# Name

* period domain
* atom domain.
* BOF61( see book Beauty Of Fractals page 61 )
* orbit trap at (0,0)
* mclosetime

# Examples
## Images

* fractalforums : cellular-coloring-of-mandelbrot-insides
* animation
* 3D view of bof61
* 2D view of BOF61

## Video

* 2D Mandelbrot zoom (4K, 60fps) ayoungblood
* You tube video : Mandelbrot, 1080p 25fps by Warren Garabrandt

# Description

Atom domains in case of the Mandelbrot set ( parameter plane) are parts of parameter plane with the same the index p.

## index

* it is positive integer $p\geq 1$
* for p=1 domain is a whole plane because in the algorithm value of complex modulus is compared to infinity
* it is equal to
** the period of hyperbolic component of the Mandelbrot set which is inside domain
** iteration at which modulus of z is minimized during iteration of critical point

# Properities

Note that :
* atom domains are overlapping
* "Atom domains surround hyperbolic components of the same period, and are generally much larger than the components themselves"
* "These domains completely enclose the hyperbolic components of the same period"
* "the atom domain is wholy within its own Newton basin, and also significantly larger than the corresponding component"
Atom domain contain :
** component of mandelbrot set with period n mv
** exterior of this component
** some other components

# Importance

It can be used for :
* fast finding ( aprioximating) of period n components of Mandelbrot set and it's centers, (domains are greater than components which makes them useful for finding components)

# Algorithm
## whole parameter plane

Shadertoy
* Shadertoy bof61

## modifications

* Filtered atom domains
* modified atom domains - which makes smaller domains more visible.

## bof61

This is the method described in the book "The Beauty of Fractals" on page 63, but the image in on page 61.
Color of point is proportional to :
* the time it takes z to reach its smallest value
* iterate of the critical point makes the closest approach
* Index (c) is the iteration when point of the orbit was closest to the origin. Since there may be more than one, index(c) is the least such.
This algorithms shows borders of domains with the same index(c) .
Fragment of code : fractint.cfrm from Gnofract4d
Cpp function
It can be used :
Note that this method can be applied to both exterior and interior. It is called atom domain. It can also be modified

# Size of the atom domain

estimation of size
Function for computing size estimation of atom domain from nucleus c and its period p :

# References

* ↑ Atom Domain by Robert P. Munafo
* ↑ Practical interior distance rendering by Claude Heiland-Allen
* ↑ The_Beauty_of_Fractals in english wikipedia
* ↑ Bof61 algorithm in wikibooks
* ↑ An orbit trap at (0,0) by hobold
* ↑ misiurewicz_domains by Claude
* ↑ mandelbrot set newton basins by Claude Heiland-Allen
* ↑ Fractint doc by Noel Giffin
* ↑ A Series of spiral bays in the Mandelbrot set by Patrick Hahn
* ↑ gnofract4d
* ↑ Practical interior distance rendering by Claude Heiland-Allen
* ↑ Atom Domain From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo
* ↑ Modified Atom Domains by Claude Heiland-Allen
* ↑ Atom domain size estimation by Claude Heiland-Allen