Mandelbrot spent much of his working life in both France and the United States and is credited with the development of fractal geometry. The word derives from the Latin frangere (to break) and fractus (broken, uneven). The word fractal was first used in 1975 by the Polish-born mathematician Benoit Mandelbrot (1924–2010). Look these up to get a nice idea of how a mathematical fractal is generated. The Sierpinski Triangle and the Cantor Set, or Cantor Dust, are two classic fractals whose algorithms (set of instructions for their construction) can be easily understood. Stick the resulting two pieces of tape to a piece of paper, and create art or poetry inspired by the fractal images. Make a fractal pattern on clear tape (see Alternative Construction above). In fact, in this kind of fractal you can’t tell the difference between the whole object and a magnification of any section you select-no matter how small! The image below shows a classic mathematical fractal, the Sierpinski Triangle, or Sierpinski Gasket. And if you were to select an even smaller piece from the first section and magnify it, this piece would also duplicate the whole. If you look at a small section of a mathematical fractal, the section will be identical to the whole object. The fractals they create are called perfect or mathematical fractals. These repeating but nonidentical patterns are called self-similar.Įven though nature can’t generate a perfect fractal pattern, mathematicians can. If you break a floret of cauliflower or broccoli off the larger head, for example, you can see that it's like a miniature version of the larger head, but it's not an exact replica. As the patterns repeat themselves at different scales, each section of the whole is similar to large- and smaller-scale structures, but are never an exact copy. This is typical of the pattern formed when fluid flows from tributaries into a central stream or flows out from a main course into several branches.Īll the fractal patterns formed in nature-including the ones you just made-are generated by random processes. The photo below shows another example of a natural fractal pattern, the branching of a river.
A few others are clouds, coastlines, jellyfish tendrils, coral reefs, and blood vessels in the lungs. The patterns created with this process often remind people of tree branches or root systems, river deltas, or lightning bolts, all of which are outstanding examples of fractal patterns in nature. By the time the two plates are separated, the fingers of air have formed intricate branching structures in the paint. Random indentations in these fingers grow as well. Small indentations of air grow and become fingers of air. When the plates are pulled apart, the less viscous air penetrates the more viscous paint, creating an unstable boundary. Your fractals are the result of a process called viscous fingering: As the paint is squeezed between the plates, the viscous paint spreads out evenly in all directions into the less viscous air layer, creating a stable, disk-shaped boundary. The tape has to be wide enough to allow the pattern to be created on a single piece two-inch-wide transparent packaging tape works well. This allows you to reuse the plastic, or to tape the pattern to a piece of paper or use it in some other creative way. What do you think the patterns look like? Do they remind you of anything? Let the paint patterns dry on the plates if you want to preserve them.Ĭover your pieces of plastic with transparent packing tape so you can peel off the fractal pattern when the paint is dry. Notice that the patterns are mirror images of each other. Once the plates are separated, observe the patterns on each one. Watch air flow into the paint as you pull the plates apart, forming a fractal pattern. It's very important that you pull the plates straight off one another. Notice the paint spreads into a disk (see photo below).Ĭarefully pull the plates apart as shown below. Squeeze the two plates together firmly, so that the paint drop forms the thinnest possible circular layer between them.
(You will be pulling the plates apart, and if the edges are lined up this may be difficult see photo below.) Place the second plate on top of the paint, but don't line up the edges of the top plate exactly with the edges of the bottom plate.
Then use the same implement to place a tiny drop of paint at the center of one of the plastic pieces, which are your plates (click to enlarge photo below). Use a straightened-out paper clip, a toothpick, a bamboo skewer, or a nail to stir the paint.
0 kommentar(er)
