Sierpinski Tetrahedra at the Morikami Museum
and Japanese Gardens

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The center image shows the Reclining Hotei, one of Seven Gods of Good Fortune. It sits in the main lobby of the Morikami. A museum is accessible from their lobby, and a beautiful tea room that looks like it is set up for attending shows or instructional talks.

Although branches usually have a fractal structure, the leaves that grow on them generally do not.  One exception is the fern family.  The leaves of all ferns display a fractal structure. Do individual petals of the Allamanda blossom have the overall shape of the entire blossom?  They don't look like it to me, and there doesn't appear to be much fractal off-branching on the plant, either. There is nice fractal branching visible on the Pentas.  It is hard to see for the leaves, but there are lots of branches, and the little branches look very similar to the big ones.

These were taken from an area in the front section of the Morikami gardens. The orange flowers are blooms of the Fire Bush (Hamelia patens). The yellow flowers are Allamanda (Allamanda cathartica) where to the right in the same picture, and a little further back, is Fountain grass (Pennisetum setaceum). In the picture on the far right, the red flowers are Pentas (Pentas lanceolata). In each picture is a family of Sierpinski Tetrahedra.

My Sierpinki tetrahedra, half-hidden here in their geometric home, are instances of a geometric fractal that uses man-made shapes. Natural fractals don't use man-made shapes.  For instance, tree branches and clouds and waves aren't made of tetrahedrons. Natural fractals have parts that look similar to the whole, called statistical self-similarity.  A piece of a tree looks similar to the entire tree.  A rough ocean wave is made up of a lot of little waves that look just like it.

The left and middle pictures were taken overlooking Morikami Lake, shown from a different angle on the Morikami Museum and Japanese Gardens home page. The left image (from Summer 2002) shows a home for 20 of my Sierpinski tetrahedron families, made of a packing of 20 tetrahedron compartments that have surface flaps which open and close. In the middle picture, I was looking almost straight down into the lake. The poster on the right has a slice of the middle picture in it, and was made (not by me) for a lecture series presented to high school students in Bremen, Germany. The background image in the poster is from a greatly lightened picture taken in Summer 2002 in the 1000 block of N. Ocean Blvd., Boca Raton, with the Sierpinski tetrahedra sitting on a concrete ledge overlooking the ocean.

I have included the poster to provide visual affirmation that the math structures in these images have meaningful connections with nature. Sierpinski tetrahedra are made up of parts that are similar to the whole, as are trees and waves, rocks, clouds, to name a few.

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The left picture shows the original museum, called the Yamato-Kan. To go inside, one must wear paper shoes provided at the entrance. I wasn't able to go inside with the tetrahedrons. The image on the far right was taken around the back of the building. You can see how beautiful it is even from the outside, the center image was taken a mere 5 feet away from it. On the opposite side is the Bonzai Garden. Right next to it, the lake comes around, and there is a bench sitting next to a feeding area for fish and turtles swimming in the lake, shown further down on the page.

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In the left image, look at the intricate branching. Can you see how the smaller branches resemble the larger branches? These are fractal properties. The center image shows the same tree, in perspective. It is a Bonzai, probably very old. My pictures don't do the Bonzai justice. The sun was up too high when I got to them.

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I had to be extremely careful not to let the Sierpinski tetrahedra touch the Bonzai, as they are very delicate, well-tended, and well-protected by Morikami employees.

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The above photos are of Morikami Lake right outside the Yamato-Kan (the original museum). Waterlife congregates here because it is a feeding spot. Food for the fish and turtles is available for purchase at the museum entrance for a very low cost, fifty cents when I was there in Summer 2003.

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Funny story about the image on the left. I was at the Morikami early on Sunday morning, before they opened to the public. While the tetrahedrons were sitting on this big rock, I walked away for a moment, and an employee cleaning the area with an industrial leaf blower ran it across the area without thinking, sending the tetrahedrons flying. He said he always cleans that rock automatically because people leave things on it. Some serious repairs were needed on the spot, but the tetrahedrons survived. :)

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All of the images in this row bring up different enlargements. The images are from two different Zen gardens. The garden shown in the center and far-right photos is huge. The image on the left was used for a poster for the Lehrerakademie Bremen's 2005 National German Teacher Enhancement Program.

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Over time, as I would set the tetrahedra in nature, I began to notice that much of nature is quietly iridescent. Life glows, much of it. The paint for the peach-toned set of structures is loaded with iridescence. They were the only set of structures with me in Summer 2003. Most of the earlier-made, more colorful sets of structures (like the yellow, orange, and red sets seen earlier on this page) are painted with pure pigments but they aren't translucent and there is no iridescence added to the paint. The first image on the page, however, shows a set of light blue structures that is from another iridescent set, made especially for the ocean, also shown in the poster several rows up.

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