Utente:Spock/Sandbox: differenze tra le versioni
Contenuto cancellato Contenuto aggiunto
Nessun oggetto della modifica |
Nessun oggetto della modifica |
||
Riga 1:
In matematica, un [[frattale]] è un oggetto geometrico in cui la [[dimensione di Hausdorff]] (δ) è strettamente superiore alla [[dimensione topologica]]. Qui di seguito è presentata una lista di frattali per dimensione di Haudorff crescente, con lo scopo di visualizzare che cosa significhi per un frattale possedere una dimensione bassa o alta.
== Frattali deterministici ==
{| border="0" cellpadding="4" rules="all" style="border: 1px solid #999; background-color:#FFFFFF"
|- align="center" bgcolor="#cccccc"
! δ<br />(valore esatto) || δ<br />(valore) || Nome || Illustrazione || width="40%" | Commenti
|-
| <math>\textstyle{\frac {ln(2)} {ln(\delta)}?}</math> || align="right" | 0.4498? || Biforcazioni dell'eqauzione logistica || align="center" |[[Image:Logistic map bifurcation diagram.png|150px]] || Nel [[diagramma di biforcazione]], quando ci si avvicina alla zona caotica, a succession of period doubling appears, in a geometric progression tending to 1/δ. (δ=[[Feigenbaum constant]]=4.6692).
|-
| <math>\textstyle{\frac {ln(2)} {ln(3)}}</math> || align="right" | 0.6309 || [[Insieme di Cantor]] || align="center" |[[image:Cantor_set_in_seven_iterations.svg|200px]] || Costruito eliminando la terza parte centrale ad ogni iterazione. Insieme [[Insieme mai denso|mai denso]], né [[Insieme numerabile|numerabile]].
|-
| <math>\textstyle{\frac {ln(6)} {ln(8)}}</math> || align="right" | 0.8617 || [[Insieme di Smith-Volterra-Cantor set]] || align="center" |[[Image:Smith-Volterra set.png|150px]] || (In bianco nella figura) costruito eliminando la quarta parte centrale ad ogni iterazione. Insieme mai denso, ma avente [[misura di Lebesgue]] ½.
|-
| <math>\textstyle{\frac {ln(8)} {ln(7)}}</math> || align="right" | 1.0686 || [[Isola di Gosper]] || align="center" |[[Image:Ile_de_Gosper.gif|100px]] ||
|-
| || align="right" | 1.26 || [[Attrattore di Hénon]] || align="center" |[[Image:Henon attractor.png|100px]] || L'attrattore di Hénon canonico (con parametri a = 1.4 and b = 0.3) possiede dimensione di Haussdorf δ = 1.261 ± 0.003. Parametri differenti conducono a differenti valori di δ.
|-
| <math>\textstyle{\frac {ln(4)} {ln(3)}}</math> || align="right" | 1.2619 || [[Curva di Koch]] || align="center" | [[Image:Koch curve.png|200px]] || 3 di queste curve formano il fiocco o l'antifiocco di Koch.
|-
| <math>\textstyle{\frac {ln(4)} {ln(3)}}</math> || align="right" | 1.2619 || Bordo della [[Curva del Drago]] || align="center" |[[Image:Terdragon boundary.png|150px]] || L-sistema: same as dragon curve with angle=30°. The Fudgeflake is based on 3 initial segments placed in a triangle.
|-
| <math>\textstyle{\frac {ln(4)} {ln(3)}}</math> || align="right" | 1.2619 || 2D [[Polvere di Cantor ]] in 2D || align="center" |[[Image:Carre_cantor.gif|100px]] || Insieme di Cantor in due dimensioni .
|-
| || align="right" | 1.3057 || [[Setaccio di Apollonio]] || align="center" |[[Image:Apollonian gasket.gif|100px]] ||
|-
| <math>\textstyle{\frac {ln(5)} {ln(3)}}</math>|| align="right" | 1.4649 || [[Box fractal]] || align="center" |[[Image:Box fractal.png|100px]] || Costruito sostituendo iterativamente ciascun quadrato con una croce di 5 quadrati.
|-
| <math>\textstyle{\frac {ln(5)} {ln(3)}}</math>|| align="right" | 1.4649 || [[Curva di Koch quadratica (type 1)]]|| align="center" |[[Image:Quadratic Koch 2.png|150px]] || Si può riconoscere il motivo della scatola frattale (vedi sopra).
|-
|<math>\textstyle{\frac {ln(8)} {ln(4)}}</math>|| align="right" | 1.5000 || [[Curva di Koch qaudratica (type 2)]] || align="center" |[[Image:Quadratic Koch.png|150px]] || Chiamata anche "Salsiccia di Minkowski".
|-
| || align="right" | 1.5236 || [[Curva del Drago]] boundary || align="center" | [[Image:Boundary dragon curve.png|150px]]|| Cf Chang & Zhang<ref> [http://www.poignance.com/math/fractals/dragon/bound.html Fractal dimension of the boundary of the dragon fractal]</ref>.
|-
| <math>\textstyle{\frac {ln(3)} {ln(2)}}</math> || align="right" | 1.5850 || Albero a 3 rami || align="center" | [[Image:Arbre 3 branches.png|110px]][[Image:Arbre 3 branches2.png|110px]] || Ogni ramo si divide in altri 3 rami. (qui i casi a 90° e 60°). La dimensione frattale dell'intero albero è quella dei rami terminali. NB: l'albero a 2 rami possiede dimensione frattale 1.
|-
| <math>\textstyle{\frac {ln(3)} {ln(2)}}</math> || align="right" | 1.5850 || [[Triangolo di Sierpinski ]] || align="center" | [[Image:SierpinskiTriangle.PNG|100px]] || Esso è anche il triangolo di Pascal modulo 2.
|-
| <math>\textstyle{\frac {ln(3)} {ln(2)}}</math> || align="right" | 1.5850 || [[Curva di Sierpinski a punta di freccia]] || align="center" | [[Image:Pfeilspitzen_fraktal.png|100px]] || Stesso limite del triangolo (vedi sopra) ma costruito con una curva unidimensionale.
|-
| <math>\textstyle{1+log_3(2)}</math> || align="right" | 1.6309 || [[Triangolo di Pascal]] modulo 3 || align="center" | [[Image:Pascal triangle modulo 3.png|150px]] || per un triangolo modulo k, se k è primo, la dimensione frattale è <math>\scriptstyle{1 + log_k(\frac{k+1}{2})}</math>(Cf Stephen Wolfram <ref>[http://www.stephenwolfram.com/publications/articles/ca/84-geometry/1/text.html Fractal dimension of the Pascal triangle modulo k]</ref>)
|-
| <math>\textstyle{1+log_5(3)}</math> || align="right" | 1.6826 || [[Triangolo di Pascal]] modulo 5 || align="center" | [[Image:Pascal triangle modulo 5.png|150px]] || Come sopra.
|-
| <math>\textstyle{\frac {ln(7)} {ln(3)}}</math> || align="right" | 1.7712 || [[Fiocco esagonale]] || align="center" | [[Image:Flocon_hexagonal.gif|100px]] || Costruito sostituendo iterativamente ogni esagono con un fiocco di 7 esagoni. Il suo bordo è il fiocco di Koch. Contiene infiniti fiocchi di Koch (bianchi e neri).
|-
| <math>\textstyle{\frac {ln(4)} {ln(2(1+cos(85^\circ))}}</math> || align="right" | 1.7848 || [[Von Koch curve 85°]], [[Frattale di Cesaro]] || align="center" | [[Image:Koch_Curve_85degrees.png|150px]] || Generalizzazione della curva di Koch con un anogolo a scelta tra 0 e 90°. La dimensione frattale è allora <math>\scriptstyle{\frac{ln(4)}{ln(2(1+cos(a))}}</math>. Il [[Frattale di Cesaro]] è basato su questo motivo.
|-
| <math>\textstyle{\frac {ln(6)} {ln(1+\phi)}}</math> || align="right" | 1.8617 || [[Fiocco pentagonale]] || align="center" | [[Image:Penta plexity.png|100px]] || Costruito sostituendo iterativamente ogni pentagono con un fiocco di 6 pentagoni. <math>\phi</math> = sezione aurea = <math>\scriptstyle{\frac{1+\sqrt{5}}{2}}</math>
|-
| <math>\textstyle{\frac {ln(8)} {ln(3)}}</math> || align="right" | 1.8928 || [[Tappeto di Sierpinski]] || align="center" | [[Image:Sierpinski6.png|100px]] ||
|-
| <math>\textstyle{\frac {ln(8)} {ln(3)}}</math> || align="right" | 1.8928 || [[Polvere di Cantor]] in 3D || align="center" | [[Image:Cube_Cantor.png|100px]]|| Insieme di Cantor in 3 dimensioni.
|-
|Estimated || align="right" | 1.9340 || Bordo del [[Frattale do Lévy]] || align="center" | [[image:LevyFractal.png|100px]] || Stimato da Duvall and Keesling (1999). La curva di per se stessa possiede dimensione frattale 2.
|-
| || align="right" | 1.974 || [[Tassellatura di Penrose]] || align="center" |[[image:pen0305c.gif|100px]] || Vedi Ramachandrarao, Sinha & Sanyal<ref>[http://www.ias.ac.in/currsci/aug102000/rc80.pdf Fractal dimension of a penrose tiling]</ref>
|-
| <math>\textstyle{2}</math> || align="right" | 2 || [[Mandelbrot set]] || align="center" | [[Image:Mandelbrot-similar1.png|100px]] || Any plane object containing a disk has Hausdorff dimension δ = 2. However, note that the boundary of the Mandelbrot set also has Hausdorff dimension δ = 2.
|-
| <math>\textstyle{2}</math> || align="right" | 2 || [[Sierpiński curve]] || align="center" | [[Image:Sierpinski-Curve-3.png|100px]] || Every [[Peano curve]] filling the plane has a Hausdorff dimension of 2.
|-
| <math>\textstyle{2}</math> || align="right" | 2 || [[Hilbert curve]] || align="center" | [[Image:Hilbert-Curve-3.png|100px]]|| Built in a similar way: the [[Moore curve]]
|-
| <math>\textstyle{2}</math> || align="right" | 2 || [[Peano curve]] || align="center" | [[Image:Peano curve.png|100px]]|| And a familly of curves built in a similar way, such as the [[Wunderlich curves]].
|-
| || align="right" | 2 || [[z-order (curve)|Lebesgue curve or z-order curve]] || align="center" | [[Image:z-order curve.png|100px]]|| Unlike the previous ones this space-filling curve is differentiable almost everywhere.
|-
| <math>\textstyle{\frac {ln(2)} {ln(\sqrt{2})}}</math> || align="right" | 2 || [[Dragon curve]] || align="center" | [[Image:Courbe du dragon.png|150px]]|| And its boundary has a fractal dimension of 1,5236.
|-
| || align="right" | 2 || [[Dragon curve|Terdragon curve]] || align="center" | [[Image:Terdragon curve.png|150px]]|| L-System : F-> F+F-F. angle=120°.
|-
| <math>\textstyle{\frac {ln(4)} {ln(2)}}</math> || align="right" | 2 || [[T-Square (fractal)|T-Square]] || align="center" | [[Image:T-Square fractal (evolution).png|200px]]||
|-
| <math>\textstyle{\frac {ln(4)} {ln(2)}}</math> || align="right" | 2 || [[Gosper curve]] || align="center" | [[Image:Gosper curve 3.png|100px]]|| Its boundary is the Gosper island.
|-
| <math>\textstyle{\frac {ln(4)} {ln(2)}}</math> || align="right" | 2 || [[Sierpinski tetrahedron]] || align="center" | [[Image:Tetraedre Sierpinski.png|80px]]||
|-
| <math>\textstyle{\frac {ln(4)} {ln(2)}}</math> || align="right" | 2 || [[H-fractal]] || align="center" |[[Image:H fractal.png|150px]]|| Also the « Mandelbrot tree » which has a similar pattern.
|-
| <math>\textstyle{\frac {ln(4)} {ln(2)}}</math> || align="right" | 2 || [[2D greek cross fractal]] || align="center" | || Each segment is replaced by a cross formed by 4 segments.
|-
| || align="right" | 2.06 || [[Lorenz attractor]] || align="center" |[[Image:Lorenz attractor.png|100px]] || For precise values of parameters.
|-
| <math>\textstyle{\frac {ln(20)} {ln(2+\phi)}}</math> || align="right" | 2.3296 || [[Dodecaedron fractal]] || align="center" |[[Image:Dodecaedron fractal.jpg|100px]]|| Each dodecaedron is replaced by 20 dodecaedrons.
|-
| <math>\textstyle{\frac {ln(13)} {ln(3)}}</math> || align="right" | 2.3347 || [[3D quadratic Koch surface (type 1)]] || align="center" |[[Image:Quadratic Koch 3D (type1).png|150px]]|| Extension in 3D of the quadratic Koch curve (type 1). The illustration shows the second iteration.
|-
| || align="right" | 2.4739 || [[Apollonian sphere packing]] || align="center" |[[Image:Apollonian spheres.jpg|100px]] || The interstice left by the apollolian spheres. Apollonian gasket in 3D. Dimension calculated by M. Borkovec, W. De Paris, and R. Peikert <ref>[http://graphics.ethz.ch/~peikert/papers/apollonian.pdf Fractal dimension of the apollonian sphere packing]</ref>.
|-
| <math>\textstyle{\frac {ln(32)} {ln(4)}}</math> || align="right" | 2.50 || [[3D quadratic Koch surface (type 2)]] || align="center" |[[Image:Quadratic Koch 3D.png|150px]]|| Extension in 3D of the quadratic Koch curve (type 2). The illustration shows the first iteration.
|-
| <math>\textstyle{\frac {ln(16)} {ln(3)}}</math> || align="right" | 2.5237 || [[Cantor hypercube]] || align="center" | || Cantor set in 4 dimensions. Generalization : in a space of dimension n, the cantor set has a hausdorff dimension of <math>\scriptstyle{n\frac{ln(2)}{ln(3)}}</math>
|-
| <math>\textstyle{\frac {ln(12)} {ln(1+\phi)}}</math> || align="right" | 2.5819 || [[Icosaedron fractal]] || align="center" |[[Image:Icosaedron fractal.jpg|100px]]|| Each Icosaedron is replaced by 12 icosaedrons.
|-
| <math>\textstyle{\frac {ln(6)} {ln(2)}}</math> || align="right" | 2.5849 || [[3D greek cross fractal]] || align="center" |[[Image:Greek cross 3D.png|200px]]|| Each segment is replaced by a cross formed by 6 segments.
|-
| <math>\textstyle{\frac {ln(6)} {ln(2)}}</math> || align="right" | 2.5849 || [[Octaedron fractal]] || align="center" |[[Image:Octaedron fractal.jpg|100px]]|| Each octaedron is replaced by 6 octaedrons.
|-
| <math>\textstyle{\frac {ln(20)} {ln(3)}}</math> || align="right" | 2.7268 || [[Menger sponge]] || align="center" | [[image:Gasket14.png|100px]] || And its surface has a fractal dimension of <math>\scriptstyle{\frac{ln(12)}{ln(3)} = 2.2618}</math>.
|-
| <math>\textstyle{\frac {ln(8)} {ln(2)}}</math> || align="right" | 3 || [[3D Hilbert curve]] || align="center" | [[Image:Hilbert512.gif|100px]]|| A Hilbert curve extended to 3 dimensions.
|}
== Random and natural fractals ==
{| border="0" cellpadding="4" rules="all" style="border: 1px solid #999; background-color:#FFFFFF"
|- align="center" bgcolor="#cccccc"
! δ<br />(exact value) || δ<br />(value) || Name || Illustration || width="40%" | Remarks
|-
|Measured||align="right"|1.24||[[How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension|Coastline of Great Britain]]||align="center"| [[Image:Gb4dot.svg|100px]] ||
|-
|<math>\textstyle{\frac {4}{3}}</math> || align="right" | 1.33 || [[Boundary of Brownian motion]] || align="center" |[[Image:Front mouvt brownien.png|150px]] || (Cf [[Wendelin Werner]])<ref>[http://www.citebase.org/fulltext?format=application%2Fpdf&identifier=oai%3AarXiv.org%3Amath%2F0010165 Fractal dimension of the brownian motion boundary]</ref>.
|-
|<math>\textstyle{\frac {4}{3}}</math> || align="right" | 1.33 || [[2D Polymer]] || align="center" | || Similar to the brownian motion in 2D with non self-intersection. (Cf Sapoval).
|-
|<math>\textstyle{\frac {4}{3}}</math> || align="right" | 1.33 || [[Percolation front in 2D]], [[Corrosion front in 2D]] || align="center" | [[Image:Front de percolation.png|150px]] || Fractal dimension of the percolation-by-invasion front, at the percolation threshold (59.3%). It’s also the fractal dimension of a stopped corrosion front (Cf Sapoval).
|-
| || align="right" | 1.40 || [[diffusion-limited aggregation|Clusters of clusters 2D]] || align="center" | || When limited by diffusion, clusters combine progressively to a unique cluster of dimension 1.4. (Cf Sapoval)
|-
| Measured|| align="right" | 1.52|| [[Coastline of Norway]] || align="center" |[[Image:Norgeskart.png|100px]] ||
|-
| Measured|| align="right" | 1.55 || [[Random walk with no self-intersection]] || align="center" | [[Image:2D self-avoiding random walk.png|150px]]|| Self-avoiding random walk in a square lattice, with a « go-back » routine for avoiding dead ends.
|-
| <math>\textstyle{\frac {5} {3}}</math>|| align="right" | 1.66|| [[3D Polymer]] || align="center" | || Similar to the brownian motion in a cubic lattice, but without self-intersection (Cf Sapoval).
|-
| || align="right" | 1.70 || [[Diffusion-limited aggregation|2D DLA Cluster]] || align="center" | [[Image:Agregation limitee par diffusion.png|150px]]|| In 2 dimensions, clusters formed by diffusion-limited aggregation, have a fractal dimension of around 1.70 (Cf Sapoval).
|-
| <math>\textstyle{\frac {91} {48}}</math> || align="right" | 1.8958 || [[2D Percolation cluster]] || align="center" | [[Image:Amas de percolation.png|150px]] || Under the percolation threshold (59.3%) the percolation-by-invasion cluster has a fractal dimension of 91/48 (Cf Sapoval). Beyond that threshold, le cluster is infinite and 91/48 becomes the fractal dimension of the « clearings ».
|-
| <math>\textstyle{\frac {ln(2)} {ln(\sqrt{2})}}</math> || align="right" | 2 || [[Brownian motion]] || align="center" | [[Image:Mouvt_brownien2.png|150px]]|| Or random walk. The Hausdorff dimensions equals 2 in 2D, in 3D and in all greater dimensions (K.Falconer "The geometry of fractal sets").
|-
| <math>\textstyle{\frac {ln(13)} {ln(3)}}</math> || align="right" | 2.33 || [[Cauliflower]] || align="center" | [[Image:Blumenkohl-1.jpg|100px]]|| Every branch carries around 13 branches 3 times smaller.
|-
| || align="right" | 2.5 || Balls of crumpled paper || align="center" | [[Image:Paperball.jpg|100px]] || When crumpling sheets of different sizes but made of the same type of paper and with the same aspect ratio (for example, different sizes in the [[ISO 216]] A series), then the diameter of the balls so obtained elevated to a non-integer exponent between 2 and 3 will be approximately proportional to the area of the sheets from which the balls have been made. [http://classes.yale.edu/fractals/FracAndDim/BoxDim/PowerLaw/CrumpledPaper.html] Creases will form at all size scales (see [[Universality (dynamical systems)]]).
|-
| || align="right" | 2.50 || [[diffusion-limited aggregation|3D DLA Cluster]] || align="center" | [[Image:3D diffusion-limited aggregation2.jpg|100px]] || In 3 dimensions, clusters formed by diffusion-limited aggregation, have a fractal dimension of around 2.50 (Cf Sapoval).
|-
| || align="right" | 2.97 || Lung surface || align="center" |[[Image:Thorax Lung 3d (2).jpg|100px]] || The alveoli of a lung form a fractal surface close to 3 (Cf Sapoval).
|}
==References==
<references/>
== See also ==
=== Bibliography ===
* <sup>1</sup>Kenneth Falconer, ''Fractal Geometry'', John Wiley & Son Ltd; ISBN 0-471-92287-0 (March 1990)
* Benoît Mandelbrot, ''The Fractal Geometry of Nature'', W. H. Freeman & Co; ISBN 0-7167-1186-9 (September 1982).
*Heinz-Otto Peitgen, ''The Science of Fractal Images'', Dietmar Saupe (éditeur), Springer Verlag, ISBN 0-387-96608-0 (August 1988)
*Michael F. Barnsley, ''Fractals Everywhere'', Morgan Kaufmann; ISBN 0-12-079061-0
*Bernard Sapoval, « Universalités et fractales », collection Champs, Flammarion.
=== Internal links ===
{{Commons|Fractal|fractals}}
* [[Fractal]]
* [[Fractal dimension]]
* [[Hausdorff dimension]]
* [[Scale invariance]]
=== External links ===
* [http://mathworld.wolfram.com/search/index.cgi?q=fractal The fractals on Mathworld]
* [http://local.wasp.uwa.edu.au/~pbourke/fractals/ Other fractals on Paul Bourke's website]
* [http://soler7.com/Fractals/FractalsSite.html Soler's Gallery]
* [http://www.mathcurve.com/fractals/fractals.shtml Fractals on mathcurve.com]
* [http://1000fractales.free.fr/index.htm 1000fractales.free.fr - Project gathering fractals created with various softwares]
* [http://library.thinkquest.org/26242/full/index.html Fractals unleashed]
[[Category:Fractals]]
[[Category:Fractal curves]]
[[Category:Mathematics-related lists|Fractals by Hausdorff dimension]]
[[fr:Liste de fractales par dimension de Hausdorff]]
= Lavori in corso =
| |||