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The first photo shows assorted collapsibles floating in a marine tank at the Gumbo Limbo Environmental Center at 1801 North Ocean Blvd. in Boca Raton, Florida. In this particular tank there are several varieties of fish, a sea star, a few stingrays, and a poisonous spiny looking creature that I didn't get a picture of (but plan to this summer 2003). The tank has a huge circumference, is fairly shallow, with sand covering the bottom, probably because stingrays like sand. The middle photo shows a stingray with the collapsibles. The fish darted from the stuctures, while the stingrays stayed close. The photo on the right shows a permutahedron for n=4 floating in a fountain at Arizona State University. Its 24 vertices are the 24 permutations of the set {1,2,3,4} or any other set of 4 distinct elements. It happens that the permutahedron for n=4 is either one and the same structure as the snub octahedron, or is isomorphic to the snub octahedron, one of the Archimedean solids. The permutahedron for n=3 is a hexagon, whose vertices are the 6 permutations of {1,2,3}. The permutahedron for n=2 is a line with endpoints being the 2 permutations of {1,2}, and the permutahedron for n=1 is a single point. Shelly Smith, a graduate student with the Department of Mathematics and Statistics at ASU, has done research on the permutahedron, and helped me with wording/notation to ensure accuracy/clarity of the information presented here. |
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These were taken in summer 2002 with my inflatable dodecahedron floating in shallow ocean waves before sunrise in the block of 1000 North Ocean Blvd. in Boca Raton, Florida. I went again a couple of mornings later, this time tossing in all of the collapsibles instead of just the dodecahedron, and they floated away. I hadn't checked the tide schedule; it was on its way out. By the time I realized what was happening, it was too late to do anything about it. The images look grainy, especially enlarged, because they had to be lightened considerably in order to see them. It was still almost dark at the time these were taken. |
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The first two photos are a continuation from the row above, a little later on. The picture on the right was taken one afternoon in Boca Raton. I've included it because the tetrahedron is underrepresented on the page, as is the octahedron, with the exception that there are some nice views of the octahedron in the pictures in the top row. |
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The photo on the left was taken at the Phoenix Zoo in Springtime 2001. The icosahedron has inside of it an interior structure of an intersection of 3 golden rectangles. The center photo shows the same structure in a different plant, also taken at the Phoenix Zoo. The picture on the right was taken at the Desert Botanical Gardens in Phoenix. The blue structure at the top of the image is a stage-2 Sierpinski tetrahedron. |
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The above two photos are accompanied by coloring sheets. They are hand traces, with the aid of a straight edge to make the math structures. In the photo to the left, the icosahedron is sitting in Purple Fountain Grass (Pennisetum setaceum v. 'Cupreum'). The photo on the right shows an exceptionally clear view of how the vertices of the interior structure have the same vertices as the icosahedron. The rectangles intersect at 90 degree angles. I have, however, seen the rectangles intersect at angles other than 90 degrees and still share the vertices of the icosahedron.
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On the left is my favorite of all the icosahedron pictures, because it looks like the rose bush "grew" an icosahedron, so well does it fit into the picture. To the right are two views of a stage-1 dodecahedron fractal that is sitting inside of a dodecahedron (the stage-0), all made from transparency film. The stage-1 is made up of 20 dodecahedra connected together to form another dodecahedron with volume removed. My construction of it is very crude, embarrassingly so, but it might be enough to get a sense of the structure. The advantage of using transparency film was being able to see through it, yet this left the interior with very little definition. It is hard to see the stage-1 inside! A solution might be to leave the outer dodecahedron transparent, but add color to the 20 miniature dodecahedra patterns (leaving the tabs clear so not to double up on the color) and print them out on transparency film. This would let the stage-1 structure stand out vividly inside the outer clear dodecahedron, and would maintain at least a semi-transparent quality to the overall structure. In the image on the far right, the little purple flowers have 5-fold symmetry (n-fold symmetry where n=5), set off nicely against the 5-fold symmetry of the dodecahedron. |
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The first two images show some miniature structures inside a sphere of radius 2 inches: six tiny stage-1 Sierpinski tetrahedra and a transparent dodecahedron (of the same size used for the stage-1 dodecahedron fractal above). No reason for doing this except for fun. In the middle picture, I left this same sphere to sit in a tree nook while it was snowing in Flagstaff, Arizona, during the winter of 2002. The picture on the right is a home for 20 tetrahedron families made from a packing of 20 empty tetrahedron compartments with surface flaps that open and close. It packs nicely into a box and enables transport of my delicate Sierpinski tetrahedron families in the baggage compartment of a plane. It also comes in handy for transporting the structures when out taking photos, keeping them safe, contained, and manageable. The measuring stick used in the photo is 12 inches long (~31 centimeters). |
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