The Dodecahedron

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The dodecahedron has 12 pentagon faces, 20 vertices, and 30 edges. The dual of the dodecahedron is the icosahedron (by interchanging the number of faces and vertices), with 20 faces, 12 vertices, and 30 edges. The ocean pictures taken in daylight are from this summer 2003 in Boca Raton. Then come photos taken last summer 2002, also in Boca Raton, with my collapsibles floating in a marine tank at the Gumbo Limbo Nature Center. Following those are photos from Summer 2002 taken at the ocean as the sun was rising

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The Gumbo Limbo photos and dodecahedron in darkness photos have extra information in the alt tags about the platonic solids, and natural fractals. This is something I intended to do for all the photos on the site, but have since backed off on the idea.

Ocean waves exhibit statistical self-similarity, in that a wave is made up of a lot of little waves that have the same basic shape as the bigger wave. Ocean waves are not fractal when the waters are too calm or too rowdy.  Visit: Statistical self-similarity in ocean waves can be seen from the window seat of a plane while it is descending over a rough ocean.  The waves will look the same as smaller and smaller sections of the ocean are in view, in spite of horizontal shifts in location of the plane.

The left and middle pictures are the same picture except the middle picture is lightened. The version on the left is what the camera took, taken before dawn, with the dodecahedron floating in the ocean waves. I couldn't see anything in the camera lens. The dodecahedron was barely visible in the water, and maintaining eye contact, I'd hurry up snap a picture as though it were visible in the lens. It was surprizing how many of them came out, even though the dodecahedron was moving with the water. In the far-right image, the flash illuminated a single face of the dodecahedron, and it looks rather eerie.

Setting math structures in nature can serve to highlight the structure in nature itself. Nature is filled with natural fractals. It is not the case, however, that all of nature is fractal.

The picture on the left needs to be enlarged to see it. The middle and right pictures have been lightened, as it was still near total darkness when these were taken.

The dodecahedron by itself is not a fractal. There is, however, such a thing as a dodecahedron fractal. The dodecahedron fractal is made from 20 dodecahedra put together in such a way to form another dodecahedron with volume removed.

None of these are lightened beyond the flash of the camera.

Not only do ocean waves have natural fractal properties, the edge of the water forms a fractal coastline in the sand. The water's edge creates a constantly changing natural fractal coastline, similar to actual coastlines that define the boundaries of continents.  The length of a coastline increases as the unit of measure is decreased. Visit this url to explore fractal coastlines:

The dodecahedron fractal is an instance of a geometric fractal.  Recall, the dodecahedron by itself is not a fractal, it is one of the five platonic solids. All five of the platonic solids can be put together to form their own fractal.  This is a very special property of the platonic solids.

The clouds in the sky are natural fractals.  A cloud is made up of a lot of little clouds. Clouds are independent of scale.  If just a piece of one were shown, one couldn't know if it were tiny, or huge. Notice the rocks my friend, Anna, is standing on.  Isn't a piece similar to the entire structure?  Rugged rocks are natural fractals.

The sequence of ocean pictures that begins in darkness was taken on July 19, 2002, in the 1600 block of North Ocean Blvd in Boca Raton. The last picture is of my friend, Anna, just before we left at about 6:55a.m. Two mornings later, I went back to repeat this with all of the inflatables, thinking it would be safe since the dodecahedron hadn't floated away. However, the tide must have been on its way out that morning. I tossed them all into the ocean in near darkness, turned around to do something with the camera, and when I turned back around, they were all floating away. By the time the sun came up, they were so far away as to be almost unrecognizable, and the entire set was lost to me. The photos taken in daylight at the top of the page are another set that I made to take with me to Florida this summer, 2003, and THIS TIME, by keeping track of the tides and a close watch on the structures, I didn't lose any of them to the ocean.

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