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Skip to content Home » Numeri magici: come la matematica compone la natura Già le popolazioni primitive, i greci, i romani o i popoli del periodo rinascimentale ritenevano determinati numeri capaci di esercitare influenze, negative o positive, sul mondo circostante , quelli che vengono conosciuti come “numeri magici “. Addirittura alcuni di essi rivestono un’importanza maggiore rispetto ad altri. Ad esempio i numeri pari erano considerati imperfetti, a differenza di quelli dispari che invece erano perfetti. Il numero 1 era per tradizione l’origine di tutti gli altri (quindi ne pari ne dispari), seguito dal numero 2 che è pari e quindi imperfetto. Il numero 3 è invece il primo numero perfetto. Il numero 4, nonostante sia pari, era di notevole importanza in quanto sommato al 3, 2 e 1 dava il numero 10 (numero perfetto in assoluto, aurea dei pitagorici ). Il numero 5 era riconducibile si primi sistemi di numerazione basati sulle dita delle mani. In modo analogo tutti i numeri successivi possiedono un significato perché visti come combinazioni di quelli iniziali.
Pitagora e i numeri magici Il famoso teorema di Pitagora Secondo Pitagora , e quindi anche i suoi adepti, il numero è tutto, in quanto permette di spiegare ogni cosa: l’ordine delle stagioni , i cicli della vegetazione , il moto degli astri. Nasce dunque l’aritmetica (arithmòs infatti significa numero in greco). Pitagora classificava i numeri in pari (sinonimo di male e perfezione) e dispari (indice di ordine e perfezione), positivi e negativi, maschili e femminili.
La sequenza di Fibonacci Esistono anche particolari sequenze di numeri come quella del matematico toscano di nome Leonardo Pisano , meglio noto come Fibonacci. Studiando la proliferazione di una coppia di conigli, Fibonacci notò la seguente progressione numerica: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, … e così vi a. La caratteristica principale di questa serie di numeri sta nel fatto che ogni numero a partire dal terzo è la somma dei due precedenti. Questa sequenza è oggi nota come la sequenza di Fibonacci.
Tali numeri ricorrono spesso nelle configurazioni tipiche del mondo naturale, come ad esempio nelle margherita da campo . La maggior parte di questi fiori, infatti, ha 13, 21 o 34 petali. I girasoli, invece, oscillano tra 34, 55, 89 o 144. Dunque il conteggio riporta sempre a multipli dei numeri di Fibonacci. Inoltre nel girasole i semi seguono un andamento spiraleggiante. Solitamente ci sono 34 spirali avvolte in senso orario o antiorario, e 55 avvolte nel senso opposto, però esistono girasoli con rapporti del numero di spirali 89/55, 144/89 e perfino 233/144.
Sequenze analoghe esistono anche per le configurazioni degli alberi o per le squame delle pigne che sono disposte in due famiglie di spirali intrecciate, ed ogni famiglia contiene un numero di Fibonacci di squame.
Il pi greco Il pi greco (π) è dato dal rapporto tra la circonferenza ed il diametro di un cerchio qualsiasi; il suo valore è 3.14159… La caratteristica principale di questo numero è che la sua rappresentazione decimale non termina mai e non è periodica. I numeri con questa caratteristica sono detti numeri irrazionali.
Il phi e la sezione aurea tra i numeri magici Caratteristiche analoghe al pi greco sono quelle del numero phi (φ ), corrispondente a 1.6180339887. Questo numero, meno famoso del 3.14, esprime una relazione geometrica chiamata sezione aurea o numero aureo . L’astronomo polacco Giovanni Keplero trovò un legame tra phi e la sequenza di Fibonacci, infatti i numeri della sua successione hanno come caratteristica che il rapporto tra un termine e il suo precedente oscilla intorno ad un numero che si avvicina sempre di più a Phi, il Rapporto Aureo.
Il Rapporto Aureo Il rapporto aureo è spesso indice di perfezione, bellezza, grazia e armonia . Questa proporzione geometrica individua un rapporto specifico tra le parti e l’intero. In ambito artistico si ricorre spesso ai cosiddetti rettangoli aurei, ovvero rettangoli i cui lati danno come rapporto il numero aureo phi. I greci ad esempio li utilizzavano per la realizzazione delle facciate dei loro templi. In natura è possibile trovare il rapporto aureo nella forma di alcune conchiglie, negli stami dei fiori, nella forma delle galassie o addirittura nelle proporzioni del corpo umano.
I numeri magici in fisica nucleare Anche la fisica non è esente da numeri particolari. Ad esempio in fisica nucleare si sente spesso parlare di numeri magic i, ovvero una particolare combinazione del numero di nucleoni (protoni o neutroni) in corrispondenza della quale i nuclei risultano particolarmente stabili (o comunque nuclei che presentano un’instabilità molto piccola e dunque una quasi-stabilità). I numeri sono: 2, 8, 20, 28, 50, 82, 126.
Si vede che per certi valori di Z (ovvero numero di protoni) e di N (ovvero numero di neutroni), i nuclei sono particolarmente stabili. In particolare l’energia di estrazione di un neutrone (protone) da un nucleo che ha N (Z) che è un «numero magico» è molto grande rispetto a quella del nucleo che ha N (Z) uguale a Nmagico + 1 (Zmagico + 1).
Questo effetto è analogo a quello che si osserva confrontando l’energia di ionizzazione di un gas nobile con quella del metallo alcalino adiacente (Z gas nobile + 1). Pertanto i nuclei magici (Z e/o N numero magico) sono l’analogo nella fisica nucleare dei gas nobili in fisica atomica. In entrambi i casi le proprietà di questi sistemi (gas nobili e nuclei magici) si interpretano come effetti di chiusura di shell.
Il modello a shell dei nuclei Il modello a shell nucleare fu proposto per la prima volta da Dimitry Ivanenko e successivamente sviluppato da E.Wigner, M. Goeppert-Mayer e J.Hans Jensen, i quali ottennero il Nobel per la fisica nel 1963. Il modello prevede che la disposizione degli elettroni in un atomo, seguendo il principio di esclusione di Pauli, tende a riempire la shell (ovvero il livello). Quando la shell risulta piena si osserva maggiore stabilità. Man mano che si aggiungono nucleoni al nucleo si registrano valori di energia di legame più bassa rispetto alla precedente situazione. Da queste configurazioni particolarmente stabili è nata l’espressione numeri magici.
A cura di Mario Liuzzo
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