The value, after every iteration of the algorithm at the origin, is always zero.
The line across the middle of [Mandelbrot Set] is drawn with exaggerated width to enable it to be more visual.
The line has been magnified at the boundary region between the point of capture and the escaping numbers. This shows coloured bands. Each colour represents the number of iterations it takes before deciding wether the value is caputered or escapes.
The Mandelbrot Set
The Complex Plane
The image is produced by taking the values from a coordinate in the complex plane and squaring them then adding the original values. This gives new values and with these the instructions are repeated till the value gets too large or until it has been repeated enough times
The Sequential lines are drawn by taking a value from the coordinate of the line and the other value is determined by the distance from the real axis and putting the values into the [algorithm] and then colour coordinate according to the result.
All these images are a compromise between resolution, magnification the maximum number of iterations and the escape value. By increasing all these values a more accurate image can be drawn.
The Self Similarity of the shape is maintained though out the set. It is possible to find similar looking copies of the set at countless magnifications of the set.
The miniature M-Sets are all connected by strands, that have no width and therefore when magnified these stands appear no thicker. These stands are unmagnifiable.
Complex numbers came from the solution ( x² + 1 ) = 0
To explain further 2 × 2 = 4 and so does -2 × -2 = 4.
Therefore the 4 is 2 or -2.
So what then is the solution to the -4 ?
The answer was the Complex Number 2 × i, where i × i = -1
The complex number, i, can be seen as a transformation of a 90° rotation.
|| So then i² would be a rotation of a 180°|
Displaying Complex Numbers
Complex numbers can be drawn on a complex plane with the Real numbers on a number line across the plane and the Imaginary number on the vertical.
The Mandelbrot algorithm is defined as:-
zk+1 = zk² + z0
where z0 = a + bi a : real part
b : imaginary part
k : the iteration number
0 < k < 100
if |zk| > 256 then exit the loop
and colour the point according
to the value of k
The [Mandelbrot set] is produce by taking a coordinate from the grid and putting the values into the algorithm, which gives the colour of the coordinate according to the result.
I feel that I need to mention Quaternion Complex numbers, because they are seen as the next step in complex maths in dimensions greater than two. I was unsatisfied with the models they produced when I used them to look for the 3D Mandelbrot Set. Using Quaternion maths gave me a 3D set but it was as if it had been turned on a lathe, with all cross sectional planes though the diameter the same, the Mandelbrot set. I was unsatisfied because it was to regular, and I was expecting bubbling cloud-like shape, irregularly complex in all dimensions of viewing.
Different Complex Numbers
With the understanding that [imaginary number i] can be represented by rotation of 90° about the origin. I defined [imaginary number j] as a rotation of 90° about the origin and a rotation of 90° with respect to [i].
So with this in mind I picked up a cube, to help me visualise the rotations, and then recorded the results.
I rotated it upwards 90° [i], then following it by turning it a different 90° backwards [j], I defined the result:
i × j = -j
Then starting again
I rotated 90° backwards [j], followed by 90° upwards [i].
This gave me the result:
j × i = i
It is the use of these results when applied into the [Mandelbrot algorithm], that produce the [Bristor set]. Imaginary numbers in additional dimensions can be derived by substituting in the new number in the above equations.
The Bristor Set or the Generalized Mandelbrot Set
I have always found it a bit strange and a bit egocentric to call this set after myself, but I feel this is one of my main life changing acheivements. The set is described by the [M-Set algorithm] with the application of my derived numbers. It is though the sets poetry that for me justifies this branch of complex mathematics for me.
The title 'the Generalized Mandelbrot Set' is given to M-Set drawn with [Quaternion] numbers.
The Bristor Set
As a 3D model the image is produced by ray tracing. This is done by taking the x,y coordinate then moving through the z planes testing for the set. On contact a ray is then aimed at different coloured light sources. The point is coloured according to the number of rays that reach the light sources.
The Sequential planes are drawn by taking a coordinates from the grid as well as the layer position and putting these values into the [algorithm] and then colouring the coordinate according to the result.
The islands are drawn in the same way as the 3D model with the sea being drawn at sequential levels.This gives the view of an island at high and low tides.
The parallel variations of the [Mandelbrot set] are drawn by taking a series of rotated planes though a tube that pass the center axis. I have called them parallel from a visual perspective, by the fact that they are all generally the same shape, yet all unique bar for the exception when a plane is the planar axis. On these two planes, when i = 0 and j = 0 the Mandelbrot Set is drawn.
The Sequential planes are examined for the first occurrence of the set. The plane is then magnified showing a complex fractal. The sequential plane is then stepped back and a new first occurrence is displayed. A magnification is made and this process is repeated.
||z = 1.0||
|z = 1.1|
|z = 1.25|
Self Similarity of the shape is maintained though out the set although the reproductions have more pronounced distortions than the reproductions of the M-Set. It is possible to find similar looking copies of the set at countless magnifications of the set.
The surface of set is an evolving repeated texture.
Magnification of the set gives increased detail. Magnification is limited by the parameters of the computer rather than the set.
The miniature B-Sets are connected by unmagnifiable strands. The strands vary in type from seeming to be like and evolving mesh or net, to being like lightning forks.
Thoughout the B-Set when looking in any dimension there is a sense of evolving from one pattern to the next. The surface of the shape has a textured pattern, like tree bark.
The cross sections give what seems to be a natural growth, like a flick of flame, or a cloud evapourating.
The 4D Bristor Set
The 3D sets are [ray traced] as in the same way as the [3D Shape], with the 4th Dimension running across the diagonal, the models are grayed, the further back they are positioned. Each model is layered over the previous models. The appearance of this image is not too dissimilar to the 3D Shape because so much information is lost; every layer is in effect flattened, and placed over the previous layers.
The sequential images are [ray traced] in the same manner as the [3D shape], with each image determined by its position along the 4th Dimension. The image appears to evolve rapidly, then becomes the [3D Bristor Set] at the origin, then decays in a similar yet different pattern to its growth. The [sequential planes] make the middle cross sections to this set.
The parallel images are produce in the same manner as the [parallel planes], by taking a series of rotated planes that pass the center axis and using the 4th dimension to add depth to each plane. So each plane is the middle cross section of each image. I have called them parallel from a visual perspective, by the fact that they are all generally the same shape, yet all unique bar for the exception when a shape is drawn on the planar axis and therefore its values equals zero. These produce the [Bristor set].
|The Sequential boxes are examined for the first occurrence of the set. The box is then magnified showing a complex fractal. The sequential box is then stepped back and a new first occurrence is displayed. A magnification is made and this process is repeated.
||z = 1.0||
|z = 1.1|| |
|z = 1.135|| |
The 4D maps are drawn by producing small 2D cells which are placed in a 2D grid. So each image is surrounded by its sequential cousins.
In this map when the 2D cells become very small in size, the [M-Set] outline becomes inceasingly visible. Also in this map is when the cells that run across the middle are stacked together, they would produce the [3D B-Set].
Another alternative to this map is to view this map from a different axis, so that
there is the [M-Set] in the cell at the origin and when the cells become very small the outline of the perpendicular cross section of the B-Set becomes more visible.
Vix and Mæve - for their loving support
Fractint - An superbly well written fractal generator
Uklinux.net - hosting my site
http://www.escati.com - suppling a free web counter
About Doug Bristor
I made this discovery early 1995, whilst planning to re-sit my second year electrionics course at Bath University. I thought it would revolutionize mordern mathematics. To me it does look so complete and whole. And I felt it demonstrates so many theroies. What I in fact found was little interest from the people I thought would be most interested.
Every once in a while I meet someone who thinks it's great and I great re-energised, and produce some more defined images.
So here I am still trying to improve my programming skills, and get used to Linux. Whilst earning a living as a teaching assistant, at the Red Balloon Learner Center.
Any comments please email them to me.