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



Several stages of growth of a fern viewed through a stage-4 Sierpinski tetrahedron, taken on Kachina Trail outside Flagstaff, Arizona, USA, ~8,500 ft. elevation, August 26, 2006.
stage-4 Sierpinski tetrahedron, taken at Slide Rock State Park, Arizona, USA.
Cathedral Rock in Sedona, Arizona, USA viewed through tetras, taken at dusk on 10/29/05.
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 Yaqui Point along Desert View Drive en route to Grand Canyon's East Rim
Taken on Hermit's Drive at Powell Memorial overlook on December 1, 2002.
Mather Point at dawn on 12/29/04 seen through tetras.
One of my first pictures, taken in Winter 2001 in Flagstaff, Arizona

      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, these visitors do not seem to want to see
their favorite plant sitting with a geometric fractal :-). I have no
proof that anyone has been swayed to view math differently as a
result of their visit here. Goals are on-track and progressing, but
the site has yet to hit a home run for Math, much work is needed.




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 rely on Brian Radspinner for answers to technical questions
about images and html code. 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!