of a Fractal Nature

photographic Math-Art essays highlighting mathematics in the natural world

(geometric fractals mimic magnification/dilitational symmetry in Nature)

inspired by the teachings and scientific investigations of Heinz-Otto Peitgen and Richard F. Voss

Stage 4 with a nested Stage 3 against Slide Rock in Oak Creek Canyon outside Sedona
stage-4 Sierpinski tetrahedron, taken at Slide Rock State Park, Arizona, USA.
Real image of stage-4 sierpinski tetrahedron against Scarlett Macaws at Phoenix Zoo.
the Stage-4 was really sitting on those rocks, it stayed completely dry, it had to, as it is made of cardstock.  Taken at Slide Rock in Autumn 2002
Stage-3 (count the number of sizes of openings) with turning Maple leaves in Oak Creek Canyon outside Sedona, Arizona, USA, Autumn 2002.
Stage-4 with boulders at Slide Rock State Park outside Sedona, Arizona, USA.
taken 04/20/06 at Mather Point
Taken on Hermit's Drive at Powell Memorial overlook on December 1, 2002.
One of my first pictures, taken in Winter 2001 in Flagstaff, Arizona
Mather Point at dawn on 12/29/04 seen through tetras.

      Shown here, the stage-4 Sierpinski tetrahedron provides a powerful visual introduction to fractal geometry and the concept of "self-similarity", in which a shape can be broken into smaller copies of the whole. Each new stage is composed of 4 smaller copies of the previous stage. As the number of stages increases, the Sierpinski tetrahedron approaches "exact self-similarity".
      Such geometric fractals provide an important scientific model for characterizing many of the complex processes and shapes found in the natural world, that are echoed in the settings of the tetrahedra. Trees, land formations, clouds, and their images exhibit "statistical self-similarity" in which a small part "looks like", but not "exactly like" the whole. Just as a part of the Sierpinski tetrahedron reminds one of the whole, a small branch of a tree reminds one of the entire tree.
      The mathematics of fractal geometry and the science of chaos are now bridging the gaps between math, science, art, and culture. They treat the messiness of the everyday world. They are based on natural self-similarity and observations of complicated behavior from simple equations. They provide a new mathematical language for capturing, manipulating, and simulating nature.

(Richard F. Voss)

Locations of visitors to this page
The map records visits to this page only, and only a small
number of visitors ever make it here, since traffic comes in
on so many different pages across the site. Visits come from
culturally/geographically diverse places, about 20% from schools
and universities worldwide, a fair amount of traffic comes from
outside of mathematics. I have made a concerted effort to provide
aesthetic math experiences to try to connect with people outside
the math community. Incorporating botanical names/information
for things such as plants regularly brings visitors here from
the outside; however, I don't know how they accept seeing
their plant of interest sitting with a geometric fractal :-).

Sathya Pillutla performed the initial groundwork for this site more
than four years ago in Spring 2002. Thank you for your investment
of time and effort, Sathya. It certainly has meant a lot to me.

Carlton VanLeuven designed a pattern I use to build component stage-1
blocks that significantly minimizes error in the Sierpinski tetrahedra.

I frequently relied on Brian Radspinner for answers to technical questions
about images and html code. I have lost track of Brian over the years, hope he is doing well, and would to find out what is happening with him. Thanks, Brian, for years of ongoing help!

Two professional friends whom I love to copycat are Paul Bourke and
Reimund Albers, usually on math stuff, but most recently, I copied
Reimund by putting up the map!