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What Is A Fractal (and what are they good for)? | Chia sẻ có ích nhất về thiết kế

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46 Comments

  1. The maths of fractals are not exactly easy:
    One way that fractals are different from finite geometric figures is how they scale. Doubling the edge lengths of a polygon multiplies its area by four, which is two (the ratio of the new to the old side length) raised to the power of two (the dimension of the space the polygon resides in). Likewise, if the radius of a sphere is doubled, its volume scales by eight, which is two (the ratio of the new to the old radius) to the power of three (the dimension that the sphere resides in). However, if a fractal's one-dimensional lengths are all doubled, the spatial content of the fractal scales by a power that is not necessarily an integer. This power is called the fractal dimension of the fractal, and it usually exceeds the fractal's topological dimension.

    Analytically, most fractals are nowhere differentiable. An infinite fractal curve can be conceived of as winding through space differently from an ordinary line – although it is still topologically 1-dimensional, its fractal dimension indicates that it also resembles a surface…

  2. Using the snowflake example, the snowflake's area wouldn't be finite if you didn't draw a circle around it. It would keep growing indefinitely wouldn't it? Not just the perimeter but the inside (area), too.

  3. It is the other way around. Math is made out of everything (we observe). If we would not have the impression of space then geometry would not even exist. That said. All though math and observed world have huge similarities, difference remain to exist. Although some natural patterns repeat really a lot in the world, they always stop at some point. There seem to be no never ending patterns. I like the video it is very clear and well made.

  4. I think its not valid to say it has finite area and the analogy given here seems flawed to me,

    If you consider a rubber-band that is circular in nature and try to stretch it, because its perimeter is finite, it will not go beyond a particular expandable area.

    On the other hand, if you have a rubberband that is a mathematical model for koch-snowflake and try to expand it using your hand, you will notie that it will keep on expanding and won't reach at any final unstretchable stage, also you will notice that by applying this deformation action on the koch snowflake, you are actually trying to convert it into a circle, however now because the perimeter is infinite, the area too will end up infinite.

    Also i wonder how this stretching-action being performed on koch-snowflake would feel like ? i assume initially it will be very easy to stretch it but later on it will resist this stretching action a little or maybe the speed at which the stretching will take place will keep on decreasing and the manual power required to stretch it further will keep on increasing ?? hmmm great content (Y)

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