{"id":10735,"date":"2021-05-01T07:00:00","date_gmt":"2021-05-01T11:00:00","guid":{"rendered":"https:\/\/mathemalchemy.org\/?p=10735"},"modified":"2024-09-24T22:45:28","modified_gmt":"2024-09-25T02:45:28","slug":"lumiere-sur-les-projections-stereographiques","status":"publish","type":"post","link":"https:\/\/mathemalchemy.org\/fr\/2021\/05\/01\/lumiere-sur-les-projections-stereographiques\/","title":{"rendered":"Lumi\u00e8re sur&#8230; les projections st\u00e9r\u00e9ographiques"},"content":{"rendered":"\n<h2 class=\"wp-block-heading has-large-font-size\">Voir la chronique vid\u00e9o d&rsquo;Henry Segerman<\/h2>\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<span class=\"embed-youtube\" style=\"text-align:center; display: block;\"><iframe loading=\"lazy\" class=\"youtube-player\" width=\"900\" height=\"507\" src=\"https:\/\/www.youtube.com\/embed\/8QQywzbGAU8?version=3&#038;rel=1&#038;showsearch=0&#038;showinfo=1&#038;iv_load_policy=1&#038;fs=1&#038;hl=fr-FR&#038;autohide=2&#038;wmode=transparent\" allowfullscreen=\"true\" style=\"border:0;\" sandbox=\"allow-scripts allow-same-origin allow-popups allow-presentation allow-popups-to-escape-sandbox\"><\/iframe><\/span>\n<\/div><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\">Lire la transcription de la vid\u00e9o<\/h2>\n\n\n\n<h3 class=\"wp-block-heading has-large-font-size\">A propos de Mathemalchemy<\/h3>\n\n\n\n<p>Bonjour, je suis Henry Segerman,<\/p>\n\n\n\n<p>Depuis un an, je travaille avec une \u00e9quipe de <a href=\"https:\/\/mathemalchemy.org\/team-23-mathematicians-artists-of-mathemalchemy\/\">24 autres math\u00e9maticiens et artistes en math\u00e9matiques<\/a> pour cr\u00e9er une grande installation artistique multim\u00e9dia c\u00e9l\u00e9brant la cr\u00e9ativit\u00e9 et la beaut\u00e9 des math\u00e9matiques. Le projet est dirig\u00e9 par <a href=\"https:\/\/mathemalchemy.org\/team-23-mathematicians-artists-of-mathemalchemy\/#Ingrid-Daubechies\">Ingrid Daubechies<\/a>, math\u00e9maticienne \u00e0 l&rsquo;universit\u00e9 de Duke, et <a href=\"https:\/\/mathemalchemy.org\/team-23-mathematicians-artists-of-mathemalchemy\/#Dominique-Ehrmann\">Dominique Ehrmann<\/a>, artiste textile bas\u00e9e au Canada.<\/p>\n\n\n\n<p>Tout a commenc\u00e9 lorsque Ingrid a vu l&rsquo;installation de Dominique <a href=\"http:\/\/dominiquehrmann.com\/en\/timetobreakfree\/\" target=\"_blank\" rel=\"noreferrer noopener\">\u00ab\u00a0Time to Break Free<\/a>\u00ab\u00a0, dans laquelle une sorte de \u00ab\u00a0machine\u00a0\u00bb transforme les figures plates d&rsquo;une courtepointe en personnages enti\u00e8rement tridimensionnels.<\/p>\n\n\n\n<p>Ingrid s&rsquo;est alors demand\u00e9 s&rsquo;il \u00e9tait possible de donner vie \u00e0 la beaut\u00e9 et \u00e0 la cr\u00e9ativit\u00e9 des id\u00e9es math\u00e9matiques de la m\u00eame mani\u00e8re.<\/p>\n\n\n\n<p>Ingrid et Dominique ont fait une pr\u00e9sentation \u00e0 la conf\u00e9rence <a href=\"https:\/\/jointmathematicsmeetings.org\/jmm\">Joint Mathematics Meetings<\/a> en janvier 2020, et un groupe s&rsquo;est form\u00e9 autour du projet, baptis\u00e9 \u00ab\u00a0Mathemalchemy\u00a0\u00bb, qui compte aujourd&rsquo;hui <a href=\"https:\/\/mathemalchemy.org\/team-23-mathematicians-artists-of-mathemalchemy\/\">24 math\u00e9maticiens et artistes<\/a>.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large coblocks-animate\" data-coblocks-animation=\"fadeIn\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"900\" height=\"499\" src=\"https:\/\/i0.wp.com\/mathemalchemy.org\/wp-content\/uploads\/2020\/09\/tour-67-e1601858479125-1024x568.jpg?resize=900%2C499&#038;ssl=1\" alt=\"\" class=\"wp-image-694\" srcset=\"https:\/\/i0.wp.com\/mathemalchemy.org\/wp-content\/uploads\/2020\/09\/tour-67-e1601858479125.jpg?resize=1024%2C568&amp;ssl=1 1024w, https:\/\/i0.wp.com\/mathemalchemy.org\/wp-content\/uploads\/2020\/09\/tour-67-e1601858479125.jpg?resize=300%2C166&amp;ssl=1 300w, https:\/\/i0.wp.com\/mathemalchemy.org\/wp-content\/uploads\/2020\/09\/tour-67-e1601858479125.jpg?resize=768%2C426&amp;ssl=1 768w, https:\/\/i0.wp.com\/mathemalchemy.org\/wp-content\/uploads\/2020\/09\/tour-67-e1601858479125.jpg?resize=1536%2C852&amp;ssl=1 1536w, https:\/\/i0.wp.com\/mathemalchemy.org\/wp-content\/uploads\/2020\/09\/tour-67-e1601858479125.jpg?resize=1200%2C666&amp;ssl=1 1200w, https:\/\/i0.wp.com\/mathemalchemy.org\/wp-content\/uploads\/2020\/09\/tour-67-e1601858479125.jpg?w=1564&amp;ssl=1 1564w\" sizes=\"auto, (max-width: 900px) 100vw, 900px\" \/><figcaption class=\"wp-element-caption\">La maquette finale de Mathemalchemy<\/figcaption><\/figure>\n\n\n\n<p>Voici la <a href=\"https:\/\/mathemalchemy.org\/2020\/11\/12\/the-maquettes-creation-process\/\">maquette actuelle \u00e0 l&rsquo;\u00e9chelle 1\/4<\/a> de l&rsquo;installation &#8211; une \u00eele magique remplie de sc\u00e8nes fantaisistes, fabriqu\u00e9e \u00e0 l&rsquo;aide des techniques et mat\u00e9riaux:<\/p>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-28f84493 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<ul class=\"wp-block-list\">\n<li>le crochet,<\/li>\n\n\n\n<li>le tricot,<\/li>\n\n\n\n<li>la courtepointe,<\/li>\n\n\n\n<li>la couture,<\/li>\n\n\n\n<li>le tissage,<\/li>\n\n\n\n<li>le perlage,<\/li>\n\n\n\n<li>la laine feutr\u00e9e \u00e0 l&rsquo;aiguille,<\/li>\n<\/ul>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<ul class=\"wp-block-list\">\n<li>la broderie,<\/li>\n\n\n\n<li>le travail du bois,<\/li>\n\n\n\n<li>la c\u00e9ramique,<\/li>\n\n\n\n<li>la p\u00e2te \u00e0 modeler,<\/li>\n\n\n\n<li>l&rsquo;impression 3D,<\/li>\n\n\n\n<li>la peinture,<\/li>\n\n\n\n<li>le vitrail,<\/li>\n<\/ul>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<ul class=\"wp-block-list\">\n<li>le soudage de l&rsquo;acier,<\/li>\n\n\n\n<li>l&rsquo;acrylique,<\/li>\n\n\n\n<li>la sculpture de fil de fer,<\/li>\n\n\n\n<li>l&rsquo;origami,<\/li>\n\n\n\n<li>le temari,<\/li>\n\n\n\n<li>le cuir,<\/li>\n\n\n\n<li>et la lumi\u00e8re.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n\n\n\n<div class=\"wp-block-cover alignfull is-light has-parallax\" style=\"min-height:900px;aspect-ratio:unset;\"><span aria-hidden=\"true\" class=\"wp-block-cover__background has-background-dim-0 has-background-dim\"><\/span><div class=\"wp-block-cover__image-background wp-image-2547 has-parallax\" style=\"background-position:50% 50%;background-image:url(https:\/\/mathemalchemy.org\/wp-content\/uploads\/2021\/04\/mathemalchemy-techniques-gallery.jpg)\"><\/div><div class=\"wp-block-cover__inner-container is-layout-flow wp-block-cover-is-layout-flow\">\n<p class=\"has-text-align-center has-primary-color has-background-background-color has-text-color has-background has-huge-font-size\">Chaque fois qu&rsquo;une id\u00e9e ou une r\u00e9f\u00e9rence math\u00e9matique pouvait \u00eatre int\u00e9gr\u00e9e, nous l&rsquo;utilisions.<\/p>\n<\/div><\/div>\n\n\n\n<p><br>L&rsquo;exposition Math\u00e9mAlchimie sera install\u00e9e en permanence \u00e0 l&rsquo;<a href=\"https:\/\/www.duke.edu\/\" target=\"_blank\" rel=\"noreferrer noopener\">universit\u00e9 de Duke<\/a>, apr\u00e8s \u00eatre partie en tourn\u00e9e, avec des arr\u00eats pr\u00e9vus \u00e0 la <a href=\"http:\/\/www.nasonline.org\/\" target=\"_blank\" rel=\"noreferrer noopener\">National Academy of Sciences<\/a> \u00e0 Washington DC, ainsi qu&rsquo;\u00e0 l&rsquo;international (sauf \u00e9ventuelles perturbations dues \u00e0 la pand\u00e9mie).<\/p>\n\n\n\n<h2 class=\"wp-block-heading has-huge-font-size\">Projection st\u00e9r\u00e9ographique<\/h2>\n\n\n\n<p>L&rsquo;une des sc\u00e8nes math\u00e9matiques auxquelles j&rsquo;ai particip\u00e9 est celle du phare. Le sommet du phare sera dot\u00e9 de deux lumi\u00e8res, l&rsquo;une projet\u00e9e horizontalement depuis un <a href=\"https:\/\/mathemalchemy.org\/2021\/01\/19\/dodecahedral-trajectory\/\">dod\u00e9ca\u00e8dre<\/a> en vitrail r\u00e9alis\u00e9 par <a href=\"https:\/\/mathemalchemy.org\/team-23-mathematicians-artists-of-mathemalchemy\/#Bronna-Butler\">Bronna Butler<\/a>, et l&rsquo;autre projet\u00e9e vers le plafond.<\/p>\n\n\n\n<p>Il s&rsquo;agit de la reprise d&rsquo;un ancien projet illustrant ce que l&rsquo;on appelle la projection st\u00e9r\u00e9ographique. La projection st\u00e9r\u00e9ographique est une repr\u00e9sentation de la sph\u00e8re sur le plan. Ainsi, tout comme la carte de Mercator permet de repr\u00e9senter les continents du globe sur une feuille de papier plane, la projection st\u00e9r\u00e9ographique est un autre moyen d&rsquo;y parvenir.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large coblocks-animate\" data-coblocks-animation=\"fadeIn\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"900\" height=\"663\" src=\"https:\/\/i0.wp.com\/mathemalchemy.org\/wp-content\/uploads\/2021\/04\/stereographic-projection-mathemalchemy-art-project.jpg?resize=900%2C663&#038;ssl=1\" alt=\"\" class=\"wp-image-2555\" srcset=\"https:\/\/i0.wp.com\/mathemalchemy.org\/wp-content\/uploads\/2021\/04\/stereographic-projection-mathemalchemy-art-project.jpg?w=1000&amp;ssl=1 1000w, https:\/\/i0.wp.com\/mathemalchemy.org\/wp-content\/uploads\/2021\/04\/stereographic-projection-mathemalchemy-art-project.jpg?resize=300%2C221&amp;ssl=1 300w, https:\/\/i0.wp.com\/mathemalchemy.org\/wp-content\/uploads\/2021\/04\/stereographic-projection-mathemalchemy-art-project.jpg?resize=768%2C566&amp;ssl=1 768w\" sizes=\"auto, (max-width: 900px) 100vw, 900px\" \/><figcaption class=\"wp-element-caption\">Projection st\u00e9r\u00e9ographique<\/figcaption><\/figure>\n\n\n\n<p>Dans le mod\u00e8le, le rayon lumineux part du p\u00f4le nord de la sph\u00e8re, se propage \u00e0 l&rsquo;int\u00e9rieur de la sph\u00e8re, traverse la sph\u00e8re vers l&rsquo;ext\u00e9rieur et continue ensuite son chemin jusqu\u00e0 atteindre le plan.<\/p>\n\n\n\n<p>Ainsi les rayons lumineux rep\u00e9sente cette transformation, qui fait correspondre l\u00e0 ou la lumi\u00e8re traverse la sph\u00e8re et l\u00e0 ou elle traverse le plan.<\/p>\n\n\n\n<p>Il suffit de regarder les ombres pour se rendre compte de l&rsquo;effet de la projection st\u00e9r\u00e9ographique. Par exemple, les triangles du milieu du plan sont \u00e0 peu pr\u00e8s de la m\u00eame taille que les triangles de la sph\u00e8re elle-m\u00eame, mais au fur et \u00e0 mesure que l&rsquo;on s&rsquo;\u00e9loigne, ils commencent \u00e0 devenir de plus en plus grands. On voit ici beaucoup d&rsquo;autres caract\u00e9ristiques int\u00e9ressantes de la projection st\u00e9r\u00e9ographique, par exemple les angles sont pr\u00e9serv\u00e9s. Tous les angles de quatre-vingt-dix, soixante degr\u00e9s et ainsi de suite sur la sph\u00e8re forment aussi des angles d&rsquo;ombres de quatre-vingt-dix et soixante degr\u00e9s. De m\u00eame, les cercles correspondent \u00e0 d&rsquo;autres cercles. Il existe tous ces grands cercles sur la sph\u00e8re qui s&rsquo;associent \u00e0 des cercles sur le plan. <\/p>\n\n\n\n<p>L&rsquo;ombre est tr\u00e8s sensible \u00e0 la position de la lumi\u00e8re par rapport \u00e0 la sph\u00e8re, mais elle n&rsquo;est pas tr\u00e8s sensible \u00e0 la position de la table. Ainsi, si j&rsquo;\u00e9loigne la sph\u00e8re et la lumi\u00e8re de la table, tout ce qui se passe, c&rsquo;est que l&rsquo;ombre s&rsquo;agrandit, sans que l&rsquo;image ne change pour autant. C&rsquo;est une bonne chose pour nous, car le phare \u00e9clairera vers le haut, projetant les ombres sur le plafond, et comme l&rsquo;installation se fera dans de nombreux endroits diff\u00e9rents, il n&rsquo;y a aucun moyen de savoir quelle sera la hauteur du plafond.<\/p>\n\n\n\n<p>Nous avons pr\u00e9vu une deuxi\u00e8me version au cas o\u00f9 cela s&rsquo;av\u00e9rerait n\u00e9cessaire. Si le plafond est trop haut dans une salle, les motifs risquent d&rsquo;\u00eatre trop agrandis, et la majeure partie des motils sera projet\u00e9e sur les murs et non le plafond. Les motifs serait toujours visibles, mais les ombres sur les murs n&rsquo;aurait pas toutes les propri\u00e9t\u00e9s agr\u00e9ables de la projection st\u00e9r\u00e9ographique. Dans ce cas, nous pr\u00e9voyons de r\u00e9utiliser une autre \u0153uvre d&rsquo;art d&rsquo;ombre de lumi\u00e8re, qui projette le mod\u00e8le du disque de Poincar\u00e9 du plan hyperbolique sur un disque. Ce motif n&rsquo;utilise qu&rsquo;un c\u00f4ne de lumi\u00e8re de 90 degr\u00e9s, qui se projette sur une zone plus petite, au lieu des 180 degr\u00e9s utilis\u00e9s dans la projection st\u00e9r\u00e9ographique de la sph\u00e8re compl\u00e8te.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large coblocks-animate\" data-coblocks-animation=\"fadeIn\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"900\" height=\"669\" src=\"https:\/\/i0.wp.com\/mathemalchemy.org\/wp-content\/uploads\/2021\/04\/poincare-stereographic-projection-mathemalchemy-art-project.jpg?resize=900%2C669&#038;ssl=1\" alt=\"\" class=\"wp-image-2557\" srcset=\"https:\/\/i0.wp.com\/mathemalchemy.org\/wp-content\/uploads\/2021\/04\/poincare-stereographic-projection-mathemalchemy-art-project.jpg?w=1000&amp;ssl=1 1000w, https:\/\/i0.wp.com\/mathemalchemy.org\/wp-content\/uploads\/2021\/04\/poincare-stereographic-projection-mathemalchemy-art-project.jpg?resize=300%2C223&amp;ssl=1 300w, https:\/\/i0.wp.com\/mathemalchemy.org\/wp-content\/uploads\/2021\/04\/poincare-stereographic-projection-mathemalchemy-art-project.jpg?resize=768%2C571&amp;ssl=1 768w, https:\/\/i0.wp.com\/mathemalchemy.org\/wp-content\/uploads\/2021\/04\/poincare-stereographic-projection-mathemalchemy-art-project.jpg?resize=200%2C150&amp;ssl=1 200w\" sizes=\"auto, (max-width: 900px) 100vw, 900px\" \/><figcaption class=\"wp-element-caption\">Mod\u00e8le du disque de Poincar\u00e9<\/figcaption><\/figure>\n\n\n\n<p>Voil\u00e0, c&rsquo;est la sph\u00e8re au sommet du phare. L&rsquo;assemblage des parties de l&rsquo;installation construites ind\u00e9pendamment les unes des autres doit avoir lieu en juillet de cette ann\u00e9e &#8211; avec un peu de chance, il y en aura d&rsquo;autres !<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Regardez la chronique vid\u00e9o d&rsquo;Henry Segerman. <\/p>\n<p>L&rsquo;une des sc\u00e8nes math\u00e9matiques auxquelles j&rsquo;ai particip\u00e9 est celle du phare. Le sommet du phare sera dot\u00e9 de deux lumi\u00e8res, l&rsquo;une projet\u00e9e horizontalement depuis un dod\u00e9ca\u00e8dre en vitrail r\u00e9alis\u00e9 par Bronna Butler, et l&rsquo;autre projet\u00e9e vers le plafond.<\/p>\n<p>Il s&rsquo;agit de la reprise d&rsquo;un ancien projet illustrant ce que l&rsquo;on appelle la projection st\u00e9r\u00e9ographique. La projection st\u00e9r\u00e9ographique est une repr\u00e9sentation de la sph\u00e8re sur le plan. Ainsi, tout comme la carte de Mercator permet de repr\u00e9senter les continents du globe sur une feuille de papier plane, la projection st\u00e9r\u00e9ographique est un autre moyen d&rsquo;y parvenir.<\/p>\n","protected":false},"author":44503789,"featured_media":9220,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_coblocks_attr":"","_coblocks_dimensions":"","_coblocks_responsive_height":"","_coblocks_accordion_ie_support":"","jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":true,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[6678],"tags":[6803,6680,7153,6728,6835],"class_list":["post-10735","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathemalchemy-en-theories","tag-carte","tag-impression-3d-fr","tag-la-lumiere","tag-phare","tag-projection-stereographique","entry"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.3 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Lumi\u00e8re sur... les projections st\u00e9r\u00e9ographiques - Mathemalchemy<\/title>\n<meta name=\"description\" content=\"Regardez la chronique vid\u00e9o d&#039;Henry Segerman sur son \u00e9tonnante projection st\u00e9r\u00e9ographique. 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