{"id":9983,"date":"2024-11-09T08:54:44","date_gmt":"2024-11-09T07:54:44","guid":{"rendered":"https:\/\/www2.ual.es\/neotrie\/?p=9983"},"modified":"2026-05-27T11:48:17","modified_gmt":"2026-05-27T10:48:17","slug":"experimentos-con-la-botella-de-klein","status":"publish","type":"post","link":"https:\/\/www2.ual.es\/neotrie\/experimentos-con-la-botella-de-klein\/","title":{"rendered":"Experimentos con la botella de Klein"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">Os dejamos una experiencia topol\u00f3gica con la botella de Klein, que ha requerido algunas mejoras en Neotrie, y que est\u00e1n ya disponibles en la <a href=\"https:\/\/docs.google.com\/document\/d\/1n5Zi8xAzuDlrFhnmtjQesDN78D8P4VJXg6dO5QSiNns\/edit?usp=sharing\" title=\"\">nueva versi\u00f3n 4.7.4<\/a>.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"576\" src=\"https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2024\/11\/screen_1920x1080_2024-10-16_11-51-46-1024x576.png\" alt=\"\" class=\"wp-image-9984\" srcset=\"https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2024\/11\/screen_1920x1080_2024-10-16_11-51-46-980x551.png 980w, https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2024\/11\/screen_1920x1080_2024-10-16_11-51-46-480x270.png 480w\" sizes=\"(min-width: 0px) and (max-width: 480px) 480px, (min-width: 481px) and (max-width: 980px) 980px, (min-width: 981px) 1024px, 100vw\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">Esta figura se ha construido con el modo extrusi\u00f3n y la nueva opci\u00f3n \"edit\" que permite rotar un elemento de la figura, si se tiene en la otra mano \"grab\". Os mostramos primero esta nueva opci\u00f3n y despu\u00e9s c\u00f3mo extruir los pol\u00edgonos para construir la botella de Klein.<\/p>\n\n\n\n<figure class=\"wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube wp-embed-aspect-4-3 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<iframe loading=\"lazy\" title=\"Rotar elemento dentro de una figura\" width=\"1080\" height=\"810\" src=\"https:\/\/www.youtube.com\/embed\/NO9m9gKLWb8?feature=oembed\"  allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n<\/div><\/figure>\n\n\n\n<figure class=\"wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube wp-embed-aspect-4-3 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<iframe loading=\"lazy\" title=\"Construcci\u00f3n de botella de Klein\" width=\"1080\" height=\"810\" src=\"https:\/\/www.youtube.com\/embed\/gdd8jbWuWJE?feature=oembed\"  allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n<\/div><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">En el siguiente ejercicio, realizaremos un agujero a la botella de Klein y pasaremos a un modelo sencillo de la misma:<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"1024\" src=\"https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2024\/11\/handleModelKleinBottle-1024x1024.jpg\" alt=\"\" class=\"wp-image-9990\" srcset=\"https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2024\/11\/handleModelKleinBottle-980x980.jpg 980w, https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2024\/11\/handleModelKleinBottle-480x480.jpg 480w\" sizes=\"(min-width: 0px) and (max-width: 480px) 480px, (min-width: 481px) and (max-width: 980px) 980px, (min-width: 981px) 1024px, 100vw\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">Otro ejercicio interesante consiste en ver que la botella de Klein es la uni\u00f3n de dos cintas de Moebius por su borde. Para ello la cortaremos por la mitad transversalmente (creando dos copias y borrando la mitad correspondiente en cada una de ellas).<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"1024\" src=\"https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2024\/11\/klein2-1024x1024.jpg\" alt=\"\" class=\"wp-image-9992\" srcset=\"https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2024\/11\/klein2-980x980.jpg 980w, https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2024\/11\/klein2-480x480.jpg 480w\" sizes=\"(min-width: 0px) and (max-width: 480px) 480px, (min-width: 481px) and (max-width: 980px) 980px, (min-width: 981px) 1024px, 100vw\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-video\"><video height=\"720\" style=\"aspect-ratio: 720 \/ 720;\" width=\"720\" controls src=\"https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2024\/11\/2half-Klein.mp4\"><\/video><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">Para el que no conozca que la botella de Klein no tiene interior ni exterior, ah\u00ed va un juego divertido de bolas en nuestro barco:<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"1024\" src=\"https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2024\/11\/kleinInShip-1024x1024.jpg\" alt=\"\" class=\"wp-image-9998\" srcset=\"https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2024\/11\/kleinInShip-980x980.jpg 980w, https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2024\/11\/kleinInShip-480x480.jpg 480w\" sizes=\"(min-width: 0px) and (max-width: 480px) 480px, (min-width: 481px) and (max-width: 980px) 980px, (min-width: 981px) 1024px, 100vw\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<figure class=\"wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube wp-embed-aspect-4-3 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<iframe loading=\"lazy\" title=\"Bolas en la botella de Klein\" width=\"1080\" height=\"810\" src=\"https:\/\/www.youtube.com\/embed\/5SzKbF_meTs?feature=oembed\"  allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n<\/div><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">Visita nuestra actividad completa en la caja de actividades dentro del juego:<\/p>\n\n\n\n<figure class=\"wp-block-embed is-type-wp-embed is-provider-neotrie-vr wp-block-embed-neotrie-vr\"><div class=\"wp-block-embed__wrapper\">\n<blockquote class=\"wp-embedded-content\" data-secret=\"XAyBW2nmEG\"><a href=\"https:\/\/www2.ual.es\/neotrie\/nueva-experiencia-multijugador-con-alumnado-de-topologia-algebraica\/\">Nueva experiencia multijugador con alumnado de Topolog\u00eda Algebraica<\/a><\/blockquote><iframe loading=\"lazy\" class=\"wp-embedded-content\" sandbox=\"allow-scripts\" security=\"restricted\" style=\"position: absolute; visibility: hidden;\" title=\"\u00abNueva experiencia multijugador con alumnado de Topolog\u00eda Algebraica\u00bb \u2014 Neotrie VR\" src=\"https:\/\/www2.ual.es\/neotrie\/nueva-experiencia-multijugador-con-alumnado-de-topologia-algebraica\/embed\/#?secret=wfX6tnmAS7#?secret=XAyBW2nmEG\" data-secret=\"XAyBW2nmEG\" width=\"600\" height=\"338\" frameborder=\"0\" marginwidth=\"0\" marginheight=\"0\" scrolling=\"no\"><\/iframe>\n<\/div><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">Si te ha gustado puedes darle me gusta y compartir en tu red favorita:<\/p>\n\n\n\n<ul class=\"wp-block-social-links is-layout-flex wp-block-social-links-is-layout-flex\"><li class=\"wp-social-link wp-social-link-instagram wp-block-social-link\"><a href=\"https:\/\/www.instagram.com\/p\/DCJZFSZM535\/\" class=\"wp-block-social-link-anchor\"><svg width=\"24\" height=\"24\" viewBox=\"0 0 24 24\" version=\"1.1\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" aria-hidden=\"true\" focusable=\"false\"><path d=\"M12,4.622c2.403,0,2.688,0.009,3.637,0.052c0.877,0.04,1.354,0.187,1.671,0.31c0.42,0.163,0.72,0.358,1.035,0.673 c0.315,0.315,0.51,0.615,0.673,1.035c0.123,0.317,0.27,0.794,0.31,1.671c0.043,0.949,0.052,1.234,0.052,3.637 s-0.009,2.688-0.052,3.637c-0.04,0.877-0.187,1.354-0.31,1.671c-0.163,0.42-0.358,0.72-0.673,1.035 c-0.315,0.315-0.615,0.51-1.035,0.673c-0.317,0.123-0.794,0.27-1.671,0.31c-0.949,0.043-1.233,0.052-3.637,0.052 s-2.688-0.009-3.637-0.052c-0.877-0.04-1.354-0.187-1.671-0.31c-0.42-0.163-0.72-0.358-1.035-0.673 c-0.315-0.315-0.51-0.615-0.673-1.035c-0.123-0.317-0.27-0.794-0.31-1.671C4.631,14.688,4.622,14.403,4.622,12 s0.009-2.688,0.052-3.637c0.04-0.877,0.187-1.354,0.31-1.671c0.163-0.42,0.358-0.72,0.673-1.035 c0.315-0.315,0.615-0.51,1.035-0.673c0.317-0.123,0.794-0.27,1.671-0.31C9.312,4.631,9.597,4.622,12,4.622 M12,3 C9.556,3,9.249,3.01,8.289,3.054C7.331,3.098,6.677,3.25,6.105,3.472C5.513,3.702,5.011,4.01,4.511,4.511 c-0.5,0.5-0.808,1.002-1.038,1.594C3.25,6.677,3.098,7.331,3.054,8.289C3.01,9.249,3,9.556,3,12c0,2.444,0.01,2.751,0.054,3.711 c0.044,0.958,0.196,1.612,0.418,2.185c0.23,0.592,0.538,1.094,1.038,1.594c0.5,0.5,1.002,0.808,1.594,1.038 c0.572,0.222,1.227,0.375,2.185,0.418C9.249,20.99,9.556,21,12,21s2.751-0.01,3.711-0.054c0.958-0.044,1.612-0.196,2.185-0.418 c0.592-0.23,1.094-0.538,1.594-1.038c0.5-0.5,0.808-1.002,1.038-1.594c0.222-0.572,0.375-1.227,0.418-2.185 C20.99,14.751,21,14.444,21,12s-0.01-2.751-0.054-3.711c-0.044-0.958-0.196-1.612-0.418-2.185c-0.23-0.592-0.538-1.094-1.038-1.594 c-0.5-0.5-1.002-0.808-1.594-1.038c-0.572-0.222-1.227-0.375-2.185-0.418C14.751,3.01,14.444,3,12,3L12,3z M12,7.378 c-2.552,0-4.622,2.069-4.622,4.622S9.448,16.622,12,16.622s4.622-2.069,4.622-4.622S14.552,7.378,12,7.378z M12,15 c-1.657,0-3-1.343-3-3s1.343-3,3-3s3,1.343,3,3S13.657,15,12,15z M16.804,6.116c-0.596,0-1.08,0.484-1.08,1.08 s0.484,1.08,1.08,1.08c0.596,0,1.08-0.484,1.08-1.08S17.401,6.116,16.804,6.116z\"><\/path><\/svg><span class=\"wp-block-social-link-label screen-reader-text\">Instagram<\/span><\/a><\/li>\n\n<li class=\"wp-social-link wp-social-link-tiktok wp-block-social-link\"><a href=\"https:\/\/www.tiktok.com\/@neotrievr\/video\/7435227584364875041\" class=\"wp-block-social-link-anchor\"><svg width=\"24\" height=\"24\" viewBox=\"0 0 32 32\" version=\"1.1\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" aria-hidden=\"true\" focusable=\"false\"><path d=\"M16.708 0.027c1.745-0.027 3.48-0.011 5.213-0.027 0.105 2.041 0.839 4.12 2.333 5.563 1.491 1.479 3.6 2.156 5.652 2.385v5.369c-1.923-0.063-3.855-0.463-5.6-1.291-0.76-0.344-1.468-0.787-2.161-1.24-0.009 3.896 0.016 7.787-0.025 11.667-0.104 1.864-0.719 3.719-1.803 5.255-1.744 2.557-4.771 4.224-7.88 4.276-1.907 0.109-3.812-0.411-5.437-1.369-2.693-1.588-4.588-4.495-4.864-7.615-0.032-0.667-0.043-1.333-0.016-1.984 0.24-2.537 1.495-4.964 3.443-6.615 2.208-1.923 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Esta figura se ha construido con el modo extrusi\u00f3n y la nueva opci\u00f3n &quot;edit&quot; que permite rotar un elemento de la figura, si se tiene en la otra\" \/>\n\t<meta name=\"robots\" content=\"max-image-preview:large\" \/>\n\t<meta name=\"author\" content=\"Jos\u00e9 Luis Rodr\u00edguez\"\/>\n\t<meta name=\"google-site-verification\" content=\"google-site-verification=yECM5sD2Ii3xsSoDBArqBmNdlDrmXh-FAAOQ40keSr4\" \/>\n\t<link rel=\"canonical\" href=\"https:\/\/www2.ual.es\/neotrie\/experimentos-con-la-botella-de-klein\/\" \/>\n\t<meta name=\"generator\" content=\"All in One SEO (AIOSEO) 4.9.8\" \/>\n\t\t<meta property=\"og:locale\" content=\"es_ES\" \/>\n\t\t<meta property=\"og:site_name\" content=\"Neotrie VR - NeoTrie VR es un software de realidad virtual y mixta, multijugador, que permite al usuario crear, manipular e interactuar con objetos geom\u00e9tricos y modelos 3D en general.\" \/>\n\t\t<meta property=\"og:type\" content=\"article\" \/>\n\t\t<meta property=\"og:title\" content=\"Experimentos con la botella de Klein \ufeff \ufeff\ufeff-\ufeff\ufeff\ufeff\ufeff\ufeff\ufeff Neotrie VR\" \/>\n\t\t<meta property=\"og:description\" content=\"Os dejamos una experiencia topol\u00f3gica con la botella de Klein, que ha requerido algunas mejoras en Neotrie, y que est\u00e1n ya disponibles en la nueva versi\u00f3n 4.7.4. 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Esta figura se ha construido con el modo extrusi\u00f3n y la nueva opci\u00f3n &quot;edit&quot; que permite rotar un elemento de la figura, si se tiene en la otra\" \/>\n\t\t<meta name=\"twitter:creator\" content=\"@NeotrieVR\" \/>\n\t\t<script type=\"application\/ld+json\" class=\"aioseo-schema\">\n\t\t\t{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"BlogPosting\",\"@id\":\"https:\\\/\\\/www2.ual.es\\\/neotrie\\\/experimentos-con-la-botella-de-klein\\\/#blogposting\",\"name\":\"Experimentos con la botella de Klein \\ufeff \\ufeff\\ufeff-\\ufeff\\ufeff\\ufeff\\ufeff\\ufeff\\ufeff Neotrie VR\",\"headline\":\"Experimentos con la botella de 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