These questions come from 1) people
who have approached me while out taking pictures, and 2) visitors to the site. I have phrased
the questions consistent with the way they are posed.
[Although some direction is provided, my site is here as an experiential site. The
Yale Fractal Geometry website provides
expert information about fractals: take a look at their fractal lab exercises page
here. Also view and download these
fractal games from Peitgen
and Voss' NSF institutes.]
Do you set triangles in nature because nature is filled with them?
Nature is not filled with triangles, or tetrahedrons, or any particular shape. The Sierpinski tetrahedra
share with certain aspects of Nature a property called Similarity, where little parts resemble
big parts over two or more levels of magnification. Similarity takes on different shapes in different objects.
Sometimes it is revealed in a pattern in a graph that repeats on different scales. It is also revealed in the
distribution of galaxies
in the universe. Try not to associate any particular "shape" with the concept of
similarity, except as it applies to specific objects, for instance, the shape of the branches of a particular tree.
You might enjoy viewing a
Grand Canyon presentation I've put up that visually outlines
fractals in nature at Grand Canyon National Park, or a
Symmetry presentation that focuses on
the magnification symmetry of fractals while highlighting distinctions/intersections between fractals and chaos.
Is everything in Nature fractal?
Some aspects of nature are fractal, and others are not. The list of things that are NOT fractal includes light, which
is made of elementary particles and has no substructure. Another example is the shape of leaves and outer foliage. These are typically
nonreproductive and do not reveal self-similar substructure as do roots, stems, and other reproductive structures (although maple leaves and the entire
fern family are exceptions, since their outer shapes are fractal). Very calm bodies of water aren't fractal, nor are water droplets; however, ocean
waves and river currents are typically fractal.
When considering aspects of Nature that are fractal, two important points to keep in mind are: 1) there is always a stopping point
with Nature, the number of sublevels can often be counted on one hand with a finger or two left over, and 2) fractals only address
properties of shape and complexity, they say very little about the physics of something; for instance, part of what is true about clouds
is that they are fractal, but clouds have a great many properties that being fractal tells us nothing about.
What should I look for in the pictures/when out in Nature?
Look for sublevels that reveal similarity. Try to discern whether an object is made up of little copies of
itself. Can a little piece be blown up to look just like or highly similar to a larger piece of the same
object? Look for resemblances between the small and large branches of a tree. Notice how a small piece of a
cloud is still a cloud. A piece of a mountain, when magnified, still looks like a mountain. A tiny
section of lightning, when magnified, reveals more lightning.
What does it mean for something to be fractal?
It means that there exists a substructure showing similarity across scales which allows it to be described using fractal geometry. This
is often accomplished by using a computer and an Iterated Function System (IFS) that recursively iterates a
simple equation. I have not personally generated fractal trees or landscapes or any fractal patterns using
a computer. If you go to Google (www.google.com) and type in IFS + beginner, some links with simple explanations
and examples should come up. Such information is also available on the Yale link provided above.
Are the pictures real?
Yes, they are real. When a mathematical model is in a picture on this website, i.e., in one of the pictures I took,
it was really there when the picture was taken. **One exception**: on the homepage, the child in the
Menger Sponge is a composite image that isn't mine. The child tossing her stage-1 sponge comes from
a digital photograph that I took of my granddaughter in the kitchen. Paul Bourke later extracted them from my image and set them
inside of one of his computer-generated stage-6 sponges. I am not able to do such things myself or there
would be more of it on the site. Kudos to Paul for this inspiring image! You must get his permission in writing
if you want to use it.
Why are you doing this?
I'm doing this partially because I like math and want people to think better of it. I show it off in
beautiful, evocative settings to make the point that Sierpinski tetrahedra in particular can stand up to the
beauty of Nature. Another reason is to frame and highlight the mathematical structure of Nature, especially
the fractal parts, by juxtaposing a geometric fractal with natural fractals.
What are your triangles made of?
They are made from cardstock that I heavily enamel for durability. Then they are painted with artist quality
pigments that often have iridescents mixed in. If you look closely at Nature, much of it is gently iridescent.
It has a glowing quality. For the Sierpinski tetrahedra to emulate the vibrance of Nature, they need to, in a
sense, glow, and react to light.
Do you leave the triangles there?
I never leave my Sierpinski tetrahedra or any other math models sitting in the environment. On occasion, I
have left them with people. Professor Peitgen has the Ocean set, and Professor Voss has a set I made for
Yellowstone National Park. Even though strong enough to set in Nature to take pictures, the structures are
still paper and are quite delicate. They get damaged falling out of trees and such, and have to be fixed constantly.
Are the platonic solids fractal?
I have devoted a section on my website to the Platonic solids even though they are not fractal. These are the
five regular polyhedra: the cube, octahedron, dodecahedron, icosahedron, and tetrahedron. Since pictures of
the geometric fractal Sierpinski's tetrahedron are seen all over my website, and they are made of tetrahedra,
it seemed important to address the tetrahedron and its roots. Each Platonic solid has its own fractal structure. This
is not true of any other polyhedra. The two famous ones are the Sierpinski tetrahedron for the tetrahedron, and
the Menger Sponge for the cube. The three others are distinguished by their respective names: the Octahedron
fractal, Icosahedron fractal, and Dodecahedron fractal.
How did you make your platonic solids?
I made the inflatable structures seen on the platonic solids pages using heavy-guage, transparent vinyl sheeting
purchased at JoAnn's Etc stores. It comes in 4 nicely distinct and lightly iridescent colors (except that as of 06/2006
it is being sold out and discontinued, I am crushed to think it will not be readily accessible for purchase any longer!).
The iridescence only shows up when light hits the structures at dusk, dawn, or with flash in darkness. I use a vinyl sealant
for the seams, called Vinyl Repair, found in the notions section of any JoAnn's
store. Ideally, the vinyl faces should be melded together using heat, but I use the sealant because it is what I
have access to. The fumes of the sealant are highly noxious. In my opinion, no child should be allowed to work
with it or even be in the vicinity while it is being used.
Are you a mathematician?
I graduated from Arizona State University in Summer 2000 with a Bachelor of Arts in Mathematics. A bachelors in math
is not nearly enough to be distinguished as a mathematician.
Are you a math teacher?
Nor am I a math teacher. I am an individual in the community carrying forward a personal project.
Are you a photographer?
A little bit, I am an amateur photographer. This was not planned and I had no prior experience
with photography. For the 3 months between November 2000 and January 2001, I used a borrowed camera, a Canon Sure
Shot. In late January 2001, I purchased a Nikon Lite Touch QD in the $100 to $200 range and used it until June
2002. My last purchase, in June 2002, was a Fujifilm Finepix 2.1 MP digital camera for $279.00. It has been a good
camera, but I now need a camera with better resolution that will pick up finer details at close range. It would
also be nice to have a waterproof camera. Mine isn't waterproof and I nearly ruined it several times standing
out in the waves to get pictures of the platonic solids. One time in particular, I got unexpectedly plastered by a
wave and the camera blitzed out for several hours.
How did you get started doing this?
After attending a 3-week NSF Pattern Exploration Institute in Florida in summer 2000, out of frustration, I began
building Sierpinski tetrahedra, flimsy ones out of copy paper. Things started coming together for me. Somehow, through
using my hands, I began understanding the information from the institute. I continued building Sierpinski tetrahedra
and am building them to this day, in different forms, although my focus has largely turned to color and light.