Mandelbulb
Encyclopedia
The Mandelbulb is a three-dimensional analogue of the Mandelbrot set
, constructed by Daniel White and Paul Nylander using spherical coordinates
.
A canonical 3-dimensional Mandelbrot set does not exist, since there is no 3-dimensional analogue of the 2-dimensional space of complex numbers. It is possible to construct Mandelbrot sets in 4 dimensions using quaternions. However, this set does not exhibit detail in all dimensions like the 2D Mandelbrot set.
White and Nylander's formula for the "nth power" of the 3D vector
is
where
They use the iteration
where z^n is defined as above and a+b is a vector addition. For n > 3, the result is a 3-dimensional bulb-like structure with fractal
surface detail and a number of "lobes" controlled by the parameter n. Many of their graphic renderings use n = 8.
which we can think of as a way to square a triplet of numbers so that the modulus is squared. So this gives, for example:
or various other permutations. This 'quadratic' formula can be applied several times to get many power-2 formula.
which we can think of as a way to cube a triplet of numbers so that the modulus is cubed. So this gives:
or other permutations.
for example. Which reduces to the complex fractal
when z=0 and 
when y=0.
There are several ways to combine two such `cubic` transforms to get a power-9 transform which has slightly more structure.
for some integer m and adding terms to make it symmetrical in 3 dimensions but keeping the cross-sections to be the same 2 dimensional fractal. (The 4 comes from the fact that 
.) For example, take the case of 
. In two dimensions where 
this is:
This can be then extended to three dimensions to give:
for arbitrary constants A,B,C and D which give different Mandelbulbs (usually set to 0). The case
gives a Mandelbulb most similar to the first example where n=9. An more pleasing result for the fifth power is got basing it on the formula: 
.
These formula can be written in a shorter way:
and equivalently for the other coordinates.
where
where f,g and h are nth power rational trinomials and n is an integer. The cubic fractal above is an example.
and are considered less appealing.
Mandelbrot set
The Mandelbrot set is a particular mathematical set of points, whose boundary generates a distinctive and easily recognisable two-dimensional fractal shape...
, constructed by Daniel White and Paul Nylander using spherical coordinates
Spherical coordinate system
In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance of that point from a fixed origin, its inclination angle measured from a fixed zenith direction, and the azimuth angle of...
.
A canonical 3-dimensional Mandelbrot set does not exist, since there is no 3-dimensional analogue of the 2-dimensional space of complex numbers. It is possible to construct Mandelbrot sets in 4 dimensions using quaternions. However, this set does not exhibit detail in all dimensions like the 2D Mandelbrot set.
White and Nylander's formula for the "nth power" of the 3D vector
iswhere
They use the iteration
where z^n is defined as above and a+b is a vector addition. For n > 3, the result is a 3-dimensional bulb-like structure with fractalFractal
A fractal has been defined as "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole," a property called self-similarity...
surface detail and a number of "lobes" controlled by the parameter n. Many of their graphic renderings use n = 8.
Quadratic formula
Other forumulae come from identities which parametrise the sum of squares to give a power of the sum of squares such as:which we can think of as a way to square a triplet of numbers so that the modulus is squared. So this gives, for example:
or various other permutations. This 'quadratic' formula can be applied several times to get many power-2 formula.
Cubic formula
Other forumulae come from identities which parametrise the sum of squares to give a power of the sum of squares such as:which we can think of as a way to cube a triplet of numbers so that the modulus is cubed. So this gives:
or other permutations.
for example. Which reduces to the complex fractal
when z=0 and when y=0.There are several ways to combine two such `cubic` transforms to get a power-9 transform which has slightly more structure.
Quintic formula
Another way to create Mandelbulbs with cubic symmetry is by taking the complex iteration formula for some integer m and adding terms to make it symmetrical in 3 dimensions but keeping the cross-sections to be the same 2 dimensional fractal. (The 4 comes from the fact that .) For example, take the case of . In two dimensions where this is:This can be then extended to three dimensions to give:
for arbitrary constants A,B,C and D which give different Mandelbulbs (usually set to 0). The case
gives a Mandelbulb most similar to the first example where n=9. An more pleasing result for the fifth power is got basing it on the formula: .Power Nine formula
This fractal has cross-sections of the power 9 Mandelbrot fractal. It has 32 small bulbs sprouting from the main sphere. It is defined by, for example:These formula can be written in a shorter way:
and equivalently for the other coordinates.
Spherical Formula
A perfect spherical formula can be defined as a formula:where
where f,g and h are nth power rational trinomials and n is an integer. The cubic fractal above is an example.
Taffy
Shapes with long strands are commonly described as looking like taffyTaffy
Taffy can refer to any of the following:* Taffy , a type of chewy, often colored, candy* Taffy is a sometimes pejorative term for a Welsh person or thing...
and are considered less appealing.