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Fractals & Chaos In Biology
By: Daniel
Introduction
Have you ever awed at the fact that the heart
beats more than 30 million times a year? Perhaps you
have wondered how part of our lungs can have a surface
area the size of a tennis court. You’re in luck! Thanks to
fractals and chaos we can accurately describe and
understand various parts of the human body.
Researchers are working around the clock to try to solve
biological problems such as cancer and cystic fibrosis.
Instead of some new dangerous method, wouldn’t it be
wonderful if we could cure diseases by the knowledge we
have gained in fractals and chaos theory? Stem cell
research, radiation treatment, and any other uncertain
cures may all be ruled out if fractals and chaos can take a
large role in the field of biology. Come, let’s find out
more!
Picture from:
http://web.ukonline.co.uk/members/j
ill.lawson/
What exactly is Fractals and Chaos and
why is it so important in Biology?
Chaos is traditionally thought of as being confusion, hysteria, and
turmoil. Chaos, however, in the sense of chaos theory is the idea that
the final outcome of something can be extremely sensitive upon initial
conditions. Amazingly, you can actually find brief organized patterns
within chaotic systems. Chaotic systems have the three main
properties of sensitivity, mixing, and periodicity. As chaos theory is
still a fairly new field of research, the properties may be apt to change
in the near future. Fractals are the models generated by math
equations resulting in chaotic systems. Fractals are very artistic,
complex, and intricate. They also have properties which include
having a fine structure, being defined by a recursive process, being too
irregular to be described by traditional geometry, having selfsimilarity, and having fractal dimension. In Biology, Chaotic systems
can be used to show the rhythms of heartbeats, walking strides, and
even the biological changes of aging. Fractals can be used to model
the structures of nerve networks, circulatory systems, lungs, and even
DNA.
(Click to see full view of background)
When was fractals and chaos first researched?
When was chaos theory relating to biology first
researched?
Henri Poincaré, a notable French mathematician, was doing research
on celestial mechanics in 1887 when he stumbled upon chaos. A
discrete error in one of his solutions later gained fame as the
foundation of all chaos theory. Since then numerous notable scientists
such as Robert Shaw and Edward Lorenz have researched the subject of
chaos. (Lorenz had the butterfly-effect idea)
The butterfly effect- A butterfly
flapping its wings in China can cause
tornadoes in Texas.
Picture from:
www.emsf.rai.it/interviste/
interviste.asp?d=502
Henri Poincaré
When was fractals and chaos first researched?
When was chaos theory relating to biology first
researched?
The father of fractals is often considered to be a man by the name of
Gaston Julia. In the early 1900’s, Julia did much research on iterated
functions, and even drew some of his famous Julia sets by hand. True,
there were some other works out there, such as Sierpinski’s triangle
and Koch’s curve, but Julia’s work was a major breakthrough. Until the
1960’s much of the work with fractals was abandoned due to lack of
technology. That changed in the 1970’s when Mandelbrot used
computers to create what we now know today as the Mandelbrot Set.
Pictures from:
http://www.fractalus.ch/index_
pic/Julia.jpg
http://www.spsu.edu/math/edw
ards/mandel/bigman.jpg
The Mandelbrot Set
Gaston Julia,
who sadly,
lost his nose
in WWI
When was fractals and chaos first researched?
When was chaos theory relating to biology first
researched?
Chaos theory relating to biology was first researched in the early 70’s.
Researchers were looking at how chaos theory could be used to model
population trends. Several researchers, such as George Oster, Robert
May, and Jim Yorke, looked at equations such as this one in their effort
to model population: xt+l = l xt(1 –xt). As for human biology, shortly
after the Mandelbrot Set was discovered this also took off. Dynamical
diseases, a term coined in this era, described diseases that show
chaotic systems. Researchers such as Leon Glass and Michael Mackey
did research in this field. Now, there are organizations dedicated to
research with chaos theory and fractals in the field of biology.
Leon Glass
Picture from:
www.physionet.org/
Onward…
Select a topic
The human heart
Brain, DNA
Lungs
Click when
finished…
The Human Heart
The rate of the beating of the human
heart can be described as chaotic. The
time between heart beats fluctuates
due to a variety of factors including
exercising, stress, and physical
activities.
A good way to listen to your heart is
with a heart rate monitor or just place
your hand on your chest or another
place where you can measure your
pulse. Then try a variety of exercises
that cause your heart rate to change
and see how rapidly and slowly your
heart goes.
Click to go
to the main
menu
Order and Chaos:
Healthy hearts vs.
Diseased hearts
The Sound of the Heart
Fractal Geometry of the
Heart and Circulatory
Structures
Order and Chaos:
Healthy hearts vs. Diseased hearts
Healthy heart
Diseased heart
 Has slight variations in
the time between one
beat to the next.
 Has a heart-rate that is a
chaotic pattern that is
self-similar.
 Doesn’t exhibit slight
variations in time
between each beat
 Has a heart-rate that is
steady, constant, and
predictable or either
extremely random
Interesting Fact:
This concept also applies to human
walking. When humans walk with
variation in their step they are normal.
With the onset of a disease, such as
Parkinson’s disease, the human stride
is more constant.
Evidence for this…
Evidence for chaotic healthy hearts
Dr. Ary L. Goldberger is director of
electrocardiography at a hospital in Boston, MA.
In the years gone past, Mr. Goldberger has done
extensive research on heart rates and chaotic
patterns.
As Dr. Goldberger was in medical school he was learning the
traditional taught methods of how hearts should be in equilibrium and Dr. Goldberger
should be constant. Yet, after listening to countless heart rates of his
patients, he began to notice variations in completely healthy hearts.
By researching with his colleagues, Goldberger was able to discover
that heart rates show fractal patterns. This is not because of physical
reasons, as many might believe, but because of physiological reasons.
As for why the fractal patterns break down in diseased hearts is still
inconclusive. Goldberger has helped establish a resource center at
http://www.physionet.org/ to help in the discovery process through
the share of ideas and data.
Next page…
Picture
from:http://focus.hms.harvard.e
du/2002/March8_2002/cardiolo
gy.html
Evidence for chaotic healthy hearts
As you can see the top graph shows a
normal healthy heart. The graph of the
healthy heart has more complexity then
the bottom graph. Complexity = healthy
in many physiological aspects. The
bottom time scale graph shows a heart
with CHF (congestive heart failure). CHF
is just one of the many diseases that
causes the heart rate to lose it’s chaotic
property. Click on the image for a further
view of the normal heart-rate.
Picture from:http://www.physionet.org/tutorials/ndc/
As you can see the heart rate is very complex
and does show signs of self-similarity. What
is more surprising is how the heart-rate
seems to lose its long range correlation as
the heart becomes diseased or break down.
(Click to continue…)
Picture from:http://www.physionet.org/tutorials/ndc
Picture from:http://www.physionet.org/tutorials/ndc
BACK TO THE
HUMAN HEART
Both of these images show
what happens as the heart
goes out of its normal state.
The bottom left graph shows a
subject with heart failure.
This graph has highly periodic
values with little variation.
The bottom right graph shows
a subject with atrial
fibrillation. This heart rate is
very erratic jumping from the
high end of heart rate to the
low end, with no particular
pattern.
Fractal Heart Music
 The son of Dr. Goldberger, a
man by the name of Zach
Davids, actually recorded the
melodies of cardiograms on
piano. He used much of his
father’s research to add the
right theme of music to the
cardiograms and give them
the tempo they needed.
BACK TO THE
HUMAN HEART
This may take a moment
to load. It should open
up a web page and load.
(must have realplayer to
view)
Fractal Geometry of the Heart and
Circulatory Structures
Fractals are very useful in modeling the heart and circulatory structures,
and they play an important role in maintaining homeostasis.
First, the main areas where fractal geometry can
be seen in the circulatory system are:
 Arteries and veins
-Their cells and organization display the properties of
fractals, such as the power-law distribution in the
diameter distribution of arteries and veins.
 Organization of heart muscle groups
- Show properties of self-similarity, fine structure, etc.

Branching of certain muscles inside the heart
- resemble the bifurcations seen in fractals such as
the Feigenbaum plot

His-Purkinje network
- The branches and bifurcation of this electrical
system are essential to human biology and
resilience.
 The tendons that connect the tricuspid valve to
the papillary muscles.
-These again show bifurcation along with other fractal
properties.
 The aortic valve leaflets
- These are layered providing a huge surface area,
while keeping a small volume
How does the fractal structure
help?
-The fractal structure of the veins,
arteries, and heart muscles help
protect the circulatory system from
the strong, violent pumping of the
human heart.
-The fractal structure, which is
usually unnecessary, can come into
play when the His-Purkinje network
is damaged. This helps the heart be
resilient and resistant to damage.
- The fractal geometry of the heart
could possibly save us everyday.
Fractal Geometry of the Heart and
Circulatory Structures
Arteries and
veins
Aortic valve
leaflets
Pictures from:
http://webvision.med.utah.edu/imageswv/ARTERIE
S.jpg,
www.synecor.com/images/
www.medtronic.com/cardsurgery/images/fix4.jpg
The Lungs
What you are hearing is a ventilator,
which hopefully you will never have to
use.
Thanks to the fractal geometry of our
lungs, most of us don’t have to use
ventilators or other breathing devices.
As Benoit Mandelbrot was doing his
research on Fractals and Chaos, he
proposed that the lung shows signs of
fractal geometry. Since then several
studies have been done to prove this
as being true. Another notable scientist
to research fractals in the lungs was
Sergey V. Buldyrev.
Picture from: polymer.bu.edu/ ~sergey/home.html
Sergey V. Buldryev
Continue…
The Lungs
The fractal geometry of
the lungs helps the lungs
resist and overcome
problems and physical
stress during their growth.
Also, In the lungs, there are
small air sacs, called alveoli,
that are responsible for the
diffusion of oxygen into the
blood.
Continue…
By the lungs having fractal geometry
they are more efficient. But how? The
rate at which the diffusion of air through
the alveoli occurs is directly proportional
to the surface area of the alveoli and
lungs.
All of the alveoli in a human adult, have
a total surface area of about 750 sq ft.!
Yet, they have a tiny volume.
Thus, the only way to model these
alveoli is through fractals. As imagined,
the fractal geometry of the alveoli is
very high, usually around 2.9 or so.
So what other parts of the lungs show fractal
geometry?
As a whole, the lungs
show much fractal
geometry. Also, in the
bronchi and bronchiole
tubes, there can be
greatly seen the property
of bifurcation. If you
look back to the
Feigenbaum plot, you
will find similar such
bifurcations.
A model of the lungs as a
whole.
The branching bronchiole tubes
(top). The bronchiole tubes
and the branching arteries
(bottom).
A digital representation of
lung tissue.
Pictures from:
http://classes.yale.edu/fractals/Panorama/Biology
/Physiology/Physiology.html
Main Menu
Fractals and Chaos in the Brain,
DNA
Out of the three topics given on the previous page, this one is the most new and
waiting to be researched. As research is in its early stages, there have been
few discoveries made in how fractals and chaos relates to the brain and DNA.
Some discoveries that have been made:
• By modeling electrical signals in the brain and nerves, scientists have been
able to model the *alpha rhythm using chaos. Due to this discovery, the
teaching of what areas in the brain control which neurological functions may
be in need of a change. If wave analysis in the brain is indeed modeled by
time series graphs it will indeed show that chaotic dynamics is important in
neurology.
• The Large surface area where neurons are packed in layers in the brain can
only be modeled by using fractal geometry.
*Image from:
http://www.geocities.
• Brain oscillations and waves is where most of the chaos theory and time
com/Omegaman_UK/
fractal.html
series graphs come into play. The Brain itself may very well be organized
strictly by the laws of chaos.
• DNA sequences can be very similar to fractals in that they display qualities of
Brownian self-similarity, which involve small random lines making up lines
instead of small, more patterned lines making up lines.
• DNA and the Brain both can be modeled by fractal geometry.
*Alpha rhythm- frequency of brainwaves between 8 -12 Hz.
Fractals and Chaos in the Brain,
DNA
•
By modeling electrical signals in the brain and nerves, scientists have been able to
model the *alpha rhythm using chaos. Alpha rhythm, along with other wave
frequencies, can often be modeled using time series graphs.
Images
from:http://www.crossroadsinstitute.org/eeg.html
If alpha rhythm is
between 8 -12 Hz and
has the graph shown
try to guess the
following.
Alpha rhythm – Does this look
familiar to any time plots seen in
the study of fractals and chaos?
Match the graph to the type
of rhythm (frequency band).
Delta: .1 – 3 Hz
Theta: 4 – 8 Hz
Beta: above 12 Hz
*Alpha rhythm- frequency of brainwaves between 8 -12 Hz.
Click to see answer
Fractals and Chaos in the Brain,
DNA
Beta: above 12 Hz
Delta: .1 – 3 Hz
Theta: 4 – 8 Hz
Images
from:http://www.crossroadsinstitute.org/eeg.html
Applications of Fractals and
Chaos in the Brain
The brain is one of the most intricate
parts of the human body. It is very
unlikely that science will ever be able to
grasp every part of its perplex design,
however chaos theory and dynamical
equations may very well be a starting
ground. In addition to being able to
model electrical signals of nerves and
the brain, chaos theory may help solve
neurological diseases and progress the
invention of artificial intelligence.
The hit movie, “AI” displayed many
types of neurosurgery and inspired
feelings about how artificial intelligence
would be viewed in the future.
Examples of Fractal Geometry of
DNA and of the Brain
A Brain Fractal.
A fractal of DNA.
Images from: www.sgeier.net/ fractals/flam3/, www. sprott.physics.wisc.edu
,.
Main Menu
See you later!
Gaston Julia,
displaying one of his
many talents.
Dr. Goldberger, with
his dog, Willy.
Pictures from: www.elnidodelescorpion.com/
N23/ilustraciones/..., reylab.bidmc.harvard.edu,
http://sv.wikipedia.org/wiki/Helge_von_Koch
www.colorcube.com/ illusions/dither.htm
http://math.youngzones.org/Fractal%20webpages/histor
y_fractals.html
Helge Von Koch, with
a comical friend.
Benoit
Mandelbrot.
Credits





Slayden, Cameron. American Association for the Advancement of Science.
“Welcome to Modern Biology” Science 291, 1177 (2001).
http://web.jjay.cuny.edu/~mwallace/ModernBiology/ (image)
May, Robert. “The chaotic rhythms of life” 10/2/2002
http://members.fortunecity.com/templarser/rhythm.html
Microsoft PowerPoint 2002 SP3, Microsoft Clipart Gallery
Rae, Greg.“Chaos Theory: A Brief Introduction” Jan. 31st, 2003
http://www.imho.com/grae/chaos/chaos.html
With sources of:
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–
–
–
–
"Bach to Chaos: Chaotic Variations on a Classical Theme", Science News, Dec. 24, 1994, pg. 428.
Gleick, James, Chaos - Making a New Science, Penguin Books Ltd, Harmondsworth, Middlesex,
1987.
Lowrie, Peter, personal interview over the Internet, May 17, 1995.
Rae, Kevin, "Chaos", unpublished paper, submitted to Professor Gould, Modern Physics class,
Claremont McKenna College, December 5, 1994.
Stewart, Ian, Does God Play Dice? The Mathematics of Chaos, Penguin Books Ltd, Harmondsworth,
Middlesex, 1989.


Browne, Malcolm. “The Fractal Heart.” New York Times, unknown date
http://www.nyu.edu/classes/neimark/FRACT1.HTM
Landau, Misia. “Healthy Heart Keeps Polyrhythmic Beat”, March 8th, 2002.
Harvard Medical, Dental, and Public Health Schools