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title="Switch to /sparql endpoint"> Sparql Endpoint </a> </li> </ul> </div> </div> </nav> <div style="margin-bottom: 60px"></div>   <section> <div class="container-xl"> <div class="row"> <div class="col"> <h1 id="title" class="display-6">About: <a href="http://dbpedia.org/resource/Beckman鈥換uarles_theorem">Beckman鈥換uarles theorem</a> </h1> </div> </div> <div class="row"> <div class="col"> <div class="text-muted"> An Entity of Type: <a href="http://dbpedia.org/class/yago/WikicatTheoremsInGeometry">WikicatTheoremsInGeometry</a>, from Named Graph: <a href="http://dbpedia.org">http://dbpedia.org</a>, within Data Space: <a href="http://dbpedia.org">dbpedia.org</a> </div> </div> </div> <div class="row pt-2"> <div class="col-xs-9 col-sm-10"> In geometry, the Beckman鈥換uarles theorem, named after Frank S. Beckman and Donald A. Quarles Jr., states that if a transformation of the Euclidean plane or a higher-dimensional Euclidean space preserves unit distances, then it preserves all Euclidean distances. Equivalently, every homomorphism from the unit distance graph of the plane to itself must be an isometry of the plane.Beckman and Quarles published this result in 1953; it was later rediscovered by other authors, and re-proved in multiple ways. Analogous theorems for rational subsets of Euclidean spaces, or for non-Euclidean geometry, are also known. </div> <div class="col-xs-3 col-sm-2"> <a href="#" class="thumbnail"> <img src="http://commons.wikimedia.org/wiki/Special:FilePath/Hadwiger-Nelson.svg?width=300" alt="thumbnail" class="img-fluid" /> </a> </div> </div> </div> </section>   <section> <div class="container-xl"> <div class="row"> <div class="table-responsive"> <table class="table table-hover table-sm table-light"> <thead> <tr> <th class="col-xs-3 ">Property</th> <th class="col-xs-9 px-3">Value</th> </tr> </thead> <tbody> <tr class="odd"><td class="col-2"><a class="uri" href="http://dbpedia.org/ontology/abstract">dbo:abstract</a> </td><td class="col-10 text-break"><ul> <li>In geometry, the Beckman鈥換uarles theorem, named after Frank S. Beckman and Donald A. Quarles Jr., states that if a transformation of the Euclidean plane or a higher-dimensional Euclidean space preserves unit distances, then it preserves all Euclidean distances. Equivalently, every homomorphism from the unit distance graph of the plane to itself must be an isometry of the plane.Beckman and Quarles published this result in 1953; it was later rediscovered by other authors, and re-proved in multiple ways. Analogous theorems for rational subsets of Euclidean spaces, or for non-Euclidean geometry, are also known. (en)</li> <li style="display:none;">Der Satz von Beckman und Quarles ist ein Satz 眉ber geometrische Transformationen. Er wurde im Jahr 1953 von und erstmals publiziert und unabh盲ngig davon von mehreren anderen Autoren bewiesen. Der Satz besagt, dass eine beliebige Selbstabbildung des n-dimensionalen euklidischen Raumes, die s盲mtliche Punktpaare mit Abstand 1 in ebensolche 眉berf眉hrt, bereits eine Isometrie ist, also s盲mtliche Abst盲nde unver盲ndert l盲sst. Dies ist 盲quivalent zu der Aussage, dass jeder Automorphismus des Einheitsdistanz-Graphen eine Isometrie ist. (de)</li> </ul></td></tr><tr class="even"><td class="col-2"><a class="uri" href="http://dbpedia.org/ontology/thumbnail">dbo:thumbnail</a> </td><td class="col-10 text-break"><ul> <li><a class="uri" rel="dbo:thumbnail" resource="http://commons.wikimedia.org/wiki/Special:FilePath/Hadwiger-Nelson.svg?width=300" href="http://commons.wikimedia.org/wiki/Special:FilePath/Hadwiger-Nelson.svg?width=300">wiki-commons:Special:FilePath/Hadwiger-Nelson.svg?width=300</a></li> </ul></td></tr><tr class="odd"><td class="col-2"><a class="uri" href="http://dbpedia.org/ontology/wikiPageID">dbo:wikiPageID</a> </td><td class="col-10 text-break"><ul> <li>7580572 (xsd:integer)</li> </ul></td></tr><tr class="even"><td class="col-2"><a class="uri" href="http://dbpedia.org/ontology/wikiPageLength">dbo:wikiPageLength</a> </td><td class="col-10 text-break"><ul> <li>18267 (xsd:nonNegativeInteger)</li> </ul></td></tr><tr class="odd"><td class="col-2"><a class="uri" 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</ul></td></tr><tr class="odd"><td class="col-2"><a class="uri" href="http://www.w3.org/1999/02/22-rdf-syntax-ns#type">rdf:type</a> </td><td class="col-10 text-break"><ul> <li><a class="uri" rel="rdf:type" resource="http://dbpedia.org/class/yago/WikicatMathematicalTheorems" href="http://dbpedia.org/class/yago/WikicatMathematicalTheorems">yago:WikicatMathematicalTheorems</a></li> <li><a class="uri" rel="rdf:type" resource="http://dbpedia.org/class/yago/WikicatTheoremsInGeometry" href="http://dbpedia.org/class/yago/WikicatTheoremsInGeometry">yago:WikicatTheoremsInGeometry</a></li> <li><a class="uri" rel="rdf:type" resource="http://dbpedia.org/class/yago/Abstraction100002137" href="http://dbpedia.org/class/yago/Abstraction100002137">yago:Abstraction100002137</a></li> <li><a class="uri" rel="rdf:type" resource="http://dbpedia.org/class/yago/Communication100033020" href="http://dbpedia.org/class/yago/Communication100033020">yago:Communication100033020</a></li> <li><a class="uri" 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Beckman and Donald A. Quarles Jr., states that if a transformation of the Euclidean plane or a higher-dimensional Euclidean space preserves unit distances, then it preserves all Euclidean distances. Equivalently, every homomorphism from the unit distance graph of the plane to itself must be an isometry of the plane.Beckman and Quarles published this result in 1953; it was later rediscovered by other authors, and re-proved in multiple ways. Analogous theorems for rational subsets of Euclidean spaces, or for non-Euclidean geometry, are also known. (en)</li> <li style="display:none;">Der Satz von Beckman und Quarles ist ein Satz 眉ber geometrische Transformationen. Er wurde im Jahr 1953 von und erstmals publiziert und unabh盲ngig davon von mehreren anderen Autoren bewiesen. Der Satz besagt, dass eine beliebige Selbstabbildung des n-dimensionalen euklidischen Raumes, die s盲mtliche Punktpaare mit Abstand 1 in ebensolche 眉berf眉hrt, bereits eine Isometrie ist, also s盲mtliche Abst盲nde unver盲ndert l盲sst. Dies ist 盲quivalent zu der Aussage, dass jeder Automorphismus des Einheitsdistanz-Graphen eine Isometrie ist. (de)</li> </ul></td></tr><tr class="odd"><td class="col-2"><a class="uri" href="http://www.w3.org/2000/01/rdf-schema#label">rdfs:label</a> </td><td class="col-10 text-break"><ul> <li style="display:none;">Satz von Beckman und Quarles (de)</li> <li>Beckman鈥換uarles theorem (en)</li> </ul></td></tr><tr class="even"><td class="col-2"><a class="uri" href="http://www.w3.org/2002/07/owl#sameAs">owl:sameAs</a> </td><td class="col-10 text-break"><ul> <li><a class="uri" rel="owl:sameAs" resource="http://rdf.freebase.com/ns/m.0265w3g" href="http://rdf.freebase.com/ns/m.0265w3g">freebase:Beckman鈥換uarles theorem</a></li> <li><a class="uri" rel="owl:sameAs" resource="http://www.wikidata.org/entity/Q2226606" href="http://www.wikidata.org/entity/Q2226606">wikidata:Beckman鈥換uarles theorem</a></li> <li><a class="uri" rel="owl:sameAs" resource="http://de.dbpedia.org/resource/Satz_von_Beckman_und_Quarles" href="http://de.dbpedia.org/resource/Satz_von_Beckman_und_Quarles">dbpedia-de:Beckman鈥換uarles 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