{"id":3780,"date":"2021-10-16T12:01:56","date_gmt":"2021-10-16T11:01:56","guid":{"rendered":"http:\/\/www2.ual.es\/neotrie\/?post_type=project&#038;p=3780"},"modified":"2026-07-02T14:46:33","modified_gmt":"2026-07-02T13:46:33","slug":"introduccion-a-la-topologia-algebraica","status":"publish","type":"project","link":"https:\/\/www2.ual.es\/neotrie\/project\/introduccion-a-la-topologia-algebraica\/","title":{"rendered":"Introducci\u00f3n a la Topolog\u00eda Algebraica"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">En este proyecto elaboramos videos en Neotrie tanto en vista normal como estereosc\u00f3pica para las clases de Introducci\u00f3n de la Topolog\u00eda Algebraica de la Universidad de Almer\u00eda, durante el curso 2020-21. Para poder realizarlos se han implementado mejoras en la calculadora gr\u00e1fica 3d. <\/p>\n\n\n\n<figure class=\"wp-block-image alignright is-resized\"><a href=\"https:\/\/link.springer.com\/chapter\/10.1007\/978-3-030-86909-0_15\" target=\"_blank\" rel=\"noopener\"><img decoding=\"async\" src=\"https:\/\/media.springernature.com\/w306\/springer-static\/cover-hires\/series\/10170\" alt=\"\"\/><\/a><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">Parte de este material aparece referenciado en el cap\u00edtulo \"Exploring Dynamic Geometry Through Immersive Virtual Reality and Distance Teaching\", en el libro \u201c<a href=\"https:\/\/link.springer.com\/chapter\/10.1007\/978-3-030-86909-0_15\" target=\"_blank\" rel=\"noopener\" title=\"\">Mathematics Education in the Age of Artificial Intelligence<\/a>\u201d, de la serie \u201cMathematics Education in the Digital Era 17\u201d. Berlin, Germany: Springer. <br><br>Abstract: Virtual reality provides an interesting environment to teach and learn 3D geometry. In this article, we discuss the use of Neotrie VR as a 3D whiteboard for distance teaching that we have carried out during the 2020\u201321 academic year, with students of the Mathematics degree at the University of Almer\u00eda. We describe a concrete case on parametric equations of surfaces, for which a 3D graphing calculator has been implemented, as well as a stereoscopic view camera to show 3D videos, which the students can view with cheap stereoscopic glasses for mobile phones. From the side of the teacher, it is certainly much easier to explain 3D concepts on a 3D whiteboard like Neotrie than to use paper and pencil, blackboard, or any 2D digital tablet. Student feedback is also analyzed after using various supports for manipulating and observing learning, including GeoGebra, which can also serve to know how to use virtual reality for distance learning.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Pedimos disculpas por el sonido defectuoso en algunos videos, que se grabaron en directo durante las clases, en la plataforma BlackBoard Collaborate Ultra.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"666\" height=\"1024\" src=\"http:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/10\/Screenshot_2021-10-22-12-36-51-26_e2d5b3f32b79de1d45acd1fad96fbb0f-666x1024.jpg\" alt=\"\" class=\"wp-image-4831\"\/><\/figure>\n\n\n\n<div data-wp-interactive=\"core\/file\" class=\"wp-block-file\"><object data-wp-bind--hidden=\"!state.hasPdfPreview\" hidden class=\"wp-block-file__embed\" data=\"http:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/10\/Encuentro_topologia_2021.pdf\" type=\"application\/pdf\" style=\"width:100%;height:200px\" aria-label=\"Incrustado de Poster Encuentro_topologia_2021.\"><\/object><a id=\"wp-block-file--media-2c9d294a-181e-4262-8a09-0fd396ab509d\" href=\"http:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/10\/Encuentro_topologia_2021.pdf\">Poster Encuentro_topologia_2021<\/a><a href=\"http:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/10\/Encuentro_topologia_2021.pdf\" class=\"wp-block-file__button wp-element-button\" download aria-describedby=\"wp-block-file--media-2c9d294a-181e-4262-8a09-0fd396ab509d\">Descarga<\/a><\/div>\n\n\n\n<figure class=\"wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<iframe loading=\"lazy\" title=\"Parametrizations in Neotrie VR\" width=\"1080\" height=\"608\" src=\"https:\/\/www.youtube.com\/embed\/aTFu6RqC2ZQ?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-16-9 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<iframe loading=\"lazy\" title=\"Cintas\" width=\"1080\" height=\"608\" src=\"https:\/\/www.youtube.com\/embed\/2irYPYOCkr4?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><figcaption class=\"wp-element-caption\">Se muestra c\u00f3mo obtener con las herramientas de Neotrie (paralela, perpendicular, comp\u00e1s y deslizador), una cinta cil\u00edndrica o de Moebius dependiendo del n\u00famero de medias vueltas, el radio y la anchura de la misma. Esta superficie puede obtenerse tambi\u00e9n introduciendo su parametrizaci\u00f3n en la calculadora gr\u00e1fica 3d y variando los par\u00e1metros correspondientes. Para ver el video en 3d, se requieren gafas VR para m\u00f3viles.<\/figcaption><\/figure>\n\n\n\n<figure class=\"wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<iframe loading=\"lazy\" title=\"Parametrizaci\u00f3n de la cinta de M\u00f6bius en vista esterosc\u00f3pica\" width=\"1080\" height=\"608\" src=\"https:\/\/www.youtube.com\/embed\/oq4Xk99Sa64?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><figcaption class=\"wp-element-caption\">Se construye la cinta de M\u00f6bius donde se han utilizado los 3 ejes, con restricci\u00f3n de movimiento, y las herramientas de paralelas, perpendiculares, y deslizador circular. Despu\u00e9s se deduce la parametrizaci\u00f3n correspondiente a dicha construcci\u00f3n. Para ver el video en 3d, se requieren gafas VR para m\u00f3viles.<\/figcaption><\/figure>\n\n\n\n<figure class=\"wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<iframe loading=\"lazy\" title=\"Nudos t\u00f3ricos en Neotrie VR\" width=\"1080\" height=\"608\" src=\"https:\/\/www.youtube.com\/embed\/rlbr-s0wg4M?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><figcaption class=\"wp-element-caption\">Se describen nudos t\u00f3ricos con la nueva calculadora gr\u00e1fica 3d implementa en Neotrie VR.<\/figcaption><\/figure>\n\n\n\n<figure class=\"wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<iframe loading=\"lazy\" title=\"Interpolatation of curves and surfaces\" width=\"1080\" height=\"608\" src=\"https:\/\/www.youtube.com\/embed\/ALr5DdcAvYY?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><figcaption class=\"wp-element-caption\">Interpolaci\u00f3n de curvas y superficies implementada en la pr\u00f3xima versi\u00f3n de Neotrie.<\/figcaption><\/figure>\n\n\n\n<figure class=\"wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<iframe loading=\"lazy\" title=\"Deformaci\u00f3n de la banda de Moebius en su circunferencia central\" width=\"1080\" height=\"608\" src=\"https:\/\/www.youtube.com\/embed\/uSHsSBdvgCU?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><figcaption class=\"wp-element-caption\">En este video mostramos c\u00f3mo introducir la homotop\u00eda entre el lazo que da dos vueltas a la circunferencia central de la cinta de Moebius y su borde.<\/figcaption><\/figure>\n\n\n\n<figure class=\"wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<iframe loading=\"lazy\" title=\"Poliedro de Cs\u00e1sz\u00e1r en 3D\" width=\"1080\" height=\"608\" src=\"https:\/\/www.youtube.com\/embed\/VECAq3eTzZo?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><figcaption class=\"wp-element-caption\">El poliedro de Cs\u00e1sz\u00e1r es la primera realizaci\u00f3n poli\u00e9drica de un toro de 7 v\u00e9rtices, sin diagonales y sin autointersecciones (Cs\u00e1sz\u00e1r, 1949), con 21 aristas y 14 caras triangulares. M\u00e1s informaci\u00f3n en <a href=\"http:\/\/www2.ual.es\/neotrie\/csaszars-polyhedron\">http:\/\/www2.ual.es\/neotrie\/csaszars-polyhedron<\/a>. Para ver el video en 3d, se requieren gafas VR para m\u00f3viles.<\/figcaption><\/figure>\n\n\n\n<figure class=\"wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<iframe loading=\"lazy\" title=\"Introducci\u00f3n a diagramas de nudos y trenzas en Neotrie VR\" width=\"1080\" height=\"608\" src=\"https:\/\/www.youtube.com\/embed\/blHpePXDxo4?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><figcaption class=\"wp-element-caption\">Se describen los movimientos de Reidemeister y el grupo de trenzas en Neotrie VR, con la c\u00e1mara estereosc\u00f3pica. Para ver el video en 3d, se requieren gafas VR para m\u00f3viles.<\/figcaption><\/figure>\n\n\n\n<div class=\"wp-block-group is-layout-flow wp-block-group-is-layout-flow\">\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-8f761849 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\" style=\"flex-basis:100%\">\n<figure class=\"wp-block-image size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"523\" src=\"https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/12\/WhatsApp-Image-2020-12-17-at-23.06.28-1024x523.jpeg\" alt=\"\" class=\"wp-image-5218\" style=\"width:819px;height:416px\" srcset=\"https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/12\/WhatsApp-Image-2020-12-17-at-23.06.28-980x500.jpeg 980w, https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/12\/WhatsApp-Image-2020-12-17-at-23.06.28-480x245.jpeg 480w\" sizes=\"(min-width: 0px) and (max-width: 480px) 480px, (min-width: 481px) and (max-width: 980px) 980px, (min-width: 981px) 1024px, 100vw\" \/><figcaption class=\"wp-element-caption\"><img decoding=\"async\" src=\"https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/12\/b26d20e8-294e-4cb2-b1c9-b199049db9fa-1-1024x531.jfif\" alt=\"\" style=\"width: undefinedpx;\"><br>Ilustraci\u00f3n del teorema de Seifert-Van Kampen aplicado al teorema de Wintinger que da la relaci\u00f3n en el grupo fundamental del exterior de un nudo.<\/figcaption><\/figure>\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\/2021\/12\/bfcc3043-33b3-4baa-bebe-f25a482641e1-1024x576.jfif\" alt=\"\" class=\"wp-image-5224\" srcset=\"https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/12\/bfcc3043-33b3-4baa-bebe-f25a482641e1-1024x576.jfif 1024w, https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/12\/bfcc3043-33b3-4baa-bebe-f25a482641e1-300x169.jfif 300w, https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/12\/bfcc3043-33b3-4baa-bebe-f25a482641e1-768x432.jfif 768w, https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/12\/bfcc3043-33b3-4baa-bebe-f25a482641e1-1536x864.jfif 1536w, https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/12\/bfcc3043-33b3-4baa-bebe-f25a482641e1-1080x608.jfif 1080w, https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/12\/bfcc3043-33b3-4baa-bebe-f25a482641e1-1280x720.jfif 1280w, https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/12\/bfcc3043-33b3-4baa-bebe-f25a482641e1-980x551.jfif 980w, https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/12\/bfcc3043-33b3-4baa-bebe-f25a482641e1-480x270.jfif 480w, https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2021\/12\/bfcc3043-33b3-4baa-bebe-f25a482641e1.jfif 1600w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><figcaption class=\"wp-element-caption\">Pueden importarse como ficheros STL \/ OBJ nudos de otros softwares.<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/div>\n\n\n\n<p class=\"wp-block-paragraph\">El proyecto contin\u00faa en versi\u00f3n multijugador:<\/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=\"d8OgWOltjv\"><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=mWuIqnKZZe#?secret=d8OgWOltjv\" data-secret=\"d8OgWOltjv\" 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\">Durante el curso 2024-25, ampliamos esta actividad con m\u00e1s experimentos sobre la botella de Klein:<\/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=\"F2b2iq3yW0\"><a href=\"https:\/\/www2.ual.es\/neotrie\/experimentos-con-la-botella-de-klein\/\">Experimentos con la botella de Klein<\/a><\/blockquote><iframe loading=\"lazy\" class=\"wp-embedded-content\" sandbox=\"allow-scripts\" security=\"restricted\" style=\"position: absolute; visibility: hidden;\" title=\"\u00abExperimentos con la botella de Klein\u00bb \u2014 Neotrie VR\" src=\"https:\/\/www2.ual.es\/neotrie\/experimentos-con-la-botella-de-klein\/embed\/#?secret=yLaRIBiMcp#?secret=F2b2iq3yW0\" data-secret=\"F2b2iq3yW0\" 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\"><\/p>\n","protected":false},"excerpt":{"rendered":"<p>En este proyecto elaboramos videos en Neotrie tanto en vista normal como estereosc\u00f3pica para las clases de Introducci\u00f3n de la Topolog\u00eda Algebraica de la Universidad de Almer\u00eda, durante el curso 2020-21. Para poder realizarlos se han implementado mejoras en la calculadora gr\u00e1fica 3d. Parte de este material aparece referenciado en el cap\u00edtulo &#8220;Exploring Dynamic Geometry [&hellip;]<\/p>\n","protected":false},"author":3,"featured_media":4818,"comment_status":"open","ping_status":"closed","template":"","meta":{"om_disable_all_campaigns":false,"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"project_category":[],"project_tag":[],"class_list":["post-3780","project","type-project","status-publish","has-post-thumbnail","hentry"],"aioseo_notices":[],"aioseo_head":"\n\t\t<!-- All in One SEO 4.9.10 - aioseo.com -->\n\t<meta name=\"description\" content=\"En este proyecto elaboramos videos en Neotrie tanto en vista normal como estereosc\u00f3pica para las clases de Introducci\u00f3n de la Topolog\u00eda Algebraica de la Universidad de Almer\u00eda, durante el curso 2020-21. Para poder realizarlos se han implementado mejoras en la calculadora gr\u00e1fica 3d. Parte de este material aparece referenciado en el cap\u00edtulo &quot;Exploring Dynamic Geometry\" \/>\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\/project\/introduccion-a-la-topologia-algebraica\/\" \/>\n\t<meta name=\"generator\" content=\"All in One SEO (AIOSEO) 4.9.10\" \/>\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=\"Introducci\u00f3n a la Topolog\u00eda Algebraica \ufeff \ufeff\ufeff-\ufeff\ufeff \ufeff\ufeffNeotrie VR\" \/>\n\t\t<meta property=\"og:description\" content=\"En este proyecto elaboramos videos en Neotrie tanto en vista normal como estereosc\u00f3pica para las clases de Introducci\u00f3n de la Topolog\u00eda Algebraica de la Universidad de Almer\u00eda, durante el curso 2020-21. Para poder realizarlos se han implementado mejoras en la calculadora gr\u00e1fica 3d. 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Para poder realizarlos se han implementado mejoras en la calculadora gr\u00e1fica 3d. 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