real time design and animation of fractal plants and trees
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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~
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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.
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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)
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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.
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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)
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S I G G R A P H 86
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