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File Format: Microsoft Powerpoint - View as HTML Here is a link to a PowerPoint presentation “created by Mrs. Gamache using the collection of ... Photo by Gayla Chandler. Post Processing by Kim Letkeman ... www.public.asu.edu/~starlite/Fractals_a_Symmetry_Approach/MathEd_PPP_PresentIt.ppt - Similar pages

Fractals:
an Introduction through Symmetry
for beginners to fractals,
highlights magnification symmetry
and fractals/ chaos connections.This presentation is
copyright © Gayla Chandler.
All borrowed images are
copyright © their owners or
creators. While no part of it is in
the public domain, it is placed on
the web for individual viewing
and presentation in classrooms,
provided no profit is involved..I hope viewers enjoy this gentle approach to math
education. The focus is Math- Art. It is sensory,
heuristic , with the intent of imparting perspective as
opposed to strict knowledge per se. Ideally,
viewers new to fractals will walk away with an
ability to recognize some fractals in everyday
settings accompanied by a sense of how fractals
affect practical aspects of our lives, looking for
connections between math and nature.
This personal project was put together with the
input of experts from the fields of both fractals and
chaos:.Academic friends who provided input
:
Don Jones
Department of Mathematics &
Statistics
Arizona State University Reimund Albers
Center for Complex Systems &
Visualization (CeVis)
University of Bremen
Paul Bourke
Centre for Astrophysics &
Supercomputing
Swinburne University of Technology
A fourth friend who has
offered feedback,
whose
path I followed in putting
this together,
and whose
influence has been
tremendous prefers not to
be named yet must be
acknowledged.
You know
who you are.
Thanks.
..First discussed will be three
common types of symmetry:
• Reflectional (Line or Mirror)
• Rotational (N- fold)
• Translational
and then: the Magnification
(Dilatational a. k. a. Dilational)
symmetry of fractals..Reflectional (aka Line or Mirror) Symmetry
A shape exhibits reflectional symmetry if the shape can be
bisected by a line L, one half of the shape removed, and the
missing piece replaced by a reflection of the remaining piece
across L, then the resulting combination is (approximately) the
same as the original. 1
From “An Intuitive Notion of Line Symmetry”
In simpler words, if you
can fold it over and it
matches up, it has
reflectional symmetry.
This leaf, for instance,
and the butterfly
caterpillar sitting on it, are
roughly symmetric. So
are human faces. Line
symmetry and mirror
symmetry mean the same
thing..The butterfly and the
children have lines of
reflection symmetry
where one side
mirrors the other.
Taken at the same time at
the Desert Botanical Gardens
Butterfly Pavilion, the little
butterfly is a Painted Lady
(Vanessa, cardui ). Its host
plant is (Thistles, cirsium)..Here is a link to a PowerPoint
presentation “created by Mrs. Gamache
using the collection of web pages by the
Adrian Bruce and students of 6B.”
This site lets you create your own
symmetry patterns! Choose your type
and color, then start moving the mouse
and clicking.
These images have
lines of symmetry
at the edge of the
water..Rotational (N- fold) Symmetry
A shape exhibits rotational symmetry if rotation about
some center point returns the shape to its original
configuration. 2
Libbrecht of
Caltech discusses
symmetry of
ice crystals
and snowflakes
Real snowflake image taken by Kenneth
Libbrecht using a special photo microscope
Real snowflake image taken by Kenneth
Libbrecht using a special photo microscope.These blooms have 5- fold rotational
symmetry. They can be turned
5 times to leave the figure unchanged
before starting over again.
These blooms have 5- fold rotational
symmetry. They can be turned
5 times to leave the figure unchanged
before starting over again.
The butterfly is a
Spicebush Swallowtail
(Papillo, troilus).
The butterfly is a
Spicebush Swallowtail
(Papillo, troilus).
A pentagon also has 5- fold symmetry. A pentagon also has 5- fold symmetry..An example of 4- fold rotational symmetry,
a property shared by the square.
An example of 4- fold rotational symmetry,
a property shared by the square..The tiny blooms have 4- fold
symmetry. Question: does
the spherical bloom they sit
on have n- fold symmetry?
This flower has
21- fold rotational
symmetry.
5- fold or 6- fold
symmetry here.The tiny blooms have 4- fold
symmetry. Question: does
the spherical bloom they sit
on have n- fold symmetry?
This flower has
21- fold rotational
symmetry.
5- fold or 6- fold
symmetry here
Julia, (Dryas, julia) Julia, (Dryas, julia)
Great Southern Whites
(Ascia monuste)
Great Southern Whites
(Ascia monuste) Zebra Longwings
(Heliconius, charitonius)
Zebra Longwings
(Heliconius, charitonius)
Butterfly
names.The bricks in the image have
translational symmetry.
The bricks in the image have
translational symmetry.
Translational Symmetry A shape exhibits translational symmetry if displacement
in some direction - horizontal or vertical, for example -
returns the shape to (approximately) its original
configuration. 3
Also, the image of the bricks will
have translation symmetry when
sliding, provided there is no
rotation during the move.
Also, the image of the bricks will
have translation symmetry when
sliding, provided there is no
rotation during the move.
Orientation must be preserved
while translating.
Orientation must be preserved
while translating..Magnification (Dilatational) Symmetry
Less familiar is symmetry under magnification:
zooming in on an object leaves the shape
approximately unaltered . 4
Zooming in on a fractal object
leaves the shape
approximately unaltered .
Fractals exhibit magnification symmetry..Types of Fractals highlighted:
• Natural
• Geometric
• Complex and Random
Brief discussion: Frames of Reference
and examples of:
Exponential Growth
Fractals Across the Disciplines.Natural Fractals
Multifractals
Chaos
Natural fractals have a limited
number of stages of growth, and
the growth between stages
shows variation. They have
connections to Multifractals and
Chaos theory..Fractals in the Biological Sciences
by N. C. Kenkel and D. J. Walker
University of Manitoba
Quantitative Plant Ecology Laboratory
below is an excerpt from their web page
Section 5.7 Chaos and Time Series Analysis:
“Chaos, which is closely related to fractal
geometry, refers to a kind of constrained
randomness (Stone and Ezrati 1996).
Wherever a chaotic process has shaped an
environment, a fractal structure is left behind.”.Photo by Gayla Chandler
Post Processing by Kim Letkeman.“Fractal geometry was designed
to handle shapes that appear
complicated, but with complexity
arranged in some hierarchical
fashion. So at a minimum,
fractals must have some
substructure.”
(Michael Frame, Yale University ).Image from Paul Bourke’s
One necessary
condition for fractal
substructure is the
same or a highly
similar shape between
a minimum of 3 stages
of growth (there exists
disagreement on this,
especially between
disciplines).
The entire fern
family reveals
self- similarity:
successive
stages of growth
that closely
resemble earlier
stages.
Self- Similarity page..Photo courtesy of Pbase artist Ville Vels of Estonia.Photo courtesy of Pbase artist Ville Vels of Estonia
Icicles made of icicles.Photo courtesy of Pbase artist Ville Vels of Estonia
Repeating
shapes on
different scales..Photo courtesy of Pbase artist Ville Vels of Estonia.Branching Patterns Branching Patterns
Corrosion cast of the liver
used in medical research.
This image is property of
Professor Dr. Heinz- Otto
Peitgen, Director of MeVis
of the University of Bremen,
It may not be used for ANY
purpose without explicit
written permission from
Professor Peitgen or MeVis.
This corrosion cast of the liver used in medical research reveals fractal branching as
do several body organs . Most branching in nature is fractal: leaf veins and rivers,
our circulatory system and lightning, to name a few..Branching Patterns Branching Patterns
Yellow: the portal
vein brings blood
in from the
intestinal
system
Black: the hepatic
vein drains blood
from the liver
Corrosion cast of the liver
used in medical research.
This image is property of
Professor Dr. Heinz- Otto
Peitgen, Director of MeVis
of the University of Bremen,
It may not be used for ANY
purpose without explicit
written permission from
Professor Peitgen or MeVis.
This corrosion cast of the liver used in medical research reveals fractal branching as
do several body organs . Most branching in nature is fractal: leaf veins and rivers,
our circulatory system and lightning, to name a few..1 1
2 2
3 3
There are several
stages of scaling
not visible in this
image, see next
image where the
top of the plant is
visible.
There are several
stages of scaling
not visible in this
image, see next
image where the
top of the plant is
visible.
Butterfly is a
Zebra Swallowtail
(Eurytides, marcellus)
Butterfly is a
Zebra Swallowtail
(Eurytides, marcellus).The scaled
branching extends
upward throughout
the plant. A small
branch, if magnified,
would look like a
larger branch.
The scaled
branching extends
upward throughout
the plant. A small
branch, if magnified,
would look like a
larger branch.
The butterfly is a
Great Southern White
The butterfly is a
Great Southern White.Also notice how entire
sections of the plant
resemble each other on
different scales. The
structure of smaller
sections dictates the
shape of larger sections.
Also notice how entire
sections of the plant
resemble each other on
different scales. The
structure of smaller
sections dictates the
shape of larger sections..Fractal branching is
captured in shadow
below. From this
view, again notice
how the parts
resemble the
whole.
Fractal branching is
captured in shadow
below. From this
view, again notice
how the parts
resemble the
whole.
A Painted Lady is present! A Painted Lady is present!.The plant in the
previous slides
resembles this
computer-
generated binary
fractal tree .
Image by Don West.Photograph courtesy of Clive Pierson, taken in Ireland. ..
Photograph courtesy of Clive Pierson, taken in Ireland.
Similarity in tree branches can
be seen in the shapes of
clumps of tree, intersections of
branches, the flow of limbs...
Repeating a simple pattern
throughout growth yields a
complex structure with
magnification symmetry, a
structure made of parts
that are similar
to the whole: a
*fractal* structure.
..Photograph courtesy of Clive Pierson, taken in Ireland. ..Photograph courtesy of Clive Pierson,
 taken in Ireland. ..Photograph courtesy of Clive Pierson, taken in Ireland. ..Photograph courtesy
of Clive Pierson, taken in Ireland. ..Leaf veins have fractal properties.
Image courtesy of Paul Bourke.With fractals, the structure behind small sections
dictates overall shape.
We have seen empirical verification of this in
previous examples, how bigger shapes were
aggregations of the smaller shapes that made them
up. This is also true of clouds, mountains, ocean
waves, lightning, and many other aspects of nature.
An ocean wave is made up of a lot of little waves,
which are in turn made up of yet smaller waves.
This is why fractal equations tend to be simple.
Tremendous complexity can result from iterating
simple patterns..Of those aspects that have an embedded fractal
structure, their fractal aspect only describes
properties of shape and complexity. Read this
Word of Caution from Nonlinear Geoscience:
Fractals. They refer to randomness that is taken
into account in Multifractal theory , which has ties to
Chaos theory and Nonlinear Dynamics.
The Yale Fractal Geometry website points out
Common Mistakes in Finding Fractals. Also view
this Introduction to Fractals PowerPoint presentation
out of Florida Atlantic University by Liebovitch and
Shehadeh that makes many fractal/ nonfractal
comparisons. Read a paper from Complexity
International about language issues with regard to
fractals: Is There Meaning In Fractal Analysis?.This is a Sweet Acacia (Acacia, smallii) tree. Its
unbloomed flower appears to be a sphere made
up of smaller- scale spheres, but take a closer
look:.This is a Sweet Acacia (Acacia, smallii) tree. Its
unbloomed flower appears to be a sphere made
up of smaller- scale spheres, but take a closer
look:.Outer foliage (leaves and petals) on
plants are usually terminal organs,
and are non- reproductive. (There are
exceptions, though, like the entire fern
family.) Reproductivity is neither
necessary nor sufficient for fractility,
but it can play a role. Root Gorelick of
the ASU biology department explains:
Outer foliage (leaves and petals) on
plants are usually terminal organs,
and are non- reproductive. (There are
exceptions, though, like the entire fern
family.) Reproductivity is neither
necessary nor sufficient for fractility,
but it can play a role. Root Gorelick of
the ASU biology department explains:.Outer foliage (leaves and petals) on
plants are usually terminal organs,
and are non- reproductive. (There are
exceptions, though, like the entire fern
family.) Reproductivity is neither
necessary nor sufficient for fractility,
but it can play a role. Root Gorelick of
the ASU biology department explains:
Outer foliage (leaves and petals) on
plants are usually terminal organs,
and are non- reproductive. (There are
exceptions, though, like the entire fern
family.) Reproductivity is neither
necessary nor sufficient for fractility,
but it can play a role. Root Gorelick of
the ASU biology department explains:
“Leaves are terminal
organs, hence don't
reproduce miniature
copies of themselves as
do stems, roots, and
many reproductive
structures. Therefore, I
expect leaves to be least
fractal of these organs.”
“Leaves are terminal
organs, hence don't
reproduce miniature
copies of themselves as
do stems, roots, and
many reproductive
structures. Therefore, I
expect leaves to be least
fractal of these organs.”
(Root Gorelick).Geometric Fractals
I like to compare Geometric fractals to
objects/ systems in a vacuum in physics. They are,
as their name suggests, geometric constructs,
perfect (Ideal) systems exempt from internal
deviations or potential changes from outside
influences (other than human error in constructing
them).
I haven’t included Complex fractals such as the
Mandelbrot Set and Julia Sets in the Geometric
fractals category. Complex fractals are mentioned
later..The Sierpinski Tetrahedron
Fractal type: Geometric
Tetrahedra are increasing in number in powers of 4
Tetrahedra are decreasing in edge- length in powers of ½
Volume is decreasing in powers of ½

Image created using MathCad by Byrge Birkeland of Agder University College, Kristiansand,
Norway.To consider this fractal, it is important to know something about
a tetrahedron .- Start with an equilateral triangle..To consider this fractal, it is important to
 know something about a tetrahedron .- Start with an equilateral triangle.- Divide it into 4
 equilateral triangles by marking the midpoints of all three sides and drawing lines to
 connect the midpoints..To consider this fractal, it is important to know something about
a tetrahedron .
- Start with an equilateral triangle.
- Divide it into 4 equilateral triangles by marking the midpoints
of all three sides and drawing lines to connect the midpoints.
Fold on the midpoint lines and
bring the tips of the equilateral
triangle together to make a
tetrahedron..To build a stage- 1:
Start with a regular tetrahedron.
It is called the stage- 0 in the
Sierpinski tetrahedron fractal
family.
Reduce it by a
factor of 1/ 2
Replicate (4 are needed). The
tetrahedra are kept transparent on
this slide to reinforce that these
are tetrahedra and not triangles..Rebuild the 4 stage- 0’s into a
stage- 1 Sierpinski tetrahedron.
The line is a handy frame of
reference for construction..Rebuild the 4 stage- 0’s into a
stage- 1 Sierpinski tetrahedron.
The line is a handy frame of
reference for construction..Rebuild the 4 stage- 0’s into a
stage- 1 Sierpinski tetrahedron.
The line is a handy frame of
reference for construction..Rebuild the 4 stage- 0’s into a
stage- 1 Sierpinski tetrahedron.
The line is a handy frame of
reference for construction..Revisiting the earlier image, notice that each tetrahedron is
replaced by 4 tetrahedra in the next stage......Determine the stage by counting the number of sizes of openings,
the stage- 1 has one size of opening, the stage- 2 two sizes of
openings, etc….What is happening to all that removed volume in
Sierpinski’s tetrahedron? You can view it on my
Sierpinski Tetrahedron and its Complement page.
This and more can be seen on on Paul Bourke’s
Platonic Solids Fractals and their Complements
page.
The Sierpinski tetrahedron is a volume analog of
the Sierpinski triangle:.The Sierpinski Triangle:
Geometric fractals are typically filling or emptying something, whether it is
length, surface area, or volume. The key points are that dimension is: 1)
changing, and 2) generally fractional.
With this fractal, it is
surface area instead of
volume that is decreasing
at each stage.
The face of a Sierpinski
tetrahedron is a same-
stage Sierpinski triangle.
The stage can be
determined by the
number of different
sizes of openings.
1
2 3
4
grows in Powers of 3
Notice how each triangle
becomes three triangles
in the next stage.
Reduce
by ½
Replicate
& Rebuild
Reduce
by ½
again
Replicate
& Rebuild.Even though most
fractals have non-
integer dimension,
there are exceptions:.For exactly self- similar shapes made of N copies,
each scaled by a factor of r, the dimension is
Log( N)/ Log( 1/ r)
The Sierpinski tetrahedron is made of N = 4 copies,
each scaled by a factor of r = 1/ 2, so its dimension is
Log( 4)/ Log( 2) = 2
So the Sierpinski tetrahedron is a shape that is
manifestly fractal, but has integer dimension!.Contrast this with the Sierpinski triangle, made of
N = 3 copies, each scaled by a factor of r = ½.
Its dimension is
Log( 3)/ Log( 2) ~= 1.58496…..
The Sierpinski triangle has fractional dimension,
more typical of fractals.
The exact answer above is Log( 3)/ Log( 2). The approximate
answer is the decimal approximation 1.58496… Rule of
thumb: keep answers in exact form unless a decimal
approximation is requested, and when requested, wait until the
very end to convert to a decimal to avoid rounding error..Self- similarity: this is a big
idea, and it only truly
applies to geometric
fractals; however, it is
used as a concept to talk
about all types of fractals..Something is self- similar when
every little part looks exactly like
the whole. The only place this can
really happen is in a perfect (Ideal)
system at infinity; however, in
order to speak about fractals
generally, one must embrace the
concept of self- similarity in a
broad way..(Everything on this slide links to relevant
websites; url’s are included since links aren’t
coming thru the conversion to PDF.)
(Everything on this slide links to relevant
websites; url’s are included since links aren’t
coming thru the conversion to PDF.)

Chaos
http:// classes. yale. edu/ fractals/ Chaos/ ChaosIntro/ ChaosIntro. html
 
Chaos
http:// classes. yale. edu/ fractals/ Chaos/ ChaosIntro/ ChaosIntro. html

Multifractals
 http:// classes. yale. edu/ fractals/ MultiFractals/ welcome. html

Multifractals
 
http:// classes. yale. edu/ fractals/ MultiFractals/ welcome. html
Random Fractals
 
http:// classes. yale. edu/ fractals/ RandFrac/ welcome. html

Random Fractals
http:// classes. yale. edu/ fractals/ RandFrac/ welcome. html

Complex Fractals http:// www. geom. uiuc. edu/~ zietlow/ defp1.html

Complex Fractals http:// www. geom. uiuc. edu/~ zietlow/ defp1.html

Mandelbrot discusses fractals

http:// www. yale. edu/ opa/ v31.n20/ story6. html

Mandelbrot discusses fractals

http:// www. yale. edu/ opa/ v31.n20/ story6. html

Mandelbrot Set
http:// www. ddewey. net/ mandelbrot/ noad. html

Mandelbrot Set
http:// www. ddewey. net/ mandelbrot/ noad. html

Julia Sets
http:// www. ibiblio. org/ e- notes/ MSet/ Period. htm
Julia Sets
http:// www. ibiblio. org/ e- notes/ MSet/ Period. htm

.Is there anything in this image
to indicate the size of the clouds?
Is there anything in this image
to indicate the size of the clouds?
Frames of Reference Frames of Reference.
This image is scale- independent. It has
no frame of reference to indicate the
size of the clouds, such as an airplane,
or the horizon.
This image is scale- independent. It has
no frame of reference to indicate the
size of the clouds, such as an airplane,
or the horizon.
Magnification symmetry
requires a frame of
reference to determine size
because zooming in
reveals approximately the
same shape( s).
Magnification symmetry
requires a frame of
reference to determine size
because zooming in
reveals approximately the
same shape( s).
Taken by Ralph Kresge.
National Weather Service
(NOAA) photo library
Fractals are scale independent. Recall that small parts
aggregate to dominate overall shape.
Fractals are scale independent. Recall that small parts
aggregate to dominate overall shape..Within a fractal system, the smallest
scale is present in multitudinous
numbers. The medium scale has a
significant presence, with a
comparative handful of giants.
Within a fractal system, the smallest
scale is present in multitudinous
numbers. The medium scale has a
significant presence, with a
comparative handful of giants.
We see examples of this in bugs
and galaxies , also in stars within
galaxies. The small are
proliferate while the huge are few
and far between.
We see examples of this in bugs
and galaxies , also in stars within
galaxies. The small are
proliferate while the huge are few
and far between..Another image by Ralph Kresge
of the NOAA. This time a tree
provides a frame of reference,
and it looks like some birds are
flying around. Visit one of their
clouds photo libraries ..Examine exponential growth in
a geometric fractal: the Menger Sponge.
Examine exponential growth in
a geometric fractal: the Menger Sponge.
The Menger Sponge is part
of a series of fractals, in that
while it is Volumetric, it has
Length and Area analogs.
The Area analog, the
Sierpinski Carpet (seen in
image), is used by Fractal
Antenna Systems as an
antenna in cell phones. The
number of scales allows for a
wide range of receptions.
Image from Yale’s Fractal Antennas page.Length analog, the Cantor Set
Area analog, the Sierpinski Carpet
Volume analog, the Menger Sponge
Images courtesy of Paul Bourke..Colorful sponge is a
stage- 1: 20 cubes put
together to form
another cube with
center parts
removed,
corresponds with
tiniest openings in
large structure..Using the same pattern,
20 stage- 1’s can be put
together to form a stage-
2 with 20x20 = 20 2 = 400
cubes. A cube is being
emptied of its volume.
Watch how quickly this
exponential growth gets
out of control..At each stage, the edge-
length of the last cube is
reduced by 1/ 3, and
replicated 20 times. So
the Menger Sponge has
fractal dimension:
log (20)/ log (3) =
approximately 2.7268….20 stage- 3’s form a
stage- 4 with
20x20x20x20 =20 4 =
160,000 cubes.
The volume is
approaching zero and
the surface area is
approaching infinity..Pick any stage, and to see
how many cubes it is
made of, raise 20 to that
power: 20 stage . This is a
stage 5: 20 5 =3,200,000
The Menger Sponge
reveals self- similarity. It
has the same structure
when magnified..This is a stage- 6. It is made up of
20 6 = 64 million cubes.
There is no uncertainty about the
way it will grow or what it will look
like after any number of stages of
growth.
How far can this go?
As far as you want it to.
There is no reason to stop here.
Image courtesy of Paul Bourke..Fractals Across the Disciplines Fractals Across the Disciplines
A selection of topics from the Yale Fractal Geometry web
page A Panorama of Fractals and Their Uses :
Art & Nature Music
Architecture Nature & Fractals
Astronomy Physiology
Finance Poetry
History Psychology
Industry Social Sciences
Literature
(The categories all link to their respective pages.).Recapping the main fractal theme addressed in this
presentation:
Fractals operate under a Symmetry of
Magnification (called Dilatation or Dilation in
literature). Different types of fractals share a
common ground of parts that are similar to the
whole. Even though self- similar substructure
must technically be present all the way to infinity
for something to be called fractal, the concept of
fractility is loosened to apply to forms (esp.
natural) with only a handful of levels of
substructure present..The simplification of complexity leading to useful results that we
have been looking at is not unique to the field of fractals, it is a
theme that runs throughout mathematics, although the methods
of simplification vary.
“Mathematics is about making clean “simplified” concepts out
of things that we notice in the world around us. In the world
[staying with fractals as an example], when it is applicable we
make a clean concept by assuming the existence of self-
similarity— infinite levels of substructure— when there are only
a few, pushing beyond reality.” (Priscilla Greenwood,
Statistician and Mathematical Biologist at Arizona State
University)
Math “works” because these simplified systems “work”.
Mathematicians could well be called The Great Simplifiers..
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Top Web Results for "heuristic"
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View results from: Dictionary Thesaurus Encyclopedia All Reference the Web
Dictionary.com Unabridged (v 1.0.1) - Cite This Source
heuÂrisÂtic /hyÊŠË̂rɪs tɪk or, often, yÊŠ-/ Pronunciation Key - Show Spelled
Pronunciation[hyoo-ris-tik or, often, yoo-] Pronunciation Key - Show IPA Pronunciation
–adjective
1. serving to indicate or point out; stimulating interest as a means of furthering investigation.
2. encouraging a person to learn, discover, understand, or solve problems on his or her own, as by
experimenting, evaluating possible answers or solutions, or by trial and error: a heuristic teaching
method.
3. of, pertaining to, or based on experimentation, evaluation, or trial-and-error methods.
heuristic - Definitions from Dictionary.com
http://dictionary.reference.com/search?q=heuristic (1 of 4) [12/14/2006 10:00:56 PM]
heuristic.4. Computers, Mathematics. pertaining to a trial-and-error method of problem solving used when an
algorithmic approach is impractical.
–noun
5. a heuristic method of argument.
6. the study of heuristic procedure.
[Origin: 1815–25; < NL heuristicus, equiv. to Gk heur(Ã-skein) to find out, discover + L -isticus -istic
]
—Related forms
heuÂrisÂtiÂcalÂly, adverb
Dictionary.com Unabridged (v 1.0.1)
Based on the Random House Unabridged Dictionary, © Random House, Inc. 2006.
American Heritage Dictionary - Cite This Source
heuÂrisÂtic   (hy -rÄ-s'tÄ-k)  Pronunciation Key      
adj.  
Of or relating to a usually speculative formulation serving as a guide in the investigation or
solution of a problem: "The historian discovers the past by the judicious use of such a heuristic
device as the 'ideal type'" (Karl J. Weintraub).
1.
Of or constituting an educational method in which learning takes place through discoveries that
result from investigations made by the student.
2.
Computer Science Relating to or using a problem-solving technique in which the most appropriate
solution of several found by alternative methods is selected at successive stages of a program for
use in the next step of the program.
3.
n.  
A heuristic method or process. 1.
heuristics (used with a sing. verb) The study and application of heuristic methods and processes. 2.
[From Greek heuriskein, to find.]
heuÂris'tiÂcalÂly adv.
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Free On-line Dictionary of Computing - Cite This Source
heuristic - Definitions from Dictionary.com
http://dictionary.reference.com/search?q=heuristic (2 of 4) [12/14/2006 10:00:56 PM].heuristic 1. A rule of thumb, simplification, or educated guess that reduces or limits the search for
solutions in domains that are difficult and poorly understood. Unlike algorithms, heuristics do not
guarantee optimal, or even feasible, solutions and are often used with no theoretical guarantee.
2. approximation algorithm.
(2001-04-12)
The Free On-line Dictionary of Computing, © 1993-2006 Denis Howe
WordNet - Cite This Source
heuristic
adjective
1. of or relating to or using a general formulation that serves to guide investigation [ant: algorithmic]
noun
1. a commonsense rule (or set of rules) intended to increase the probability of solving some problem
WordNet® 2.1, © 2005 Princeton University
On-line Medical Dictionary - Cite This Source
heuristic
heuristic: in CancerWEB's On-line Medical Dictionary
On-line Medical Dictionary, © 1997-98 Academic Medical Publishing & CancerWEB
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heuristic - Definitions from Dictionary.com
http://dictionary.reference.com/search?q=heuristic (4 of 4) [12/14/2006 10:00:56 PM].Large-Scale Distribution of Galaxies
How big is the universe? What is our place in it?
These questions have fascinated people since the first observations of the night sky. Possible answers
have been fiercely debated, occasionally at the cost of lives.
Here we are interested in a more specific question: is the universe fractal, at least over some range of
distances?
First, how was the scalse of distances to the galaxies established?
Historical precedents for some aspects of a fractal galaxy distribution are found in the hierarchical
cosmologies of Kant, Swedenborg, and Lambert.
The cosmologies of Newton and of Einstein are homogeneous, not hierarchical.
In the early 80s, Mandelbrot presented fractal forgeries of the distribution of galaxies.
Small-scale fractal distributions.
Large-scale fractal distributions.
De Vaucouleurs law shows hierarchical cosmologies need have no preferred observer.
One model of galaxy formation is based on percolation.
Summarizing:
* There is general agreement about the existence of fractal structures out to about 50 million light years.
* Recent evidence suggests there is no upper cut-off of the fractal structure, and that the large-scale
distribution of mass in the universe has d = 2. Both the absence of an upper cut-off and the value of the
dimension are controversial.
* Isotropic fractal distributions are compatible with the equivalence of all observers.
* The Standard Cosmological Principle, that matter is distributed in a homogeneous and isotropic
fashion can be replaced by the Conditional Cosmological Principle, that matter is distributed in a
hierarchical and isotropic fashion.
Fractal Geometry
http://classes.yale.edu/fractals/Panorama/Astronomy/Galaxies/Galaxies.html
(1 of 2) [12/14/2006 10:08:21 PM].*
Observationally, hierarchical distribution implies long-range departures from Hubble's Law.
However, the distance scales to the galaxies is difficult to determine.
Twenty-five years ago, concerning the distance scale to the farthest galaxies one of my astronomy
teachers remarked,"If you get the right order of magnitude in the exponent, you're doing very well."
In the quarter-century that has passed since then, observations have improved significantly.
But still, there is much uncertainty.
The range of fractality in the universe is not likely to be resolved soon to everybody's satisfaction.
Fractal Geometry