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998:{\displaystyle {\begin{aligned}f_{1}(x,y)&={\begin{bmatrix}0.00&0.00\\0.00&0.16\end{bmatrix}}{\begin{bmatrix}x\\y\end{bmatrix}}\\f_{2}(x,y)&={\begin{bmatrix}0.85&0.04\\-0.04&0.85\end{bmatrix}}{\begin{bmatrix}x\\y\end{bmatrix}}+{\begin{bmatrix}0.00\\1.60\end{bmatrix}}\\f_{3}(x,y)&={\begin{bmatrix}0.20&-0.26\\0.23&0.22\end{bmatrix}}{\begin{bmatrix}x\\y\end{bmatrix}}+{\begin{bmatrix}0.00\\1.60\end{bmatrix}}\\f_{4}(x,y)&={\begin{bmatrix}-0.15&0.28\\0.26&0.24\end{bmatrix}}{\begin{bmatrix}x\\y\end{bmatrix}}+{\begin{bmatrix}0.00\\0.44\end{bmatrix}}\end{aligned}}}
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smaller scale. V-variable fractals allow for such randomness and variability across scales, while at the same time admitting a continuous dependence on parameters which facilitates geometrical modelling. These factors allow us to make the hybrid biological models... ...we speculate that when a V -variable geometrical fractal model is found that has a good match to the geometry of a given plant, then there is a specific relationship between these code trees and the information stored in the genes of the plant.
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This transformation encodes the self-similarity relationship of the entire fern with the sub-structure which consists of the fern with the removal of the section which includes the bottom two leaves. In the matrix representation, it can be seen to be a slight clockwise rotation, scaled to be slightly
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This coordinate transformation is chosen 1% of the time and just maps any point to a point in the first line segment at the base of the stem. In the iterative generation, it acts as a reset to the base of the stem. Crucially it does not reset exactly to (0,0) which allows it to fill in the base stem
103:
IFSs provide models for certain plants, leaves, and ferns, by virtue of the self-similarity which often occurs in branching structures in nature. But nature also exhibits randomness and variation from one level to the next; no two ferns are exactly alike, and the branching fronds become leaves at a
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Though
Barnsley's fern could in theory be plotted by hand with a pen and graph paper, the number of iterations necessary runs into the tens of thousands, which makes use of a computer practically mandatory. Many different computer models of Barnsley's fern are popular with contemporary
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direction. In the iterative generation, this transformation is applied with probability 85% and is intuitively responsible for the generation of the main stem, and the successive vertical generation of the leaves on either side of the stem from their "original" leaves at the base.
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This transformation encodes the self-similarity of the entire fern with the bottom left leaf. In the matrix representation, it is seen to be a near-90° counterclockwise rotation, scaled down to approximately 30% size with a translation in the positive
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Similarly, this transformation encodes the self-similarity of the entire fern with the bottom right leaf. From its determinant it is easily seen to include a reflection and can be seen as a similar transformation as
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68:, the Barnsley fern shows how graphically beautiful structures can be built from repetitive uses of mathematical formulas with computers. Barnsley's 1988 book
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direction. In the iterative generation, it is applied with probability 7% and is intuitively responsible for the generation of the lower-left leaf.
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By playing with the coefficients, it is possible to create mutant fern varieties. In his paper on V-variable fractals, Barnsley calls this trait a
263:{\displaystyle f_{w}(x,y)={\begin{bmatrix}a&b\\c&d\end{bmatrix}}{\begin{bmatrix}x\\y\end{bmatrix}}+{\begin{bmatrix}e\\f\end{bmatrix}}}
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One experimenter has come up with a table of coefficients to produce another remarkably naturally looking fern however, resembling the
1408:-axis. In the iterative-generation, it is applied with probability 7% and is responsible for the generation of the bottom right leaf.
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The number of copies that the fractal has can be expressed with the formula: 2·∞ where n is the number of iterations the fractal has.
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mathematicians. As long as math is programmed correctly using
Barnsley's matrix of constants, the same fern shape will be produced.
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which is translated and serves as a kind of "kernel" from which all other sections of the fern are generated via transformations
99:. He has used fractals to model a diverse range of phenomena in science and technology, but most specifically plant structures.
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1062:) and then the new points are iteratively computed by randomly applying one of the following four coordinate transformations:
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sets, i.e. it is a mathematically generated pattern that can be reproducible at any magnification or reduction. Like the
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is based on the course which he taught for undergraduate and graduate students in the School of
Mathematics,
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Fractal fern in four states of construction. Highlighted triangles show how the half of one
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fern fractal as a matrix of values shown in a table. In the table, the columns "
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80:. After publishing the book, a second course was developed, called
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131:. The formula for one transformation is the following:
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The fern code developed by
Barnsley is an example of an
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attempting to imitate nature with mathematical models.
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84:. Barnsley's work has been a source of inspiration to
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493:These correspond to the following transformations:
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95:(IFS) to create a fractal. This follows from the
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46:. He made it to resemble the black spleenwort,
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289:" are the coefficients of the equation, and "
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2131:""V-variable fractals and superfractals""
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60:The fern is one of the basic examples of
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1045:The first point drawn is at the origin (
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2813:List of fractals by Hausdorff dimension
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1224:smaller and translated in the positive
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293:" represents the probability factor.
2102:, Boston, MA: Academic Press, 1993,
1029:is transformed to half of one whole
2153:, table III.3, IFS code for a fern.
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1404:albeit with a reflection about the
42:who first described it in his book
13:
3363:Los Angeles Center for Digital Art
3075:Generative artificial intelligence
14:
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2795:How Long Is the Coast of Britain?
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2169:. Morgan Kaufmann. p. 86.
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74:Georgia Institute of Technology
16:Fractal which resembles a plant
2819:The Fractal Geometry of Nature
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1420:Barnsley fern mutated into a
3348:Austin Museum of Digital Art
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7:
2995:Artificial intelligence art
2965:Fractal-generating software
2835:Chaos: Making a New Science
485:Largest right-hand leaflet
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2219:with supplied coefficients
2163:Barnsley, Michael (2000).
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449:Largest left-hand leaflet
127:Barnsley's fern uses four
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35:named after the British
3383:V&A Digital Futures
3120:Photograph manipulation
2827:The Beauty of Fractals
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82:Fractal Measure Theory
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3216:Lynn Hershman Leeson
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3025:Digital illustration
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2768:Hamid Naderi Yeganeh
2558:Burning Ship fractal
2490:Weierstrass function
2217:Other fern varieties
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3115:Music visualization
3040:Digital photography
2531:Space-filling curve
2508:Multifractal system
2391:Space-filling curve
2376:Sierpinski triangle
2166:Fractals everywhere
2151:Fractals Everywhere
2100:Fractals Everywhere
1017:Computer generation
273:Barnsley shows the
70:Fractals Everywhere
66:Sierpinski triangle
44:Fractals Everywhere
3261:Ben Rubin (artist)
3206:Desmond Paul Henry
3005:Computer art scene
2970:Animation software
2758:Aleksandr Lyapunov
2738:Desmond Paul Henry
2702:Self-avoiding walk
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341:Portion generated
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108:—Michael Barnsley
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3221:Zachary Lieberman
3191:Stephanie Dinkins
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2801:Coastline paradox
2778:Wacław Sierpiński
2763:Benoit Mandelbrot
2687:Fractal landscape
2595:Misiurewicz point
2500:Strange attractor
2381:Apollonian gasket
2371:Sierpinski carpet
2198:"Barnsley's Fern"
2196:Weisstein, Eric.
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1459:fern. These are:
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3319:Remember To Rise
3293:Edmond de Belamy
3285:Notable artworks
3276:Pindar Van Arman
3211:Mario Klingemann
3080:Generative music
3060:Evolutionary art
3055:Electronic music
3035:Digital painting
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2718:Michael Barnsley
2585:Lyapunov fractal
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78:Fractal Geometry
40:Michael Barnsley
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3404:Affine geometry
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3159:Notable artists
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3145:Texture mapping
3110:Motion graphics
3100:Interactive art
3030:Digital imaging
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3226:Margot Lovejoy
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3045:Digital poetry
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3010:Computer music
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2680:Brownian motor
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2651:Pickover stalk
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2607:Newton fractal
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2590:Mandelbrot set
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2573:Newton fractal
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2361:Koch snowflake
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3329:Organizations
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3299:Barnsley fern
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3246:Trevor Paglen
3244:
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3236:Eric Millikin
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3231:Mauro Martino
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3171:Cory Arcangel
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277:code for his
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41:
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37:mathematician
34:
30:
29:Barnsley fern
23:Barnsley fern
21:
3317:
3310:
3303:
3298:
3291:
3181:Harold Cohen
3166:Refik Anadol
3135:Software art
3105:Internet art
2847:Chaos theory
2842:Kaleidoscope
2833:
2825:
2817:
2743:Gaston Julia
2723:Georg Cantor
2548:Escape-time
2480:Gosper curve
2428:Lévy C curve
2413:Dragon curve
2350:
2292:Box-counting
2224:
2212:
2201:. Retrieved
2191:
2180:. Retrieved
2165:
2158:
2150:
2146:
2125:
1609:
1576:
1543:
1510:
1454:
1448:
1446:
1442:superfractal
1441:
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1397:
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492:
455:
419:
383:
347:
278:
274:
272:
126:
115:Construction
109:
107:
102:
90:
81:
77:
69:
62:self-similar
59:
47:
43:
28:
26:
3368:Lumen Prize
3256:Anna Ridler
3186:Char Davies
3150:Virtual art
3140:Systems art
3065:Fractal art
2932:3D graphics
2920:2D graphics
2889:Digital art
2838:(1987 book)
2830:(1986 book)
2822:(1982 book)
2808:Fractal art
2728:Bill Gosper
2692:Lévy flight
2438:Peano curve
2433:Moore curve
2319:Topological
2304:Correlation
285:" through "
3398:Categories
3373:onedotzero
3251:Casey Reas
3196:Jake Elwes
3130:Render art
3090:Glitch art
3015:Crypto art
2944:3D printer
2646:Orbit trap
2641:Buddhabrot
2634:techniques
2622:Mandelbulb
2423:Koch curve
2356:Cantor set
2203:2010-01-07
2182:2010-01-07
2084:References
1727:iterations
1646:Pseudocode
1450:Cyclosorus
3409:L-systems
3338:Artfutura
3266:Karl Sims
3125:Pixel art
3095:Immersion
3020:Cyberarts
3000:ASCII art
2753:Paul Lévy
2632:Rendering
2617:Mandelbox
2563:Julia set
2475:Hexaflake
2406:Minkowski
2326:Recursion
2309:Hausdorff
2075:increment
906:−
781:−
658:−
76:, called
3378:SIGGRAPH
3343:Artmedia
3201:David Em
2990:Art game
2953:Software
2910:Computer
2903:Hardware
2663:fractals
2550:fractals
2518:L-system
2460:T-square
2268:Fractals
1338:= −0.15
1198:= −0.04
3085:GIF art
2612:Tricorn
2465:n-flake
2314:Packing
2297:Higuchi
2287:Assouad
1745:between
1724:maximum
1564:−0.002
1558:−0.005
1378:+ 0.24
1369:= 0.26
1347:+ 0.28
1293:+ 0.22
1284:= 0.23
1262:− 0.26
1207:+ 0.85
1176:+ 0.04
1167:= 0.85
1100:= 0.16
1027:leaflet
1009:Formula
56:History
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2711:People
2661:Random
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2536:H tree
2455:String
2343:system
2173:
2138:
2126:et al.
2106:
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1630:0.083
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1555:0.005
1387:+ 0.44
1253:= 0.2
464:−0.15
431:−0.26
398:−0.04
110:et al.
2983:Forms
2939:Xerox
2896:Tools
2787:Other
2134:(PDF)
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482:0.07
479:0.44
473:0.24
470:0.26
467:0.28
446:0.07
443:1.60
437:0.22
434:0.23
428:0.20
410:0.85
407:1.60
401:0.85
395:0.04
392:0.85
377:Stem
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365:0.16
31:is a
2925:2.5D
2171:ISBN
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1982:0.28
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909:0.15
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27:The
2915:CGI
1952:1.6
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1874:1.6
1778:0.0
1751:and
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1703:0.0
1694:0.0
1685:0.0
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1655:all
1453:or
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1336:+ 1
1282:+ 1
1251:+ 1
1196:+ 1
1165:+ 1
1098:+ 1
1086:= 0
1084:+ 1
1060:= 0
1033:or
275:IFS
3400::
2797:"
2124:,
2114:^
2092:^
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2045:xn
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371:0
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52:.
2881:e
2874:t
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2793:"
2260:e
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2206:.
2185:.
2136:.
2128:,
2078:t
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2066:y
2060:=
2057:x
2054:)
2048:,
2042:(
2018:+
2015:y
2012:*
2006:+
2003:x
2000:*
1994:=
1988:y
1985:*
1979:+
1976:x
1973:*
1967:-
1964:=
1958::
1949:+
1946:y
1943:*
1937:+
1934:x
1931:*
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1037:.
987:]
968:[
963:+
958:]
952:y
945:x
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932:]
900:[
895:=
888:)
885:y
882:,
879:x
876:(
871:4
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838:[
833:+
828:]
822:y
815:x
809:[
802:]
770:[
765:=
758:)
755:y
752:,
749:x
746:(
741:3
737:f
727:]
708:[
703:+
698:]
692:y
685:x
679:[
672:]
640:[
635:=
628:)
625:y
622:,
619:x
616:(
611:2
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597:]
591:y
584:x
578:[
571:]
542:[
537:=
530:)
527:y
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518:(
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232:+
227:]
221:y
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201:]
195:d
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178:a
172:[
167:=
164:)
161:y
158:,
155:x
152:(
147:w
143:f
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