Q: As a fractal the Mandelbrot Set has fractional dimension?

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A fractal is a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole (self similar). The term "fractal" was coined by Benoît Mandelbrot in 1975 and was derived from the Latin fractus meaning "broken" or "fractured." A mathematical fractal is based on an equation that undergoes iteration, a form of feedback based on recursion.

The Mandelbrot set is named after Benoît B. Mandelbrot.The Julia set is named after Gaston Maurice Julia.

The Mandelbrot set is a set of points satisfying a particular criterion (discussed in more detail below). It doesn't "represent" anything, it's just a set of points. The colorful images you sometimes see are not just the Mandelbrot set (a point is either in the set or it isn't), but also points outside the set colored in a particular way which can be thought of as representing how long it took to decide that the point was not in the set.The way to generate a Mandelbrot set is this:For each point c in some region of the complex plane (a cartesian coordinate system where the X value represents the "real" part of a complex number and the Y value represents the "imaginary" part of the complex number), a mathematical operation is performed. This operation is simply to iterate the following equation:zn+1 = zn2 + c(where z0 = 0).If the absolute value of zn remains bounded, the point c is in the Mandelbrot set. If, however, the value of zn goes to infinity as n goes to infinity, then the point is not in the set.The coloring is generally based on the number of calculations (basically, the value of n) before the absolute value got larger than some cutoff (often the cutoff is 2; once the absolute value reaches 2, the z value is certain to go to infinity eventually).The interesting thing about the Mandelbrot set is that it's not a simple shape, as you might initially expect, but a highly irregular shape. Benoit Mandelbrot, for whom the set is named, coined the term "fractal" for such complicated shapes.

The Mandelbrot graph is generated iteratively and so is a function of a function of a function ... and in that sense it is a composite function.

Triangles, spheres, pentagons, cylinders, circles, ellipses, the Mandelbrot Set, etc.

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In mathematics, a fractal is a subset of Euclidean space with a fractal dimension that strictly exceeds its topological dimension. Fractals appear the same at different scales, as illustrated in successive magnifications of the Mandelbrot set.

A Mandelbrot set is a mathematical set. Its boundaries are two-dimensional, easy recognizable fractal shapes. It is named after Benoit Mandelbrot, a Polish-born mathematician.

A fractal is a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole (self similar). The term "fractal" was coined by Benoît Mandelbrot in 1975 and was derived from the Latin fractus meaning "broken" or "fractured." A mathematical fractal is based on an equation that undergoes iteration, a form of feedback based on recursion.

The father of the Fractal Trigeometry is Jules Ruis. He developed the so called Julius Ruis Set, being a smart presentation of 400 Julia Sets, indicating that the Mandelbrot Set is the parameter basin of all closed Julia Sets.

The Mandelbrot set is named after Benoît B. Mandelbrot.The Julia set is named after Gaston Maurice Julia.

It really depends on the fractal, and there are many possible ways to define them. As an example, for the Mandelbrot set, a number of calculations involving complex numbers are done for each point in the complex plane, to determine whether a point is part of the set or not. However, other definitions are possible as well.

The Mandelbrot set is a set of points satisfying a particular criterion (discussed in more detail below). It doesn't "represent" anything, it's just a set of points. The colorful images you sometimes see are not just the Mandelbrot set (a point is either in the set or it isn't), but also points outside the set colored in a particular way which can be thought of as representing how long it took to decide that the point was not in the set.The way to generate a Mandelbrot set is this:For each point c in some region of the complex plane (a cartesian coordinate system where the X value represents the "real" part of a complex number and the Y value represents the "imaginary" part of the complex number), a mathematical operation is performed. This operation is simply to iterate the following equation:zn+1 = zn2 + c(where z0 = 0).If the absolute value of zn remains bounded, the point c is in the Mandelbrot set. If, however, the value of zn goes to infinity as n goes to infinity, then the point is not in the set.The coloring is generally based on the number of calculations (basically, the value of n) before the absolute value got larger than some cutoff (often the cutoff is 2; once the absolute value reaches 2, the z value is certain to go to infinity eventually).The interesting thing about the Mandelbrot set is that it's not a simple shape, as you might initially expect, but a highly irregular shape. Benoit Mandelbrot, for whom the set is named, coined the term "fractal" for such complicated shapes.

The Mandelbrot graph is generated iteratively and so is a function of a function of a function ... and in that sense it is a composite function.

Triangles, spheres, pentagons, cylinders, circles, ellipses, the Mandelbrot Set, etc.

The set of rational numbers is the union of the set of fractional numbers and the set of whole numbers.

Sierpinski's Triangle Sierpinski's Carpet The Wheel of Theodorus Mandelbrot Julia Set Koch Snowflake ...Just to name a few(:

By definition, a mixed number contains a fractional part. Also, by definition, an integer does not have a fractional part.