Title: Parallel Fractal Computation and Algorithmic Art: An ArtScience Collaboration
1Parallel Fractal Computation and Algorithmic Art
An Art/Science Collaboration
- Jennifer Burg
- Department of Computer Science
- Wake Forest University
- Winston-Salem, NC
2Purpose of this talk
- To introduce you to fractal computation and other
forms of algorithmic art, implemented both
sequentially and in parallel - To help you decide if you want to take on this
project as a graduate thesis, graduate project,
or undergraduate senior honors project.
3Dance, poetry, digital media, and concepts of
science and mathematics
- What happens when you put these together?
- Fibonacci and Phi, December 2003
- The Fibonacci Sequence and the Golden Ratio, Phi
- Poetry and digital images
- High-performance computation for real-time
generation of fractals choreographed with dance
4Collaborators
- Choreographer Karola Lüttringhaus alban
elved dance company Winston-Salem, NC
(http//albanelved.com) - Jennifer Burg
Department of Computer Science Wake
Forest University - Tim Miller
- Parallel Systems Administrator
- Wake Forest University
5Parallel Computation for Real-time Fractal
Generation Choreographed with Dance
6What taking on this project entails
- Learn how Mandelbrot and Julia fractals are
computed, both sequentially and in parallel. - Create a portable exhibit of the fractal
computation for the iDMAa 2006 conference (Oxford
OH, April 6-8). - Think of and implement new creative twists on the
images, algorithm, setup, etc. - This project could be spread out over one year or
two (ending either May 2006 or May 2007). That
is, you could make it a project or a full-blown
thesis.
7Mandelbrot Fractal
8Mandelbrot Fractal
9Julia Fractal
10Another Julia Fractal
11Another Julia Fractal, up close
12Another Julia Fractal, up close
13Another Julia Fractal, up close
14Mandelbrot Fractal Computation
c and z are complex numbers. To compute a
pixels color, map the pixels horizontal and
vertical coordinates to (cr, ci) on a real-number
plane. cr is the real component of c and ci is
the coefficient of the imaginary component. The
span of the real number plane is determined
experimentally. cr ? -1.5 1.5 and ci ? -1.5
1.5 works. z is computed repeatedly for a
maximum number of iterations or until it is
determined whether or not the computation is
bounded.
15(No Transcript)
16- Demo of Mandelbrot and Julia Fractals Written in
Macromedia Director
17Simulation of driving through a Mandelbrot
fractal
18Mandelbrot Fractal Computation
c and z are complex numbers. To compute a
pixels color, map the pixels horizontal and
vertical coordinates to (cr, ci) on a real-number
plane. cr is the real component of c and ci is
the coefficient of the imaginary component. The
span of the real number plane is determined
experimentally. cr ? -1.5 1.5 and ci ? -1.5
1.5 works. z is computed repeatedly for a
maximum number of iterations or until it is
determined whether or not the computation is
bounded.
19Computing z2 c with Complex Numbers
The computation is done as follows z is given
by (zr, zi) and c is given by (cr, ci). Let
z_new be the value of z that will become the
input to the subsequent computation, with
components z_newr and z_newi. Consider first
what z2 c yields.
20Computing z2 c with Complex Numbers (continued)
To do this computation using only real number
values (with implicit in the imaginary
component of each complex number), we need to
separate out the real from the imaginary
components (the ones with a factor of i), as we
have done in the last line of the previous slide.
This gives
and
21Computing Julia fractals
Computation of a Julia fractal is very similar to
computation of a Mandelbrot fractal. c is a
constant. z is initially mapped from the pixel's
position. Different constants for c produce
fractals of different shapes.
22Simulation of driving through a Julia fractal
23- Demo of Mandelbrot and Julia Fractals Written in
Macromedia Director
24The Night Sky(closing scene)
- Created as a canned movie
- 2000 image files were generated by the parallel
fractal-generating program and written to files - The image files were put together into a
Quicktime movie.
25The Closing Night Sky (go to 11830)
26Parallel implementation of Mandelbrot fractal
- Standard MPI implementation with the following
additions - The breaking of laser beams on stage triggers a
signal to client computer. Client sends a
message across the network to server a Linux
cluster of 16 nodes. - MPI program standard master/slave structure, but
all pixel data has to be funneled through the
master for display. Slaves dont send their
pixel data individually to the client.
27Phi by Jennifer Burg
What are these haunting messages Coded in cryptic
languages Through sights and sounds and
senses Speaking to us without words? And how do
we decipher The beauty as it strikes us, By
saying it or counting it Make it finally our
own? Who hears the golden music best, Who sees
with clearest vision? And can they tell me what
they see And write down every note? A child has
eyes still bright and true An ear open to
voices. A child hears shouts of tiny Whos On
dust specks in the air.
A child knows worlds hold worlds inside Each
world leads to another And angels dance on heads
of pins When you reach infinity. But children
dont have words for this And insufficient
numbers And as we age we want to say Or count
what we have known. The ancients mathematicians
sought A language most eternal And found in pure
proportions A number timeless and divine. The
golden ratio they called it And Phi in Greek we
named it. And everywhere we find its mark In
nature and in art.
28A message whispers softly In the angles of a
seashell And calls my soul to trace a curve Down
paths that never end. I am told that these mute
messages All have Phi locked within them. What is
this magic number? And what secrets does it
hold? We cannot write the number, So irrational
by nature. Never ending, always changing As it
steps toward the sublime. What would happen if
we could Know the endless perfection? Say in
words and in numbers What we dont yet understand?
I see a message etched In a ragged rocky
coastline And the pattern is repeated In the
ripples at my feet. A fern unfurls its growing
leaves Like natures own fresh fractal So I paint
a fractal of my own To find the world inside. I
try to read a message In the face of a
sunflower But Im blinded by the spirals Spinning
left, and spinning right. The spirals leave my
dazzled grasp A galaxy is born And sends to me
through heavens time A metaphor of stars.
29If we mark the musical intervals With infinite
precision Can we make a human symphony From the
harmony of the spheres? If we trace the
seashells spiral Down endless perfect
angles Will we finally find the center And meet
the eye of God? We cannot take the measure Of
Your exquisite beauty Though its woven in the
fabric Of our world and our flesh. We cannot say
Your name Though its written in the sky Still we
silently rejoice to read The messages You send.
30web links
- Technical details of the real-time fractal
computation were published in the 2004 Conference
on Parallel and Distributed Computing. See
http//www.cs.wfu.edu/burg/papers/ClusterComputat
ionRealTimeDance.pdf - A description of the entire Fibonacci and Phi
production can be found at http//www.cs.wfu.edu/
burg/papers/DancingWithFractals.pdf - A description of both Wake Forest/alban elved
collaborations can be found at http//www.cs.wfu.e
du/burg/papers/CSAndDance.pdf - A description the parallel Mandelbrot calculation
can be found at http//www.cs.appstate.edu/can/cl
asses/5530/mpiman/node35.html - A description of an efficient sequential
implementation of Mandelbrot fractals can be
found at - http//xaos.sourceforge.net/english.php
- Another description of fractal computation can be
found at - http//www.cs.wfu.edu/burg/nsf-due-0340969/works
heets/MandelbrotFractalsProgrammingAssignment.pdf - Another type of algorithmic art (Koch snowflakes)
is described at - http//www.cs.wfu.edu/burg/nsf-due-0340969/works
heets/KochSnowflakeProgrammingAssignment.pdf - Another type of algorithmic art (spirals) is
described at - http//www.cs.wfu.edu/burg/nsf-due-0340969/inter
active/spiral.htm
31What taking on this project entails
- Learn how Mandelbrot and Julia fractals are
computed, both sequentially and in parallel. - Create a portable exhibit of the fractal
computation for the iDMAa 2006 conference (Oxford
OH, April 6-8). - Think of and implement new creative twists on the
images, algorithm, setup, etc. - This project could be spread out over one year or
two (ending either May 2006 or May 2007). That
is, you could make it a project or a full-blown
thesis.
32Additional References Briggs, J. (1992).
Fractals The Patterns of Chaos. New York
Touchstone Book/Simon Schuster. Livio, M.
(2002). The Golden Ratio The Story of Phi, the
Worlds Most Astonishing Number. New York,
Broadway Books. Mandelbrot, B. B. (1988).
Fractal Geometry of Nature. New York, W. H.
Freeman and Co. Pickover, C. (1995). Keys to
Infinity. New York, John Wiley Sons. For
other books by Clifford Pickover, see
http//sprott.physics.wisc.edu/pickover/bookscp.h
tml