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The usage of the word "gasket" to refer to the Sierpiński triangle refers to gaskets such as are found in motors, and which sometimes feature a series of holes of decreasing size, similar to the fractal; this usage was coined by Benoit Mandelbrot, who thought the fractal looked similar to "the part that prevents leaks in motors". [23]
Sierpiński curves are a recursively defined sequence of continuous closed plane fractal curves discovered by Wacław Sierpiński, which in the limit completely fill the unit square: thus their limit curve, also called the Sierpiński curve, is an example of a space-filling curve.
The diagram shows the construction on an IFS from two affine functions. The functions are represented by their effect on the bi-unit square (the function transforms the outlined square into the shaded square). The combination of the two functions forms the Hutchinson operator. Three iterations of the operator are shown, and then the final image ...
Sierpiński demonstrated that this fractal is a universal curve, in that any possible one-dimensional graph, projected onto the two-dimensional plane, is homeomorphic to a subset of the Sierpinski gasket. For curves that cannot be drawn on a 2D surface without self-intersections, the corresponding universal curve is the Menger sponge.
Certain fractal metric spaces such as the Sierpiński gasket can be equipped with the α-dimensional Hausdorff measure where α is the Hausdorff dimension. In general, however, a metric space may not have an "obvious" choice of measure. One application of metric measure spaces is generalizing the notion of Ricci curvature beyond Riemannian ...
An n-flake, polyflake, or Sierpinski n-gon, [1]: 1 is a fractal constructed starting from an n-gon.This n-gon is replaced by a flake of smaller n-gons, such that the scaled polygons are placed at the vertices, and sometimes in the center.
An object whose irregularity is constant over different scales ("self-similarity") is a fractal (examples include the Menger sponge, the Sierpiński gasket, and the Koch curve or snowflake, which is infinitely long yet encloses a finite space and has a fractal dimension of circa 1.2619).
Sierpiński triangle generated by Rule 90. The time-space diagram of Rule 90 is a plot in which the i th row records the configuration of the automaton at step i. When the initial state has a single nonzero cell, this diagram has the appearance of the Sierpiński triangle, a fractal formed by combining triangles into larger triangles. Rules 18 ...