"[…] the world is not complete chaos. Strange attractors often do have structure: like the Sierpinski gasket, they are self-similar or approximately so. And they have fractal dimensions that hold important clues for our attempts to understand chaotic systems such as the weather." (Manfred Schroeder, "Fractals, Chaos, Power Laws Minutes from an Infinite Paradise", 1990)
"The basic geometric construction of the Sierpinski gasket goes as follows. We begin with a triangle in the plane and then apply a repetitive scheme of operations to it (when we say triangle here, we mean a blackened, ‘filled-in’ triangle). Pick the midpoints of its three sides. Together with the old vertices of the original triangle, these midpoints define four congruent triangles of which we drop the center one. This completes the basic construction step. In other words, after the first step we have three congruent triangles whose sides have exactly half the size of the original triangle and which touch at three points which are common vertices of two contiguous triangles. Now we follow the same procedure with the three remaining triangles and repeat." (Heinz-Otto Peitgen et al, "Chaos and Fractals: New Frontiers of Science" 2nd Ed., 2004)
"The second surprise is that the copy machine paradigm is not just a way to recover ‘mathematical monsters’ like the Sierpinski gasket or its relatives (soon we will see many of them). Let us ask what the images are which we can obtain this way. What can they look like? The answer is simply incredible. For many more natural pictures there is a copy machine of the above kind which generates the desired picture. However, it is a difficult problem to design the machine for a given picture." (Heinz-Otto Peitgen et al, "Chaos and Fractals: New Frontiers of Science" 2nd Ed., 2004)
"[...] the Sierpinski gasket is absolutely tame when compared with the carpet, though visually there seems to be not much of a difference. The Sierpinski gasket is a hotel which can accommodate only a few (one-dimensional, compact) very simple species from flatland. Thus, there is, in fact, a whole world of a difference between these two fractals." (Heinz-Otto Peitgen et al, "Chaos and Fractals: New Frontiers of Science" 2nd Ed., 2004)
"[...] we may be led to believe that the secret to the tendency toward the formation of the Sierpinski gasket is our choice of an appropriately dimensioned rectangle as the initial image in starting the feedback process. To show that this is not the case, let us assume that instead of a rectangle as the initial image, we choose a triangle or any arbitrary image, which may be represented well enough by the letters NCTM. The question is: What will then evolve in the process? [...] The same final structure is approximated as we run the machine. Each step produces a composite of images which rapidly decrease in size. It doesn’t matter in the least whether these images are rectangles, triangles, or the letters NCTM; the same final composite image is approached in each case - namely, the Sierpinski gasket. In other words, the machine produces one - and only one - final image in the process, and that final image is totally independent from the image with which we start! This magnificent behavior seems to be a miracle. But in mathematical terms it just means that we have a process which produces a sequence of results tending toward one final object which is independent from how we start the process. This property is called stability." (Heinz-Otto Peitgen et al, "Chaos and Fractals: New Frontiers of Science" 2nd Ed., 2004)
"The chaos game may seem like a special trick that can produce Sierpínski triangle but little else. However, the procedure of using randomness to make fractals turns out to be quite general. For example, if one plays the chaos game with four points arranged in a square, not surprisingly one obtains the Sierpínski carpet [...]. Slightly more complicated versions of the chaos game can yield many other fractal shapes. In fact, it turns out that almost any shape - fractal or nonfractal - can be generated by a version of the chaos game. A generalized version of the chaos game turns out to be remarkably powerful and flexible." (David P Feldman,"Chaos and Fractals: An Elementary Introduction", 2012)
"Once you look at the Sierpinski triangle for a very long time you see more consequences of the construction, but they are rather short consequences, they don't require a very long sequence of thinking. In a certain sense, the most surprising, the richest sciences are those in which we start from simple rules and then go on to very, very long trains of consequences and very long trains of consequences, which you are still predicting correctly." (Benoît Mandelbrot, [Segment 70 - Peoples Archive interview])
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