{"id":2866,"date":"2018-12-09T17:07:00","date_gmt":"2018-12-09T16:07:00","guid":{"rendered":"http:\/\/www2.ual.es\/neotrie\/?p=2866"},"modified":"2026-05-27T11:45:37","modified_gmt":"2026-05-27T10:45:37","slug":"csaszars-polyhedron","status":"publish","type":"post","link":"https:\/\/www2.ual.es\/neotrie\/csaszars-polyhedron\/","title":{"rendered":"Cs\u00e1sz\u00e1r\u2019s polyhedron"},"content":{"rendered":"\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"576\" src=\"http:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2020\/05\/screen_1920x1080_2018-12-03_19-21-54-1024x576.png\" alt=\"\" class=\"wp-image-2867\" srcset=\"http:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2020\/05\/screen_1920x1080_2018-12-03_19-21-54-980x551.png 980w, http:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2020\/05\/screen_1920x1080_2018-12-03_19-21-54-480x270.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\">The Cs\u00e1sz\u00e1r polyhedron is the first polyhedral realization of a torus with 7 vertices, without diagonals and without self-intersections (Cs\u00e1sz\u00e1r, 1949), with 21 edges and 14 triangular faces.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The 1-skeleton is the complete graph on 7 vertices. So we can manipulate it to get the graph of the Csaszar\u2019s polyhedron. One can construct this model of the torus as the quotient of a square, and build step by step the construction. There is a last option to say Csaszar in high voice (or Csaszar with faces (on PC version)) to get the polyhedron automatically on the scene, thanks to the Speech Recognition System of Neotrie.\u00a0 One can enlarge the figure, and visit it from the inside, fly through the hole of the torus and see its interior too.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"576\" src=\"http:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2020\/05\/screen_1920x1080_2018-12-03_20-43-22-1024x576.png\" alt=\"\" class=\"wp-image-2870\" srcset=\"http:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2020\/05\/screen_1920x1080_2018-12-03_20-43-22-980x551.png 980w, http:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2020\/05\/screen_1920x1080_2018-12-03_20-43-22-480x270.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<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"576\" src=\"http:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2020\/05\/screen_1920x1080_2018-12-06_14-52-18-1024x576.png\" alt=\"\" class=\"wp-image-2872\" srcset=\"http:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2020\/05\/screen_1920x1080_2018-12-06_14-52-18-980x551.png 980w, http:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2020\/05\/screen_1920x1080_2018-12-06_14-52-18-480x270.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<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"576\" src=\"http:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2020\/05\/screen_1920x1080_2018-12-06_14-33-47-1024x576.png\" alt=\"\" class=\"wp-image-2871\" srcset=\"http:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2020\/05\/screen_1920x1080_2018-12-06_14-33-47-980x551.png 980w, http:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2020\/05\/screen_1920x1080_2018-12-06_14-33-47-480x270.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<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"576\" src=\"http:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2020\/05\/screen_1920x1080_2018-12-06_14-56-21-1024x576.png\" alt=\"\" class=\"wp-image-2873\" srcset=\"http:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2020\/05\/screen_1920x1080_2018-12-06_14-56-21-980x551.png 980w, http:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2020\/05\/screen_1920x1080_2018-12-06_14-56-21-480x270.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<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"576\" src=\"http:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2020\/05\/screen_1920x1080_2018-12-03_19-35-161-1024x576.png\" alt=\"\" class=\"wp-image-2868\" srcset=\"http:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2020\/05\/screen_1920x1080_2018-12-03_19-35-161-980x551.png 980w, http:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2020\/05\/screen_1920x1080_2018-12-03_19-35-161-480x270.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<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"576\" src=\"http:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2020\/05\/screen_1920x1080_2018-12-03_19-52-43-1024x576.png\" alt=\"\" class=\"wp-image-2869\" srcset=\"http:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2020\/05\/screen_1920x1080_2018-12-03_19-52-43-980x551.png 980w, http:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2020\/05\/screen_1920x1080_2018-12-03_19-52-43-480x270.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<h3 class=\"wp-block-heading\">Demonstrations of Neotrie in Paris<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Last week we visited the \u201cPalais de la D\u00e9couverte\u201d and&nbsp; the \u201cCit\u00e9 des Sciences\u201d in Paris, invited by Guillaume Reuiller.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">During two days we were testing Neotrie VR and planning future collaborations with members and responsibles of Universcience, Science Ouverte, teachers of GIPTIP, and \u201cComit\u00e9 de Culture Math\u00e9matique de l\u2019Institut Henry Poincar\u00e9\u201d. It was a motivating and great experience! I would like to thank them here for their interest on Neotrie VR and its future applications and uses.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The activity of the Cs\u00e1sz\u00e1r\u2019s polyhedron was one of the VR experiences started by&nbsp;<a href=\"https:\/\/twitter.com\/neotrie\/status\/1068955957407956992\">Roger&nbsp;Mansuy<\/a>&nbsp;(second of the right in the next picture). We&nbsp; have completed this in this post.<\/p>\n\n\n\n<figure class=\"wp-block-gallery columns-3 is-cropped wp-block-gallery-1 is-layout-flex wp-block-gallery-is-layout-flex\"><ul class=\"blocks-gallery-grid\"><li class=\"blocks-gallery-item\"><figure><img loading=\"lazy\" decoding=\"async\" width=\"768\" height=\"576\" src=\"http:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2020\/05\/img_20181201_110200.jpg\" alt=\"\" data-id=\"2874\" data-full-url=\"http:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2020\/05\/img_20181201_110200.jpg\" data-link=\"http:\/\/www2.ual.es\/neotrie\/?attachment_id=2874\" class=\"wp-image-2874\" srcset=\"http:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2020\/05\/img_20181201_110200.jpg 768w, http:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2020\/05\/img_20181201_110200-480x360.jpg 480w\" sizes=\"(min-width: 0px) and (max-width: 480px) 480px, (min-width: 481px) 768px, 100vw\" \/><figcaption class=\"blocks-gallery-item__caption\">ptr<\/figcaption><\/figure><\/li><li class=\"blocks-gallery-item\"><figure><img loading=\"lazy\" decoding=\"async\" width=\"724\" height=\"1024\" src=\"http:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2020\/05\/img_20181201_113944-724x1024.jpg\" alt=\"\" data-id=\"2875\" data-full-url=\"http:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2020\/05\/img_20181201_113944.jpg\" data-link=\"http:\/\/www2.ual.es\/neotrie\/?attachment_id=2875\" class=\"wp-image-2875\" srcset=\"http:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2020\/05\/img_20181201_113944-724x1024.jpg 724w, http:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2020\/05\/img_20181201_113944-480x679.jpg 480w\" sizes=\"(min-width: 0px) and (max-width: 480px) 480px, (min-width: 481px) 724px, 100vw\" \/><figcaption class=\"blocks-gallery-item__caption\">ptr<\/figcaption><\/figure><\/li><li class=\"blocks-gallery-item\"><figure><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"768\" src=\"http:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2020\/05\/img_20181201_114951-1024x768.jpg\" alt=\"\" data-id=\"2876\" data-full-url=\"http:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2020\/05\/img_20181201_114951-scaled.jpg\" data-link=\"http:\/\/www2.ual.es\/neotrie\/?attachment_id=2876\" class=\"wp-image-2876\" srcset=\"http:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2020\/05\/img_20181201_114951-980x735.jpg 980w, http:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2020\/05\/img_20181201_114951-480x360.jpg 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=\"blocks-gallery-item__caption\">cof<\/figcaption><\/figure><\/li><li class=\"blocks-gallery-item\"><figure><img loading=\"lazy\" decoding=\"async\" width=\"768\" height=\"576\" src=\"http:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2020\/05\/img_20181201_123451.jpg\" alt=\"\" data-id=\"2877\" data-full-url=\"http:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2020\/05\/img_20181201_123451.jpg\" data-link=\"http:\/\/www2.ual.es\/neotrie\/?attachment_id=2877\" class=\"wp-image-2877\" srcset=\"http:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2020\/05\/img_20181201_123451.jpg 768w, http:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2020\/05\/img_20181201_123451-480x360.jpg 480w\" sizes=\"(min-width: 0px) and (max-width: 480px) 480px, (min-width: 481px) 768px, 100vw\" \/><\/figure><\/li><li class=\"blocks-gallery-item\"><figure><img loading=\"lazy\" decoding=\"async\" width=\"768\" height=\"576\" src=\"http:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2020\/05\/img_20181201_124120.jpg\" alt=\"\" data-id=\"2878\" data-full-url=\"http:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2020\/05\/img_20181201_124120.jpg\" data-link=\"http:\/\/www2.ual.es\/neotrie\/?attachment_id=2878\" class=\"wp-image-2878\" srcset=\"http:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2020\/05\/img_20181201_124120.jpg 768w, http:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2020\/05\/img_20181201_124120-480x360.jpg 480w\" sizes=\"(min-width: 0px) and (max-width: 480px) 480px, (min-width: 481px) 768px, 100vw\" \/><figcaption class=\"blocks-gallery-item__caption\">edf<\/figcaption><\/figure><\/li><\/ul><\/figure>\n\n\n\n<h3 class=\"wp-block-heading\">Downloads:<\/h3>\n\n\n\n<ul class=\"wp-block-list\"><li><a href=\"https:\/\/drive.google.com\/a\/ual.es\/file\/d\/11ka0BgyF35fAOGiXsU_FOLBEywqABm_F\/view?usp=sharing\">csaszar.neot<\/a><br>Download this file to your local pc folder c:\/documents\/Neotrie\/NeotrieSaves, and open it from the windows file system inside the temple. Try to move and glue the edges of the torus to build the Cs\u00e1sz\u00e1r\u2019s polyhedron.<\/li><li><a href=\"http:\/\/www2.ual.es\/neotrie\/csaszars-polyhedron\/\">csazar4.neot<\/a> this is the final result, where you can enlarge the polyhedron and fly through the narrow hollow of the figure.<\/li><\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">References:<\/h3>\n\n\n\n<ol class=\"wp-block-list\"><li>J. Bokowski and A. Eggert:<em><a href=\"https:\/\/upcommons.upc.edu\/bitstream\/handle\/2099\/1067\/st17-10-a7-ocr.pdf?sequence=1&amp;isAllowed=y\">Toutes les r\u00e9alisations du tore de Moebius avec sept sommets<\/a><\/em>, Topologie Struct.&nbsp;<strong>17<\/strong>&nbsp;(1991), 59-78.<\/li><li>A. Cs\u00e1sz\u00e1r:&nbsp;<em>A polyhedron without diagonals<\/em>, Acta Sci. Math., Szeged&nbsp;<strong>13<\/strong>&nbsp;(1949-1950), 140-142.<\/li><li>F.H. Lutz,&nbsp;&nbsp;<a href=\"http:\/\/www.eg-models.de\/models\/Classical_Models\/2001.02.069\/_direct_link.html\">Cs\u00e1sz\u00e1r\u2019s Torus<\/a>,&nbsp;<em>Electronic Geometry Models<\/em>: 2001.02.069.<\/li><li><a href=\"https:\/\/en.wikipedia.org\/wiki\/Cs%C3%A1sz%C3%A1r_polyhedron\">https:\/\/en.wikipedia.org\/wiki\/Cs%C3%A1sz%C3%A1r_polyhedron<\/a><\/li><li><a href=\"https:\/\/www.gaussianos.com\/el-sorprendente-poliedro-de-csaszar\/\">https:\/\/www.gaussianos.com\/el-sorprendente-poliedro-de-csaszar\/<\/a><\/li><\/ol>\n\n\n\n<h3 class=\"wp-block-heading\">Updated (4 January, 2019):<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Here is the review by&nbsp;<a href=\"https:\/\/twitter.com\/neotrie\/status\/1068955957407956992\">Roger&nbsp;Mansuy<\/a>&nbsp;on the next nr 111 of&nbsp;<a href=\"https:\/\/www.quadrature.info\/?fbclid=IwAR1wxW6w87xVaQINKgoCuo0ixgU-lE9cr1lNjaYx2qA5LTcIR2VQ_W6Kmek\" target=\"_blank\" rel=\"noreferrer noopener\">Quadrature<\/a>.<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/topologia.files.wordpress.com\/2018\/12\/dwfw-wtwoaalosi.jpg\" alt=\"DwFW-WTWoAALOSI.jpg\" class=\"wp-image-14818\"\/><\/figure>\n\n\n\n<figure class=\"wp-block-image size-large\"><a href=\"https:\/\/youtu.be\/VECAq3eTzZo\" rel=\"https:\/\/youtu.be\/VECAq3eTzZo\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"553\" src=\"http:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2020\/10\/csaszar-1024x553.png\" alt=\"\" class=\"wp-image-3400\" srcset=\"https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2020\/10\/csaszar-1024x553.png 1024w, https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2020\/10\/csaszar-300x162.png 300w, https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2020\/10\/csaszar-768x414.png 768w, https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2020\/10\/csaszar-1536x829.png 1536w, https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2020\/10\/csaszar-1080x583.png 1080w, https:\/\/www2.ual.es\/neotrie\/wp-content\/uploads\/2020\/10\/csaszar.png 1920w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/a><figcaption>Updated: October 6th, 2020. 3d view of the scene. See a video: <a href=\"https:\/\/youtu.be\/VECAq3eTzZo\">https:\/\/youtu.be\/VECAq3eTzZo<\/a><\/figcaption><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":3,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"om_disable_all_campaigns":false,"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"categories":[43],"tags":[53,78],"class_list":["post-2866","post","type-post","status-publish","format-standard","hentry","category-universidad","tag-poliedros","tag-superficies"],"aioseo_notices":[],"aioseo_head":"\n\t\t<!-- All in One SEO 4.9.10 - aioseo.com -->\n\t<meta name=\"description\" content=\"The Cs\u00e1sz\u00e1r polyhedron is the first polyhedral realization of a torus with 7 vertices, without diagonals and without self-intersections (Cs\u00e1sz\u00e1r, 1949), with 21 edges and 14 triangular faces. The 1-skeleton is the complete graph on 7 vertices. So we can manipulate it to get the graph of the Csaszar\u2019s polyhedron. One can construct this model\" \/>\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\/csaszars-polyhedron\/\" \/>\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=\"Cs\u00e1sz\u00e1r\u2019s polyhedron - Neotrie VR\" \/>\n\t\t<meta property=\"og:description\" content=\"The Cs\u00e1sz\u00e1r polyhedron is the first polyhedral realization of a torus with 7 vertices, without diagonals and without self-intersections (Cs\u00e1sz\u00e1r, 1949), with 21 edges and 14 triangular faces. The 1-skeleton is the complete graph on 7 vertices. So we can manipulate it to get the graph of the Csaszar\u2019s polyhedron. One can construct this model\" \/>\n\t\t<meta property=\"og:url\" content=\"https:\/\/www2.ual.es\/neotrie\/csaszars-polyhedron\/\" \/>\n\t\t<meta property=\"article:published_time\" content=\"2018-12-09T16:07:00+00:00\" \/>\n\t\t<meta property=\"article:modified_time\" content=\"2026-05-27T10:45:37+00:00\" \/>\n\t\t<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/neotrie\" \/>\n\t\t<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n\t\t<meta name=\"twitter:site\" content=\"@NeotrieVR\" \/>\n\t\t<meta name=\"twitter:title\" content=\"Cs\u00e1sz\u00e1r\u2019s polyhedron - Neotrie VR\" \/>\n\t\t<meta name=\"twitter:description\" content=\"The Cs\u00e1sz\u00e1r polyhedron is the first polyhedral realization of a torus with 7 vertices, without diagonals and without self-intersections (Cs\u00e1sz\u00e1r, 1949), with 21 edges and 14 triangular faces. The 1-skeleton is the complete graph on 7 vertices. So we can manipulate it to get the graph of the Csaszar\u2019s polyhedron. 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