Natural Fractals in Grand Canyon National Park
by Gayla Chandler
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This presentation may not be altered or used for profit.
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I gave this as a special program at Grand Canyon National Park
on New Years Eve 2004 and News Years Day 2005. The slideshow
and notes are now accessible from the web for viewing by individuals and/or use
by teachers in classrooms. It should project well provided the projector is set
for 800x600 image size. While several fractal topics are addressed, the main topic
(Canyon section: slides 12-55) highlights different ways to recognize a prominent pattern of three mutually
orthogonal joints that is present throughout the walls and rim of Grand Canyon on a massive number
of scales. The pattern, caused by rock uplift and expansion, was shown to me by Paul Knauth,
Geology Professor at Arizona State University. Due to the long length of the Canyon section, some other sections
(like clouds and lightning) were shortened in the interests of viewing time and file download size.
If for any reason, links are inaccessible during viewing, they are listed in a Word
I have put up a page of Follow-Up Images,
selected especially for post-presentation viewing. At least one and usually several aspects of similarity discussed in the
presentation are prominent in each image. I put these up full-sized (1600 x 1200) at medium quality,
so they may be used as screensavers or as desktop backgrounds for any size monitor.
I also have a new web-based presentation up, interdisciplinary, made for
math education, called Fractals: an Introduction through Symmetry. It is standards-rich.
Please forgive the ...question mark symbols littered throughout the page. I just found them on May 4, 2012. They seem to be in place of
commas, and apostrophes, or maybe just apostrophes. I don't know what is wrong, but please try to overlook them until I can fix the problem.
Slide Notes Icon
Clicking on this icon on any slide in the presentation should take you to the appropriate place in the
notes. However, I have found that this often doesn't work properly. Thumbnail images have been included
as a map to find one's place in the notes when they need to be accessed outside of the presentation.
Slide 1—(Introduction 1):
"Natural Fractals" is hyperlinked to the Earth Monitoring System website's
Nonlinear Geoscience—Fractals page. This site provides easy to read information
about fractal systems in nature, focusing on geology. I think it is an appropriate
introductory link for a
presentation about natural fractals in the Grand Canyon! It is a friendly site to explore. The
link was originally to the main page of the Yale Fractal Geometry website,
the most comprehensive web source that I know of for technical information
Many links to Yale's pages are scattered throughout the presentation, plus,
the Snow section is taken verbatim from their site.
Slide 2—(Introduction 2):
Although all images in this presentation were taken at Grand Canyon,
the intended focus is on similarity in nature generally, i.e., not
just at Grand Canyon. I often use the term similarity instead of self-similarity.
Natural fractals have self-similar properties. Keep in mind, however,
that the exact self-similarity in geometric fractals and the approximate
self-similarity in natural fractals are not identical. Even though not
identical, they share significant properties that include scale-invariance
and similar shapes that repeat on different scales.
The first link on this slide is terrific for kids. It goes to Cynthia
Lanius’ main fractals page. If you use the search term “fractals” in
Google, Cynthia's page is the first hit, it is the number one fractals
page, is very friendly and upbeat—is geared toward children—and the
information presented is credible.
The second link is to Paul Bourke's “Self Similarity” page. Paul's site
is diverse in content, and is technologically advanced. It is a high-end
site geared toward adults. It has to be one of the most fun sites to
explore, if you like math and the sciences. On this page, he does a
great job of showing similarity on different scales, and scale-invariance
(aka scale independence). This is a fine page for beginners to the subject
from both a words and visual standpoint.
“What shape the parts are depends entirely on the object.” This cannot
be overstated. Let me breeze ahead to Slide 56 in the presentation,
the first slide where a Sierpinski tetrahedron is sitting in an image.
If you see a Sierpinski tetrahedron in one of my pictures, it doesn’t
imply that nature is filled with triangles and/or tetrahedra or any
like thing. Repeated experiences around people while out taking photographs
with the structures over the past three
years have shown me that this is a common assumption to jump to, and,
once jumped to it is very hard for people—adults and children alike—to
undo. Nature has its own shapes that are typically completely different
from familiar shapes in Euclidean geometry, although we will see some
intersections of nature with Euclidean shapes in the Canyon section.
In the Canyon, we will be looking for shapes like lines and boxes, blocks,
and rectangles. The Canyon has a lot of rectilinear structure.
Slide 3—(Introduction 3):
This slide provides links to information about scale-invariance,
the magnification symmetry of fractals, connections between
fractals and chaos, and some perspective
about what is fractal and what is not.
Usually when we're walking around,
there are objects around to indicate how big something is. We can see
a lot of scaling in tree limbs, for instance, several stages of substructure
are present. Tree limbs are scale-invariant even when there is no question about
the size of the limbs. The point is that if there were no frame of reference,
a tiny section of tree could be magnified and passed off as a bigger section:
symmetry of magnification.
The Earth Monitoring System page gives a good description of
chaotic processes. One example of things fractal but not chaotic: geometric
fractals—because they are completely deterministic. There is nothing uncertain
about the way they are going to grow or the shape they will become at infinity.
An example of something chaotic but not fractal: frenzied ocean waves.
Perhaps similar substructure isn't observable once a
certain level of chaotic turbulence is reached—when the process gets too wild, randomness
may take over to a point that underlying repeating patterns are
undetectable. (I'm only trying to give a sense of the situation, am not speaking
with authority about this.)
The galaxies link goes to a thesis detailing an attempt to use the fractal dimension
of galaxies as a method of comparison and classification. The research produced
inconclusive results, largely because of noise in the image data. I have
included the link regardless because it is a good example of how fractals are being sought
out for answers to difficult problems, many times with success.
also included a link to Yale's page about "things that look like fractals
but aren't", to reinforce that everything isn't fractal,
and to give a sense about what is fractal and what is not. Now,
some of the examples on the page don't connect with me personally, I
would prefer some examples that are more concrete, but hopefully the
page will add some perspective to your experience.
Slide 4—(Topics 1):
This slide provides valuable information about the presentation, and
should be read. The topic slides follow. On slides 5-11, the topic introduced
will be underlined, which signifies a link. Several of these links go
to pages with artificial counterparts to the real world systems addressed
in the topics.
Slide 5—(Topics 2):
The “Canyon”, meaning the Canyon itself, is the longest section of the
presentation. The link will take you to Paul Bourke's main page for
fractal terrains. Right underneath the Canyon link is a another link
to a page Paul wrote about how to make a fractal landscape. The Grand
Canyon has a fractal dimension of ~2.25.
Slide 6—(Topics 3):
The topic introduced is Clouds, with a link to a url that gives different
perspectives about them. One of the sections in the link is entitled:
“Clouds are not fractal” and makes the point that although fractal geometry
describes certain aspects of clouds (including complexity, and general
shape), there are many aspects of clouds and cloud behavior that fractal
geometry doesn't explain at all.
Slide 7—(Topics 4) :
Trees are the topic. The link goes to a site with some fractal trees
and bushes. The branches in trees and bushes are almost always fractal,
whereas their outer foliage rarely is. There is good reason for this,
referenced in slides 78 and 86.
Slide 8—(Topics 5):
Lightning. The link goes to a Yale Fractal Geometry page with a series
of pictures of natural fractals, one of which is lightning.
Slide 9—(Topics 6):
"Boundaries" links to a set of fractal games from an NSF-funded program
that I attended for several summers out of Florida Atlantic University
(FAU) in Boca Raton. "Coastlines" is the fractal boundaries game. It
is very easy to play, and it is fun to change the parameters to arrive
at different coastline appearances. IMPORTANT NOTE: these games
were created using Java language. Windows no longer comes with the ability
to run Java. If your computer is fairly new, you might have to download
the necessary software to run these games and other Java programs, available
at no charge, at: http://java.sun.com. I am told it is a fairly time-consuming
Slide 10—(Topics 7):
Rocks. I have linked to some general information about geology and fractals.
It is good from a perspective standpoint and fits in well with my presentation.
I disagree with one thing said on the page. It is stated that the detail
in (natural) fractals goes on ad infinitum. In nature, however, this
really isn’t the case. Greater detail that goes on forever is the idea
behind fractals, yes, but in nature this idea can’t play out all the
way. For instance, you can’t keep breaking a rock into smaller rocks,
forever, eg., the fractal structure of a natural object doesn’t go all
the way to its atomic structure. Whatever the specific similarity aspect
you are observing in a natural object, that object will have pre-fractal
states where the shape-similarity aspect is not present.
Slide 11—(Topics 8):
Snowflakes. This link takes you to Kenneth Libbrecht's snowflake site
with all-real snowflake images taken with a special photo microscope.
Libbrecht also makes snowflakes using an electric needle of some sort.
It is a fabulous site to explore. His focus is not on fractals, you
won’t hear him even say the word fractal, even though he says much of
the same things. The section on snow in my presentation, however—it
is taken verbatim from the Yale Fractal Geometry website—is presented
from a fractals perspective and uses all the proper lingo. I have included
a link to the Yale Snow page on slide 120.
Slide 12—(Canyon 1):
I took this picture on Desert View Drive on the temperamental day of
November 30, 2002. (Weather changes can affect the interior of the Canyon
dramatically. Small changes in the weather can quickly cause big changes
in visibility inside the Canyon—do you recognize the familiar ring of
The Canyon section deals with visible patterns in the Canyon walls and
rim. After viewing the presentation, a few things in this image should
jump out and have meaning.
Slides 13-15—(Canyon 2, 3 and
“Look for the same shape on different scales.” I’ve used arrows to point
out some shapes that remind me of scallops, also outlined one of them.
On slide 14, I outlined a smaller scallop from the edge of the bigger
scallop. If we could zoom in, we would probably also notice scallops
on the smaller scallop. Slide 15 shows the image with no words. This
provides an opportunity to examine the image without animations present.
The different masses of land shadow each other. They are all in the
same vicinity, probably have about the same rock content, have been
exposed to approximately the same weather conditions over a long period
Slides 16-17—(Canyon 5 and
I've included this image because the two dogleg sections in the foreground
of the Canyon stand out like beacons. They look a lot alike, and there
are probably dogleg extensions at their edges.
Slides 18-19—(Canyon 7 and
This is the first slide showing the main pattern theme that I have chosen
to address in this section. It is a pattern of mutually orthogonal joints
that run throughout the Canyon, pointed out to me by a geology professor
at ASU. The joints are present on a massive number of scales. The rock
is permeated with them, but we only see them after they have been exposed
to weathering. They were caused by rock uplift and expansion. It has
to do with the way rock releases pressure. The joints begin forming
way underground while the rock is uplifting, and expansion continues
above ground until all pressure is released. The first release is in
the direction of uplift, the second release is mutually orthogonal to
the first, and the third and final direction is mutually orthogonal
to both the first and second releases of pressure. But pressure is releasing
in concert, in an interconnected way, as opposed to: 1) all one
direction, 2) all the next direction, and 3) all the final direction.
The professor who revealed this to me is introduced in the next slide.
For the moment, in this slide, start looking for the pattern highlighted
in this image.
Slides 20-23—(Canyon 9, 10,
11, and 12):
Slide 20 shows Paul Knauth, Professor of Geology at ASU, pictured with
his granddaughter. He has been observing this joint pattern for about
Once you learn to recognize the joints in their different forms, looking
at them both in images and in person at the Canyon is very easy. You
can see them from anywhere, but they stand out (as a thematic pattern)
more clearly from a distance. The joints in the top half of the Canyon
are at the same orientation. (All information I am providing is from
what Knauth relayed to me, a brief synopsis follows.)
In the bottom half of the Canyon: multiple sets of joints
are present at different orientations. This has happened because lower sections
have been recompressed and later reuplifted,
reintroducing the expansion process where sometimes the uplift has
been at an angle.
From a joints standpoint, the geology from the half-point down is
more complex than the geology from the half-point to the top, so I decided
to stick with the top half. Plus, I have never even been to the bottom
of the Canyon to see these more complex patterns for myself.
The “mutually orthogonal” link provided on this slide goes to a definition
and explanation of what it means to be mutually orthogonal. The Grand
Canyon link goes to the home page for Grand Canyon National Park.
Slides 24-27—(Canyon 13, 14,
15, and 16):
You don't have to look only for blocks, you can look for lines. There
are horizontal lines in the form of ridges or cracks, vertical lines
in the form of ridges or cracks, and blunt faces. Blunt faces are outermost
sections of Canyon, behind or underneath which are many cracks. Imagine
looking at a loaf of bread head on. While you can only see the heel,
you know there are a lot of slices behind it. This is how it is with
blunt faces in the Canyon.
Slides 28-30—(Canyon 28, 29,
This image was taken on Grandview Trail on Desert View Drive, December
1, 2002. I’ve used it because it shows so many right angles and lines,
blocks and faces. On the left top-third you can see the edge of a wide,
weathered, horizontal crack.
Slides 31-33—(Canyon 20, 21,
A typical wide, vertical crack is seen here at the rim. Notice the soft
edges of the horizontal ridges that run throughout the image along the
rim. Slide 32 (Canyon 21) shows a place Knauth pointed out where this
section will almost surely break away. We have already noticed that
the walls of the Canyon tend toward bluntness. The piece of rock immediately
below this section has already released, and the topmost section will
someday follow. It will be a terrible thing if someone happens to be
standing on it when it falls. Slide 33 shows the image with no words
or animation. Notice the shadowing in the rear of the image, and the common
ridges shared throughout this area of the Canyon. We are looking at
sections of Canyon in the same vicinity that probably have a highly
similar rock concentration that have been exposed to similar weather
conditions over a long period of time. Is it any wonder that the Canyon
looks the same in localized regions?
Slide 34—(Canyon 23):
This image begins a Canyon Rim focus on the joints, to make a connection
between the rim and walls of the Canyon. I find it to be an amazing
image because the blocks in the foreground have such a “regular” appearance.
It was taken in the Mather Point area. It looks to me like someone actually
carved the blocks, but it is nature’s own carvings that we see here.
The rim will recede with time, as more and more joints are weathered
and fall, new joints are revealed and weathered, and more blocks fall
from the Rim into the Canyon.
Slides 35-36—(Canyon 24 and
This image, like the previous image, is from Mather Point. I have outlined
some exposed, weathered cracks on the rim. They continue further along in hairline
cracks that are at the surface but are not yet weathered. I am highlighting
barely enough to bring a few patterns out visually. There is a great
deal more to glean from these images than the little bit I highlight.
For instance, look at the section leading out into the Canyon in the
center of the image: the lines, the blocks, the weathering. We can look
at the images and see where blocks are, but oftentimes, we can also
see where blocks were and are no longer. Broken extensions of Canyon
used to be sections of solid rock with Canyon joints running through
Slide 37—(Canyon 26):
This is another instance of what is shown in the previous few images.
At this point, try to bring to mind the previous slides. Look at the
horizontal and vertical cracks and ridges. Look at the bluntness of
the Canyon walls overall, and the myriad individual blunt pieces that
make them up. Focusing now on the closest Canyon wall in this image,
it is a big, blunt face that is made of blunt faces that is made of
blunt faces. There is, furthermore, much more detail than we can see,
even if we were scaling the wall with our nose to it. At any given point
in time, we can only see what has been weathered. The rest is hidden
to our eyes.
Slides 38-40—(Canyon 27, 28,
This image is from Desert View Drive. It shows a weathered section of
rim with lots of joints. The cracks highlighted in yellow, while orthogonal
to each other, are off-orientation to the other localized cracks. Everywhere
you look, there are exceptions of one kind or another to prominent patterns.
But now look at the lines in the background sections of Canyon. There
are four big background sections, and the joint pattern stands out clearly
in all of them.
Slides 41-42—(Canyon 30 and
Here is another rim example, this one is from the beginning of Desert
View Drive en route from the South Rim, a long way yet from the East
Slides 43-45—(Canyon 32, 33,
Even after Knauth told me about the joints, I got lost in big views
of the Canyon, there is so much information to take in, so many ways
to look at the Canyon and experience it. Originally, this image was
meant to point out the falling block highlighted in Slide 45 (Canyon
34). I had accidentally placed a line around the text box, but left
it there, because the line bordering the box appealed to me. It looked
so "right." It took a long time to realize that the rectangular outline
on the box looked good because the image is itself filled with rectangular
shapes. I later added a slide [Slide 44 (Canyon 33)] to highlight some
of the rectangular shapes in the image.
Forget about the joints for a moment, just enjoy the rectangles, or
whatever it is you see. Perhaps instead of rectangles, you noticed the
pyramid-like shapes below the rectangles, pyramidal shapes, by the way,
that are made of smaller pyramidal shapes made of yet smaller pyramidal
shapes. How many repeating patterns can you find?
On Slide 45 (Canyon 34), the arrow is pointed toward a block that is
poised at an angle and will someday fall.
Slides 46-47—(Canyon 35 and
I have turned the focus to falling blocks. The block in this image has
started its way down the Canyon. Winter 2005 has brought a lot of rain
and snow to the high country in Arizona that will help blocks like this
weather and slip.
Slides 48-49—(Canyon 37 and
The joint pattern isn't as obvious in this image, camoflauged by lots
of things going on in the rock. One particular spot in the image is
reminiscent of the previous image, but without a telltale block, a clear
space left by a fallen block that is no longer around.
Slide 50—(Canyon 39):
This shows a heavily weathered section with a lot of fallen blocks.
Look at the section in the far background, too, where only a piece of
the (now) topmost section remains. Some areas of Canyon stay together
better than others. I wonder, is it because localized areas have seen
harsher weather over time, or the rock in this area is more susceptible
to weathering? Whatever the reasons are, this section of Canyon looks
like it is falling away faster than some neighboring sections. As blocks
fall, joints at their surface weather. Big blocks weather into smaller
blocks, and by the time they arrive in the lower part of the Canyon,
many are tiny rectilinear rocks. I asked Knauth: if these blocks are
all falling down into the Canyon, won't the Canyon eventually fill up
with blocks? He said many arrive at the bottom in little pieces and
are carried away by the river, that the river acts as a big conveyor
belt that transports the remains of fallen blocks out of the Canyon.
Slide 51—(Canyon 40):
I have called this image field of fallen blocks because of all
the blocks that have built up on that single ridge, as if the plateau
is holding on to them and won’t let them follow their 'bigger' path.
Also notice how flat yet knarled the closest section of Canyon wall
is. In an up-close view, the joint pattern would hardly be visible, if at all,
present as it is in concert with so many other processes.
Slides 52-53—(Canyon 41 and
Several areas are highlighted here that show similarity. Notice the
ovals that highlight a section of Canyon boundary. Look at the Canyon
boundary inside the little oval and then look at the boundary inside
the big oval. The smaller section of boundary looks very much like a
miniature version of the larger section. If we took a smaller section
within the small oval, the same thing would happen.
Slide 54—(Canyon 43):
The images used thus far in this presentation have had a frame of reference
present, like the skyline, or trees, something that has revealed the
relative size of the landmasses in the images. This image has no external
frame of reference. It is difficult to determine the size of what we
are seeing. We could isolate a smaller section of the image and pass
it off as a more massive section of Canyon or a smaller piece than it actually is. This
is an example of scale-independence, or scale-invariance.
Scale-invariance is crucial to being fractal, no matter the type of
fractal structure. I’m isolating on natural and geometric fractals in
this presentation, but there are a lot of types. Scale-invariance is
an inherent property of all types of fractals.
Slide 55—(Canyon 44):
Taken the same day as the previous image, at the East Rim, this image
is here because I like it, and it seemed like a nice ending place for
the Canyon section. The next sections are shorter. The joints demanded
a lot of attention, first of all, to look at them in different ways
(looking for blocks versus looking for lines, far away views versus
underfoot views on the Canyon rim), and secondly, to consider effects
from joints (like falling blocks). This was put together for presentation
at Grand Canyon National Park. The Interpretive Education Division of
the GCNPS expected that every image and all content should focus on
Grand Canyon, so it made sense to have the biggest section be on the
Canyon itself. Paul Knauth made the Canyon section worth looking at
by telling me about the joint pattern, and taking time out of his busy
schedule to answer many questions.
Slides 56-58—(Clouds 1, 2,
Clouds. How big is a given cloud? You can’t know unless there is some
external frame of reference in the scene. If you break a cloud into
smaller and smaller pieces, at some point you will have something that
isn’t a cloud. I’m not sure at what point that occurs, but self-similar
detail doesn’t go on forever in nature. It has a lower limit. What is
the upper limit of a cloud, i.e., is there a limit to how big a cloud
can get? Limits are assumed to exist with nature, however, a reasonable
question to ask might be: how big could the biggest cloud possibly be?
Is there such a thing as the biggest possible cloud? There are some
huge versions in space made of dust and interstellar gas. A vapor cloud
on earth would probably be limited in size by the boundary of our atmosphere.
But I don't know, am making no claims about it, am rather throwing this
out as a question.
Slide 57 (Clouds 2) briefly addresses the appearance of the Sierpinski
tetrahedron and provides three links: the first is to my Sierpinski
tetrahedron page; the second is to an explanation about geometric fractals
on the Arcytech website; the third is to a section on L-systems on the
Archytech website, it is a really great site to learn about fractals
(the applets are written in Java, and Windows no longer comes with the
ability to run it; either the images will come up, or if they don't,
a link is on the image to download necessary software). An important
thing is to realize that we are not looking for triangles and tetrahedrons
in nature. Nature has its own, very different shapes that repeat on
different scales and are scale independent. The tetrahedron is in the
image 1) because it looks beautiful there, and 2) to highlight and draw
attention to the mathematical structure of nature.
Slides 59-60—(Clouds 4 and
Almost everything in the image is fractal: the snow, the boundary the
snow makes on the rock ledge, the ledge (made as it is, of rock), the
clouds, the Canyon, the stage-4 Sierpinski tetrahedron, the branches
of the tree. The needles on the tree aren’t fractal.
Slide 61—(Clouds 6):
This image is a great complement to some artificial landscape slides
created by Richard F. Voss, called making clouds out of mountains.
I have a copy of the slides to use for in-person presentations, but
cannot use them in the web version. His artificial slides look like
a re-creation of a section of this real-world image. Voss created an
artificial mountain/canyon landscape, and by changing the lighting,
made it into clouds, from the same file, the same Iterated Function
System. I haven’t said much about clouds, but the point is to look at
them and realize that if a piece of a cloud breaks away, it becomes
a whole cloud in its own right. The parts are similar to the whole.
Slide 62—(Trees 1):
This is the second largest section of the presentation. It might better
be called “Branching”, except that it focuses on the branching in trees.
It is a large section because I wanted to consider trees with bare branches,
and also trees with leaves on them. It can be difficult to see the fractal
structure of tree branches when they are covered by leaves. To add to
the confusion, leaves are typically not fractal. You have to separate
them in your mind from the fractal structure of the tree while looking
at the tree, and this can be hard when almost all you see is leaves.
They do have fractal properties, however.
that run through
them are fractal almost without exception, and there are times when their
edges have a fractal
boundary, but leaves are generally not made up of little leaves that look
just like them. Here is an example of a computer generated
Slides 63-65—(Trees 2, 3, and
This is the easy way to look at trees, is when they have bare branches.
I believe one of the Interpretative Rangers said this tree is a Juniper,
seen frequently in the Canyon.
From tree to tree, there are different
characteristics between the limbs, such as the shapes
of the intersections, from wide U shapes to sharp V's and everything
in-between. Pretty much, whatever the shape at an intersection in a
given tree, that shape will hold in the intersections throughout the
entire tree. The tree in this image also has curly branches. Notice
that there isn't just an isolated branch or two with a curly limb, this
is a characteristic pattern of the limbs throughout the tree.
Slide 66—(Trees 5):
This image covers a lot of ground, across several slides. I have worked on them
a lot, and they probably still need more work.
In both the tree and the (Sierpinski) tetrahedron there is fractal structure:
scale invariance, small parts that look like big parts, and exponential
growth—an overall fractal pattern. In spite of commonalities, there
are differences between the way these two objects grow that are representative
of differences between geometric fractals and natural fractals generally.
Some differences include that the Sierpinski tetrahedron uses an exact
shape (a regular tetrahedron), the growth pattern between stages is
exact, and the growth can feasibly go on forever. Contrast this with
the tree, where both the similarity between the shape of the limbs and
the growth pattern between stages is approximate rather than exact,
plus, the tree has a limited number of stages of growth. It "tips out".
The tree shows approximate, or statistical self-similarity. Geometric
fractals, on the other hand, show exact self-similarity. One qualification:
self-similarity only really happens in the limit, i.e., at infinity.
Somewhere, at infinity, the Sierpinski tetrahedron will achieve self-similarity,
where every little part will be identical to the whole structure. It
is a kind of "out-there" thought!
Slide 67—(Trees 6):
This step begins the process of following a connected path through
4 stages of growth of a Sierpinski tetrahedron. There is a single
opening highlighted in this slide, the one and only biggest opening.
At any given stage of growth, there is a single biggest opening. Notice
there are a few choices for the second biggest opening, 4 to be exact.
Whatever path is chosen, much detail and growth of the structure will
be left behind along the way, and the numbers will exponentially explode
as growth progresses.
The link on this slide is to the Yale fractal geometry labs page (suggested
classroom activities with fractals that include history and instructions). The Sierpinski
tetrahedron lab is a short way down the page. It is a great lab for older grades,
say 6th and up. It addresses more than building the structures, it lends perspective
to what is happening. It might also be the case that Yale's building
method (using envelopes to make the tetrahedrons) would work great
for grades as low as 2nd or 3rd, I just don't know.
Slide 68—(Trees 7):
There were 4 same-size choices for the second largest opening, all
were on a connected path, I chose one, leaving 3 openings behind.
There are 4x4=16 next largest openings, where only four are touching
a connected path. One will be chosen, leaving 15 same-size openings
behind. Next will be 4x4x4=64 smallest openings, again with four touching
our path. One opening will be chosen and 63 same-size openings left
Slide 69—(Trees 8):
Following what appears to be a small path—only 4 stages of growth
so far—we have already left behind 3+15+63=81 openings total. Beyond
this image, the next stage—the stage-5—has 128 openings to choose
from, with (as always with Sierpinski's tetrahedron) 4 openings touching
a connected path.
Sierpinski's tetrahedron grows in an identical form from stage to
stage. The shape is a regular tetrahedron, and it always grows in
powers of 4. Each stage is 4 of the previous stage set tip-to-tip.
The edge-length either doubles at each stage, or the lengths of the
individual tetrahedra are halved at each stage, depending on the growth
method chosen. To see pictures of this, visit my Sierpinski tetrahedron
page at www.public.asu.edu/~starlite/sierpinskitetrahedron.html.
The link on this slide is to the Yale Fractal Geometry website's Self-Similarity
Slide 70—(Trees 9):
Now we've switched from the tetrahedron to the tree. We're following
a connected path through the tree.
Five massive branches lead off the trunk at the first intersection
(one branch is hidden from view by the frontmost branch). Each intersection
along the path is marked by a yellow oval, called a node. The possible
paths leading off the node are highlighted, where only one limb is
followed to the next node. With the first path choice, 4/5’s of the
tree is left behind. The next intersection has 4 branches leading
off it, one path is taken, leaving (4/5)x(3/4) of tree behind. At
the 3rd intersection, there are only three limbs. One limb is chosen,
the other two abandoned, leaving (4/5)x(3/4)x(2/3)’s behind. Next
intersection, there are only 2 limbs, leaving a total so far of (4/5)x(3/4)x(2/3)x(1/2)=~80%
of the detail of the tree behind. When stages 5 ane 6 are included,
plus two additional stages that were too small to highlight on the
slide, each with two branches to choose from in this way-oversimplified
example, approximately 99% of the tree is left behind.
The slide includes a link to one of Yale's pages about natural fractals
and approximate self-similarity.
Slide 71—(Trees 10):
Notice that the tree isn’t growing in exactly the same way from stage
to stage. Although the number of limbs at each stage is growing exponentially, the
rate of growth between stages is decreasing, growth is surging forward
with limits, probably because nature is efficient. The limbs
have to fit into a real-world space that includes constraints of size
and proportion. Leaves have to fit on its many branches. If 4 or 5
limbs had branched off at every single intersection along the way,
there wouldn’t be room for all the limbs much less any leaves! Assuming
that similar growth is taking place throughout the tree, there will be
5x4x3x2x2x2x2x2 stage-8 limbs, or approximately 2,000 of the tiniest sized
limbs (found by multiplying the number of limbs at each stage along
the path, and including the two stages at the top that were too small
I can't tell, there may be an intersection of branches hidden behind
the tetrahedron, a 3-branch node. If this is correct and the number
of intersections is really 9, then the number of tiny limbs increases to
somewhere around 5,000-6,000 in the real tree in this image. Suppose
the tree had grown at a consistent rate exponentially, like the tetrahedron.
At the first node there were 5 branches. If it had continued at a
rate of 5 branches per intersection, the number of smallest limbs (stage-
8 limbs) would be 5^8 = 5x5x5x5x5x5x5 = 390,625. And if there are really
9 stages to this tree, 5^9 = 5x5x5x5x5x5x5x5x5 would be 1,953,125 tiny
limbs! It simply couldn’t be. So real trees in nature show exponential
growth, yes, but it shows up in a different way than in the Sierpinski
tetrahedron, and geometric fractals in general. This doesn't invalidate
the fractal structure that they share. A small piece of a tree closely
resembles a bigger piece of the same tree. Both Sierpinski’s tetrahedron
and the tree look very much like the parts that make them up, even
though they reveal different patterns of fractal growth.
It should be noted that some biological systems, bacteria growth is
one example, model classic exponential growth, but when that happens,
to whatever degree it happens, it is because the natural system can
handle it. Otherwise, nature would make adjustments, similar to the
tree in this example.
Slide 72—(Trees 11):
Try to see the content of the previous slides without any animations.
Or maybe just enjoy the image without thinking about anything. I’m
trying to make mathematical connections in nature—not to take away
the aesthetic experience of nature, rather to blend the two.
Slides 73-74—(Trees 12 and
This image points out the similarity in shape between a small and large
section of bare branches, and further compares a bare and branched section
of limbs. It is one of my favorite images from the Canyon, taken December
1, 2002, on Hermit’s Route.
Slides 75-77—(Trees 14, 15,
"When leaves hide branches." When tree branches have leaf coverage,
it can be difficult to examine the shapes of the branches. You have
to look for little sections of tree that resemble each other, and sections
within sections that resemble each other. You also need to satisfy yourself
that this is true not because of the leaves but because of the branches
underneath them that the leaves are growing on. This was a sticking
point for me. For at least two years after learning about fractals,
I had terrible difficulties with leafed trees. It is due to this long
personal difficulty that I am including this section. If it was a stumbling
block for me, it might be a stumbling block for other people, too.
Slides 78-80—(Trees 17, 18,
The words in parenthesis on this slide are from Root Gorelick, a faculty
research associate with the School of Life Sciences at Arizona State University.
There was nothing about terminal organs in this presentation until he
pointed out the connection.
These slides show a tree with brushy features where the branches underneath
are almost not visible. The slides for this image 1) reinforce the message
of the slides from the previous two images, and 2) focus on leaves not
typically being fractal. They are at the tips of the tree. Leaves are
terminal organs and as such do not reproduce little copies of themselves.
On the other hand, the
vein structure in leaves reveal fractal branching
patterns, and some leaves have fractal boundaries, meaning their
reveal the same pattern at smaller scales of detail, i.e. when zooming in.
There are so many distinctions to make, or so it seems to me. The point
here is that leaves are generally not made up of little leaves that
look just like them. The fern,
however, is a big, leafy plant with leaves that are made of leaves that
are made of leaves that have the same shape. This holds with the
asparagus fern too, although the pattern is more subtle and harder to see.
You could think of an asparagus fern as a plant made of wide-bristled
brushes made of smaller wide-bristled brushes made of smaller
wide-bristled brushes that circle around whatever stem they are coming off of.
are not part of the Canyon decor (that I have seen), they were not included
in the slideshow, I bring them up only in these notes.
Slides 81-85—(Trees 20, 21,
22, 23, and 24):
Nothing new is introduced in this slide. It serves to support the previous
slides. The image was taken in the schoolyard of the Grand Canyon on-site
school. To me, these branches resemble a boot, and I’ve followed the
boot shape through some smaller sections of branches. The huge branch
is made up of little branches that have a similar appearance, that when
put together, form a bigger shape that resembles the smaller shapes.
The branches are pretty much hidden by leaves that clothe them. People
wear clothes every day. We don't have to see people without their clothes
on in order to recognize them, and we don't have to see trees without
leaves on them to recognize the fractal branching of the limbs underneath!
Slides 86-90—(Trees 25, 26,
27, 28, and 29):
This set of slides is important. It follows the theme of leaves being
terminal organs and not fractal. The words on Slides 87 and 88 are Root
Gorelick's, not mine, and how grateful I am to have his perspective to
include in this presentation. I took the image, liked the image, and
wanted to use it, but Root made it meaningful.
The really important part of this group
of slides is that the image shows a fractal structure in a male cone
containing bracts with only one level of substructure. The male cone
reproduces copies of itself on one scale in the form of bracts that
the terminal organs (the needles on the tree) grow out of. It is frequently
said that nature shows a limited number of similar scales. Here is the
bottom rung, the least you can have, one stage of similarity. The pattern
of mutually orthogonal joints running throughout Grand Canyon, on the
other hand, is present on a magnificent, uncountable number of scales
(it is finite, but it is a huge number). We have natural systems like
clouds and the Canyon with a magnificent number of scales, systems like
trees and bushes with a nice handful of scales, and here are these male
cones with only a single scale of similar substructure. (I should point
out that some people might take issue with whether or not one stage
of similarity in this male cone is enough to say it is fractal. There
will probably be dissenting views.)
Slides 91-94—(Lightning 1,
2, 3, and 4):
I’ve included lightning because it is enigmatic to look at and exciting
to children. In my in-person (i.e., not a web version) presentation,
I have an image of lightning striking the North Rim of Grand Canyon
taken by professional photographer Mike McFadden. Slide 91 (Lightning
1) links to the image on McFadden’s web page. The structure of lightning
is self-similar. As you zoom into it, what you see is more lightning.
As an aside, lightning strikes produce beautiful branching patterns.
Peter Ledlie took the picture shown on these slides. Ledlie’s url at
the bottom of Slide 91 (Lightning 1) shows pictures of him using equipment
to produce lightning strikes in his laboratory.
Slide 95—(Boundaries 1):
This is one of my favorite images, and it is only seen in this one slide.
I like the way the Rim boundary stretches into the distance. Fractal
boundaries is a terrific topic for Grand Canyon. It is the topic that
the Rangers seemed to identify with the most, especially relating to
hiking and maps in the bottom of the Canyon. I hope to include some
of their perspectives and experiences as they apply to fractal geometry.
I haven’t even been to the bottom of the Canyon, have only seen it from
the top, aside from a couple of very short walks in. This section shows
the greatest potential for expansion.
Slides 96-99—(Boundaries 2,
3, and 4):
An extended fractal boundary has been created along the Canyon rim by
a block that is getting ready to fall. The block has separated and tilted
forward, and I have followed the big crack it has made with an imaginary
measuring stick. In Slide 97 (Boundaries 3), the measuring stick is
halved, and more detail is captured. In Slide 98 (Boundaries 4), the
measuring stick is halved again, and now we are beginning to closely
follow the path of this separation. If the measuring stick were shortened
again, and again, we would continue to follow the path more closely.
As a rough path is increasingly "hugged", the distance around it gets
longer, as more detail is captured. The rim boundary of Grand Canyon
is constantly changing.
On another note, look at the saw-tooth edge of the dark section of Canyon
in the background. The pattern is tiered: in smaller scale above, in
larger scale below.
Slide 100—(Boundaries 5):
I applied a measuring stick to a Hawkwatch International map of Grand
Canyon, with Hawkwatch’s permission.
6, 7, and 8):
A section of rim boundary is outlined and a measuring stick applied
to it. Look at the numbers on Slide 101 (Boundaries 6) that compare
relative unit measure to coastline length. These numbers tell a compelling story.
The length of the Coast of Britain is a problem that really occurred,
recounted in the link provided.
9, 10, 11, and 12):
Tree trunks have fractal boundaries, where the fractal dimension is dependent
upon the roughness of the trunk. A rougher trunk has a higher fractal
dimension. This looks like a trunk with considerable roughness. The
pattern of the trunk itself, the shape of the pieces that make it up,
also shows statistical self-similarity, in that small sections of bark
resemble larger sections of bark. The link on the slide goes to a website
at the University of Manitoba called Fractals in the Biological Sciences.
Tree trunks are mentioned in their section 1.3. The link, however, goes
to section 5 where habitats are discussed. There is a lot of space for
tiny bugs on this trunk that bigger bugs can’t even get to. Tree trunks
play an important role in supporting biological diversity. I recommend
reading two of their sections entirely: sections 1 and 5, just skim
through the many fractal applications in biology that are briefly described in them.
13 and 14):
This trunk is much smoother than the last one and will have smaller
fractal dimension. It will still be a longer distance around it for
a bug with shorter legs.
Slides 110-112—(Frame of
Reference 1, 2, and 3):
A quarter or a dollar bill on this rock would give a concrete idea about
the quantity of rock in the image, because it would be an exact, external
frame of reference. We have to guess about the size of the lizard, but
at the same time, we know it isn't a Godzilla. At least the lizard gives
us a general idea about how much rock we are seeing in the image.
Slides 113-116—(Frame of
Ref. 4, Rocks/Mountains 1, 2, and 3):
This is a plain little image that has been very useful.
Slides 117-118—(Snow 1
This picture was taken on New Year’s Day 2005 at around 9:00 a.m. in
the Mather Point area. It had snowed overnight. I couldn’t find any
links that talk about the fractal properties of accumulated snow, but
there are a lot of links that talk about snowflakes. Going backward
to trees for a moment, look at the wide forks formed by the branches
in the tree on the left, and how the pattern follows all the way through
the tree. It looks as though the limbs are reaching way out to the side
Slide 119—(Snow 3):
This is a favorite snowy image taken on Desert View Drive on November
Slides 120-122—(Snow 4,
5, and 6):
This section is verbatim from the Yale Fractal Geometry pages, with
a link provided to the page it came from. If you click on “snow” at
the bottom of the page, it takes you to a group of links for a few more
of their pages on snow. It is a big site to maneuver, but they are very
good about keeping individual pages within the site small. I keep linking
to pages on the Yale site in order to introduce and reintroduce people
to it, because the information provided is trustworthy. It is such a
massive site, if you want to seek something out on the Yale Fractal
Geometry web pages, my advice is to go to Google (www.google.com), and
in the search box, type the desired term and then add: + Yale + fractals.
This way, Google will bring up results on their website first. It’s
what I do most of the time.
Slide 123—(Snow 7):
This is one of Wilson Bentley’s images. The fractal structure is triangles
made of triangles.
1, 2, and 3):
Hopefully at this point, an overall sense has been developed about fractals
in nature. If you have looked at the entire presentation and have visited
the links provided, perhaps a precursory, comparative understanding
has formed with regard to geometric fractals as well. It’s a huge topic.
My wish for viewers is that the journey be enjoyable while perspective
In order of sequence of events:
Summer 2003—Heinz-Otto Peitgen
and Richard F. Voss held a two-week NSF
on working with PowerPoint, which I attended.
Spring 2003 to Winter 2004—knowing ahead of time that I would
learn to use PowerPoint at the above-referenced institute,
I began to consider introducing a presentation about natural
fractals into a national park, to make a case for bringing math education
about fractals into what I think is a very appropriate environment for
it, from multiple angles. Not until January 2004 did I actually suggest this to the Grand
Canyon National Park Service. Their response, while noncommittal,
was open-ended. Preparatory conversations with the
Division of Interpretive
and Resource Education ensued, and it was decided that I would give
a trial run presentation at the Grand Canyon on-site school.
February 2004—Marilyn Carlson
provided the loan of a laptop.
February 2004—Richard F. Voss loaned me the use of his making clouds
of of mountains images for in-person presentations. His images aren't
included in any web version, but they were a huge contribution to the project
overall and helped it to go forward, making connections between artificial
and real-world fractals in such a beautiful way. Whatever is on the
internet is unfortunately out there for the taking and there would be
no way to keep his images safe.
March 15, 2004—I gave the trial run fractals in nature presentation at the
Grand Canyon on-site school. That presentation was considerably different from this one. At
that time I didn't even know about the joints. The decision was made
to go forward on something totally canyon-focused.
July 2004—Paul Knauth
told me about
the Canyon joints in a series of conversations, after which I began putting
the presentation together. He took time out of his schedule to speak with me more than
once when he was wildly busy with grant writing and other pressing projects.
September 10, 2004—Don Jones
through the initial draft and offered several content-related suggestions,
all of which were incorporated. Off and on since then he has offered additional suggestions and feedback.
September 12, 2004—Charles Wahler
offered suggestions, all incorporated.
October 26, 2004—I gave a trial run presentation at the
and Cognition Seminar. Suggestions offered by professors from
mathematics, psychology, biology, human and family studies, and philosophy
brought about a number of changes/deletions/additions.
One example in particular: the insight offered about terminal organs
on slides 80 and 86-90 came from
December 2004—The presentation was given as a special program at
Canyon National Park on New Years Eve 2004 and New Year's Day 2005.
Several interpretive rangers had interesting comments related
to their experiences with the Canyon which I plan to follow up on and
incorporate at a later time.
February-March 2005—Considerations for placing this on the web were
size of download, providing slide notes, and generally trying to put
it up in such a way that people unfamiliar with the subject matter could
navigate it alone and choose to project it in a classroom if so desired.
has helped all along, but even more than usual
with the web version, answering questions about images and html code.
March 2005—Reimund Albers
suggestions for the web version related to animations on some slides
and incorporating higher quality images. It was around 7MB's at that
time, and Reimund found that it didn't project well, so I replaced 480x640
medium quality images with 600x800 high quality images, increasing the
file size to 15MB's. Striking a balance between file download size and
viewing quality has been a struggle.
March 2005—Paul Bourke
that focused on my animations and the appearance
of the text. All of his suggestions were incorporated, after which both
of us made the presentation available from our websites.
If the presentation were accessible from my website only,
it would have stayed relatively obscure for a long time. By putting it on Paul's
site, too, it went straight to the top of some big search terms in
Google. Plus, it is accessible from Paul's fractals page which comes
up at or near the top of a great many searches. The difference is like
being in a rowboat in a lake with no breeze versus white-water rafting
on the Zambesi.
April 6, 2005—it is now available in pdf form, thanks in large part to advice
and assistance from Information Technology personnel at ASU. Plus, unexpected
problems with the PowerPoint version have been resolved.
answered questions about equipment and
computer processes on many occasions when their time was spread thin.