Math illustrations key1/8/2023 ![]() Īnalytically, many fractals are nowhere differentiable. This power is called the fractal dimension of the fractal, and it usually exceeds the fractal's topological dimension. 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. 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). 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). One way that fractals are different from finite geometric figures is how they scale. Fractal geometry lies within the mathematical branch of measure theory. Fractals often exhibit similar patterns at increasingly smaller scales, a property called self-similarity, also known as expanding symmetry or unfolding symmetry if this replication is exactly the same at every scale, as in the Menger sponge, it is called affine self-similar. Fractals appear the same at different scales, as illustrated in successive magnifications of the Mandelbrot set. ![]() In mathematics, a fractal is a subset of Euclidean space with a fractal dimension that strictly exceeds its topological dimension. Zoom-in of the boundary of the Mandelbrot Set
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