
<ul><li><a rel="nofollow" class="external text" href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.17.1329">Projective Geometry for Machine Vision</a> — tutorial by Joe Mundy and Andrew Zisserman.</li>
<li><a rel="nofollow" class="external text" href="http://xahlee.info/projective_geometry/projective_geometry.html">Notes</a> based on Coxeter's <i>The Real Projective Plane</i>.</li>
<li><a rel="nofollow" class="external text" href="http://lear.inrialpes.fr/people/triggs/pubs/isprs96/isprs96.html">Projective Geometry for Image Analysis</a> — free tutorial by Roger Mohr and Bill Triggs.</li>
<li><a rel="nofollow" class="external text" href="http://www.geometer.org/mathcircles/projective.pdf">Projective Geometry.</a> — free tutorial by Tom Davis.</li>
<li><a rel="nofollow" class="external text" href="http://neo-classical-physics.info/uploads/3/0/6/5/3065888/burali-forti_-_grassman_and_proj._geom..pdf">The Grassmann method in projective geometry</a> A compilation of three notes by Cesare Burali-Forti on the application of exterior algebra to projective geometry</li>
<li><a rel="nofollow" class="external text" href="http://neo-classical-physics.info/uploads/3/0/6/5/3065888/burali-forti_-_diff._geom._following_grassmann.pdf">C. Burali-Forti, "Introduction to Differential Geometry, following the method of H. Grassmann"</a> (English translation of book)</li>
<li><a rel="nofollow" class="external text" href="http://neo-classical-physics.info/uploads/3/0/6/5/3065888/kummer_-_rectilinear_ray_systems.pdf">E. Kummer, "General theory of rectilinear ray systems"</a> (English translation)</li>
<li><a rel="nofollow" class="external text" href="http://neo-classical-physics.info/uploads/3/0/6/5/3065888/pasch_-_focal_and_singularity_surfaces.pdf">M. Pasch, "On the focal surfaces of ray systems and the singularity surfaces of complexes"</a> (English translation)</li></ul>
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</style></div><div role="navigation" class="navbox" aria-labelledby="Mathematics_and_art295" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3"><div id="Mathematics_and_art295" style="font-size:114%;margin:0 4em"><a href="Mathematics_and_art" title="Mathematics and art">Mathematics and art</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Algorithm" title="Algorithm">Algorithm</a></li>
<li><a href="Catenary" title="Catenary">Catenary</a></li>
<li><a href="Fractal" title="Fractal">Fractal</a></li>
<li><a href="Golden_ratio" title="Golden ratio">Golden ratio</a></li>
<li><a href="Hyperboloid_structure" title="Hyperboloid structure">Hyperboloid structure</a></li>
<li><a href="Minimal_surface" title="Minimal surface">Minimal surface</a></li>
<li><a href="Paraboloid" title="Paraboloid">Paraboloid</a></li>
<li><a href="Perspective_(graphical)" title="Perspective (graphical)">Perspective</a>
<ul><li><a href="Camera_lucida" title="Camera lucida">Camera lucida</a></li>
<li><a href="Camera_obscura" title="Camera obscura">Camera obscura</a></li></ul></li>
<li><a href="Plastic_ratio" title="Plastic ratio">Plastic ratio</a></li>
<li>Proportion
<ul><li><a href="Proportion_(architecture)" title="Proportion (architecture)">Architecture</a></li>
<li><a href="Body_proportions" title="Body proportions">Human</a></li></ul></li>
<li><a href="Symmetry" title="Symmetry">Symmetry</a></li>
<li><a href="Tessellation" title="Tessellation">Tessellation</a></li>
<li><a href="Wallpaper_group" title="Wallpaper group">Wallpaper group</a></li></ul>
</div></td><td class="noviewer navbox-image" rowspan="9" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Forms</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Algorithmic_art" title="Algorithmic art">Algorithmic art</a></li>
<li><a href="Anamorphosis" title="Anamorphosis">Anamorphic art</a></li>
<li><a href="Mathematics_and_architecture" title="Mathematics and architecture">Architecture</a>
<ul><li><a href="Geodesic_dome" title="Geodesic dome">Geodesic dome</a></li>
<li><a href="Pyramid" title="Pyramid">Pyramid</a></li>
<li><a href="Vastu_shastra" title="Vastu shastra">Vastu shastra</a></li></ul></li>
<li><a href="Computer_art" title="Computer art">Computer art</a></li>
<li><a href="Mathematics_and_fiber_arts" title="Mathematics and fiber arts">Fiber arts</a></li>
<li><a href="Fourth_dimension_in_art" title="Fourth dimension in art">4D art</a></li>
<li><a href="Fractal_art" title="Fractal art">Fractal art</a></li>
<li><a href="Islamic_geometric_patterns" title="Islamic geometric patterns">Islamic geometric patterns</a>
<ul><li><a href="Girih" title="Girih">Girih</a></li>
<li><a href="Jali" title="Jali">Jali</a></li>
<li><a href="Muqarnas" title="Muqarnas">Muqarnas</a></li>
<li><a href="Zellij" title="Zellij">Zellij</a></li></ul></li>
<li><a href="Knot" title="Knot">Knotting</a>
<ul><li><a href="Celtic_knot" title="Celtic knot">Celtic knot</a></li>
<li><a href="Croatian_interlace" title="Croatian interlace">Croatian interlace</a></li>
<li><a href="Interlace_(art)" title="Interlace (art)">Interlace</a></li></ul></li>
<li><a href="Music_and_mathematics" title="Music and mathematics">Music</a></li>
<li><a href="Origami" title="Origami">Origami</a>
<ul><li><a href="Mathematics_of_paper_folding" title="Mathematics of paper folding">Mathematics</a></li></ul></li>
<li><a href="Mathematical_sculpture" title="Mathematical sculpture">Sculpture</a></li>
<li><a href="String_art" title="String art">String art</a></li>
<li><a href="String_figure" title="String figure">String figure</a></li>
<li><a href="Tessellation" title="Tessellation">Tiling</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Artworks</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="List_of_works_designed_with_the_golden_ratio" title="List of works designed with the golden ratio">List of works designed with the golden ratio</a></li>
<li><i><a href="Continuum_(sculpture)" title="Continuum (sculpture)">Continuum</a></i></li>
<li><i><a href="Mathemalchemy" title="Mathemalchemy">Mathemalchemy</a></i></li>
<li><i><a href="Mathematica%3A_A_World_of_Numbers..._and_Beyond" title="Mathematica: A World of Numbers... and Beyond">Mathematica: A World of Numbers... and Beyond</a></i></li>
<li><i><a href="Octacube_(sculpture)" title="Octacube (sculpture)">Octacube</a></i></li>
<li><i><a href="Pi_(art_project)" title="Pi (art project)">Pi</a></i></li>
<li><i><a href="Pi_in_the_Sky" title="Pi in the Sky">Pi in the Sky</a></i></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Mathematics_and_architecture" title="Mathematics and architecture">Buildings</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Cathedral_of_Saint_Mary_of_the_Assumption_(San_Francisco)" title="Cathedral of Saint Mary of the Assumption (San Francisco)">Cathedral of Saint Mary of the Assumption</a></li>
<li><a href="Hagia_Sophia" title="Hagia Sophia">Hagia Sophia</a></li>
<li><a href="Kresge_Auditorium" title="Kresge Auditorium">Kresge Auditorium</a></li>
<li><a href="Pantheon%2C_Rome" title="Pantheon, Rome">Pantheon</a></li>
<li><a href="Parthenon" title="Parthenon">Parthenon</a></li>
<li><a href="Great_Pyramid_of_Giza" title="Great Pyramid of Giza">Pyramid of Khufu</a></li>
<li><a href="Sagrada_Fam%C3%ADlia" title="Sagrada Família">Sagrada Família</a></li>
<li><a href="Sydney_Opera_House" title="Sydney Opera House">Sydney Opera House</a></li>
<li><a href="Taj_Mahal" title="Taj Mahal">Taj Mahal</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="List_of_mathematical_artists" title="List of mathematical artists">Artists</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Renaissance</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Paolo_Uccello" title="Paolo Uccello">Paolo Uccello</a></li>
<li><a href="Piero_della_Francesca" title="Piero della Francesca">Piero della Francesca</a></li>
<li><a href="Leonardo_da_Vinci" title="Leonardo da Vinci">Leonardo da Vinci</a>
<ul><li><i><a href="Vitruvian_Man" title="Vitruvian Man">Vitruvian Man</a></i></li></ul></li>
<li><a href="Albrecht_D%C3%BCrer" title="Albrecht Dürer">Albrecht Dürer</a></li>
<li><a href="Parmigianino" title="Parmigianino">Parmigianino</a>
<ul><li><i><a href="Self-portrait_in_a_Convex_Mirror" class="mw-redirect" title="Self-portrait in a Convex Mirror">Self-portrait in a Convex Mirror</a></i></li></ul></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">19th–20th<br>Century</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="William_Blake" title="William Blake">William Blake</a>
<ul><li><i><a href="The_Ancient_of_Days" title="The Ancient of Days">The Ancient of Days</a></i></li>
<li><i><a href="Newton_(Blake)" title="Newton (Blake)">Newton</a></i></li></ul></li>
<li><a href="Jean_Metzinger" title="Jean Metzinger">Jean Metzinger</a>
<ul><li><i><a href="Dancer_in_a_Caf%C3%A9" title="Dancer in a Café">Danseuse au café</a></i></li>
<li><i><a href="L'Oiseau_bleu_(Metzinger)" class="mw-redirect" title="L'Oiseau bleu (Metzinger)">L'Oiseau bleu</a></i></li></ul></li>
<li><a href="Giorgio_de_Chirico" title="Giorgio de Chirico">Giorgio de Chirico</a></li>
<li><a href="Man_Ray" title="Man Ray">Man Ray</a></li>
<li><a href="M._C._Escher" title="M. C. Escher">M. C. Escher</a>
<ul><li><i><a href="Circle_Limit_III" title="Circle Limit III">Circle Limit III</a></i></li>
<li><i><a href="Print_Gallery_(M._C._Escher)" title="Print Gallery (M. C. Escher)">Print Gallery</a></i></li>
<li><i><a href="Relativity_(M._C._Escher)" title="Relativity (M. C. Escher)">Relativity</a></i></li>
<li><i><a href="Reptiles_(M._C._Escher)" title="Reptiles (M. C. Escher)">Reptiles</a></i></li>
<li><i><a href="Waterfall_(M._C._Escher)" title="Waterfall (M. C. Escher)">Waterfall</a></i></li></ul></li>
<li><a href="Ren%C3%A9_Magritte" title="René Magritte">René Magritte</a>
<ul><li><i><a href="The_Human_Condition_(Magritte)" title="The Human Condition (Magritte)">La condition humaine</a></i></li></ul></li>
<li><a href="Salvador_Dal%C3%AD" title="Salvador Dalí">Salvador Dalí</a>
<ul><li><i><a href="Crucifixion_(Corpus_Hypercubus)" title="Crucifixion (Corpus Hypercubus)">Crucifixion</a></i></li>
<li><i><a href="The_Swallow's_Tail" title="The Swallow's Tail">The Swallow's Tail</a></i></li></ul></li>
<li><a href="Crockett_Johnson" title="Crockett Johnson">Crockett Johnson</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Contemporary</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Max_Bill" title="Max Bill">Max Bill</a></li>
<li><a href="Martin_Demaine" title="Martin Demaine">Martin</a> and <a href="Erik_Demaine" title="Erik Demaine">Erik Demaine</a></li>
<li><a href="Scott_Draves" title="Scott Draves">Scott Draves</a></li>
<li><a href="Jan_Dibbets" title="Jan Dibbets">Jan Dibbets</a></li>
<li><a href="John_Ernest" title="John Ernest">John Ernest</a></li>
<li><a href="Helaman_Ferguson" title="Helaman Ferguson">Helaman Ferguson</a></li>
<li><a href="Peter_Forakis" title="Peter Forakis">Peter Forakis</a></li>
<li><a href="Susan_Goldstine" title="Susan Goldstine">Susan Goldstine</a></li>
<li><a href="Bathsheba_Grossman" title="Bathsheba Grossman">Bathsheba Grossman</a></li>
<li><a href="George_W._Hart" title="George W. Hart">George W. Hart</a></li>
<li><a href="Desmond_Paul_Henry" title="Desmond Paul Henry">Desmond Paul Henry</a></li>
<li><a href="Anthony_Hill_(artist)" title="Anthony Hill (artist)">Anthony Hill</a></li>
<li><a href="Charles_Jencks" title="Charles Jencks">Charles Jencks</a>
<ul><li><i><a href="Garden_of_Cosmic_Speculation" title="Garden of Cosmic Speculation">Garden of Cosmic Speculation</a></i></li></ul></li>
<li><a href="Andy_Lomas" title="Andy Lomas">Andy Lomas</a></li>
<li><a href="Robert_Longhurst" title="Robert Longhurst">Robert Longhurst</a></li>
<li><a href="Jeanette_McLeod" title="Jeanette McLeod">Jeanette McLeod</a></li>
<li><a href="Hamid_Naderi_Yeganeh" title="Hamid Naderi Yeganeh">Hamid Naderi Yeganeh</a></li>
<li><a href="Istv%C3%A1n_Orosz" title="István Orosz">István Orosz</a></li>
<li><a href="Hinke_Osinga" title="Hinke Osinga">Hinke Osinga</a></li>
<li><a href="Antoine_Pevsner" title="Antoine Pevsner">Antoine Pevsner</a></li>
<li><a href="Tony_Robbin" title="Tony Robbin">Tony Robbin</a></li>
<li><a href="Alba_Rojo_Cama" title="Alba Rojo Cama">Alba Rojo Cama</a></li>
<li><a href="Reza_Sarhangi" class="mw-redirect" title="Reza Sarhangi">Reza Sarhangi</a></li>
<li><a href="Oliver_Sin" title="Oliver Sin">Oliver Sin</a></li>
<li><a href="Hiroshi_Sugimoto" title="Hiroshi Sugimoto">Hiroshi Sugimoto</a></li>
<li><a href="Daina_Taimi%C5%86a" title="Daina Taimiņa">Daina Taimiņa</a></li>
<li><a href="Roman_Verostko" title="Roman Verostko">Roman Verostko</a></li>
<li><a href="Margaret_Wertheim" title="Margaret Wertheim">Margaret Wertheim</a></li></ul>
</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theorists</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Ancient</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Polykleitos" title="Polykleitos">Polykleitos</a>
<ul><li><i>Canon</i></li></ul></li>
<li><a href="Vitruvius" title="Vitruvius">Vitruvius</a>
<ul><li><i><a href="De_architectura" title="De architectura">De architectura</a></i></li></ul></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Renaissance</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Filippo_Brunelleschi" title="Filippo Brunelleschi">Filippo Brunelleschi</a></li>
<li><a href="Leon_Battista_Alberti" title="Leon Battista Alberti">Leon Battista Alberti</a>
<ul><li><i><a href="De_pictura" title="De pictura">De pictura</a></i></li>
<li><i><a href="De_re_aedificatoria" title="De re aedificatoria">De re aedificatoria</a></i></li></ul></li>
<li><a href="Piero_della_Francesca" title="Piero della Francesca">Piero della Francesca</a>
<ul><li><i><a href="De_prospectiva_pingendi" title="De prospectiva pingendi">De prospectiva pingendi</a></i></li></ul></li>
<li><a href="Luca_Pacioli" title="Luca Pacioli">Luca Pacioli</a>
<ul><li><i><a href="Divina_proportione" title="Divina proportione">De divina proportione</a></i></li></ul></li>
<li><a href="Leonardo_da_Vinci" title="Leonardo da Vinci">Leonardo da Vinci</a>
<ul><li><i><a href="A_Treatise_on_Painting" title="A Treatise on Painting">A Treatise on Painting</a></i></li></ul></li>
<li><a href="Albrecht_D%C3%BCrer" title="Albrecht Dürer">Albrecht Dürer</a>
<ul><li><i>Vier Bücher von Menschlicher Proportion</i></li></ul></li>
<li><a href="Sebastiano_Serlio" title="Sebastiano Serlio">Sebastiano Serlio</a>
<ul><li><i>Regole generali d'architettura</i></li></ul></li>
<li><a href="Andrea_Palladio" title="Andrea Palladio">Andrea Palladio</a>
<ul><li><i><a href="I_quattro_libri_dell'architettura" title="I quattro libri dell'architettura">I quattro libri dell'architettura</a></i></li></ul></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Romantic</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Samuel_Colman" title="Samuel Colman">Samuel Colman</a>
<ul><li><i>Nature's Harmonic Unity</i></li></ul></li>
<li><a href="Frederik_Macody_Lund" title="Frederik Macody Lund">Frederik Macody Lund</a>
<ul><li><i>Ad Quadratum</i></li></ul></li>
<li><a href="Jay_Hambidge" title="Jay Hambidge">Jay Hambidge</a>
<ul><li><i>The Greek Vase</i></li></ul></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Modern</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Owen_Jones_(architect)" title="Owen Jones (architect)">Owen Jones</a>
<ul><li><i><a href="Owen_Jones_(architect)#The_Grammar_of_Ornament" title="Owen Jones (architect)">The Grammar of Ornament</a></i></li></ul></li>
<li><a href="Ernest_Hanbury_Hankin" title="Ernest Hanbury Hankin">Ernest Hanbury Hankin</a>
<ul><li><i>The Drawing of Geometric Patterns in Saracenic Art</i></li></ul></li>
<li><a href="G._H._Hardy" title="G. H. Hardy">G. H. Hardy</a>
<ul><li><i><a href="A_Mathematician's_Apology" title="A Mathematician's Apology">A Mathematician's Apology</a></i></li></ul></li>
<li><a href="George_David_Birkhoff" title="George David Birkhoff">George David Birkhoff</a>
<ul><li><i>Aesthetic Measure</i></li></ul></li>
<li><a href="Douglas_Hofstadter" title="Douglas Hofstadter">Douglas Hofstadter</a>
<ul><li><i><a href="G%C3%B6del%2C_Escher%2C_Bach" title="Gödel, Escher, Bach">Gödel, Escher, Bach</a></i></li></ul></li>
<li><a href="Nikos_Salingaros" title="Nikos Salingaros">Nikos Salingaros</a>
<ul><li><i>The 'Life' of a Carpet</i></li></ul></li></ul>
</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Publications</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><i><a href="Journal_of_Mathematics_and_the_Arts" title="Journal of Mathematics and the Arts">Journal of Mathematics and the Arts</a></i></li>
<li><i><a href="Lumen_Naturae" title="Lumen Naturae">Lumen Naturae</a></i></li>
<li><i><a href="Making_Mathematics_with_Needlework" title="Making Mathematics with Needlework">Making Mathematics with Needlework</a></i></li>
<li><i><a href="Rhythm_of_Structure" title="Rhythm of Structure">Rhythm of Structure</a></i></li>
<li><i><a href="Viewpoints%3A_Mathematical_Perspective_and_Fractal_Geometry_in_Art" title="Viewpoints: Mathematical Perspective and Fractal Geometry in Art">Viewpoints: Mathematical Perspective and Fractal Geometry in Art</a></i></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Organizations</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Ars_Mathematica_(organization)" title="Ars Mathematica (organization)">Ars Mathematica</a></li>
<li><a href="The_Bridges_Organization" title="The Bridges Organization">The Bridges Organization</a></li>
<li><a href="European_Society_for_Mathematics_and_the_Arts" title="European Society for Mathematics and the Arts">European Society for Mathematics and the Arts</a></li>
<li><a href="Goudreau_Museum_of_Mathematics_in_Art_and_Science" title="Goudreau Museum of Mathematics in Art and Science">Goudreau Museum of Mathematics in Art and Science</a></li>
<li><a href="Institute_For_Figuring" title="Institute For Figuring">Institute For Figuring</a></li>
<li><a href="Mathemalchemy" title="Mathemalchemy">Mathemalchemy</a></li>
<li><a href="National_Museum_of_Mathematics" title="National Museum of Mathematics">National Museum of Mathematics</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Droste_effect" title="Droste effect">Droste effect</a></li>
<li><a href="Mathematical_beauty" title="Mathematical beauty">Mathematical beauty</a></li>
<li><a href="Patterns_in_nature" title="Patterns in nature">Patterns in nature</a></li>
<li><a href="Sacred_geometry" title="Sacred geometry">Sacred geometry</a></li></ul>
</div></td></tr><tr><td class="navbox-abovebelow" colspan="3"><div>
<ul><li><span class="noviewer" typeof="mw:File"><span title="Category"></span></span> <b>Category</b></li></ul>
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</style></div><div role="navigation" class="navbox authority-control" aria-labelledby="Authority_control_databases_frameless&#124;text-top&#124;10px&#124;alt=Edit_this_at_Wikidata&#124;link=https&#58;//www.wikidata.org/wiki/Q177409#identifiers&#124;class=noprint&#124;Edit_this_at_Wikidata1989" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Authority_control_databases_frameless&#124;text-top&#124;10px&#124;alt=Edit_this_at_Wikidata&#124;link=https&#58;//www.wikidata.org/wiki/Q177409#identifiers&#124;class=noprint&#124;Edit_this_at_Wikidata1989" style="font-size:114%;margin:0 4em">Authority control databases </div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">National</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="Projektive Geometrie"><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4047436-7">Germany</a></span></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="Geometry, Projective"><a rel="nofollow" class="external text" href="https://id.loc.gov/authorities/sh85054157">United States</a></span></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="Géométrie projective"><a rel="nofollow" class="external text" href="https://catalogue.bnf.fr/ark:/12148/cb11944448r">France</a></span></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="Géométrie projective"><a rel="nofollow" class="external text" href="https://data.bnf.fr/ark:/12148/cb11944448r">BnF data</a></span></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://id.ndl.go.jp/auth/ndlna/00571800">Japan</a></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="projektivní geometrie"><a rel="nofollow" class="external text" href="https://aleph.nkp.cz/F/?func=find-c&local_base=aut&ccl_term=ica=ph124640&CON_LNG=ENG">Czech Republic</a></span></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://kopkatalogs.lv/F?func=direct&local_base=lnc10&doc_number=000112019&P_CON_LNG=ENG">Latvia</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://www.nli.org.il/en/authorities/987007565327505171">Israel</a></span></li></ul></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><a rel="nofollow" class="external text" href="https://lux.collections.yale.edu/view/concept/d97065e7-5a91-4046-ace0-a44cb20b59bb">Yale LUX</a></span></li></ul></div></td></tr></tbody></table></div></div><!--htdig_noindex--><div><div class="zim-footer">
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