For further technical information see Self Similarity and Gasket & Sponge
To see more math in nature photos, visit Gayla's Fun with Fractals and the Platonic Solids pages.

[Click on the images for an enlarged version]

 The left and center photographs were taken at the South Rim of the Grand Canyon on Hermit's Rest Route at daybreak. The photo on the right was taken on Desert View Drive that leads to the East Rim of the Grand Canyon. All of the photographs are real. The movement of the page is from Winter through Autumn. Plant names will be provided when I can get them from horticulturists.

 The first image was taken in misty rain around Trail Overlook at the South Rim. I couldn't risk taking my Sierpinski tetrahedra out in that weather. The other two were taken on Hermit's Rest Route. Sometimes, there just weren't any good places to set the tetrahedrons. These photos are rich in natural fractal structures with statistical self-similarity. If uncertain about the concepts of natural fractals and statistical self-similarity, there is a link to Paul's Self-Similarity page at the top of this page. It is a short page that won't take much time to look over, and it should add meaning and context to many of the photographs. The photographs basically juxtapose the geometric fractal Sierpinski's tetrahedron with natural fractal structures in the environment. Geometric fractals use different shapes than natural fractals, and they grow in different ways. There are many types of fractals.

 The above three photos were all taken at Boyce Thompson Arboretum State Park, located outside of the small mining town of Superior, Arizona. The plant on the left is some kind of aloe or agave. In the middle photo, a stage-4 Sierpinski tetrahedron is sitting on the limb of a River Bushwillow tree. To the right, a stage-3 and a birds nest complement each other nicely.

 The left and middle photos were taken at ASU in Spring 2003. The structure on the far left is exceptionally tiny, with a total edge-length of 8 inches, made up of 256 1/2-inch tetrahedra set tip to tip. The center photo shows a nearly birds-eye view of the stage-3. The 4-color stage-4 on the right is set against Scarlet Macaws at the Phoenix Zoo, taken in Spring 2001. The macaw at the top was in full wingspan moments before I snapped the picture. The tetrahedron is painted in Cerulean Blue hue, Naphthol Red Light, Cadmium Yellow Medium, and Deep Permanent Green.

 The left image was taken in a dark, dense thicket of Everglades National Park. The middle and right images were taken at the Morikami Japanese Gardens in Delray Beach, Florida. In the center, the red flowers are Pentas (Pentas lanceolata). Moving right, the yellow flowers are Allamanda (Allamanda cathartica), and a little further back, is Fountain grass (Pennisetum setaceum). The Morikami provided the names.

 Sections from both images were used for the poster sitting between them, which I had no part in making, am merely showing it here. The image on the left was taken at the Morikami Museum and Japanese Gardens in Delray Beach, Florida. The poster was made to publicize a lecture series presented to high school students in Bremen, Germany, to stimulate interest in higher mathematics.

 All images were taken on the beach at Boca Raton. There is a paraglider in the sky in the left and middle photos. The photo on the right is included because it seems to shimmer, an effect apparently created by the sun on the wet sand combined with the fuzziness, giving a surreal effect. It was either dusk or daybreak, and I didn't use the flash. For some reason, when this happens, the camera takes a "lighter" picture without the flash but cannot capture lines sharply. These were taken in the block of 1000 North Ocean Blvd. in Boca Raton, Florida.

 Moving into Autumn now, these photos were taken in Sedona in October 2002. The middle photo shows a beautiful branching structure that looks a lot like branching in the body. At this time, I had begun experimenting with stacking the structures.

 The left image was taken at a commercial trout fishing spot just north of Sedona on Scenic Route 89A. The beautiful coppery-red bush on the right is a dead or dying Manzanita. Thriving Manzanitas stay green all year long.

 Taken in Oak Creek Canyon in November 2002 along scenic Hwy 89 in the Switchbacks between Sedona and Flagstaff, the left and middle photos show turning Maple leaves. On the right is a stage-2 Sierpinski tetrahedron poised with the brilliantly turned leaves of an African Sumak.