{"id":31513,"date":"2025-02-19T04:51:19","date_gmt":"2025-02-19T03:51:19","guid":{"rendered":"https:\/\/xmau.com\/wp\/notiziole\/?p=31513"},"modified":"2025-02-18T17:47:55","modified_gmt":"2025-02-18T16:47:55","slug":"il-rapporto-superaureo","status":"publish","type":"post","link":"https:\/\/xmau.com\/notiziole\/2025\/02\/19\/il-rapporto-superaureo\/","title":{"rendered":"Il rapporto superaureo"},"content":{"rendered":"

Un paio di mesi fa avevo parlato del rapporto argenteo<\/a>, che evidentemente vale un po’ meno di quello aureo, nel senso che \u00e8 meno piacevole all’occhio. Ma ci sono altri numeri simili? Certo che s\u00ec! Oggi per esempio vi parler\u00f2 del rapporto superaureo (supergolden ratio<\/a> in inglese). Come il rapporto aureo \u00e8 la radice (positiva) dell’equazione $x^2 = x + 1$, il rapporto superaureo \u00e8 la radice (unica reale) dell’equazione $x^3 = x^2 + 1$, ed \u00e8 indicata con la lettera greca ψ (psi). Equivalentemente, ψ \u00e8 il rapporto $\\frac{a+b}{a}$ tale per cui $\\left( \\frac{a+b}{a} \\right)^{2} = \\frac{a}{b}$. Le prime cifre decimali di ψ sono 1,465571231876768…, e vale l’uguaglianza $\\psi^{2} \\left( \\psi – 1 \\right) = 1$.
\n\"rettangoli<\/a>
\nIl rettangolo qui in figura ha area ψ; e i tre rettangoli blu, rosso e giallo hanno area rispettivamente $\\frac{1}{\\psi}, \\frac{1}{\\psi}^3, \\frac{1}{\\psi}^5$ (quello verde ha la stessa area di quello rosso, per la cronaca). Come vedete, a differenza del rapporto aureo si salta di due potenze per volta.
\nAnche il rapporto superaureo ha molte caratteristiche interessanti, che per\u00f2 vi racconter\u00f2 la prossima settimana :-). <\/p>\n

Immagine di OmegaFallon, da Wikimedia Commons<\/a>.<\/small><\/p>\n","protected":false},"excerpt":{"rendered":"

Per chi trova svalutato il rapporto aureo…<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"_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},"jetpack_post_was_ever_published":false},"categories":[1005,214],"tags":[],"class_list":["post-31513","post","type-post","status-publish","format-standard","hentry","category-matelight-2025","category-matematica_light"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/phh2yV-8ch","jetpack-related-posts":[{"id":31571,"url":"https:\/\/xmau.com\/notiziole\/2025\/02\/26\/il-rapporto-superaureo-2\/","url_meta":{"origin":31513,"position":0},"title":"Il rapporto superaureo – 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 "mate-light-2025"","block_context":{"text":"mate-light-2025","link":"https:\/\/xmau.com\/notiziole\/category\/matematica_light\/matelight-2025\/"},"img":{"alt_text":"spirale superaurea","src":"https:\/\/i0.wp.com\/xmau.com\/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\/notiziole\/wp-content\/uploads\/sites\/6\/2025\/02\/Supergolden_spiral.svg_.png?resize=350%2C200&ssl=1 1x, https:\/\/i0.wp.com\/xmau.com\/notiziole\/wp-content\/uploads\/sites\/6\/2025\/02\/Supergolden_spiral.svg_.png?resize=525%2C300&ssl=1 1.5x"},"classes":[]},{"id":30250,"url":"https:\/\/xmau.com\/notiziole\/2024\/12\/25\/il-rapporto-argenteo\/","url_meta":{"origin":31513,"position":1},"title":"Il rapporto argenteo","author":".mau.","date":"2024-12-25","format":false,"excerpt":"Vale meno del rapporto aureo, per\u00f2...","rel":"","context":"In "mate-light-2024"","block_context":{"text":"mate-light-2024","link":"https:\/\/xmau.com\/notiziole\/category\/matematica_light\/matelight-2024\/"},"img":{"alt_text":"il rapporto argenteo","src":"https:\/\/i0.wp.com\/xmau.com\/notiziole\/wp-content\/uploads\/sites\/6\/2024\/12\/silverratio.png?resize=350%2C200&ssl=1","width":350,"height":200},"classes":[]},{"id":31636,"url":"https:\/\/xmau.com\/notiziole\/2025\/03\/05\/il-rapporto-superaureo-3\/","url_meta":{"origin":31513,"position":2},"title":"Il rapporto superaureo – 3","author":".mau.","date":"2025-03-05","format":false,"excerpt":"Ultima parte della trattazione sul numero superaureo, con un (immancabile?) frattale.","rel":"","context":"In "mate-light-2025"","block_context":{"text":"mate-light-2025","link":"https:\/\/xmau.com\/notiziole\/category\/matematica_light\/matelight-2025\/"},"img":{"alt_text":"frattale di Rauzy","src":"https:\/\/i0.wp.com\/xmau.com\/notiziole\/wp-content\/uploads\/sites\/6\/2025\/03\/Supergolden_Rauzy_sqr.png?resize=350%2C200&ssl=1","width":350,"height":200},"classes":[]},{"id":31816,"url":"https:\/\/xmau.com\/notiziole\/2025\/03\/19\/il-teorema-di-lin-mcmullin\/","url_meta":{"origin":31513,"position":3},"title":"Il teorema di Lin McMullin","author":".mau.","date":"2025-03-19","format":false,"excerpt":"Il rapporto aureo spunta nei posti pi\u00f9 impensati.","rel":"","context":"In "mate-light-2025"","block_context":{"text":"mate-light-2025","link":"https:\/\/xmau.com\/notiziole\/category\/matematica_light\/matelight-2025\/"},"img":{"alt_text":"Rappresentazione grafica del teorema di McMullin","src":"https:\/\/i0.wp.com\/xmau.com\/notiziole\/wp-content\/uploads\/sites\/6\/2025\/03\/McMullin.jpg?resize=350%2C200&ssl=1","width":350,"height":200},"classes":[]},{"id":31897,"url":"https:\/\/xmau.com\/notiziole\/2025\/03\/26\/il-rapporto-plastico\/","url_meta":{"origin":31513,"position":4},"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 "mate-light-2025"","block_context":{"text":"mate-light-2025","link":"https:\/\/xmau.com\/notiziole\/category\/matematica_light\/matelight-2025\/"},"img":{"alt_text":"spirali plastiche","src":"https:\/\/i0.wp.com\/xmau.com\/notiziole\/wp-content\/uploads\/sites\/6\/2025\/03\/plastic-spirals-1.png?resize=350%2C200&ssl=1","width":350,"height":200},"classes":[]},{"id":31941,"url":"https:\/\/xmau.com\/notiziole\/2025\/04\/02\/il-rapporto-plastico-2\/","url_meta":{"origin":31513,"position":5},"title":"Il rapporto plastico – 2","author":".mau.","date":"2025-04-02","format":false,"excerpt":"Ancora sul numero plastico, con tante figure...","rel":"","context":"In "mate-light-2025"","block_context":{"text":"mate-light-2025","link":"https:\/\/xmau.com\/notiziole\/category\/matematica_light\/matelight-2025\/"},"img":{"alt_text":"icosidodecadodecaedro camuso","src":"https:\/\/i0.wp.com\/xmau.com\/notiziole\/wp-content\/uploads\/sites\/6\/2025\/04\/480px-Snub_icosidodecadodecahedron.png?resize=350%2C200&ssl=1","width":350,"height":200},"classes":[]}],"jetpack_likes_enabled":true,"jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/xmau.com\/notiziole\/wp-json\/wp\/v2\/posts\/31513","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/xmau.com\/notiziole\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/xmau.com\/notiziole\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/xmau.com\/notiziole\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/xmau.com\/notiziole\/wp-json\/wp\/v2\/comments?post=31513"}],"version-history":[{"count":4,"href":"https:\/\/xmau.com\/notiziole\/wp-json\/wp\/v2\/posts\/31513\/revisions"}],"predecessor-version":[{"id":31520,"href":"https:\/\/xmau.com\/notiziole\/wp-json\/wp\/v2\/posts\/31513\/revisions\/31520"}],"wp:attachment":[{"href":"https:\/\/xmau.com\/notiziole\/wp-json\/wp\/v2\/media?parent=31513"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/xmau.com\/notiziole\/wp-json\/wp\/v2\/categories?post=31513"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/xmau.com\/notiziole\/wp-json\/wp\/v2\/tags?post=31513"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}