![]() ![]() ![]() Starting in the 17th century with notions of recursion, fractals have moved through increasingly rigorous mathematical treatment to the study of continuous but not differentiable functions in the 19th century by the seminal work of Bernard Bolzano, Bernhard Riemann, and Karl Weierstrass, and on to the coining of the word fractal in the 20th century with a subsequent burgeoning of interest in fractals and computer-based modelling in the 20th century. Sierpinski carpet (to level 6), a fractal with a topological dimension of 1 and a Hausdorff dimension of 1.893 A line segment is similar to a proper part of itself, but hardly a fractal. 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 locally fills space more efficiently than an ordinary line. Īnalytically, many fractals are nowhere differentiable. This power is called the fractal dimension of the geometric object, to distinguish it from the conventional dimension (which is formally called the 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 and is in general greater than its conventional dimension. Likewise, if the radius of a filled 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 conventional dimension of the filled sphere). Doubling the edge lengths of a filled 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 conventional dimension of the filled polygon). One way that fractals are different from finite geometric figures is how they scale. Fractal geometry lies within the mathematical branch of measure theory. This exhibition of similar patterns at increasingly smaller scales is 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, the shape is called affine self-similar. Many fractals appear similar at various scales, as illustrated in successive magnifications of the Mandelbrot set. In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. ![]() Zooming into the boundary of the Mandelbrot set Mandelbrot set at the cardioid left boundary The Mandelbrot set: its boundary is a fractal curve with Hausdorff dimension 2 Mandelbrot set with 12 encirclements Lichtenberg figures are examples of natural phenomena which exhibit fractal properties.For other uses, see Fractal (disambiguation). Although Lichtenberg only studied two-dimensional (2D) figures, modern high voltage researchers study 2D and 3D figures (electrical trees) on, and within, insulating materials. This discovery was also the forerunner of the modern day science of plasma physics. ![]() By then pressing blank sheets of paper onto these patterns, Lichtenberg was able to transfer and record these images, thereby discovering the basic principle of modern xerography. After discharging a high voltage point to the surface of an insulator, he recorded the resulting radial patterns by sprinkling various powdered materials onto the surface. In 1777, Lichtenberg built a large electrophorus to generate high voltage static electricity through induction. When they were first discovered, it was thought that their characteristic shapes might help to reveal the nature of positive and negative electric “fluids”. Lichtenberg figures are named after the German physicist Georg Christoph Lichtenberg, who originally discovered and studied them. This Art would look spectacular on a wall or on a stand.Ī forever gift of what Mother Natures High Voltage Energy Looks Like and she is a great artist !! The Artwork has a polyurethane finish of three coats and measures 12 inches high and 3 1/2 wide by 1/2 inch thick and Weighs 8 Ounces. The Lichtenberg (Fractal) pattern was created using 3200 Volts at. This Branded Artwork 2992 is made From Red Wood. ![]()
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