7/28/2019 Mathed PDF Presentit Fractals
1/89
7/28/2019 Mathed PDF Presentit Fractals
2/89
Some
preliminarywords
7/28/2019 Mathed PDF Presentit Fractals
3/89
Some
preliminarywords
All underlined words arelinks.
I have placed a framearound every image in thepresentation that wasmade by or taken bysomeone else. Thecaptions underneath theframed images link to the
web pages they camefrom.
7/28/2019 Mathed PDF Presentit Fractals
4/89
Some
preliminarywords
All underlined words arelinks.
I have placed a framearound every image in thepresentation that wasmade by or taken bysomeone else. Thecaptions underneath theframed images link to the
web pages they camefrom.
Who isPeitgen?
http://sprott.physics.wisc.edu/fractals/chaos/http://sprott.physics.wisc.edu/fractals/chaos/http://sprott.physics.wisc.edu/fractals/chaos/http://sprott.physics.wisc.edu/fractals/chaos/7/28/2019 Mathed PDF Presentit Fractals
5/89
Fractals:
a Symmetry approach
http://www.math.umass.edu/~mconnors/fractal/fractal.htmlhttp://en.wikipedia.org/wiki/Symmetryhttp://en.wikipedia.org/wiki/Symmetryhttp://www.math.umass.edu/~mconnors/fractal/fractal.html7/28/2019 Mathed PDF Presentit Fractals
6/89
First discussed will be threecommon types of symmetry:
Reflectional (Line or Mirror)
Rotational (N-fold)
Translational
7/28/2019 Mathed PDF Presentit Fractals
7/89
First discussed will be threecommon types of symmetry:
Reflectional (Line or Mirror)
Rotational (N-fold)
Translational
and then: the Magnification(Dilatational a.k.a. Dilational)
symmetry of fractals.
http://dict.die.net/dilatation/http://dict.die.net/dilatation/7/28/2019 Mathed PDF Presentit Fractals
8/89
Reflectional (aka Line or Mirror) Symmetry
A shape exhibits reflectional symmetry if the shape can bebisected by a line L, one half of the shape removed, and themissing piece replaced by a reflection of the remaining pieceacross L, then the resulting combination is (approximately) thesame as the original.1
From An Intuitive Notion of Line Symmetry
http://classes.yale.edu/fractals/IntroToFrac/SelfSim/Symmetry/ReflSymmetry.htmlhttp://mathworld.wolfram.com/Bisection.htmlhttp://classes.yale.edu/fractals/IntroToFrac/SelfSim/Symmetry/ReflSymmetry.htmlhttp://regentsprep.org/Regents/math/symmetry/Lsymmet.htmhttp://classes.yale.edu/fractals/IntroToFrac/SelfSim/Symmetry/ReflSymmetry.htmlhttp://mathworld.wolfram.com/Bisection.htmlhttp://classes.yale.edu/fractals/IntroToFrac/SelfSim/Symmetry/ReflSymmetry.htmlhttp://regentsprep.org/Regents/math/symmetry/Lsymmet.htm7/28/2019 Mathed PDF Presentit Fractals
9/89
Reflectional (aka Line or Mirror) Symmetry
reflectional symmetryA shape exhibits if the shape can bebisected by a line L, one half of the shape removed, and themissing piece replaced by a reflection of the remaining pieceacross L, then the resulting combination is (approximately) thesame as the original.1
From An Intuitive Notion of Line Symmetry
In simpler words, if you
can fold it over and itmatches up, it has
reflectional symmetry.
This leaf, and the butterfly
caterpi llar sitt ing on it, areroughly symmetric. So are
human faces. Line
symmetry and mirror
symmetry are terms that
mean the same thing.
http://classes.yale.edu/fractals/IntroToFrac/SelfSim/Symmetry/ReflSymmetry.htmlhttp://mathworld.wolfram.com/Bisection.htmlhttp://classes.yale.edu/fractals/IntroToFrac/SelfSim/Symmetry/ReflSymmetry.htmlhttp://regentsprep.org/Regents/math/symmetry/Lsymmet.htmhttp://classes.yale.edu/fractals/IntroToFrac/SelfSim/Symmetry/ReflSymmetry.htmlhttp://mathworld.wolfram.com/Bisection.htmlhttp://classes.yale.edu/fractals/IntroToFrac/SelfSim/Symmetry/ReflSymmetry.htmlhttp://regentsprep.org/Regents/math/symmetry/Lsymmet.htm7/28/2019 Mathed PDF Presentit Fractals
10/89
The butterfly and the
children have lines ofreflection symmetry
where one sidemirrors the other.
Taken at the same time atthe Desert Botanical Gardens
Butterfly Pavilion, the littlebutterfly is a Painted Lady(Vanessa, cardui). Its host
plant is (Thistles, cirsium).
http://desertbotanical.org/index.aspx?pageID=554http://desertbotanical.org/index.aspx?pageID=554http://desertbotanical.org/index.aspx?pageID=554http://desertbotanical.org/index.aspx?pageID=554http://www.fs.fed.us/r4/htnf/resources/wildflowers/thistles.shtmlhttp://www.fs.fed.us/r4/htnf/resources/wildflowers/thistles.shtmlhttp://desertbotanical.org/index.aspx?pageID=554http://desertbotanical.org/index.aspx?pageID=554http://desertbotanical.org/index.aspx?pageID=5547/28/2019 Mathed PDF Presentit Fractals
11/89
Here is a link to a PowerPointpresentation created by Mrs. Gamache
using the collection of web pages by theAdrian Bruce and students of 6B.
This site lets you create your ownsymmetry patterns! Choose your typeand color, then start moving the mouseand clicking.
Both images have
a curved line ofsymmetry at the
edge of the water.
http://www.adrianbruce.com/Symmetry/power/LineSymmetry.ppthttp://www.adrianbruce.com/Symmetry/power/LineSymmetry.ppthttp://www.adrianbruce.com/Symmetry/power/LineSymmetry.ppthttp://www.adrianbruce.com/Symmetry/power/LineSymmetry.ppthttp://www.scienceu.com/geometry/handson/kali/index.cgi?group=w632http://www.scienceu.com/geometry/handson/kali/index.cgi?group=w632http://www.scienceu.com/geometry/handson/kali/index.cgi?group=w632http://www.scienceu.com/geometry/handson/kali/index.cgi?group=w632http://www.scienceu.com/geometry/handson/kali/index.cgi?group=w632http://www.scienceu.com/geometry/handson/kali/index.cgi?group=w632http://www.scienceu.com/geometry/handson/kali/index.cgi?group=w632http://www.adrianbruce.com/Symmetry/power/LineSymmetry.ppthttp://www.adrianbruce.com/Symmetry/power/LineSymmetry.ppthttp://www.adrianbruce.com/Symmetry/power/LineSymmetry.ppt7/28/2019 Mathed PDF Presentit Fractals
12/89
These blooms have 5 fold rotationalThese blooms have 5 fold rotational
7/28/2019 Mathed PDF Presentit Fractals
13/89
These blooms have 5-fold rotationalsymmetry. They can be turned
5 times to leave the figure unchanged
before starting over again.
These blooms have 5-fold rotationalsymmetry. They can be turned
5 times to leave the figure unchanged
before starting over again.
These blooms have 5 fold rotationalThese blooms have 5-fold rotational
7/28/2019 Mathed PDF Presentit Fractals
14/89
These blooms have 5-fold rotationalsymmetry. They can be turned
5 times to leave the figure unchanged
before starting over again.
These blooms have 5-fold rotationalsymmetry. They can be turned
5 times to leave the figure unchanged
before starting over again.
The butterfly is aSpicebush Swallowtail(Papillo, troilus).
The butterfly is aSpicebush Swallowtail(Papillo, troilus).
A pentagon also has 5-fold symmetry.A pentagon also has 5-fold symmetry.
7/28/2019 Mathed PDF Presentit Fractals
15/89
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.
7/28/2019 Mathed PDF Presentit Fractals
16/89
The tiny blooms have 4-fold
symmetry. Question: doesthe spherical bloom they sit
on have n-fold symmetry?
This flower has21-fold rotational
symmetry.
5-fold or 6-foldsymmetry here
7/28/2019 Mathed PDF Presentit Fractals
17/89
T l ti l S t
7/28/2019 Mathed PDF Presentit Fractals
18/89
The bricks in the image have
translational symmetry.
The bricks in the image have
translational symmetry.
Translational Symmetry
A shape exhibits translational symmetry if displacementin some direction - horizontal or vertical, for example -returns the shape to (approximately) its original
configuration.3
Also, the image of the bricks willhave translation symmetry when
sliding, provided there is norotation during the move.
Also, the image of the bricks willhave translation symmetry when
sliding, provided there is norotation during the move.
Orientation must be preservedwhile translating.
Orientation must be preservedwhile translating.
M ifi ti (Dil t ti l) S t
http://classes.yale.edu/fractals/IntroToFrac/SelfSim/Symmetry/TransSymmetry.htmlhttp://classes.yale.edu/fractals/IntroToFrac/SelfSim/Symmetry/TransSymmetry.htmlhttp://classes.yale.edu/fractals/IntroToFrac/SelfSim/Symmetry/TransSymmetry.htmlhttp://classes.yale.edu/fractals/IntroToFrac/SelfSim/Symmetry/TransSymmetry.html7/28/2019 Mathed PDF Presentit Fractals
19/89
Magnification (Dilatational) Symmetry
symmetry under magnificationLess familiar is :zooming in on an object leaves the shape
approximately unaltered.4
Zooming in on a fractal objectleaves the shape
approximately unaltered.
Fractals exhibit magnification symmetry.
http://classes.yale.edu/fractals/IntroToFrac/SelfSim/Symmetry/Symmetry.htmlhttp://classes.yale.edu/fractals/IntroToFrac/SelfSim/Symmetry/Symmetry.htmlhttp://classes.yale.edu/fractals/IntroToFrac/SelfSim/Symmetry/Symmetry.htmlhttp://classes.yale.edu/fractals/IntroToFrac/SelfSim/Symmetry/Symmetry.html7/28/2019 Mathed PDF Presentit Fractals
20/89
7/28/2019 Mathed PDF Presentit Fractals
21/89
Natural Fractals
Multifractals
Chaos
Natural fractals have a limited number ofstages of growth, and the growthbetween stages shows variation. They
have connections to Multifractals andChaos theory.
http://www.umanitoba.ca/faculties/science/botany/labs/ecology/fractals/applications.htmlhttp://www.matpack.de/Info/Mathematics/Multifractals.htmlhttp://www.khwarzimic.org/activities/chaos-intro.htmlhttp://www.khwarzimic.org/activities/chaos-intro.htmlhttp://www.matpack.de/Info/Mathematics/Multifractals.htmlhttp://www.umanitoba.ca/faculties/science/botany/labs/ecology/fractals/applications.html7/28/2019 Mathed PDF Presentit Fractals
22/89
Fractal geometry was designedto handle shapes that appear
complicated, but with complexityarranged in some hierarchical
fashion. So at a minimum,fractals must have some
substructure.
(Michael Frame, Yale University)
This is a Sweet Acacia (Acacia, smallii) tree. Its
http://www.enature.com/flashcard/show_flash_card.asp?recordNumber=TS0252http://www.enature.com/flashcard/show_flash_card.asp?recordNumber=TS02527/28/2019 Mathed PDF Presentit Fractals
23/89
This is a Sweet Acacia (Acacia, smallii) tree. Itsunbloomed flower appears to be a sphere madeup ofsmaller-scale spheres, but a closer look
reveals the little buds to be cylindrical.
Outer foliage (leaves and petals) onOuter foliage (leaves and petals) on
http://www.enature.com/flashcard/show_flash_card.asp?recordNumber=TS0252http://math.bu.edu/DYSYS/chaos-game/node5.htmlhttp://math.bu.edu/DYSYS/chaos-game/node5.htmlhttp://www.enature.com/flashcard/show_flash_card.asp?recordNumber=TS02527/28/2019 Mathed PDF Presentit Fractals
24/89
Ou e o age (ea es a d pe as) oplants are usually terminal organs,
and are non-reproductive. (There are
exceptions, though, like the entire fernfamily.) Root Gorelick of the ASU
biology department explains:
Ou e o age (ea es a d pe as) oplants are usually terminal organs,
and are non-reproductive. (There are
exceptions, though, like the entire fernfamily.) Root Gorelick of the ASU
biology department explains:
Leaves are terminal
organs, hence don'treproduce miniature
copies of themselves as
do stems, roots, andmany reproductive
structures. Therefore, I
expect leaves to be leastfractal of these organs.
(Root Gorelick)
Leaves are terminal
organs, hence don'treproduce miniature
copies of themselves as
do stems, roots, andmany reproductive
structures. Therefore, I
expect leaves to be least
fractal of these organs.
(Root Gorelick)
The bloom of the SweetA i f h
http://www.biology-online.org/dictionary/terminalhttp://math.bu.edu/DYSYS/dysys.htmlhttp://www.biology-online.org/dictionary/terminalhttp://math.bu.edu/DYSYS/dysys.htmlhttp://math.bu.edu/DYSYS/dysys.htmlhttp://math.bu.edu/DYSYS/dysys.htmlhttp://www.biology-online.org/dictionary/terminalhttp://www.biology-online.org/dictionary/terminal7/28/2019 Mathed PDF Presentit Fractals
25/89
Image from Paul Bourkes
Acacia tree from theprevious slide isreproductive, but that is
not enough to be fractal.Having a fractalsubstructure requiresthe same or a highly
similar shape betweena minimum of 3 stagesof growth (there existssome disagreement on
this, Im going with Yale,a good source).
The entire fernfamily reveals self-
similarity:
successive stages
of growth thatclosely resemble
earlier stages. Self-Similaritypage.
Butterfly is aButterfly is a
http://astronomy.swin.edu.au/~pbourke/fractals/selfsimilar/index.htmlhttp://astronomy.swin.edu.au/~pbourke/fractals/selfsimilar/index.html7/28/2019 Mathed PDF Presentit Fractals
26/89
11
22
33
There are severalstages of scaling
not visible in thisimage, see nextimage where the
top of the plant isvisible.
There are severalstages of scaling
not visible in thisimage, see nextimage where the
top of the plant isvisible.
Butterfly is aZebra Swallowtail
(Eurytides, marcellus)
Butterfly is aZebra Swallowtail
(Eurytides, marcellus)
The butterfly is aThe butterfly is aCl dl Gi S l h
7/28/2019 Mathed PDF Presentit Fractals
27/89
The scaled
branching extendsupward throughoutthe plant. A small
branch, if magnified,would look like a
larger branch.
The scaled
branching extendsupward throughoutthe plant. A small
branch, if magnified,would look like a
larger branch.
Cloudless Giant Sulpher(Phoebis, sennae)
Cloudless Giant Sulpher(Phoebis, sennae)
7/28/2019 Mathed PDF Presentit Fractals
28/89
Also notice how entiresections of the plant
resemble each other ondifferent scales. Thestructure of smaller
sections dictates theshape of larger sections.
Also notice how entiresections of the plant
resemble each other ondifferent scales. Thestructure of smaller
sections dictates theshape of larger sections.
7/28/2019 Mathed PDF Presentit Fractals
29/89
Fractal branching isFractal branching is
7/28/2019 Mathed PDF Presentit Fractals
30/89
captured in shadowbelow. From thisview, again noticehow the partsresemble thewhole.
captured in shadowbelow. From thisview, again noticehow the partsresemble thewhole.
A Painted Lady is present!A Painted Lady is present!
7/28/2019 Mathed PDF Presentit Fractals
31/89
The plant in the
previous slidesresembles thiscomputer-
generated binaryfractal tree.
Image by Don West
http://classes.yale.edu/http://classes.yale.edu/fractals/FracTrees/welcome.htmlhttp://faculty.plattsburgh.edu/don.west/trees/http://faculty.plattsburgh.edu/don.west/trees/http://classes.yale.edu/http://classes.yale.edu/fractals/FracTrees/welcome.html7/28/2019 Mathed PDF Presentit Fractals
32/89
With fractals, the structure behind small sections
dictates overall shape.
We saw empirical verification of this in the previousexample. We saw that the bigger shapes wereaggregations of the smaller shapes that made them
up. This is also true of clouds, mountains, oceanwaves, 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. Thisis why fractal equations tend to be simple.
Tremendous complexity can result fromiterating
simple patterns.
http://dict.die.net/empirical/http://dict.die.net/aggregation/http://mathforum.org/library/drmath/view/54531.htmlhttp://mathforum.org/library/drmath/view/54531.htmlhttp://dict.die.net/aggregation/http://dict.die.net/empirical/7/28/2019 Mathed PDF Presentit Fractals
33/89
Image courtesy of Paul Bourke
7/28/2019 Mathed PDF Presentit Fractals
34/89
Beware of complacency! Many aspects ofnature are fractal, while others are not. Ofthose aspects that have an embedded
fractal structure, their fractal aspect onlydescribes properties of shape andcomplexity. Read this Word of Caution
from Nonlinear Geoscience: Fractals.The randomness referred to in theirstatement is given consideration in
Multifractal theory, which has ties toChaos theory and Nonlinear Dynamics.
Geometric Fractals
http://journal-ci.csse.monash.edu.au/ci/vol06/jelinek/jelinek.htmlhttp://ems.gphys.unc.edu/nonlinear/fractals/geometry.htmlhttp://www.physics.mcgill.ca/~gang/multifrac/clouds/clouds.htmhttp://www.geocities.com/Athens/6398/chaos.htmhttp://amath.colorado.edu/faculty/jdm/faq-%5b2%5d.htmlhttp://amath.colorado.edu/faculty/jdm/faq-%5b2%5d.htmlhttp://www.geocities.com/Athens/6398/chaos.htmhttp://www.physics.mcgill.ca/~gang/multifrac/clouds/clouds.htmhttp://ems.gphys.unc.edu/nonlinear/fractals/geometry.htmlhttp://journal-ci.csse.monash.edu.au/ci/vol06/jelinek/jelinek.html7/28/2019 Mathed PDF Presentit Fractals
35/89
Geometric Fractals
Geometric fractals might be compared to objects/systems
in a vacuumin physics, in that abnormalities arenonexistent in them. They are, as their name suggests,geometric constructs, perfect (Ideal) systems with nointernal deviations or potential changes from outside
influences (apart from human error in constructing them).Geometric fractals have no randomness, and noconnections to Chaos theory.
Im not including Complex fractals in this category such asthe Mandelbrot Set, J ulia Sets, or any fractal that lies in the
complex plane. Complex fractals are highlighted later.
The Sierpinski Tetrahedron
http://csep10.phys.utk.edu/guidry/violence/lightspeed.htmlhttp://csep10.phys.utk.edu/guidry/violence/lightspeed.html7/28/2019 Mathed PDF Presentit Fractals
36/89
pFractal type: Geometric
The number of tetrahedra is increasing in powers of 4The edge-length of the tetrahedra is decreasing in powers of
The volume of Sierpinskis tetrahedron is decreasing in powers of
Image created using MathCad by Byrge Birkeland of Agder University College, Kristiansand, Norway
To consider this fractal, it is important to know something about
http://home.hia.no/~byrgeb/imageslinks.htmhttp://home.hia.no/~byrgeb/imageslinks.htmhttp://en.wikipedia.org/wiki/Tetrahedron7/28/2019 Mathed PDF Presentit Fractals
37/89
a tetrahedron.
To consider this fractal, it is important to know something abouth d
http://en.wikipedia.org/wiki/Tetrahedronhttp://en.wikipedia.org/wiki/Tetrahedronhttp://en.wikipedia.org/wiki/Tetrahedronhttp://en.wikipedia.org/wiki/Tetrahedron7/28/2019 Mathed PDF Presentit Fractals
38/89
a tetrahedron.
- Start with an equilateral triangle.
To consider this fractal, it is important to know something aboutt t h d
http://en.wikipedia.org/wiki/Tetrahedronhttp://en.wikipedia.org/wiki/Tetrahedronhttp://en.wikipedia.org/wiki/Tetrahedronhttp://en.wikipedia.org/wiki/Tetrahedron7/28/2019 Mathed PDF Presentit Fractals
39/89
a tetrahedron.
- Start with an equilateral triangle.- Divide it into 4 equilateral triangles by marking the midpointsof all three sides and drawing lines to connect the midpoints.
http://en.wikipedia.org/wiki/Tetrahedronhttp://www.mathwords.com/e/equilateral_triangle.htmhttp://www.mathwords.com/e/equilateral_triangle.htmhttp://en.wikipedia.org/wiki/Tetrahedron7/28/2019 Mathed PDF Presentit Fractals
40/89
7/28/2019 Mathed PDF Presentit Fractals
41/89
To construct a stage-1:
Reduce it by afactor of 1/2
Start with a regular tetrahedron.
It is called the stage-0 in theSierpinski tetrahedron fractalfamily.
Replicate (4 are needed). Thetetrahedra are kept transparent onthis slide to reinforce that these
are tetrahedra and not triangles.
Rebuild the 4 stage-0s into a
http://www.emints.org/ethemes/resources/S00001378.shtmlhttp://www.mathwords.com/t/tetrahedron.htmhttp://www.emints.org/ethemes/resources/S00001378.shtmlhttp://www.mathwords.com/t/tetrahedron.htm7/28/2019 Mathed PDF Presentit Fractals
42/89
Rebuild the 4 stage-0 s into astage-1 Sierpinski tetrahedron.
The line is a handy frame ofreference for construction.
Rebuild the 4 stage-0s into a
7/28/2019 Mathed PDF Presentit Fractals
43/89
Rebuild the 4 stage 0 s into astage-1 Sierpinski tetrahedron.
The line is a handy frame ofreference for construction.
Now repeat this process againand again.
Revisiting the earlier image, notice that each tetrahedron isl d b 4 t t h d i th t t
7/28/2019 Mathed PDF Presentit Fractals
44/89
replaced by 4 tetrahedra in the next stage.
Determine the stage by counting the number ofsizes of openings,
the stage-1 has one size of opening, the stage-2 two sizes ofopenings, etc
Revisiting the earlier image, notice that each tetrahedron isl d b 4 t t h d i th t t
7/28/2019 Mathed PDF Presentit Fractals
45/89
replaced by 4 tetrahedra in the next stage.
Determine the stage by counting the number ofsizes of openings,
the stage-1 has one size of opening, the stage-2 two sizes ofopenings, etc
Revisiting the earlier image, notice that each tetrahedron isreplaced by 4 tetrahedra in the next stage
7/28/2019 Mathed PDF Presentit Fractals
46/89
replaced by 4 tetrahedra in the next stage.
Determine the stage by counting the number ofsizes of openings,
the stage-1 has one size of opening, the stage-2 two sizes ofopenings, etc
Revisiting the earlier image, notice that each tetrahedron isreplaced by 4 tetrahedra in the next stage
7/28/2019 Mathed PDF Presentit Fractals
47/89
replaced by 4 tetrahedra in the next stage.
Determine the stage by counting the number ofsizes of openings,
the stage-1 has one size of opening, the stage-2 two sizes ofopenings, etc
Revisiting the earlier image, notice that each tetrahedron isreplaced by 4 tetrahedra in the next stage
7/28/2019 Mathed PDF Presentit Fractals
48/89
replaced by 4 tetrahedra in the next stage.
Determine the stage by counting the number ofsizes of openings,
the stage-1 has one size of opening, the stage-2 two sizes ofopenings, etc
The Sierpinski Triangle: grows in Powers of 3
http://math.rice.edu/~lanius/fractals/http://math.rice.edu/~lanius/fractals/7/28/2019 Mathed PDF Presentit Fractals
49/89
Reduceby again
Replicate& Rebuild
Reduceby
Replicate& Rebuild
Notice how each trianglebecomes three triangles
in the next stage.
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.
The stage can be
determined by thenumber of differentsizes of openings.
123
4
With this fractal, it is
surface area instead ofvolume that is decreasingat each stage.
The face of a Sierpinski
tetrahedron is a same-stage Sierpinski triangle.
http://math.bu.edu/DYSYS/chaos-game/node6.htmlhttp://math.bu.edu/DYSYS/chaos-game/node6.html7/28/2019 Mathed PDF Presentit Fractals
50/89
A quick word about the Chaos Game. It is a neat game thatuses a randomprocess to create the Sierpinski Triangle You
http://math.bu.edu/DYSYS/applets/chaos-game.htmlhttp://math.bu.edu/DYSYS/applets/chaos-game.html7/28/2019 Mathed PDF Presentit Fractals
51/89
uses a random process to create the Sierpinski Triangle. You
can play it at the above link.
The Sierpinski triangle and other Geometric fractals can becreated using random processes like the Chaos Game; however,as stated in this link, the fractals created by such games are notchaotic. Geometric fractals typically map into the Cartesian
and have no connections with mathematical Chaos.plane
A quick word about the Chaos Game. It is a neat game thatuses a randomprocess to create the Sierpinski Triangle You
http://www.geocities.com/ResearchTriangle/System/8956/Fractal/intro.htmhttp://mathworld.wolfram.com/CartesianPlane.htmlhttp://mathworld.wolfram.com/CartesianPlane.htmlhttp://mathworld.wolfram.com/CartesianPlane.htmlhttp://mathworld.wolfram.com/CartesianPlane.htmlhttp://mathworld.wolfram.com/CartesianPlane.htmlhttp://mathworld.wolfram.com/CartesianPlane.htmlhttp://mathworld.wolfram.com/CartesianPlane.htmlhttp://mathworld.wolfram.com/CartesianPlane.htmlhttp://www.geocities.com/ResearchTriangle/System/8956/Fractal/intro.htmhttp://math.bu.edu/DYSYS/applets/chaos-game.htmlhttp://math.bu.edu/DYSYS/applets/chaos-game.html7/28/2019 Mathed PDF Presentit Fractals
52/89
uses a random process to create the Sierpinski Triangle. You
can play it at the above link.
The Sierpinski triangle and other Geometric fractals can becreated using random processes like the Chaos Game; however,as stated in this link, the fractals created by such games are notchaotic. Geometric fractals typically map into the Cartesian
plane and have no connections with mathematical Chaos.
The Chaos Game was named more than 20 years ago whenthere was no real definition of chaos. One qualification: I donot include the Mandelbrot Set and J ulia SetsComplex fractals
that lie in the complex plane and have significant connections toChaosin the Geometric fractals category.
It is easier to talk about fractal
http://www.geocities.com/ResearchTriangle/System/8956/Fractal/intro.htmhttp://mathworld.wolfram.com/CartesianPlane.htmlhttp://mathworld.wolfram.com/CartesianPlane.htmlhttp://mathworld.wolfram.com/CartesianPlane.htmlhttp://mathworld.wolfram.com/CartesianPlane.htmlhttp://mathworld.wolfram.com/CartesianPlane.htmlhttp://mathworld.wolfram.com/ComplexPlane.htmlhttp://mathworld.wolfram.com/ComplexPlane.htmlhttp://mathworld.wolfram.com/CartesianPlane.htmlhttp://mathworld.wolfram.com/CartesianPlane.htmlhttp://mathworld.wolfram.com/CartesianPlane.htmlhttp://www.geocities.com/ResearchTriangle/System/8956/Fractal/intro.htmhttp://www.cs.cornell.edu/Courses/cs312/2000fa/handouts/fractals.htmlhttp://www.cs.cornell.edu/Courses/cs312/2000fa/handouts/fractals.htmlhttp://www.cs.cornell.edu/Courses/cs312/2000fa/handouts/fractals.htmlhttp://www.cs.cornell.edu/Courses/cs312/2000fa/handouts/fractals.htmlhttp://www.cs.cornell.edu/Courses/cs312/2000fa/handouts/fractals.html7/28/2019 Mathed PDF Presentit Fractals
53/89
dimension in Geometric fractalsbecause the property is more exact
in them, even though it applies toall types of fractal objects and
systems.
Even though most fractals havenon-integer dimension, there are
exceptions:
For exactly self-similar shapes made of N copies,each scaled by a factor of r, the dimension is
http://www.cs.cornell.edu/Courses/cs312/2000fa/handouts/fractals.htmlhttp://www.cs.cornell.edu/Courses/cs312/2000fa/handouts/fractals.htmlhttp://www.cs.cornell.edu/Courses/cs312/2000fa/handouts/fractals.htmlhttp://www.cs.cornell.edu/Courses/cs312/2000fa/handouts/fractals.htmlhttp://www.cs.cornell.edu/Courses/cs312/2000fa/handouts/fractals.html7/28/2019 Mathed PDF Presentit Fractals
54/89
each scaled by a factor of r, the dimension is
Log(N)/Log(1/r)
For exactly self-similar shapes made of N copies,each scaled by a factor of r, the dimension is
7/28/2019 Mathed PDF Presentit Fractals
55/89
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
http://classes.yale.edu/fractals/Labs/SierpTetraLab/SierpTetraLab.htmlhttp://classes.yale.edu/fractals/Labs/SierpTetraLab/SierpTetraLab.html7/28/2019 Mathed PDF Presentit Fractals
56/89
Contrast this with the Sierpinski triangle, made ofN = 3 copies each scaled by a factor of r =
7/28/2019 Mathed PDF Presentit Fractals
57/89
N = 3 copies, each scaled by a factor of r = .
Its dimension is
Log(3)/Log(2) ~= 1.58496..
Contrast this with the Sierpinski triangle, made ofN = 3 copies each scaled by a factor of r =
7/28/2019 Mathed PDF Presentit Fractals
58/89
N 3 copies, each scaled by a factor of r .
Its dimension is
Log(3)/Log(2) ~= 1.58496..
The Sierpinski triangle has fractional dimension,
more typical of fractals.
Contrast this with the Sierpinski triangle, made ofN = 3 copies, each scaled by a factor of r = .
http://math.rice.edu/~lanius/fractals/dim.htmlhttp://math.rice.edu/~lanius/fractals/dim.html7/28/2019 Mathed PDF Presentit Fractals
59/89
N 3 copies, each scaled by a factor of r .
Its dimension is
Log(3)/Log(2) ~= 1.58496..
The Sierpinski triangle has fractional dimension,
more typical of fractals.
The exact answer is Log(3)/Log(2). Theapproximate answer (often more useful in realworld applications) is the decimal approximation1.58496
http://math.rice.edu/~lanius/fractals/dim.htmlhttp://www.trottermath.net/probsolv/toplug.htmlhttp://www.trottermath.net/probsolv/toplug.htmlhttp://math.rice.edu/~lanius/fractals/dim.html7/28/2019 Mathed PDF Presentit Fractals
60/89
Self-similarity: this is a bigidea, and it only trulyapplies to geometric
fractals; however, it isused as a concept to talk
about all types of fractals.
Something is self-similar when every
http://math.rice.edu/~lanius/fractals/selfsim.htmlhttp://math.rice.edu/~lanius/fractals/selfsim.html7/28/2019 Mathed PDF Presentit Fractals
61/89
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.Only geometric fractals have a chance
of arriving at this state; however, inorder to speak about fractals
generally, one must embrace theconcept of self-similarity in a broad
way.
hCh
(All categories listed on this slidelink to relevant websites.)
(All categories listed on this slidelink to relevant websites.)
http://classes.yale.edu/fractals/Chaos/ChaosIntro/ChaosIntro.htmlhttp://classes.yale.edu/fractals/Chaos/ChaosIntro/ChaosIntro.htmlhttp://classes.yale.edu/fractals/Chaos/ChaosIntro/ChaosIntro.html7/28/2019 Mathed PDF Presentit Fractals
62/89
ChaosChaos
MultifractalsMultifractals
Random FractalsRandom Fractals
Mandelbrot SetMandelbrot Set
J ulia SetsJ ulia Sets
Image courtesy of J im Muth
Complex FractalsComplex FractalsMandelbrot discusses fractalsMandelbrot discusses fractals
ChaosChaos
http://classes.yale.edu/fractals/Chaos/ChaosIntro/ChaosIntro.htmlhttp://classes.yale.edu/fractals/Chaos/ChaosIntro/ChaosIntro.htmlhttp://classes.yale.edu/fractals/MultiFractals/welcome.htmlhttp://classes.yale.edu/fractals/MultiFractals/welcome.htmlhttp://classes.yale.edu/fractals/RandFrac/welcome.htmlhttp://classes.yale.edu/fractals/RandFrac/welcome.htmlhttp://www.ddewey.net/mandelbrot/noad.htmlhttp://www.ddewey.net/mandelbrot/noad.htmlhttp://www.ibiblio.org/e-notes/MSet/Period.htmhttp://www.ibiblio.org/e-notes/MSet/Period.htmhttp://www.geom.uiuc.edu/~zietlow/defp1.htmlhttp://www.geom.uiuc.edu/~zietlow/defp1.htmlhttp://www.yale.edu/opa/v31.n20/story6.htmlhttp://www.yale.edu/opa/v31.n20/story6.htmlhttp://classes.yale.edu/fractals/Chaos/ChaosIntro/ChaosIntro.htmlhttp://classes.yale.edu/fractals/Chaos/ChaosIntro/ChaosIntro.htmlhttp://www.geom.uiuc.edu/~zietlow/defp1.htmlhttp://www.yale.edu/opa/v31.n20/story6.htmlhttp://classes.yale.edu/fractals/Chaos/ChaosIntro/ChaosIntro.htmlhttp://classes.yale.edu/fractals/Chaos/ChaosIntro/ChaosIntro.htmlhttp://classes.yale.edu/fractals/MultiFractals/welcome.htmlhttp://www.ibiblio.org/e-notes/MSet/Period.htmhttp://www.ddewey.net/mandelbrot/noad.htmlhttp://classes.yale.edu/fractals/RandFrac/welcome.html7/28/2019 Mathed PDF Presentit Fractals
63/89
This image is scale-independent. It hasno frame of reference to indicate the
This image is scale-independent. It hasno frame of reference to indicate thei f th l d h i l
7/28/2019 Mathed PDF Presentit Fractals
64/89
size of the clouds, such as an airplane,or the horizon.
size of the clouds, such as an airplane,or the horizon.
Magnification symmetryrequires a frame ofreference to determine size
because zooming inreveals approximately thesame shape(s).
Magnification symmetryrequires a frame ofreference to determine size
because zooming inreveals approximately thesame shape(s).
Taken by Ralph Kresge.Click inside frame to visitNational Weather Service
(NOAA) photo library
Fractals are scale independent. Recall that small partsaggregate to dominate overall shape.
Fractals are scale independent. Recall that small partsaggregate to dominate overall shape.
Within a fractal system, the smallestscale is present in multitudinous
7/28/2019 Mathed PDF Presentit Fractals
65/89
scale is present in multitudinous
numbers. The medium scale has asignificant presence, with acomparative handful of giants.
We see examples of this in bugsand galaxies, also in stars within
galaxies. The small areproliferate while the huge are fewand far between.
http://www.umanitoba.ca/faculties/science/botany/labs/ecology/fractals/applications.htmlhttp://classes.yale.edu/fractals/Panorama/Astronomy/Galaxies/Galaxies.htmlhttp://classes.yale.edu/fractals/Panorama/Astronomy/Galaxies/Galaxies.htmlhttp://www.umanitoba.ca/faculties/science/botany/labs/ecology/fractals/applications.html7/28/2019 Mathed PDF Presentit Fractals
66/89
Butterfly Wing Branching PatternsButterfly Wing Branching Patterns
7/28/2019 Mathed PDF Presentit Fractals
67/89
A closer view would reveal fractal branching in the veins of the wings of this ZebraSwallowtail. Almost all branching in nature is fractal. Leaf veins are an another instance
of fractal branching. And rivers. And ourcirculatory system. And lightning.
From the Bugbios website:the Caligo genus
http://www.stat.rice.edu/~riedi/UCDavisHemoglobin/fractal.htmlhttp://www.photovault.com/Link/Orders/Flora/Leaves/OFLVolume01.htmlhttp://classes.yale.edu/fractals/Panorama/Nature/Rivers/Rivers.htmlhttp://classes.yale.edu/fractals/Panorama/Biology/Physiology/Physiology.htmlhttp://www.museum.vic.gov.au/scidiscovery/lightning/shapes.asphttp://www.museum.vic.gov.au/scidiscovery/lightning/shapes.asphttp://classes.yale.edu/fractals/Panorama/Biology/Physiology/Physiology.htmlhttp://classes.yale.edu/fractals/Panorama/Nature/Rivers/Rivers.htmlhttp://www.photovault.com/Link/Orders/Flora/Leaves/OFLVolume01.htmlhttp://www.stat.rice.edu/~riedi/UCDavisHemoglobin/fractal.htmlhttp://www.insects.org/class/patterns/index.htmlhttp://www.insects.org/class/patterns/index.htmlhttp://www.insects.org/class/patterns/index.htmlhttp://www.insects.org/class/patterns/index.htmlhttp://www.insects.org/class/patterns/index.htmlhttp://www.insects.org/class/patterns/index.htmlhttp://www.insects.org/class/patterns/index.htmlhttp://www.insects.org/class/patterns/index.html7/28/2019 Mathed PDF Presentit Fractals
68/89
Images from Bugbios
by Dexter Sear
From the Bugbios website:the Caligo genus
http://www.insects.org/class/patterns/index.htmlhttp://www.insects.org/class/patterns/index.htmlhttp://www.insects.org/class/patterns/index.htmlhttp://www.insects.org/class/patterns/caligo.htmlhttp://www.insects.org/class/patterns/caligo.htmlhttp://www.insects.org/class/patterns/index.htmlhttp://www.insects.org/class/patterns/index.htmlhttp://www.insects.org/class/patterns/index.htmlhttp://www.insects.org/class/patterns/index.htmlhttp://www.insects.org/class/patterns/index.htmlhttp://www.insects.org/class/patterns/index.htmlhttp://www.insects.org/class/patterns/index.htmlhttp://www.insects.org/class/patterns/index.htmlhttp://www.insects.org/class/patterns/index.html7/28/2019 Mathed PDF Presentit Fractals
69/89
Images from Bugbios
by Dexter Sear
This almost looks like a computer-generated fractalimage, but it isnt. It is a collage of Caligo butterfly wings.Notice the scaled rippling pattern in isolated wingsections. It is a fairly safe conjecture that these ripplesmight have fractal properties. This isnt to suggest that allcolorful butterfly wing patterns would reveal fractal scaling.
http://www.insects.org/class/patterns/index.htmlhttp://www.insects.org/class/patterns/index.htmlhttp://www.insects.org/class/patterns/index.htmlhttp://www.insects.org/class/patterns/caligo.htmlhttp://www.insects.org/class/patterns/caligo.htmlhttp://www.insects.org/class/patterns/index.html7/28/2019 Mathed PDF Presentit Fractals
70/89
http://astronomy.swin.edu.au/~pbourke/fractals/diaxialplane/index.html7/28/2019 Mathed PDF Presentit Fractals
71/89
Length analog, the Cantor Set
Area analog, the Sierpinski Carpet
Volume analog, the Menger Sponge
Images courtesy ofPaul Bourke
http://en.wikipedia.org/wiki/Cantor_sethttp://ecademy.agnesscott.edu/~lriddle/ifs/carpet/carpet.htmhttp://planetmath.org/encyclopedia/MengerSponge.htmlhttp://astronomy.swin.edu.au/~pbourke/fractals/http://astronomy.swin.edu.au/~pbourke/fractals/carpet/index.htmlhttp://astronomy.swin.edu.au/~pbourke/fractals/apollony/http://astronomy.swin.edu.au/~pbourke/fractals/diaxialplane/index.htmlhttp://ecademy.agnesscott.edu/~lriddle/ifs/carpet/carpet.htmhttp://astronomy.swin.edu.au/~pbourke/fractals/http://planetmath.org/encyclopedia/MengerSponge.htmlhttp://en.wikipedia.org/wiki/Cantor_set7/28/2019 Mathed PDF Presentit Fractals
72/89
7/28/2019 Mathed PDF Presentit Fractals
73/89
Using the same pattern,
20 stage-1s can be puttogether to form a stage-2 with 20x20 = 202 = 400cubes. A cube is beingemptied of its volume.Watch how quickly thisexponential growth getsout of control.
7/28/2019 Mathed PDF Presentit Fractals
74/89
At each stage, the edge-length of the last cube isreduced by 1/3, andreplicated 20 times. Sothe Menger sponge has
fractal dimension:
log (20)/log (3) =approximately 2.7268
7/28/2019 Mathed PDF Presentit Fractals
75/89
7/28/2019 Mathed PDF Presentit Fractals
76/89
How far can this go?
As far as you want it to.
There is no reason to stop here.
7/28/2019 Mathed PDF Presentit Fractals
77/89
This is a stage-6. It is made up of
26 = 64 million cubes.
There is no uncertainty about the
way it wil l grow or what it wil l look
like after any number of stages ofgrowth.
p
Image courtesy ofPaul Bourke.
Fractals Across the DisciplinesFractals Across the Disciplines
http://astronomy.swin.edu.au/~pbourkehttp://astronomy.swin.edu.au/~pbourke7/28/2019 Mathed PDF Presentit Fractals
78/89
a handful of topics from the Yale Fractal Geometryweb pageA Panorama of Fractals and Their Uses:
Art & Nature Music
Architecture Nature & Fractals
Astronomy PhysiologyFinance Poetry
HistoryPsychology
Industry Social Sciences
Literature
(The categories all link to their respective pages.)
http://classes.yale.edu/fractals/http://classes.yale.edu/fractals/Panorama/http://classes.yale.edu/fractals/Panorama/Art/ArtAndNature/ArtAndNature.htmlhttp://classes.yale.edu/fractals/Panorama/Music/Mus/Music.htmlhttp://classes.yale.edu/fractals/Panorama/Architecture/Arch/Arch.htmlhttp://classes.yale.edu/fractals/Panorama/Nature/NAtFracGallery/NatFracGallery.htmlhttp://classes.yale.edu/fractals/Panorama/Astronomy/Galaxies/Galaxies.htmlhttp://classes.yale.edu/fractals/Panorama/Biology/Physiology/Physiology.htmlhttp://classes.yale.edu/fractals/Panorama/SocialSciences/FinRisk/FinRisk.htmlhttp://classes.yale.edu/fractals/Panorama/Literature/Poetry/Poetry.htmlhttp://classes.yale.edu/fractals/Panorama/SocialSciences/History/History.htmlhttp://classes.yale.edu/fractals/Panorama/SocialSciences/Psychology/Psychology.htmlhttp://classes.yale.edu/fractals/Panorama/ManuFractals/ManuFract/ManuFractals.htmlhttp://classes.yale.edu/fractals/Panorama/SocialSciences/SocSci/SocSci.htmlhttp://classes.yale.edu/fractals/Panorama/Literature/Lit/Literature.htmlhttp://classes.yale.edu/fractals/Panorama/SocialSciences/SocSci/SocSci.htmlhttp://classes.yale.edu/fractals/Panorama/SocialSciences/Psychology/Psychology.htmlhttp://classes.yale.edu/fractals/Panorama/Literature/Poetry/Poetry.htmlhttp://classes.yale.edu/fractals/Panorama/Biology/Physiology/Physiology.htmlhttp://classes.yale.edu/fractals/Panorama/Nature/NAtFracGallery/NatFracGallery.htmlhttp://classes.yale.edu/fractals/Panorama/Music/Mus/Music.htmlhttp://classes.yale.edu/fractals/Panorama/ManuFractals/ManuFract/ManuFractals.htmlhttp://classes.yale.edu/fractals/Panorama/Literature/Lit/Literature.htmlhttp://classes.yale.edu/fractals/Panorama/Astronomy/Galaxies/Galaxies.htmlhttp://classes.yale.edu/fractals/Panorama/SocialSciences/FinRisk/FinRisk.htmlhttp://classes.yale.edu/fractals/Panorama/SocialSciences/History/History.htmlhttp://classes.yale.edu/fractals/Panorama/Architecture/Arch/Arch.htmlhttp://classes.yale.edu/fractals/Panorama/Art/ArtAndNature/ArtAndNature.htmlhttp://classes.yale.edu/fractals/Panorama/http://classes.yale.edu/fractals/7/28/2019 Mathed PDF Presentit Fractals
79/89
Clint Sprott
made thisimage from an
IFS written byPeitgen.
http://sprott.physics.wisc.edu/sprott.htmhttp://www.cs.wpi.edu/~matt/courses/cs563/talks/cbyrd/pres1.htmlhttp://www.cs.wpi.edu/~matt/courses/cs563/talks/cbyrd/pres1.htmlhttp://sprott.physics.wisc.edu/sprott.htmhttp://sprott.physics.wisc.edu/fractals/chaos/7/28/2019 Mathed PDF Presentit Fractals
80/89
Who isPeitgen?
Clint Sprott
made thisimage from an
IFS written byPeitgen.
http://sprott.physics.wisc.edu/fractals/chaos/http://sprott.physics.wisc.edu/fractals/chaos/http://sprott.physics.wisc.edu/sprott.htmhttp://www.cs.wpi.edu/~matt/courses/cs563/talks/cbyrd/pres1.htmlhttp://sprott.physics.wisc.edu/fractals/chaos/http://sprott.physics.wisc.edu/fractals/chaos/http://www.cs.wpi.edu/~matt/courses/cs563/talks/cbyrd/pres1.htmlhttp://sprott.physics.wisc.edu/sprott.htmhttp://sprott.physics.wisc.edu/fractals/chaos/7/28/2019 Mathed PDF Presentit Fractals
81/89
Who isPeitgen?
Clint SprottHeinz-Otto Peitgenuses fractalresearch in the arena of medicine toassist surgeons in identifying andoperating on tumors. MeVis dealswith medical research. CeViseducates teachers about fractal
geometry and math/science/art/musicconnections. FAU runs a sisterprogram of CeVis, where I learnedabout fractals, directed by Peitgen
and
made thisimage from an
IFS written byPeitgen.
Richard F.Voss.
http://sprott.physics.wisc.edu/fractals/chaos/http://sprott.physics.wisc.edu/fractals/chaos/http://sprott.physics.wisc.edu/sprott.htmhttp://weblog.science.fau.edu/info/?p=69http://www.cs.wpi.edu/~matt/courses/cs563/talks/cbyrd/pres1.htmlhttp://web.gc.cuny.edu/sciart/0102/fractals.htmlhttp://web.gc.cuny.edu/sciart/0102/fractals.htmlhttp://weblog.science.fau.edu/info/?p=69http://sprott.physics.wisc.edu/fractals/chaos/http://sprott.physics.wisc.edu/fractals/chaos/http://www.cs.wpi.edu/~matt/courses/cs563/talks/cbyrd/pres1.htmlhttp://sprott.physics.wisc.edu/sprott.htm7/28/2019 Mathed PDF Presentit Fractals
82/89
Ready to generatesome fractals using
the computer?
Visit Peitgen andVoss fractal
games website.
These gamesrequire J ava .
http://www.math.fau.edu/MLogan/Pattern_Exploration/MainDirections.htmlhttp://www.math.fau.edu/MLogan/Pattern_Exploration/MainDirections.htmlhttp://www.math.fau.edu/MLogan/Pattern_Exploration/MainDirections.htmlhttp://www.math.fau.edu/MLogan/Pattern_Exploration/MainDirections.htmlhttp://www.math.fau.edu/MLogan/Pattern_Exploration/MainDirections.html7/28/2019 Mathed PDF Presentit Fractals
83/89
To build real-worldmodels of
Sierpinskistetrahedron:
Visit mySierpinski Build
page for 5th gradeand under. Older
grades will preferYales methodusing envelopes.
http://www.public.asu.edu/~starlite/sierpinskibuild.htmlhttp://classes.yale.edu/fractals/Labs/SierpTetraLab/SierpTetraLab.htmlhttp://classes.yale.edu/fractals/Labs/SierpTetraLab/SierpTetraLab.htmlhttp://www.public.asu.edu/~starlite/sierpinskibuild.html7/28/2019 Mathed PDF Presentit Fractals
84/89
To build real-worldmodels of the
Menger Sponge:
The Business Card cubemethod should work well
for grades 7th and up. Forgrades 4th through 6th, itmight be best to instead
build paper cubes usingtape and tabs, and then
tape the cubes together to
construct the Sponge.
http://www.nedbatchelder.com/text/cardcube.htmlhttp://www.nedbatchelder.com/text/cardcube.html7/28/2019 Mathed PDF Presentit Fractals
85/89
Recapping the main fractal theme addressed in thispresentation:
7/28/2019 Mathed PDF Presentit Fractals
86/89
Fractals operate under a Symmetry ofMagnification (called Dilatation or Dilation inliterature). Different types of fractals share acommon ground of parts that are similar to thewhole. Even though self-similar substructure
must technically be present all the way to infinityfor something to be called fractal, the concept offractility is loosened to apply to forms (esp.
natural) with only a handful of levels ofsubstructure present.
The simplification of complexity leading to useful results that wehave been looking at is not unique to the field of fractals, it is atheme that runs throughout mathematics, with varying methods
f i lifi i
7/28/2019 Mathed PDF Presentit Fractals
87/89
of simplification.
The simplification of complexity leading to useful results that wehave been looking at is not unique to the field of fractals, it is atheme that runs throughout mathematics, with varying methods
f i lifi ti
7/28/2019 Mathed PDF Presentit Fractals
88/89
of simplification.
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 wemake a clean concept by assuming the existence of self-similarityinfinite levels of substructurewhen there are only
a few, pushing beyond reality. (Priscilla Greenwood,Statistician and Mathematical Biologist at Arizona State
University)
The simplification of complexity leading to useful results that wehave been looking at is not unique to the field of fractals, it is atheme that runs throughout mathematics, with varying methods
f i lifi ti
7/28/2019 Mathed PDF Presentit Fractals
89/89
of simplification.
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 wemake a clean concept by assuming the existence of self-similarityinfinite levels of substructurewhen 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, but
that is another presentation, and another day.