{"id":31941,"date":"2025-04-02T04:51:07","date_gmt":"2025-04-02T02:51:07","guid":{"rendered":"https:\/\/xmau.com\/wp\/notiziole\/?p=31941"},"modified":"2025-04-02T11:56:44","modified_gmt":"2025-04-02T09:56:44","slug":"il-rapporto-plastico-2","status":"publish","type":"post","link":"https:\/\/xmau.com\/wp\/notiziole\/2025\/04\/02\/il-rapporto-plastico-2\/","title":{"rendered":"Il rapporto plastico &#8211; 2"},"content":{"rendered":"<div class='__iawmlf-post-loop-links' style='display:none;' data-iawmlf-post-links='[{&quot;id&quot;:777,&quot;href&quot;:&quot;https:\\\/\\\/commons.wikimedia.org\\\/wiki\\\/File:Plastic_square_partitions.svg&quot;,&quot;archived_href&quot;:&quot;http:\\\/\\\/web-wp.archive.org\\\/web\\\/20251218055606\\\/https:\\\/\\\/commons.wikimedia.org\\\/wiki\\\/File:Plastic_square_partitions.svg&quot;,&quot;redirect_href&quot;:&quot;&quot;,&quot;checks&quot;:[{&quot;date&quot;:&quot;2026-02-11 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07:17:07&quot;,&quot;http_code&quot;:200},{&quot;date&quot;:&quot;2026-03-18 20:19:50&quot;,&quot;http_code&quot;:200},{&quot;date&quot;:&quot;2026-03-23 08:32:33&quot;,&quot;http_code&quot;:200},{&quot;date&quot;:&quot;2026-03-27 10:16:00&quot;,&quot;http_code&quot;:200},{&quot;date&quot;:&quot;2026-04-01 12:46:08&quot;,&quot;http_code&quot;:429},{&quot;date&quot;:&quot;2026-04-06 03:07:07&quot;,&quot;http_code&quot;:200},{&quot;date&quot;:&quot;2026-04-09 18:31:49&quot;,&quot;http_code&quot;:429},{&quot;date&quot;:&quot;2026-04-13 18:06:41&quot;,&quot;http_code&quot;:200}],&quot;broken&quot;:false,&quot;last_checked&quot;:{&quot;date&quot;:&quot;2026-04-13 18:06:41&quot;,&quot;http_code&quot;:200},&quot;process&quot;:&quot;done&quot;},{&quot;id&quot;:780,&quot;href&quot;:&quot;http:\\\/\\\/www.software3d.com\\\/Stella.php&quot;,&quot;archived_href&quot;:&quot;http:\\\/\\\/web-wp.archive.org\\\/web\\\/20260202145248\\\/https:\\\/\\\/www.software3d.com\\\/Stella.php&quot;,&quot;redirect_href&quot;:&quot;&quot;,&quot;checks&quot;:[{&quot;date&quot;:&quot;2026-02-11 22:54:35&quot;,&quot;http_code&quot;:200},{&quot;date&quot;:&quot;2026-02-17 06:42:21&quot;,&quot;http_code&quot;:200},{&quot;date&quot;:&quot;2026-02-21 19:53:11&quot;,&quot;http_code&quot;:200},{&quot;date&quot;:&quot;2026-02-26 07:14:38&quot;,&quot;http_code&quot;:200},{&quot;date&quot;:&quot;2026-03-04 23:23:22&quot;,&quot;http_code&quot;:200},{&quot;date&quot;:&quot;2026-03-11 06:45:33&quot;,&quot;http_code&quot;:200},{&quot;date&quot;:&quot;2026-03-14 07:17:07&quot;,&quot;http_code&quot;:200},{&quot;date&quot;:&quot;2026-03-18 20:19:50&quot;,&quot;http_code&quot;:200},{&quot;date&quot;:&quot;2026-03-23 08:32:34&quot;,&quot;http_code&quot;:200},{&quot;date&quot;:&quot;2026-03-27 10:16:01&quot;,&quot;http_code&quot;:200},{&quot;date&quot;:&quot;2026-04-01 12:46:09&quot;,&quot;http_code&quot;:200},{&quot;date&quot;:&quot;2026-04-06 03:07:08&quot;,&quot;http_code&quot;:200},{&quot;date&quot;:&quot;2026-04-09 18:31:49&quot;,&quot;http_code&quot;:200},{&quot;date&quot;:&quot;2026-04-13 18:06:37&quot;,&quot;http_code&quot;:200}],&quot;broken&quot;:false,&quot;last_checked&quot;:{&quot;date&quot;:&quot;2026-04-13 18:06:37&quot;,&quot;http_code&quot;:200},&quot;process&quot;:&quot;done&quot;}]'><\/div>\n<p>Abbiamo visto le prime propriet\u00e0 del rapporto plastico &rho;. Ma naturalmente ce ne sono molte altre. Per prima cosa, &rho; \u00e8 un <i>numero morfico<\/i>; per la precisione, uno dei due unici numeri morfici maggiori di 1. La nozione di numero morfico \u00e8 cos\u00ec di nicchia che mentre scrivo non c&#8217;\u00e8 nemmeno una voce di Wikipedia in inglese al riguardo: per\u00f2 non \u00e8 poi cos\u00ec complicata. Prendiamo il buon vecchio rapporto aureo &phi;. Sappiamo che vale la formula<\/p>\n<p style=\"text-align: center;\">$ \\begin{cases}\\varphi\\!+\\!1\\;=\\;\\varphi^{2},\\\\ \\varphi\\!-\\!1\\;=\\;\\varphi^{-1}\\end{cases} $<\/p>\n<p>Per il rapporto plastico vale una formula simile, anche se con esponenti diversi:<\/p>\n<p style=\"text-align: center;\">$ \\begin{cases}\\rho\\!+\\!1\\;=\\;\\rho^{3},\\\\ \\rho\\!-\\!1\\;=\\;\\rho^{-4}\\end{cases} $<\/p>\n<p>In generale un numero <i>x<\/i> maggiore di 1 \u00e8 morfico se sia <i>x<\/i>+1 che <i>x<\/i>&minus;1 sono potenze di <i>x<\/i>. Questa propriet\u00e0 \u00e8 condivisa solo da &phi; e &rho; e ha un interessante corollario di cui parler\u00f2 un&#8217;altra volta (devo ancora fare tutti i conti&#8230;). Per il momento, tenete solo presente che $1 +\\varphi^{-1} +\\varphi^{-2} =2$; inoltre $\\sum_{n=0}^{13} \\rho^{-n} =4$. Dalla definizione della successione di Perrin abbiamo intravisto, e magari intuito, che $\\rho^{n} =\\rho^{n-2} +\\rho^{n-3}$; abbiamo anche $ \\rho^{n} =\\rho^{n-1} +\\rho^{n-5} = \\rho^{n-3} +\\rho^{n-4} +\\rho^{n-5} $.<\/p>\n<p>Graficamente, il rapporto plastico ha delle interessanti propriet\u00e0. Tra l&#8217;altro, il primo a studiare questo numero intorno al 1960, riferendosi proprio all&#8217;architettura, \u00e8 stato l&#8217;olandese dom Hans van der Laan: il &#8220;dom&#8221; sta appunto a indicare che era un monaco benedettino. Anch&#8217;egli ha definito una successione come quelle di Perrin e Padovan dove il rapporto tra termini successivi tende a &rho;; i valori iniziali nel suo caso sono $V_1 = 0, V_0 = V_2 = 1$. Ma l&#8217;architettura non \u00e8 il mio campo, quindi passo; e soprattutto ci sono propriet\u00e0 pi\u00f9 semplici.<\/p>\n<p>Prendiamo per esempio un quadrato di lato unitario: come possiamo dividerlo in tre rettangoli simili? Una soluzione facile \u00e8 fare tre rettangoli paralleli di lati 1 e 1\/3. Una soluzione abbastanza facile \u00e8 quella di fare un rettangolo di lati 1 e 2\/3, e dividere la striscia rimanente in due rettangoli di lato 1\/3 e 1\/2. Ma c&#8217;\u00e8 una terza possibilit\u00e0, mostrata a destra nella figura qui sotto.<\/p>\n<p><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" src=\"https:\/\/i0.wp.com\/xmau.com\/wp\/notiziole\/wp-content\/uploads\/sites\/6\/2025\/04\/Plastic_square_partitions.svg_.png?resize=625%2C190&#038;ssl=1\" alt=\"tre rettangoli simili in un quadrato.\" width=\"625\" height=\"190\" class=\"aligncenter size-full wp-image-31971\" srcset=\"https:\/\/i0.wp.com\/xmau.com\/wp\/notiziole\/wp-content\/uploads\/sites\/6\/2025\/04\/Plastic_square_partitions.svg_.png?w=640&amp;ssl=1 640w, https:\/\/i0.wp.com\/xmau.com\/wp\/notiziole\/wp-content\/uploads\/sites\/6\/2025\/04\/Plastic_square_partitions.svg_.png?resize=300%2C91&amp;ssl=1 300w, https:\/\/i0.wp.com\/xmau.com\/wp\/notiziole\/wp-content\/uploads\/sites\/6\/2025\/04\/Plastic_square_partitions.svg_.png?resize=624%2C190&amp;ssl=1 624w\" sizes=\"auto, (max-width: 625px) 100vw, 625px\" \/><\/p>\n<p>In questo caso, il rettangolo a sinistra divide il quadrato in due parti le cui aree hanno rapporto &rho;, e quindi i suoi lati sono in rapporto plastico; il rapporto tra i lati del rettangolo grande e quelli del lato medio \u00e8 dunque &rho;, mentre quello tra i lati del rettangolo medio e del rettangolo piccolo \u00e8 &rho;&sup2;. Sempre con i rettangoli si pu\u00f2 costruire una spirale plastica, che assomiglia a una spirale aurea ma come vedete dalla figura spunta un po&#8217; fuori dai rettangoli. <\/p>\n<p><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" src=\"https:\/\/i0.wp.com\/xmau.com\/wp\/notiziole\/wp-content\/uploads\/sites\/6\/2025\/04\/Plastic_spiral.svg_.png?resize=625%2C469&#038;ssl=1\" alt=\"spirale plastica\" width=\"625\" height=\"469\" class=\"aligncenter size-full wp-image-31974\" srcset=\"https:\/\/i0.wp.com\/xmau.com\/wp\/notiziole\/wp-content\/uploads\/sites\/6\/2025\/04\/Plastic_spiral.svg_.png?w=960&amp;ssl=1 960w, https:\/\/i0.wp.com\/xmau.com\/wp\/notiziole\/wp-content\/uploads\/sites\/6\/2025\/04\/Plastic_spiral.svg_.png?resize=300%2C225&amp;ssl=1 300w, https:\/\/i0.wp.com\/xmau.com\/wp\/notiziole\/wp-content\/uploads\/sites\/6\/2025\/04\/Plastic_spiral.svg_.png?resize=768%2C576&amp;ssl=1 768w, https:\/\/i0.wp.com\/xmau.com\/wp\/notiziole\/wp-content\/uploads\/sites\/6\/2025\/04\/Plastic_spiral.svg_.png?resize=624%2C468&amp;ssl=1 624w\" sizes=\"auto, (max-width: 625px) 100vw, 625px\" \/><\/p>\n<p>Non poteva poi mancare il frattale di Rauzy: poich\u00e9 il rapporto plastico \u00e8 vicino a 1, \u00e8 difficile accorgersi che le tre figure colorate sono in rapporto &rho;&sup2; : &rho; : 1.<\/p>\n<p><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" src=\"https:\/\/i0.wp.com\/xmau.com\/wp\/notiziole\/wp-content\/uploads\/sites\/6\/2025\/04\/Plastic_Rauzy_cub.png?resize=549%2C480&#038;ssl=1\" alt=\"frattale di Rouzy plastico\" width=\"549\" height=\"480\" class=\"aligncenter size-full wp-image-31981\" srcset=\"https:\/\/i0.wp.com\/xmau.com\/wp\/notiziole\/wp-content\/uploads\/sites\/6\/2025\/04\/Plastic_Rauzy_cub.png?w=549&amp;ssl=1 549w, https:\/\/i0.wp.com\/xmau.com\/wp\/notiziole\/wp-content\/uploads\/sites\/6\/2025\/04\/Plastic_Rauzy_cub.png?resize=300%2C262&amp;ssl=1 300w\" sizes=\"auto, (max-width: 549px) 100vw, 549px\" \/><\/p>\n<p>Termino con una curiosit\u00e0 pi\u00f9 lessicale che altro: il raggio della sfera che circoscrive un icosidodecadodecaedro camuso di lato unitario (s\u00ec, ci sono due &#8220;dodeca&#8221; consecutivi, non \u00e8 un errore di copincolla) \u00e8 $\\frac{1}{2} \\sqrt{ \\frac{2 \\rho -1}{\\rho -1}} $. Direi per\u00f2 che non ce ne facciamo molto&#8230;<\/p>\n<p><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" src=\"https:\/\/i0.wp.com\/xmau.com\/wp\/notiziole\/wp-content\/uploads\/sites\/6\/2025\/04\/480px-Snub_icosidodecadodecahedron.png?resize=480%2C480&#038;ssl=1\" alt=\"icosidodecadodecaedro camuso\" width=\"480\" height=\"480\" class=\"aligncenter size-full wp-image-31985\" srcset=\"https:\/\/i0.wp.com\/xmau.com\/wp\/notiziole\/wp-content\/uploads\/sites\/6\/2025\/04\/480px-Snub_icosidodecadodecahedron.png?w=480&amp;ssl=1 480w, https:\/\/i0.wp.com\/xmau.com\/wp\/notiziole\/wp-content\/uploads\/sites\/6\/2025\/04\/480px-Snub_icosidodecadodecahedron.png?resize=300%2C300&amp;ssl=1 300w, https:\/\/i0.wp.com\/xmau.com\/wp\/notiziole\/wp-content\/uploads\/sites\/6\/2025\/04\/480px-Snub_icosidodecadodecahedron.png?resize=200%2C200&amp;ssl=1 200w\" sizes=\"auto, (max-width: 480px) 100vw, 480px\" \/><\/p>\n<p><small>Immagini da Wikimedia Commons: i <a href=\"https:\/\/commons.wikimedia.org\/wiki\/File:Plastic_square_partitions.svg\">rettangoli plastici<\/a> sono di David Eppstein, di pubblico dominio; la <a href=\"https:\/\/commons.wikimedia.org\/wiki\/File:Plastic_spiral.svg\">spirale plastica<\/a> e il <a href=\"https:\/\/commons.wikimedia.org\/wiki\/File:Plastic_Rauzy_cub.png\">frattale di Rauzy<\/a> sono di Zilverspreeuw, CC-BY-SA-4.0; l&#8217;<a href=\"http:\/\/480px-Snub_icosidodecadodecahedron.png\">icosidodecadodecaedro camuso<\/a> \u00e8 di Tomruen, usando il <a href=\"http:\/\/www.software3d.com\/Stella.php\">software Stella<\/a>. <\/small><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Ancora sul numero plastico, con tante figure&#8230;<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_lmt_disableupdate":"no","_lmt_disable":"no","jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"activitypub_content_warning":"","activitypub_content_visibility":"","activitypub_max_image_attachments":3,"activitypub_interaction_policy_quote":"anyone","activitypub_status":"federated","footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[1005,214],"tags":[],"class_list":["post-31941","post","type-post","status-publish","format-standard","hentry","category-matelight-2025","category-matematica_light"],"modified_by":".mau.","jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p6hcSh-8jb","jetpack-related-posts":[{"id":31897,"url":"https:\/\/xmau.com\/wp\/notiziole\/2025\/03\/26\/il-rapporto-plastico\/","url_meta":{"origin":31941,"position":0},"title":"Il rapporto plastico","author":".mau.","date":"2025-03-26","format":false,"excerpt":"Un altro numero che si trova spesso nelle successioni ricorsive","rel":"","context":"In &quot;mate-light-2025&quot;","block_context":{"text":"mate-light-2025","link":"https:\/\/xmau.com\/wp\/notiziole\/category\/matematica_light\/matelight-2025\/"},"img":{"alt_text":"spirali plastiche","src":"https:\/\/i0.wp.com\/xmau.com\/wp\/notiziole\/wp-content\/uploads\/sites\/6\/2025\/03\/plastic-spirals-1.png?resize=350%2C200&ssl=1","width":350,"height":200},"classes":[]},{"id":31571,"url":"https:\/\/xmau.com\/wp\/notiziole\/2025\/02\/26\/il-rapporto-superaureo-2\/","url_meta":{"origin":31941,"position":1},"title":"Il rapporto superaureo &#8211; 2","author":".mau.","date":"2025-02-26","format":false,"excerpt":"Continuiamo a vedere le propriet\u00e0 del rapporto superaureo, e le somiglianze e differenze con il rapporto aureo.","rel":"","context":"In &quot;mate-light-2025&quot;","block_context":{"text":"mate-light-2025","link":"https:\/\/xmau.com\/wp\/notiziole\/category\/matematica_light\/matelight-2025\/"},"img":{"alt_text":"spirale superaurea","src":"https:\/\/i0.wp.com\/xmau.com\/wp\/notiziole\/wp-content\/uploads\/sites\/6\/2025\/02\/Supergolden_spiral.svg_.png?resize=350%2C200&ssl=1","width":350,"height":200,"srcset":"https:\/\/i0.wp.com\/xmau.com\/wp\/notiziole\/wp-content\/uploads\/sites\/6\/2025\/02\/Supergolden_spiral.svg_.png?resize=350%2C200&ssl=1 1x, https:\/\/i0.wp.com\/xmau.com\/wp\/notiziole\/wp-content\/uploads\/sites\/6\/2025\/02\/Supergolden_spiral.svg_.png?resize=525%2C300&ssl=1 1.5x"},"classes":[]},{"id":31636,"url":"https:\/\/xmau.com\/wp\/notiziole\/2025\/03\/05\/il-rapporto-superaureo-3\/","url_meta":{"origin":31941,"position":2},"title":"Il rapporto superaureo &#8211; 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