real time design and animation of fractal plants and trees

8/19/2019 Real Time Design and Animation of Fractal Plants and Trees

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Dallas Au gu st 18-22 Volum e 20 Nu mb er 4 1986

R e a l T i m e D e s i gn a n d n i m a t i o n o f

F r a c t a l P l a n t s a n d T r e e s

Peter E. Oppenheimer

New York Insti tute of Technology

Computer Graphics Lab

B S T R C T

The goal of science is to understand why things are the

way they are. By emulat ing the logic of natur e, computer

simulation programs capture the essence of natural objects,

thereby serving as a tool of science. When these programs

express this essence visually, they serve as an in strument of

art as well.

This paper presents a fractal computer model of

bran ching objects. This program generates pictures of sim-

ple orderly plants, complex gnarled trees, leaves, vein sys-

tems, as well as inorganic structures such as river deltas,

snowflakes, etc. The geometry a nd topology of the model are

controlled by numerical parameters which are analogous to

the organism's DNA. By manipulating the genetic parame-

ters, one can modify the geometry of the object in real time,

using tree b sed graphics hardware. The random effects of

the e nvironment are taken into account, to produce greater

diversity and realism. Increasing the number of significant

paramete rs yields more complex and evolved species.

The program provides a study in the structure of

branching objects that is both scientific and artistic. The

results suggest that organisms and computers deal with

complexity in simila r ways.

CR CATAGORIES AND SUBJECT DESCRIPTORS: C.3

]Special-Purpose and Application-Based Systems]: Real-time sys-

tems; G.3 [Probability and Statistics]: Random number generation;

1.3.3 [Computer Graphics]: Computational Geometry and Object

Modeling; 1.3.7 [Computer Graphic@ Three-Dimensional Graphics

and Realism; 1.6.0 [Computer Graphics]: Simulation and Modeling;

£3 ILife and Medical Sciences]: Biology; J.5 [Arts and Humanities]:

Arts, fine and performing.

ADDITIONAL KEY WORDS AND PHRASES: plant, tree , fractal,

real time, DNA, genetics animation, reeursion, stocbasitie model-

ing, naturM phenomen~

Permission to copy without fee all or part of this material is granted

provided that the copies are not made or distributed for direct

commercial advantage, the ACM copyright notice and the title of the

publication and its date appear, and notice is given that copying is by

permission of the Association for Computing Machinery, To copy

otherwise, or to republish, requires a fee and/ or specific permission.

© 1986 ACM0- 89791 -196- 2/86/ 008/0 055 00.75

1. INTRODUCTION

Benoit Mandelbrot recognized that the relationship

between large scale structure and small scale detail is an

importa nt aspect of natural phenomena. He gave the name

fr ct l to objects th at exhibit increasing detail as one

zooms in closer. [12][13 If the small scale detail resembles

the large scale detail, the object is said to be self-similar.

The geometric notio n of fractal self-similarity has

become a paradigm for structure in the natural world.

Nowhere is this principle more evident than in the world of

bota ny. Recursive branchi ng at many levels of scale, is a

primary mechanism of growth in most plants. Analogously,

recursive branching algorithms, are furdamental to comput-

ers. Many high performance processing engines specialize in

tree data structures.

Computer generation of trees has been of interest for several

years now. Examples of comput er generated trees include

Benoit Mandelbro t (19777){1982) [12],[13]

Marshall,Wilson ,Car lson (19S0) [14]

Craig Reynolds , (1981) [19]

Yoichiro Kawaguchi (1982) [10]

Geoff G ard iner (1984) [9]

Aono,Kunii (1984) [2]

Ai ry Ray Smith , Bill Reeves (1984)(1985) [211[lSl

Jules Bloomenthal (1985) [4]

Demko,Hodges,Naylor (1985) [6]

Eyrolles, Francon, Viennot {1986) [8]

among others.

Not all of the above used recursive tree data structures.

Mandelbrot is largely responsible for the growing awareness

of vecursion as a process in natur e. He and Kawagnchi each

used minimal recursive topologies and added simple

geometric relationships to generate complex images of

branching plants and other objects. Airy Ray Smith

applied the theory of formal languages to generate more

complex and systematized tree topologies, while emphasizing

the separation between topology and geometry.

Each at tempt a t modeling trees takes a new approach,

and thereby captures some different aspect of branching

phenomena.

55


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~. S I G G R A P H 86

2 . T H E T R E E M O D E L

T h e

t r e e m o d e l p r e s e n t e d i n t h i s p a p e r h a s t h e f o l lo w -

ing f e a ture s :

• A d e t a i l e d p a r a m e t e r i z a t i o n o f t h e g e o m e t r i c r e l a ti o n s h i p

b e t w e e n t r e e n o d e s .

• R e a l T i m e D e s i g n a n d A n i m a t i o n o f t r e e i m a g e s u s i n g h i g h

p e r f o r m a n c e h a r d w a r e .

• A p p l i c a t i o n o f S t o c h a s t i c ( r a n d o m ) M o d e l i n g t o b o t h t o p o -

l o g ic a l a n d g e o m e t r i c p a r a m e t e r s .

• S t o c h a s t i c m o d e l i n g o f t r e e b a r k .

• H igh Re solu t ion (2024 x 1980} Sha d e d 3D Re n de r ings .

H e r e s h o w t h e m o d e l w o r k s :

T h i s p r o g r a m i m p l e m e n t s a r e c u m i v e t r e e m o d e l . E a c h

t r e e g e n e r a t e d s a t is f i es t h e f o l l o w in g r e c u r s iv e t r e e n o d e

de f in i t ion :

tr : ~

{

D r a w

B r a n c h

S e g m e n t

i f ( t o o s m a l l )

D r a w

l e a f

e l s e

{

C o n t i n u e t o

B r a n c h

T r a n s f o r m S t e m

n t r e e ~

}

r e p e a t n t i m e s

{

T r a n s f o r m B r a n c h

-

t r e e ~

}

}

}

P a r a p h r a s e d , a t r e e n o d e i s a b r a n c h w i t h o n e o r m o r e

t r e e n o d e s a t t a c h e d , t r a n s f o r m e d b y a 3 x 3 li n e a r t r a n s f o r m a -

t i o n . O n c e t h e b r a n c h e s b e c o m e s m a l l e n o u g h , t h e b r a n c h -

i n g s t o p s a n d a

le f

i s d r a w n . T h e t r e e s a r e d i f f e r e n t i at e d

b y t h e g e o m e t r y o f t h e t r a n s f o r m a t i o n s r e l a t in g t h e n o d e to

t h e b r a n c h e s a n d t h e t o p o l o g y o f t h e n u m b e r o f b r a n c h e s

c o m i n g o u t o f e a c h n o d e . T h e s e b r a n c h i n g a t t r i b u t e s a r e

c o n t r o l l a b l e b y a s e t o f n u m e r i c a l p a r a m e t e r s . E d i t i n g t h e s e

p a r a m e t e r s , c h a n g e s t h e t r e e s a p p e a r a n c e . T h e s e p a r a m e -

t e r s i n c l u d e :

• T h e a n g l e b e t w e e n t h e m a i n s t e m a n d t h e b r a n c h e s

• T h e s iz e r a t i o o f t h e m a i n s t e m t o t h e b r a n c h e s

• T h e r a t e a t w h i c h t h e s t e m t a p e r s

* T h e a m o u n t o f he l i ca l t w i s t i n t h e b r a n c h e s

• T h e n u m b e r o f b r a n c h e s p e r s t e m s e g m e n t

Figur e 1 shows a s im ple e xa m ple .

3 . R A N D O M N U M B E R S I N F R A C T A L M O D E L I N G

I f t h e p a r a m e t e r s r e m a i n c o n s t a n t t h r o u g h o u t t h e t r e e ,

one ge t s a ve ry r e gula r looking t r e e suc h a s a f e rn . Thi s t r e e

i s s t r i c t ly s e l f s im i la r ; tha t i s, the sm a l l no de s of the t r e e a re

i d e n t i c a l t o t h e t o p l e v el l a rg e s t n o d e o f t h e t r e e .

i f t h e p a r a m e t e r s v a r y t h r o u g h o u t t h e t r e e, o n e g e ts a n

i r r e g u l a r g n a r l e d t r e e s u c h a s a j u n i p e r t r e e . I n o r d e r t o

a c h i e v e t h i s , e a c h p a r a m e t e r i s g i v e n a m e a n v a l u e a n d a

s t a n d a r d d e v i a ti o n . A t e a c h n o d e o f t h e t r e e, t h e p a r a m e t e r

v a l u e is r e g e n e r a t e d b y t a k i n g t h e m e a n v a l u e a n d a d d i n g a

r a n d o m p e r t u r b a t i o n , s c al e d b y t h e s t a n d a r d d e v i a t i o n . T h e

g r e a t e r t h e s t a n d a r d d e v i a t i o n , t h e m o r e r a n d o m , i r r e g u l a r ,

a n d g n a r l e d t h e t r e e . T h e r e s u l t i n g t r e e i s

st tistic lly

self-

s im i la r ; no t

strictly

se lf-s imilar .

T h e r e a r e s e v e r a l r e a s o n s f o r t h e s t o c h a s t i c a p p r o a c h .

F i r s t , a d d i n g r a n d o m n e s s t o t h e m o d e l g e n e r a t e s a m o r e

n a t u r a l l o o k i n g i m ag e . L a r g e tr e e s h a v e a n i n t r i n s i c i r r e gu -

l a r i t y ( c a u s e d i n p a r t b y t u r b u l e n t e n v i r o n m e n t a l e f f e c t s ) .

R a n d o m p e r t u r b a t i o n s i n t h e m o d e l r e f l e c t t h i s i r r e g u l a r i t y .

S e c o n d , r a n d o m p e r t u r b a t i o n s r e f le c t t h e d i v e r s i t y i n n a t u r e .

A s i n gl e s e t o f t r e e m o d e l p a r a m e t e r s c a n g e n e r a t e a w h o l e

fore s t o f t r e e s , e a c h s l igh t ly d i f f e re n t . Thi s inc re a se d da ta -

b a s e a m p l i f i c a t i o n i s o n e o f t h e h a l l m a r k f e a t u r e s o f f r a c t a l

t e e hnique s .

W h e n g e n e r a t i n g r a n d o m n u m b e r s , o n e m u s t b e c a r e f u l

a b o u t c o n s i s t e n c y a n d r e p r o d u c i b i l i t y . S i m p l y re s e e d i n g t h e

r a n d o m n u m b e r g e n e r a t o r a t t h e o u t s e t i s n o t e n o u g h . F o r

e xa m ple , i f one de c re a se s the

minimum br nch ~ize

p a r a m e -

te r , one would l ike the ne w t r e e to be a s im ple e x te ns ion of

t h e o l d t r e e . H o w e v e r , t h e o r d e r i n w h i c h th e s e n ew b r a n c h

n o d e s a r e g e n e r a t e d i s i n t e r m i x e d w i t h t h e o l d b r a n c h n o d e s .

T h i s c o u l d p o t e n t i a l l y s c r a m b l e t h e r a n d o m n u m b e r g e n e r a -

t o r , d r a s t i c al l y c h a n g i n g t h e s h a p e o f t h e r e s u l t i n g tr e e . T o

a v o i d t h i s , t h e r a n d o m p e r t u r b a t i o n s c o r r e s p o n d i n g t o a

g i v e n n o d e , s h o u l d b e d e t e r m i n e d b y t h e

topologic l br nch

ddress o f t h e n o d e , a n d n o t t h e o r d e r i n w h i c h i t i s g e n -

e r a t e d .

o

F i g u r e 1

s t e m / s t e m r a t i o . ~ . 8

b r a n c h / s t e m r a t i o ~ . 4

b r an c h i n g an g l e = 60 °

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Dallas Aug ust 18-22 Volume 20 Nu mb er 4 1986

[

F a l l e n L e a f - A n e x a g g e r a t e d i m a g e o f v e i n

b r a n c h in g in a f a l l l e af . Th e e x t e r n a l b o u n d a r y

s h a p e i s t h e l i m i t o f g r o w t h o f t h e i n t e r n a l

veins. 512x480}

S n o w f l a k e s S e l t s imi la r b r a n c h in g o c c u r s in

th e in o r g a n ic wo r ld a s we l l , 5 1 2 x 4 8 0 r e s o lu t io n )

F r a e t a l F e r n Th e c la s s i c o r g a n ic s e l f s imi la r

f o r m. Ev e r y b r a n c h p o in t lo o k s th e s a me --

but a t d if fe ren t sca les . 1024 x 960)

57


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S I G G R A P H 86

a)

(b)

c)

d)

Figure 2

Spirals Helixes

Figure 3

The Display List

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Decreas ing the minimum branch size p a r a m e t e r c a n

al so cause branches to P O P on the t r ee , caus ing a d i s turbing

ef f ec t i n animat ions . Thi s can be avoided by drawing par t i a l

b r a n c h s e g m e n t s w h e n e v e r s c a l i n g a b r a n c h s e g m e n t w o u l d

cause i t t o be smal l er t han the minimum branch size. Using

thi s method causes the branches to grow smoothly as t he

m i n i m u m b r a n c h s iz e p a r a m e t e r i s d e c r e a se d , r a t h e r t h a n

popping an.

4 M O D E L I N G S T E M SH A P E

By s t r ipping a l l of t he branches one i s l e f t wi th jus t a

s t em. By varying the t r ans fo rmat ion between s t em seg-

ment s , one der ives the c l as s of spirals and helize~ a n d t h e i r

r andom per turbat ions . These shapes appear i n a ll f orms of

growth, organic and inorganic - - f rom the inner ear , s ea

shel ls , and plant sprouts , to spiral galaxies . Spirals and

helizes

are in some sense degenera t e s e l f s imi l ar se t s . The y

are the a tomic uni t s t ha t make up the f r ac t a l t r ees .

Figure 2 shows 4 typica l

s t e m

shapes .

a)

cylinder:

t he t r ans form at ion i s a t r ans l a t ion and a s ca le .

b)

piral:

one per forms a ro t a t ion perpendicular t o t he s t em

axis , i l l addi t ion to a scale and t ranslat ion.

c)

heliz:

one per forms an addi t ional ro t a t ion a long the s t em

axis.

d)

squiggle

B y r a n d o m l y c h a n g i n g t h e t r a n s f o r m a t i o n f r o m

~egment to s egment , ease e) becomes ease d) .

Branches ar e s imply s t em shapes a t t ached to the main s t em

and each o ther .

5 . R E N D E R I N G T H E F R A C T A L

A var i e ty of geomet r i c e l ement s can be used to r ende r

the branch segment s . The s imples t pr imi t ive is one single

vector l i ne per t r ee node. Ant i a l i ased vector l i nes wi th var i -

able th i cknes s a l low one to t aper t he branches towards the

t ip . Thi s m ethod i s s a t i s f ac tory for l eaves , f erns and o the r

s imple plants , for smal l scale detai l in complex scenes , or for

more abs t r ac t s ty l i zed images . Varying the vector color pro-

vides depth cuing and shading, and can a l so be used to

reade r b los soms or fo l iage . Ant i a l i ased vector s ace s imi l ar t o

par t icle sys tems used by Bi l l Reeves in his fores t images . [18

The th i cker branches of a t r ee r equi r e a shaded 3D pr i -

mat ive . Bump mapped polygonal .pr i sms ar e used to f l esh

out t he t r ees in 3D. The program makes sure tha t t he

polygons jo in cont inuous ly a long each l imb. The branc hes

e m a n a t i n g f r o m a li m b s im p l y in t e r p e n e t r a t e t h e l i m b . F o r

a more curvi l i near l imb shape, one can l i nk s evera l pr i sms

t o g e t h e r b e t w e e n b r a n c h p o i n t s .

Shaded polygon l imbs ar e f ar more computa t ional ly

expens ive than ant i a l i ased vectors . S ince the numbe r of

branches increases exponent i a l ly wi th branchin g depth , one

can spend mos t of an e t ern i ty r ender ing sub p ixel l imb t ips ,

where bark t exture an d shading a r en t v i s ib l e anyway. In

addi t ion to being f as t er , sub p ixel vector s ar e eas i er t o

ant i a l i as t han polygons on our avai l able r ender ing packag e.

So for complex t r ees wi th a h igh l evel of br anching deta i l ,

polygonal t ubes were used for t he l arge s ca l e det a i l s , chang-

ing over t o vector s for t he smal l s tuf f . One can not i ce the

ar t i f ac t s of t h i s t echnique. Overa l l , however , t he eye ignores

th i s i ncons i s t ency i f t he c utover l evel i s deep enou gh.

Thin ner branches r equi r e f ewer polygons around the c i r -

cumference; i n f ac t t r i angu lar t ubes wi l l do for t he smal l es t

branches .

6 B A R K

S a w t o o t h w a v e s m o d u l a t e d b y B r o w n i s h f r a e t a l n o i s e

are the source for t he s imula t ed bark t exture . Thi s t exture i s

bum p map ped onto the t r ee l imbs . More specif i ca lly ,

bar k (x ,y) = saw [ N * [x + R * noise (x,y))]

w here

22 * frac tio n(t) if frac tion (t) < .5

saw ( t ) = * (1 - f ra ct io n( t ) ) i f f ract ion ( t ) > .5

noise (x,y) is a per iodic ( ie wrapping) f ractal noise funct ion.

N i s t he num be r of bark r idges , and

R i s t he roughnes s of t he bark .

Parap hrased , bark i s genera t e d by adding f r ac t a l noise

to a r amp, t hen pas s ing the r esul t t hrough a s awtooth func-

t ion . A c lose up v i ew of t he bark would look l i ke the r idges

of a fr ac t a l moun ta in r ange. By adding the noi se before the

sawtoo th funct ion , t he cr es t s of t he s awto oth r idges become

wiggly . Note tha t bark(x ,y) wraps in x and y .

7 . R E A L T I M E F R A C T A L G E N E R A T I O N

Comp lex t r ee images can t ake 2 hour s or more to

r ender on a VAX 780. Edi t ing t r ee parameter s a t t h is r a t e

i s not very ef f ec t ive . Near r ea l t ime f eedback i s needed to

a l low one to f r ee ly explore the parameter space , and des ign

the des i r ed t r ee . S ince ver t ex t r ans form at ion cos t is h igh for

a complex f r ac t a l t r ee , hardware opt imized for l i near

t r ans form at ions was used for t he r ea l t ime edi tor . Th e

E v a n s a n d S u t h e r l a n d M u l t i P i c t u r e S y s t e m g e n e r a t e s v e c t o r

drawings of complex 3D di splay l i st s in near r ea l t ime. Th e

di splay l i s ts on the MPS look a lo t l ike our t r ee nodes : pr im-

i t i ve e l ement s , t r ans forme d by l i near t r ans format ions , and

l inked by point er s t o o the r nodes . For a s t r i c t ly se l f s imi l ar

t r ee , t he t r ans format ion i s cons t ant , t herefore the ent i r e

di splay l i s t can share a s ingle mat r ix . To m odi fy the t r ee

one only has to change th i s one mat r ix , r a ther t han an

ent i r e d i sp l ay l is t . Thi s makes up dat ing the t r ee d i splay l i st

very f as t . For non- s t r i c t ly s e l f s imi l ar t r ees , t he t r ans forma-

t ions ar e not t he s ame. However a l ookup t able of l es s t han

a dozen t r ans format ions , i s adequate to provide the neces -

sary pseudo randomness . [21]

Figure 3 i l l us t r a t es t he logic of t he d i spl ay l i s t.

The l ef t s ide of t he d i spl ay l i s t conta ins the topologica l

descr ip t ion of t he t r ee . Each node conta ins point er s t o

of f spring nodes p lus a poin t er t o t he t r ans forma t ion mat r ix

which r e l a t es t ha t node to it s paren t node. Thi s par t of t he

di splay l l s t i s pure ly topologica l : i t conta ins no geomet r i c

data . The geomet r i c i nformat ion i s conta ined in the smal l

l i st of t r ans fo rmat io n mat r i ces on the r ight . To edi t a t r ee ,

one can cr ea t e a l a rge topology l i s t once and then r apid ly

manip ula t e only the smal l geom et ry li s t. Ai ry Ray Sm i th in

his research on

9raflals

r ecognized the s epara t ion of t he

topologica l and geom et r i c aspect s of t r ees . He ca l l s t hese

c o m p o n e n t s t h e g r a p h , a n d t h e i n t e r p r e t a t i o n r e s p e c t i v e l y .

Hi s work deal s pr imar i ly wi th speci f i ca t ion of t he t r ee topol -

ogy, i gnor ing in t erpre t a t ion for t he m os t par t I 211. Th e

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i~i~ • ::i _.

i

i i

V i e w s I

By ra ndomly pe r tu rb ing the b ra nc h -

ing parameters , one genera tes a more na tura l is-

tic gnarled tree. 2048x1920)

B l o s s o m t l m e - Th e bark of th is cherry t ree i s

ma de w i th bum p m a ppe d po lygona l t ube s.

The b lossoms a re colored vec tors {par t ic le sys-

tems} Instead of model ing b lossoms, one can

simply

ip

the branches in p ink pa in t .

1024x960)

R a s p b e r r y G a r d e n a t K y o t o - T h e l ea ve s

and bran ches a re genera ted by the f rae ta l

branc hing prog ram. Berr ies a re based on syl ri -

me t ry mode l s by H a re sh La lva n i , w i th a dde d

ra ndom pe r tu rba t ions . N on-se l f s im i l ar f e a tu re s

such as berr ies , require gene t ic spec ia l iza t ion .

1024 080)

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S I G G R A P H 86

paper presented here emphasizes the geometric interpreta-

tion. The thesis of this paper is tha t t he key to realistically

modeling the diversity of trees lies in controlling the

geometric inte rpre tati on. Many different topologies were

used in this project. But by var ying the geometric interpre-

tation of a single topology, one could still generate a wide

variety of trees each with its own distinct taxonomic iden-

tity.

The real time generation of fractal trees has been pack-

aged as an interactive editing system. This multi-window

system allows one to edit the tree parameters, (both

geometric and topological) via graphically displayed sliders.

A vector image of the tree responds in real time. To see all

the parameters change at once one performs keyframe inter-

polation of the parameters. Each tree parameterization is

writ ten to a keyframe file. A cubic spline program, interpo -

lates these parameters to create the inbetween frames. In

the resulting animation, the tree metamorphoses from key to

key. A simple

tree growth

animation is achieved by interpo-

lating the trunk width parameter, and the reeursion size

cutoff parameter. Modifying additional parameters makes

the growth more complex and na tur al looking. For example,

many plants uncu rl as they grow. A metamorphosis anima-

tion is achieved by interpolat ing parameters from different

tree species. Growing and metamorphosing tree animations

appear in The alladium animation produced at NYIT. [1]

8. FRACTA LS, COMPU TERSt AND DNA

The economic advantage of this program is that a

highly complex structure is generated from a simple concise

kernel of dat a which is easy to produce. (Such large data -

base amplification is a primary advantage of fractal tech-

niques in general.) How does the representation of complex-

ity by computers compare with complex expression in nature

itselff

Presumably, complexity in nature has evolved because

it can bestow benefits on an organism. But, as with com-

puters, complexity must not be a burden. Genetic economy

demands that intricate structures be described by a limited

supply of DNA. This struggle to simplify genetic require-

ments, determines the geometric structure of the plant.

Form follows genetic economics.

This suggests a natural explanation as to why self-

similarity abounds in the natural world: evolution has

resolved the tension between complexity and simplicity in

the same way that computer science has -- with recursive

fractal algorithms. If fra~tal techniques help the computer

resolve the demands of data base amplification, then presum-

ably organisms can benefit as well. For genes and comput-

ers alike, self-similarity is the key to th rift y use of data .

[ 6 sl

The parameters of the tree program are numerical

counterpar ts to the DNA code that describes a tree s

bran chin g characteristic s. The logic of the gene is mimicked

although the mechanism is different. The early stages of the

model contained only 3 changeable parameters. The result-

lag images were of very simple fern like plants . New species

were generated by controlling the parameter values, rerolling

the dice of mutation and then selecting the forms that

would be allowed to proliferate. As the model became more

complex with the inclusion of more parameters, the program

created images of more geneticMly complex trees such as

cheery trees, higher on the evo lutio nary scale. Whereas

natural selection of organisms is based on survival value,

this aesthetic selection is based upon resemblance to the

forms of nature.

9. CONCLUSION

Of course any scientific model is simply an attempted

transl ation of nature into some quantifiable form. The suc-

cess of the model is measured by some qualifyably predictive

result. In experimental science, the success of a theory is

measured by the degree to which the predicative model

matches experimental data. Computer graphics now pro-

vides anoth er style of predictive modeling. The success of a

computer simulation is reflected in how well the image

resembles the object being modeled. If one can model a com-

plex object through simple rules, one has mastered the com-

plexity. Wh at appeared to be complex, proves to be primi-

tive in the end. And the proof (although subjective) is in

the picture.

1 0 A C K N O W L E D G E M E N T S

Thanks go to all the members of NYIT staff and

management for their help in the production of this work.

Particular mention goes to Paul Heckbert and Lance Willi-

ams for software support and consultation. Kevln Hunter s

Brownian Fractal Noise Texture was used to generate the

bark. l-laresh Lalvani contributed the original symmetry

models for the raspberries. Design and direc tion by Rebecca

Alien made the alladium Anima tio n possible. Special

thanks go to Jane Nisselson and Robert Wright for editing,

and to Ariel Shaw, Cydney Gordon, and the NYIT photo

departm ent, for film support.

Special thanks go to Benoit Mandeibrot, who has been

a tremendous inspiration and influence.

62


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Dallas August 18-22 Volume 20 Numb er 4 1986

J J

W i n t e r - S n o w A r ~ o r i t h m b y L a n c e W i l li a m s .

O n e r e n d e r s t h e t r e e t w i c e . O n c e w i t h a s i d e

l i g h t s ou r c e . O n c e w i t h a n o v e r h e a d

s n o w

s o u r c e . T h e s n o w s t i k s t o t h e b u m p - m a p p e d

t e x t u r e .

Vie ws H - 2048 x 1920)

6 3


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S I G G R A P H 86

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real time design and animation of fractal plants and trees

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