The Bristor Set
A Point A Line A Plane A Box A HyperBox
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Shape
Sequential
Parallel
Singularity
Strings
Magnification
Evolution
Similarity
Boundaries
 
Imaginary numbers
Gallary
About
Acknowledgements

The Point

The value, after every iteration of the algorithm at the origin, is always zero.

 

The Line

The Line
The line across the middle of [Mandelbrot Set] is drawn with exaggerated width to enable it to be more visual.

 

Magnification

The Magnified Line
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 Mandelbrot Set 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

 

Sequential

The sequential lines of Mandelbrot Set 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.

 

The Edge

The lower order boundary of  the Mandelbrot Set 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.

 

Self-Similarity

An enlargement of the M-Set
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.

 

String

The stands connecting the miniature M-Sets
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.

 

The Mathematics

Complex Numbers

Complex numbers came from the solution ( x + 1 ) = 0
To explain further 2 2 = 4 and so does -2 -2 = 4.
Therefore the sqrt4 is 2 or -2.
So what then is the solution to the sqrt-4 ?
The answer was the Complex Number 2 i, where i i = -1

i, a rotation of 90° The complex number, i, can be seen as a transformation of a 90° rotation. i x i So then i would be a rotation of a 180°

 

Displaying Complex Numbers

The Complex Plane 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 Algorithm

The Mandelbrot algorithm is defined as:-

	zk+1 = zk + 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.

 

Quaternions

The M-Set using Quaternions 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 x j = -j
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
j x i = i
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

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.

 

layers through the Bristor set

Sequential

Cross sections through the Bristor set
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.
Cross sectional islands though the Bristor setThe 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.

 

rotated cross sections

Parallel

Parallel variations of the Mandelbrot set
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.

 

Singularity

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 Magification 1
Magification 1

Magification 11

Magification 80
z = 1.1z = 1.0
Magification 1

Magification 62

Magification 210
z = 1.25z = 1.0
Magification 1

Magification 271

Magification 4M

 

A miniature Bristor Set

Self Similarity

The surface texture of the Bristor Set 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.

 

increasing detail of the Bristor Set

Magnification

Magnification of the set gives increased detail. Magnification is limited by the parameters of the computer rather than the set.

 

The connecting stands of the Bristor Set

String

The connecting mesh of the Bristor 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. .

 

Evolution

Surface texture of the Bristor SetThoughout 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.
Evolution within the Bristor Set The cross sections give what seems to be a natural growth, like a flick of flame, or a cloud evapourating.

 

The 4D Bristor Set

Static 4D

the static 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.

 

Sequential

Cross sections through the 4D Bristor setThe 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.

 

Parallel

parallel shapes of the Bristor 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].

 

Singularity

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 Magification 77
Magification 77
z = 1.1Magification 35
Magification 35
z = 1.135Magification 13
Magification 13

 

Maps

4D map of the Bristor set 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. 4D map of the B-set where the cells are very small
In this map when the 2D cells become very small in size, the [M-Set] outline becomes inceasingly visible. the mid-line cells of the 4D mapAlso in this map is when the cells that run across the middle are stacked together, they would produce the [3D B-Set].

4D map of the Bristor set with M-Set at origin
4D map of the Bristor set focused on the M-Set at origin
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.

 

Acknowledgements

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

Doug 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.

Doug Bristor