SherbyNet - /page/wikibook/entry.mu

Micron Document

<!DOCTYPE html>

<title>Ordered geometry</title>

</head>

Ordered geometry

</h1>

</header>

Ordered geometry is a form of <a href="Geometry" title="Geometry">geometry</a> featuring the concept of intermediacy (or "betweenness") but, like <a href="Projective_geometry" title="Projective geometry">projective geometry</a>, omitting the basic notion of <a href="Metric_(mathematics)" class="mw-redirect" title="Metric (mathematics)">measurement</a>. Ordered geometry is a fundamental geometry forming a common framework for <a href="Affine_geometry" title="Affine geometry">affine</a>, <a href="Euclidean_geometry" title="Euclidean geometry">Euclidean</a>, <a href="Absolute_geometry" title="Absolute geometry">absolute</a>, and <a href="Hyperbolic_geometry" title="Hyperbolic geometry">hyperbolic geometry</a> (but not for projective geometry).

<div class="mw-heading mw-heading2"><h2 id="History">History</h2></div>

<a href="Moritz_Pasch" title="Moritz Pasch">Moritz Pasch</a> first defined a geometry without reference to measurement in 1882. His axioms were improved upon by <a href="Giuseppe_Peano" title="Giuseppe Peano">Peano</a> (1889), <a href="David_Hilbert" title="David Hilbert">Hilbert</a> (1899), and <a href="Oswald_Veblen" title="Oswald Veblen">Veblen</a> (1904).<a href="#cite_note-Coxeter69-1">[1]</a>: 176 <a href="Euclid" title="Euclid">Euclid</a> anticipated Pasch's approach in definition 4 of The Elements: "a straight line is a line which lies evenly with the points on itself".<a href="#cite_note-2">[2]</a>

<div class="mw-heading mw-heading2"><h2 id="Primitive_concepts">Primitive concepts</h2></div>

The only <a href="Primitive_notion" title="Primitive notion">primitive notions</a> in ordered geometry are <a href="Point_(geometry)" title="Point (geometry)">points</a> A, B, C, ... and the <a href="Ternary_relation" title="Ternary relation">ternary relation</a> of intermediacy [ABC] which can be read as "B is between A and C".

<div class="mw-heading mw-heading2"><h2 id="Definitions">Definitions</h2></div>

The segment AB is the <a href="Set_(mathematics)" title="Set (mathematics)">set</a> of points P such that [APB].

The interval AB is the segment AB and its end points A and B.

The ray A/B (read as "the ray from A away from B") is the set of points P such that [PAB].

The line AB is the interval AB and the two rays A/B and B/A. Points on the line AB are said to be collinear.

An angle consists of a point O (the vertex) and two non-collinear rays out from O (the sides).

A triangle is given by three non-collinear points (called vertices) and their three segments AB, BC, and CA.

If three points A, B, and C are non-collinear, then a plane ABC is the set of all points collinear with pairs of points on one or two of the sides of triangle ABC.

If four points A, B, C, and D are non-coplanar, then a space (<a href="3-space" class="mw-redirect" title="3-space">3-space</a>) ABCD is the set of all points collinear with pairs of points selected from any of the four faces (planar regions) of the <a href="Tetrahedron" title="Tetrahedron">tetrahedron</a> ABCD.

<div class="mw-heading mw-heading2"><h2 id="Axioms_of_ordered_geometry">Axioms of ordered geometry</h2></div>

<ol><li>There exist at least two points.</li>

<li>If A and B are distinct points, there exists a C such that [ABC].</li>

<li>If [ABC], then A and C are distinct (A ≠ C).</li>

<li>If C and D are distinct points on the line AB, then A is on the line CD.</li>

<li>If AB is a line, there is a point C not on the line AB.</li>

<li>(<a href="Axiom_of_Pasch" class="mw-redirect" title="Axiom of Pasch">Axiom of Pasch</a>) If ABC is a triangle and [BCD] and [CEA], then there exists a point F on the line DE for which [AFB].</li>

<li>Axiom of <a href="Dimensionality" class="mw-redirect" title="Dimensionality">dimensionality</a>:

<ol><li>For planar ordered geometry, all points are in one plane. Or</li>

<li>If ABC is a plane, then there exists a point D not in the plane ABC.</li></ol></li>

<li>All points are in the same plane, space, etc. (depending on the dimension one chooses to work within).</li>

<li>(Dedekind's Axiom) For every partition of all the points on a line into two nonempty sets such that no point of either lies between two points of the other, there is a point of one set which lies between every other point of that set and every point of the other set.</li></ol>

These axioms are closely related to <a href="Hilbert's_axioms#II._Order" title="Hilbert's axioms">Hilbert's axioms of order</a>. For a comprehensive survey of axiomatizations of ordered geometry see Pambuccian (2011).<a href="#cite_note-3">[3]</a>

<div class="mw-heading mw-heading2"><h2 id="Results">Results</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Sylvester's_problem_of_collinear_points">Sylvester's problem of collinear points</h3></div>

The <a href="Sylvester%E2%80%93Gallai_theorem" title="Sylvester–Gallai theorem">Sylvester–Gallai theorem</a> can be proven within ordered geometry.<a href="#cite_note-4">[4]</a><a href="#cite_note-Coxeter69-1">[1]</a>: 181, 2

<div class="mw-heading mw-heading3"><h3 id="Parallelism">Parallelism</h3></div>

<a href="Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Gauss</a>, <a href="J%C3%A1nos_Bolyai" title="János Bolyai">Bolyai</a>, and <a href="Nikolai_Lobachevsky" title="Nikolai Lobachevsky">Lobachevsky</a> developed a notion of <a href="Parallel_postulate" title="Parallel postulate">parallelism</a> which can be expressed in ordered geometry.<a href="#cite_note-Coxeter69-1">[1]</a>: 189, 90

Theorem (existence of parallelism): Given a point A and a line r, not through A, there exist exactly two limiting rays from A in the plane Ar which do not meet r. So there is a parallel line through A which does not meet r.

Theorem (transmissibility of parallelism): The parallelism of a ray and a line is preserved by adding or subtracting a segment from the beginning of a ray.

The <a href="Transitive_relation" title="Transitive relation">transitivity</a> of parallelism cannot be proven in ordered geometry.<a href="#cite_note-5">[5]</a> Therefore, the "ordered" concept of parallelism does not form an <a href="Equivalence_relation" title="Equivalence relation">equivalence relation</a> on lines.

<ul><li><a href="Absolute_geometry" title="Absolute geometry">Absolute geometry</a></li>

<li><a href="Affine_geometry" title="Affine geometry">Affine geometry</a></li>

<li><a href="Cyclic_order" title="Cyclic order">Cyclic order</a></li>

<li><a href="Erlangen_program" title="Erlangen program">Erlangen program</a></li>

<li><a href="Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a>

<ul><li><a href="Hilbert's_axioms" title="Hilbert's axioms">Hilbert's axioms</a></li>

<li><a href="Tarski's_axioms" title="Tarski's axioms">Tarski's axioms</a></li></ul></li>

<li><a href="Incidence_geometry" title="Incidence geometry">Incidence geometry</a></li>

<li><a href="Lattice_(order)" title="Lattice (order)">Lattice (order)</a></li>

<li><a href="Non-Euclidean_geometry" title="Non-Euclidean geometry">Non-Euclidean geometry</a></li>

<li><a href="Point-pair_separation" title="Point-pair separation">Point-pair separation</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

/* start https://en.wikipedia.org/ */

.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}

/* end https://en.wikipedia.org/ */

</style><div class="reflist">

/* start https://en.wikipedia.org/ */

.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}

/* end https://en.wikipedia.org/ */

</style><cite id="CITEREFCoxeter1969" class="citation book cs1"><a href="Harold_Scott_MacDonald_Coxeter" title="Harold Scott MacDonald Coxeter">Coxeter, H.S.M.</a> (1969). <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoge0002coxe">Introduction to Geometry</a> (2nd ed.). <a href="John_Wiley_and_Sons" class="mw-redirect" title="John Wiley and Sons">John Wiley and Sons</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-471-18283-4</bdi>. <a href="Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0181.48101">0181.48101</a>.</cite>

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite id="CITEREFHeath1956" class="citation book cs1"><a href="Thomas_Little_Heath" class="mw-redirect" title="Thomas Little Heath">Heath, Thomas</a> (1956) [1925]. <a rel="nofollow" class="external text" href="https://archive.org/details/thirteenbooksofe00eucl/page/165">The Thirteen Books of Euclid's Elements (Vol 1)</a>. New York: <a href="Dover_Publications" title="Dover Publications">Dover Publications</a>. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/thirteenbooksofe00eucl/page/165">165</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-486-60088-2</bdi>.</cite> <code class="cs1-code">{{cite book}}</code>: ISBN / Date incompatibility (help)

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> <cite id="CITEREFPambuccian2011" class="citation journal cs1">Pambuccian, Victor (2011). "The axiomatics of ordered geometry: I. Ordered incidence spaces". Expositiones Mathematicae. 29: 24–66. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.exmath.2010.09.004">10.1016/j.exmath.2010.09.004</a>.</cite>

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite id="CITEREFPambuccian2009" class="citation journal cs1">Pambuccian, Victor (2009). <a rel="nofollow" class="external text" href="https://doi.org/10.1215%2F00294527-2009-010">"A Reverse Analysis of the Sylvester–Gallai Theorem"</a>. Notre Dame Journal of Formal Logic. 50 (3): 245–260. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1215%2F00294527-2009-010">10.1215/00294527-2009-010</a>. <a href="Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:1202.03023">1202.03023</a>.</cite>

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> <cite id="CITEREFBusemann1955" class="citation book cs1">Busemann, Herbert (1955). Geometry of Geodesics. Pure and Applied Mathematics. Vol. 6. New York: <a href="Academic_Press" title="Academic Press">Academic Press</a>. p. 139. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-12-148350-9</bdi>. <a href="Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0112.37002">0112.37002</a>.</cite> <code class="cs1-code">{{cite book}}</code>: ISBN / Date incompatibility (help)

</li>

</ol></div></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-03-04" href="https://en.wikipedia.org/wiki/?title=Ordered_geometry&oldid=1278697109">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

</div>

</div></div>

</div>

</main>

</div>

</body></html>