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Visto che è da un po' di tempo che non scrivo, anche perché in altri affari affaccendato (vedi, LA tesi), e considerando che per un po' non avrò un altro libro da recensire, ho pensato di parlarvi di un frattale. Un po' anche per ripetere tante volte «un po'» ed un po' per provare un nuovo plugin per la formattazione di formule matematiche scritte in LaTeX.

Il frattale in questione si chiama Koch Snowflake (fiocco di neve di Koch o stella di Koch).

La sua costruzione è ricorsiva e consiste semplicemente nel seguire le seguenti istruzioni a partire da un triangolo equilatero (l'immagine seguente mostra la costruzione ripetuta tre volte):

1. si divide ciascun lato della figura in tre segmenti uguali;
2. si costruisce un triangolo equilatero avente un lato coincidente con il segmento centrale di ogni lato ed il nuovo vertice esterno alla figura, poi si rimuove i suddetti segmenti centrali;
3. si prosegue dal punto 1 con la figura ottenuta al punto 2.

Ora che avete capito come si disegna, proseguirò con il dimostrare quali proprietà sorprendenti abbia questo frattale. Per fare ciò bisognerà calcolare semplicemente quale siano i limiti del perimetro e dell'area per un numero infinito di passi. È molto più facile di quanto sembri, anche se per spiegarvelo per iscritto dovrò complicare un po' le notazioni :).

Le due idee più chiare per calcolare perimetro ed area sono:

• perimetro: ad ogni passo si determina il numero di segmenti che sostituiscono un lato del triangolo iniziale, si moltiplica per la lunghezza del singolo segmento, ottenendo la lunghezza dei segmenti generati a partire da un singolo lato del triangolo iniziale e si moltiplica il tutto per tre, per ottenere il perimetro totale;
• area: ad ogni passo, si somma all'area ottenuta al passo precedente quella costituita dai triangoli che vengono, di fatto, aggiunti; come per il perimetro, si prende in considerazione un singolo lato del triangolo iniziale, si determina il numero di triangoli da aggiungere, l'area di uno di essi, si moltiplicano tali valori per ottenere l'area aggiuntiva su un singolo lato e si moltiplica, infine, il tutto per tre, per ottenere l'area totale da aggiungere a quella valutata al passo precedente.

Nelle notazioni che seguono, i numeri a pedice indicano i passi ai quali sono calcolati i valori e la lunghezza del lato del triangolo iniziare è pari ad . Il fatto di non calcolare certi valori o raggrupparli in modo inizialmente bizzarro con delle parentesi è intenzionale. Infine, si ricorda che l'area di un triangolo equilatero di lato è pari a .

Siano:

• SpL: segmenti per lato, il numero di segmenti che sostituiscono un singolo lato del triangolo iniziale;
• LSpL: la lunghezza di uno dei suddetti segmenti;
• TaddN: il numero di triangoli da aggiungere rispetto al passo precedente per un singolo lato del triangolo iniziale;
• TaddA: l'area di uno dei suddetti triangoli;
• TaddATot: l'area totale da aggiungere rispetto al passo precedente;
• P: il perimetro totale;
• A: l'area totale.

Passo 0:

• : non ha significato
• : non ha significato
• : non ha significato

Passo 1:

• :

Passo 2:

• :
• 
  

Passo 3:

• :
• 
  
  

A questo punto è il caso di fermarsi perché si vede chiaramente un pattern. Riporto i valori di perimetro ed area ad ogni passo e la generalizzazione al generico passo n>0:

• Perimetro
  • ...
• Area
  • ...
  • 
    

Ora non resta che determinare il limite per del perimetro e dell'area. Per quanto riguarda il perimetro è immediato vedere che tenda all'infinito, poiché la frazione è maggiore di 1:

Per quanto riguarda l'area bisogna osservare che la sommatoria coinvolge termini di una serie geometrica di ragione r=, il cui limite si può calcolare con la formula (chi non è convinto, clicchi qui):

Nel caso non ve ne siate ancora accorti, all'infinito il perimetro di quella figura sarà infinito, mentre l'area convergerà ad un valore ben preciso, ovvero gli dell'area del triangolo iniziale, poiché .

Ovvero, un perimetro infinito racchiude un'area finita. Sorprendente, no? 🙂

Proprietà analoghe valgono per altri frattali. Se vi interessa questo argomento potete leggere un'introduzione su cosa siano i frattali o più semplicemente vedere un'interessante lista di immagini di frattali. Vi propongo le pagine in inglese di  Wikipedia perché sono le più complete, ricordandovi però che Wikipedia è un formidabile strumento di traduzione semantica (leggasi: cliccate sulle lingue che preferite nel menu "languages" a sinistra).

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[1] La sommatoria in questione, che coinvolge termini di una serie geometrica, è della forma . Chiamando la somma ed esplicitandone qualche termine si ha:

Moltiplicando entrambi i membri per si ottiene:

Sottraendo membro a membro le due equazioni si ha:

Q.E.D.

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